OSIPP Discussion Paper: DP-2021-E-001

Multi-Player Bayesian Learning with Misspecified
ModelsJanuary 15, 2021
Takeshi Murooka*
Osaka School of International Public Policy (OSIPP), Osaka University

Yuichi Yamamoto†
Institute of Economic Research, Hitotsubashi University

Abstract: We consider strategic players who may have a misspecified view about an environment, andinvestigate their long-run behavior. Each period, players simultaneously take actions, observe a public
outcome, and then update own belief about an uncertain economic state by using Bayes' rule. We provide
a condition under which players' beliefs and actions converge to a steady state, and then characterize
how one's misspecification influences the long-run (steady-state) actions. When a player has a biased
view about the physical environment (e.g., overconfidence on own capability or prejudice on an
opponent's capability), the presence of strategic interaction influences the size of the impact of
misspecification, but not the direction. In particular, when the game is symmetric, the presence ofstrategic interaction amplifies the deviation of the long-run actions from those in the correctly specified
model. When a player misspecifies the opponent's view about the environment (e.g., the player is notaware of the opponent's bias), the strategic interaction generates a directional impact for the long-run
actions. We extensively discuss implications to a variety of applications, such as Cournot duopoly, teamproduction, tournaments, and discrimination.
JEL Codes: C73, D83, D90, D91
Keywords:

model misspecification, learning, convergence, overconfidence, bias transmission*

Email: murooka@osipp.osaka-u.ac.jp†

Email: yyamamoto@ier.hit-u.ac.jp

Multi-Player Bayesian Learning with
Misspecified Models∗Takeshi Murooka
Osaka University

Yuichi Yamamoto
Hitotsubashi University

January 15, 2021

Abstract
We consider strategic players who may have a misspecified view about
an environment, and investigate their long-run behavior. Each period, players simultaneously take actions, observe a public outcome, and then updateown belief about an uncertain economic state by using Bayes’ rule. We
provide a condition under which players’ beliefs and actions converge to asteady state, and then characterize how one’s misspecification influences thelong-run (steady-state) actions. When a player has a biased view about thephysical environment (e.g., overconfidence on own capability or prejudiceon an opponent’s capability), the presence of strategic interaction influences
the size of the impact of misspecification, but not the direction. In particular,when the game is symmetric, the presence of strategic interaction amplifies
the deviation of the long-run actions from those in the correctly specified
model. When a player misspecifies the opponent’s view about the environment (e.g., the player is not aware of the opponent’s bias), the strategicinteraction generates a directional impact for the long-run actions. We extensively discuss implications to a variety of applications, such as Cournot
duopoly, team production, tournaments, and discrimination.
∗

We are grateful to seminar and conference audiences for helpful comments. Murooka acknowledges financial support from JSPS KAKENHI (JP16K21740, JP18H03640, JP19K01568,
JP20K13451). Yamamoto acknowledges financial support from JSPS KAKENHI (JP20H00070,
JP20H01475).

1

1Introduction

Economic agents often take actions based on a subjective and potentially misspecified view of the world.1 Past work has shown that in a static game-theoretic environment, one’s misspecification can improve her equilibrium payoff. The reason
is that one’s misspecification can influence the opponent’s action through strategicinteraction. For example, in a one-shot Cournot game, if a firm is overconfident
about the demand and willing to produce more, the rival firm best-responds toit and produces less, which improves the overconfident firm’s profit (Kyle andWang, 1997). We build on this literature by considering a Bayesian-learning feature; we develop a dynamic model in which misspecified players learn an unknown economic state from public signals. Our primary interest is to understandhow misspecified players process information, and how it influences their longrun behavior.
We believe that this question is of importance for the following reason. In
many economic applications, it is assumed that players know all relevant aspectsof the environment. This assumption is often justified by the argument that even
if some economic variable is initially unknown, as long as it is fixed over time,players will eventually learn it from observed outcomes. However, when players
have a misspecified view about some aspect of the world, they may process information incorrectly and may fail to learn the true economic variable. Thus, tounderstand the long-run behavior of misspecified players, we need to consider adynamic model (rather than a one-shot model) and carefully think about how they
update the beliefs over time.
More generally, when players have a misspecified view about the world, it isnatural to expect that they will eventually change the belief about some economic
1 As experimental and empirical evidence, people exhibit overconfidence in strategic entries(Camerer and Lovallo, 1999), corporate investments (Malmendier and Tate, 2005), and merger
decisions (Malmendier and Tate, 2008). There is also recent evidence that overconfidence is persistent; Hoffman and Burks (2020) find that workers are persistently overconfident about theirown productivity, and Huffman, Raymond, and Shvets (2019) find that managers are persistently
overconfident about future performance.

2

variable, as they observe outcomes which are systematically different from the anticipation. For example, if a firm is persistently overconfident about some aspectof the demand function (e.g., the intercept of the inverse demand curve), on average, actual prices are lower than the firm’s anticipation. So it is likely that after
a long time, this firm becomes (unrealistically) pessimistic about other aspect ofthe demand (e.g., the slope of the inverse demand curve). Similarly, in tournaments, if an agent is persistently overconfident about her own capability, after aseries of unexpected losses, she may start to think that the tournament is unfair.
Our framework is useful to understand players’ long-run behavior in these cases,
and we show that this learning feature has a substantial impact on the equilibriumoutcomes in various applications.
Formally, we consider a two-player Bayesian learning problem with model
misspecification. In each period t, each player i simultaneously chooses an action
xi and then learns an unknown economic state θ from a public signal y. The distribution of the signal y is influenced by the state θ , the action profile x = (x1 , x2 ),and a parameter a which describes the environment/technology. For example, ina Cournot duopoly, the distribution of the market price y depends on the quantity
x1 + x2 , as well as the intercept a and the slope θ of the inverse demand curve.
We assume that a player may have a misspecification, in that she may havea persistently biased view about the parameter a. In Section 3, we consider thecase in which player 1 is unbiased and knows the true environment, but player 2
incorrectly believes that the true parameter is A , a. We call it first-order misspecification, as player 2 has an incorrect first-order belief about the parameter a. Thisis the simplest form of misspecification, and encompasses a variety of commonly
observed biases, including overconfidence about one’s own capability, prejudiceabout the opponent’s capability, misestimation about market demand, and so on.
We provide a condition in which players’ beliefs (about θ ) and actions converge to a steady state after a long time. Then, we characterize how player 2’s
misspecification influences this steady state. It turns out that the impact of misspecification on the biased player’s action can be written as a product of two terms,
a base misspecification effect and a multiplier effect. The base misspecification ef-

3fect measures how one’s misspecification would influence her own action if there
was no strategic interaction between the players, in that the opponent takes aconstant action every period. This essentially measures the impact of misspecification in a single-agent Bayesian learning problem, which is studied in specificcontexts in the literature (e.g., Fudenberg, Romanyuk, and Strack (2017), Heidhues, Kőszegi, and Strack (2018), and He (2019)).
The multiplier effect is new, and it describes how strategic interaction amplifies/reduces the base misspecification effect. When a biased player changes her
action due to the base misspecification effect, the opponent best-responds to it,which in turn influences the biased player’s steady-state belief and the optimal
action, and this process continues repeatedly. The multiplier effect quantifies theoverall size of this feedback process. Under a regularity condition, the multipliereffect is positive, which means that strategic interaction can influence the size ofthe impact of misspecification, but not the direction. In particular, if the game issymmetric as in common-interest games, the multiplier effect is larger than one,so both strategic complements and substitutes amplify the base misspecification
effect.
To fix ideas, consider a Cournot duopoly in which some aspect of the demand(e.g., the slope of the inverse demand curve) is unknown. Suppose that one of thefirms is persistently overconfident about other aspect of the demand (e.g., the intercept of the inverse demand curve). This overconfidence boosts the firm’s incentive
to produce, and as noted earlier, it improves the overconfident firm’s equilibrium
payoff in the static game. This is the direct effect of overconfidence. However, in
our dynamic model, the firm’s overconfidence leads to incorrect learning, which
weakens the incentive to produce; indeed, the overconfident firm finds that realized prices are systematically lower than the anticipation on average, so after
a long time, it becomes unrealistically pessimistic about the unknown economic
state. This is the indirect learning effect of overconfidence, and the base misspecification effect is the sum of these two countervailing forces. If the direct effectoutweighs the indirect effect, the base misspecification effect is positive. In thiscase, the result of Kyle and Wang (1997) persists even in the long run, that is, the4overconfident firm produces more and earns a better profit than in the correctlyspecified case. However, if the indirect effect is larger, this result is overturned:
while overconfidence improves the static equilibrium payoff, in the long run, it isdetrimental and the overconfident firm earns a lower profit.
When there are multiple players, they may not only have a bias about a physical environment, but have a bias about the opponent’s view about the environment.
For example, a junior scholar may incorrectly believe that he is undervalued by asenior colleague (in reality, the senior colleague correctly understands the juniorscholar’s capability). Or, a worker (she) may be unaware that a co-worker (he)is overconfident about his own capability. In Section 4, we consider how such“second-order misspecification” influences players’ behavior. We consider thecase in which one of the players has second-order misspecification (i.e., she correctly specifies a but has an incorrect view about the opponent’s view about a), thecase in which one of the players has “double misspecification” in that she has both
first-order and second-order misspecification, and the case in which both players
have double misspecification. The analysis is significantly more complicated thanin the case of first-order misspecification, because a player with second-order ordouble misspecification has an incorrect view about how the opponent updates thebelief, and we need to keep track of how the belief of such a “hypothetical player”
evolves over time. Nonetheless, under an additional assumption, we find that
players’ actions and beliefs still converge to a steady state. We also show that the
impact of one’s misspecification on the steady-state outcome is still represented
by a product of the base misspecification effect and the multiplier effect.
To illustrate how second-order misspecification influences players’ behavior,consider the Cournot model, and as in the case of first-order misspecification, assume that one of the firms (say, firm 2) is overconfident about some aspect of thedemand. In addition, assume now that the rival firm (say, firm 1) has second-ordermisspecification, i.e., firm 1 is not aware of firm 2’s overconfidence and incorrectly believes that firm 2 has an unbiased view about the demand. In this setup,firm 2’s overconfidence cannot directly affect the opponent’s production. Indeed,even though firm 2 is overconfident and willing to produce more, firm 1 is not5

aware of it and hence does not change the production level. Accordingly, in the
one-shot game, firm 2’s overconfidence never improves the equilibrium payoff.
Nonetheless, we find that in the long run, overconfidence may be beneficial. To
see why, note that firm 1 is not aware of firm 2’s overconfidence, and hence itmakes a wrong prediction about firm 2’s production. In particular, for some parameter values, firm 1 underestimates firm 2’s production, in which case it findsthat a market price is systematically lower than the anticipation. Then as time
goes, firm 1 becomes pessimistic about an unknown economic state and reduces
the production, which benefits the overconfident firm 2.
A key here is that even though firm 1 has an unbiased view about the physicalenvironment, incorrect learning occurs due to second-order misspecification. A
similar idea can be used to explain bias transmission. In Section 4.4.3, we consider a teacher who has a bias against female students. We show that the teacher’sbias can endogenously induce female students’ negative self-stereotypes, if thestudents are not aware of the teacher’s bias.
In Section 5, we present a general model which subsumes all of the aforementioned misspecifications as special cases, and characterize the asymptotic behaviorof players’ actions an beliefs. Section 6 summarizes the related literature.

2

Setup

There are two players i = 1, 2 and infinitely many periods t = 1, 2, · · · . At the
beginning of the game, an unobservable economic state θ ∗ is drawn from a closed
interval Θ = [θ , θ ], according to a common prior distribution µ ∈ 4Θ. In eachperiod t, each player i has a belief µit ∈ 4Θ about θ , and chooses an action xi
from a closed interval Xi = [0, xi ]. Player i’s action xi is not observable by the otherplayer j , i. Given an action profile x = (x1 , x2 ), the players observe a noisy public
signal y = Q(x1 , x2 , a, θ ∗ ) + ε, where a ∈ R is a fixed parameter which describes
a physical environment (e.g., a parameter which determines a market demand, aplayer’s capability, etc) and ε is a random noise whose distribution is N(0, 1).
Each player i receives a payoff ui (xi , y). Both Q and ui are twice continuously
6differentiable.
Crucially, players may have bias about the environment, and as a result, andthe parameter a is not common knowledge. We allow a variety of bias, including
bias about the parameter a, bias about the opponent’s bias about the parameter a,and so on. In what follows, we will consider each bias separately.

3

First-Order Misspecification

We first consider the simplest form of misspecification, called first-order misspecification.

3.1

First-Order Misspecification: Characterization

Suppose that one of the players (player 2) has a biased view about the parameter a, whereas the other player is unbiased. Specifically, consider the following
information structure:• Player 1 believes that for each parameter θ , the signal y is given by y =Q(x1 , x2 , a, θ ) + ε.
• Player 2 (incorrectly) believes that for each parameter θ , the signal y isgiven by y = Q(x1 , x2 , A, θ ) + ε, where A , a.
• The above beliefs are common knowledge (e.g., player 2 believes that player
1 believes that y = Q(x1 , x2 , a, θ ) + ε, and the like).
This is the case in which players “agree to disagree” with the level of the parameter a. Kyle and Wang (1997) consider a short-run effect of this first-ordermisspecification, i.e., they consider how it influences players’ actions in a variantof one-shot Cournot game. In this section, we consider a long-run effect of thismisspecification.
Player 1’s subjective expected stage-game payoff given an action profile x anda state θ is
U1 (x, θ ) = E[u1 (x1 , Q(x, a, θ ) + ε)]7and player 2’s subjective expected stage-game payoff is
U2 (x, A, θ ) = E[u2 (x2 , Q(x, A, θ ) + ε)],where the expectation is taken with respect to ε. Note that player 2 evaluates payoffs given her subjective signal distribution Q(x, A, θ )+ε. To economize notation,
we will write U2 (x, θ ) instead of U2 (x, A, θ ) when it does not cause a confusion.
We assume that players are myopic, so that they play a Nash equilibrium eachperiod. In period one, both players have the same belief µ11 = µ21 = µ, so a Nash)|µ]equilibrium (x11 , x21 ) solves the first-order condition ∂ E[Ui∂(x,θ
= 0 for each i,xiwhere the expectation is taken with respect to θ . At the end of period one, players
observe a public signal y1 , and update the posterior beliefs using Bayes’ rule.
Assuming that no one has deviated in period one, each player i’s posterior beliefµi2 in period two is given byµ11 (θ ) f (y − Q(x1 , a, θ )),11
Θ µ1 (θ̃ ) f (y − Q(x , a, θ̃ ))d θ̃

µ12 (θ ) = Rµ11 (θ ) f (y − Q(x1 , A, θ )),11
Θ µ1 (θ̃ ) f (y − Q(x , A, θ̃ ))d θ̃µ22 (θ ) = R

where x1 is the Nash equilibrium played in period one and f is the density functionof the noise term ε. Note that player 2’s posterior µ22 differs from player 1’s
posterior µ12 , as she incorrectly believes that the mean output is Q(x1 , A, θ ) rather
than Q(x1 , a, θ ). These posteriors are common knowledge among players.2 So inperiod two, players play a Nash equilibrium given the belief profile µ 2 = (µ12 , µ22 ),∂ E[U (x,θ )|µ 2 ]iiwhich solves= 0 for each i. Likewise, in any subsequent period t ≥ 3,
∂ xi
players play a Nash equilibrium given the belief profile µ t = (µ1t , µ2t ), where µ t iscomputed by Bayes’ rule.
Our goal is to characterize the long-run outcome in this Bayesian learningmodel, i.e., how players’ actions xt and beliefs µ t look like after a long time.
While the evolution of the biased player’s belief is governed by a complicated2 This is because the players’ information structure about the parameter a is common knowl-

edge.

8

stochastic process, we show in Section 5 that there is a sufficient condition underwhich players’ beliefs and actions converge almost surely to a steady state after along time. As will be seen, this condition is satisfied in many economic examples,such as Cournot competition and team production. So in what follows, we willassume that players’ beliefs and actions converge to a steady state, and investigate
the property of the steady state.
Formally, a steady state in this Bayesian learning model is a pair (x1∗ , x2∗ , µ1∗ , µ2∗ )of an action profile and a belief profile which satisfies the following four conditions:x1∗ ∈ arg max U1 (x1 , x2∗ , θ ∗ ),(1)

x2∗ ∈ arg max U2 (x1∗ , x2 , θ2 ),(2)µ1∗ = 1θ ∗ ,(3)µ2∗ = 1θ2 s.t. Q(x∗ , A, θ2 ) = Q(x∗ , a, θ ∗ ).

(4)

x1

x2

The first two conditions are incentive compatibility conditions, which require thateach player maximizes her payoff given some beliefs. The next two conditions
require that these beliefs satisfy consistency: (3) asserts that the unbiased player
1 correctly learns the true state θ ∗ in a steady state. (4) requires that player 2’s
belief be concentrated on a state θ2 with which her subjective signal distribution
coincides with the true distribution. This condition must be satisfied in a steadystate, because otherwise, player 2 is “surprised” by observed signals being different from what she thinks, and changes her belief about θ accordingly.
In general, this steady-state belief θ2 is different from the true state θ ∗ , i.e., thebiased player 2 cannot learn θ ∗ correctly. For example, consider the case in which
the mean output Q is increasing in a and θ . By the implicit function theorem, we
Qa< 0. So if player 2 is overconfident about the parameter a, shehave ∂∂θA2 = − Qθunderestimates the state θ . Intuitively, the overconfident player observes signals
which are lower than her anticipation, and becomes pessimistic about the state θ .
We assume that for each (x, A), there is a unique state θ2 which solves the consistency condition Q(x, a, θ ∗ ) = Q(x, A, θ ), and we denote it by θ2 (x, A). Intuitively,
9this θ2 (x, A) is player 2’s long-run belief; if players choose the same action profilex every period, then almost surely, player 2’s belief will be concentrated on thestate θ2 (x, A) after a long time (Berk (1966)). Player 1’s long-run belief is definedas θ1 (x, A) = θ ∗ for all x, because she is unbiased and can learn the true state θ ∗regardless of players’ play.
The steady-state action profile is different from a Nash equilibrium, because
player 2’s belief is endogenously determined by the consistency condition (4).
Indeed, it is characterized as an intersection of the asymptotic best response correspondences (rather than the standard best response correspondences), which isdefined as()
BRi (x−i ) = xi xi ∈ arg max
Ui xi0 , x−i , θi (x, A) .
(5)
0
xi

Intuitively, BRi (x−i ) describes player i’s optimal action in the long run, when
the opponent chooses the same fixed action x−i each period and player i updates
the belief over time. Player 1’s asymptotic best response BR1 coincides with thestandard best response correspondence given the state θ ∗ , because she is unbiased
and can learn the true state θ ∗ regardless of the opponent’s play. In contrast, player
2’s asymptotic best response BR2 is different from the standard best response,
because her long-run belief θ2 (x, A) is endogenously determined; if the opponent
chooses the same action x1 every period and if player 2’s action converges to somex2 , then the limiting belief must be concentrated on θ2 (x, A). Player 2’s limiting
action x2 must be optimal given this belief, as stated in the definition of BRi . By afixed-point theorem, BRi (x−i ) is non-empty for all x−i . Also a standard argumentshows that BRi is upper hemi-continuous in x−i .
As is well-known, the slope of the standard best-response function is given∂ 2U /∂ x ∂ xby − ∂ 2Ui /∂ i2 x j ; note that both the denominator and the numerator measure theiiimpact of one’s action on player i’s marginal utility. It turns out that the slope BR0iof the asymptotic best response function can be computed by a similar formula,
BR0i = −

10

Mi j
,
Mii

where for each i and j (possibly i = j),∂
Mi j =∂xj=∂Ui (x, θ )∂ xi
θ =θi (x,A)!

∂ 2Ui (x, θi )∂ θi (x, A)∂ 2Ui (x, θ ).
+∂ xi ∂ x j θi =θi (x,A)∂ xi ∂ θ θ =θi (x,A) ∂ x j

measures the impact of player j’s action on player i’s marginal utility in the longrun, assuming that players choose the same action every period.3 Intuitively,
player j’s action x j influences player i’s marginal utility directly and indirectly
through the long-run belief θi (x, A). The first term of Mi j represents the direct
effect, and the second term represents the indirect effect through the belief. The
indirect effect is zero for i = 1, because player 1’s long-run belief is constant anddoes not depend on the actions (i.e., θ1 (x, A) = θ ∗ for all x). Also, for i = 2, theindirect effect disappears in the limit as A → a, because θ2 (x, a) = θ ∗ for all x. So
when model misspecification is small (i.e., A is close to a), each Mi j is approx2
imated by ∂∂xi ∂Uxi j , so BR0i is approximately the same as the slope of the standard
best-response function.
Our first proposition quantifies the impact of player 2’s first-order misspecification on the steady-state action, using this slope of the asymptotic best response
curve. We assume that the steady state satisfies the following regularity condition:
Definition 1. A steady state x∗ is regular if the following conditions are satisfied
in x∗ : (i) the steady-state action xi∗ is uniquely optimal, i.e., Ui (x∗ , θi (x∗ , A)) >∗ , θ (x∗ , A)) for all i and x , x∗ , (ii) x∗ and θ (x∗ , A) are interior points,
Ui (xi , x−i
i
i
2
i
0
0
(iii) BR1 BR2 , 1, and (iv) Mii , 0 for each i.
The conditions (i) and (ii) are standard. The condition (iii) is satisfied forgeneric parameters, and it ensures that the multiplier effect appearing in Proposition 1 below is well-defined. The condition (iv) requires two inequalities, M11 =
2
2
∂ 2U1
< 0 and M22 = ∂∂ 2Ux 2 + ∂∂ θx2 ∂∂x U∂2θ < 0. The first inequality is simply the second∂ 2 x1
22
2
order condition for player 1’s incentive compatibility. The second inequality is dif2
ferent from the second-order condition, as it involves the learning effect ∂∂ θx2 ∂∂x U∂2θ .
2

3 Here we assume that M , 0, so that BR0 is well-defined.
ii
i

11

2

This inequality is satisfied in many economic applications, e.g., it is satisfied if thesecond-order condition for player 2’s incentive compatibility holds and misspecification is small (so that the learning effect is close to zero). Note that Heidhues,Kőszegi, and Strack (2018) impose a similar condition: They consider Bayesianlearning by a single agent and assume a unique steady state, which requires Mii ≤ 0
in the steady state.4 As will be seen, this condition (iv) is useful when we interpret the base misspecification effect appearing in Proposition 1. This conditionM
also ensures that the slope of the asymptotic best response curve BR0i = − Miiij iswell-defined.5
Finally, let!
∂∂U2 (x, A, θ )M2A : =
∂A∂ x2
θ =θ2 (x,A)=∂ 2U2 (x, A, θ )∂ 2U2 (x, A, θ )∂ θ2 (x, A)+∂ x2 ∂ A∂ x2 ∂ θ∂Aθ =θ2 (x,A)θ =θ2 (x,A)(6)

denote the impact of player 2’s bias A on her marginal utility in the long run.
22
2 ∂ θ2
Again, the first term ∂∂x U∂2A measures the direct effect, and the second term ∂∂x U
2
2 ∂ θ2 ∂ Ameasures the indirect effect through the belief. Now we are ready to state our first
proposition.
Proposition 1 (Steady State under First-Order Misspecification). Let x∗ be a regular steady state for some parameter A∗ .6 Then there is an open neighborhood of
4 Indeed, in a single-agent problem, if M > 0 in some steady state, then there are multipleii

steady states. A proof is akin to that of the second part of Proposition 1, and hence is omitted.
5 We conjecture that if there is a steady state with M > 0, then it is unstable in the sense that
22
players’ actions do not converge there with positive probability. At least, we show that if player 1chooses the steady-state action every period and player 2 learns an unknown state θ , then player
2’s action never converges to a steady state with M22 > 0. The proof is available upon request.
6 The regularity conditions (i) and (ii) ensure that the steady state is continuous with respect
to the parameter A and the first-order condition for the incentive compatibility is satisfied there.
The regularity conditions (iii) is to use the implicit function theorem. M11 , 0 and M22 , 0 in theregularity condition (iv) ensure that the base misspecification effect and the slope of the asymptotic
best response curve are well-defined.

12

A∗ such that for any value A in this neighborhood, there is a regular steady statex∗ which is continuous with respect to A, and we have
∂ x2∗
M2A
1
=−
·
∂A
M22 1 − BR01 BR02
∂ x1∗ ∂ x2∗=· BR01 .
∂A∂A
Suppose in addition that given the parameter A∗ , the steady state is unique andeach asymptotic best response BRi is a continuous function. Then, BR01 BR02 < 1.7
This proposition shows that the impact of first-order misspecification on the
2A
steady-state action is represented as the base misspecification effect − M
M22 times
the multiplier effect 1−BR10 BR0 . The base misspecification effect measures how
1
2
player 2’s bias influences her steady-state action x2∗ in the absence of strategicinteraction. To see what it means, consider the case in which player 1 chooses thesame fixed action each period, so player 2 faces a single-agent problem. Supposethat player 2’s bias A increases a bit. This influences player 2’s marginal utility
by M2A (note that this includes the indirect effect through the steady-state belief),
and hence her optimal long-run action changes. The base misspecification effect
2A−M
M22 measures this change. Since we assume M22 < 0, the sign of the basemisspecification effect coincides with the sign of M2A . That is, player 2’s steadystate action increases if and only if a change in A has a positive impact on her
marginal utility.
This base misspecification effect is further divided into two parts: Since M2A isa sum of the direct effect and the indirect effect (see (6)), the base misspecification
7 If these additional assumptions do not hold, there may be a steady state with BR0 BR0 >1
2

1. But it seems that such a steady state is unstable in an evolutionary sense, especially when
misspecification is small. Indeed, in a one-shot game with correctly specified model, a Nashequilibrium with BR01 BR02 > 1 is not stable under the replicator dynamics (hence it is not an ESS)or the best response dynamics. So in practice, if players’ play converge after a long time, it isnatural to expect that BR01 BR02 < 1 in the steady state.

13effect is rewritten as




 2


1 
∂ U2 (x∗ , A, θ )M2A∂ θ2 ∂ 2U2 (x∗ , A, θ )

.
−
=−
+

M22
M22 
∂
x
∂
A
∂
A
∂
x
∂
θ
∗
∗
2
2
θ =θ2 (x ,A)θ =θ2 (x ,A) 

|
{z
} |
{z
}
direct effect from A’s change

indirect leargning effect through θ2

The first term in the brackets is the direct effect, which measures how player 2’s
bias influences her action, when her belief θ2 is fixed. This coincides with theimpact of one’s bias on her action in a one-shot game with no learning, whichis studied by past work such as Kyle and Wang (1997). The second term in thebrackets is the (indirect) learning effect, which measures how player 2’s bias influences her action through the steady-state belief θ2 ; the term ∂∂θA2 measures how
2player 2’s bias influences her steady-state belief, and ∂∂x U∂2A measures how it in2
fluences her marginal utility. Intuitively, this learning effect represents how thebiased player modifies her view about the world and how it influences her action in the long run. As will be explained later, this effect is the source of theself-defeating feature in Heidhues, Kőszegi, and Strack (2018).
The multiplier effect 1−BR10 BR0 in Proposition 1 measures how strategic inter1
2
action between two players amplifies/weakens the base misspecification effect. To
better understand the nature of this multiplier effect, suppose that player 2 changes
her action by ∆. Then player 1 best-responds to it and changes her action by BR01 ∆,which in turn has a feedback effect of BR01 BR02 ∆ on player 2’s steady-state action;note that player 1’s action influences player 2’s optimal action directly and indirectly through her belief θ2 (x, A), and both these effects are taken into account in
the asymptotic best response BR02 .
This process continues multiple times; the feedback effect on player 2’s action
influences player 1’s action, which again causes a feedback effect of (BR01 BR02 )2 ∆on player 2’s action, and so on. Summing all these feedback effects, player 2’s
action changes by∞1∑ (BR01BR02)k ∆ = 1 − BR0 BR0 ∆.
1
2
k=0
So the multiplier 1−BR10 BR0 can be seen as a result of the infinite adjustment process
1

2

14

between the two strategic players.
The following corollary is an immediate consequence of Proposition 1:
Corollary 1. Suppose that all the assumptions in Proposition 1 (including theones in the second part) are satisfied. Then we have the following results:(i) The multiplier 1−BR10 BR0 is positive. So a strategic interaction influences the12
size of the impact of misspecification, but not the direction.
(ii) If sgn(BR01 ) = sgn(BR02 ), then the multiplier 1−BR10 BR0 is greater than one
12
and is increasing in |BR0i |. So both strategic substitutes and strategic complements amplify the impact of the first-order misspecification.
(iii) If sgn(BR01 ) , sgn(BR02 ), then the multiplier 1−BR10 BR0 is less than one and12decreasing in |BR0i |. So a strategic interaction reduces the impact of thefirst-order misspecification.
For the special case in which A = a, BRi reduces to the standard best response
curve, so it is fairly easy to compute the slope BR0i . In particular, when the gameis symmetric, we have sgn(BR01 ) = sgn(BR02 ) in any symmetric equilibrium, so a
strategic interaction amplifies the impact of first-order misspecification.
So far we have assumed that only player 2 is misspecified, but it is straightforward to see that a similar result holds even when both players have first-ordermisspecification. Suppose that player 1 believes that the true parameter is A1 , a,and player 2 believes that the true parameter is A2 , a. Suppose also that thesefirst-order beliefs are common knowledge. For each i, let θi (x, Ai ) denote θ which
satisfies the consistency condition Q(x, a, θ ∗ ) = Q(x, Ai , θi ). Intuitively, whenplayers play the same action profile x every period, player i’s limiting belief (ifexists) must be concentrated on θi (x, Ai ). Define each player i’s asymptotic best
response BRi by (5), and let BR0i denote its slope. Then the impact of a marginalincrease in player 2’s misspecification on steady-state actions can be represented

15

just as in Proposition 1, that is,
∂ x2∗
M2A
1
=−
·
,
∂ A2
M22 1 − BR01 BR02
∂ x1∗
∂ x∗
= 2 · BR01 .
∂ A2 ∂ A2
The only difference from Proposition 1 is that player 1’s asymptotic best response
is different from the standard best response; now it takes into account the fact that
her long-run belief θ1 (x, A1 ) is endogenously determined.

3.2

First-Order Misspecification: Applications

This subsection investigates applications in Cournot duopoly, team production,and tournament, with highlighting how first-order misspecification influences thelong-run behavior and welfare.
3.2.1

Cournot duopoly

We first consider a symmetric Cournot duopoly and study how a firm’s bias influences the equilibrium outcome. In each period, each firm i = 1, 2 simultaneously chooses its quantity xi . Then, each firm observes a market price y =Q(x1 + x2 , a, θ ) + ε, where a is a parameter which influences the demand and θ isan unknown economic state. Firm i’s payoff is ui (xi , y) = xi y − c(xi ), where xi y is
firm i’s revenue and c(xi ) is firm i’s production cost. Throughout, we assume thatthe inverse demand function Q is strictly decreasing and weakly concave in thefirst element, and the cost function c is strictly increasing and weakly convex.8
Kyle and Wang (1997), Heifetz, Shannon, and Spiegel (2007), and Englmaier(2010) study (a variant of) one-shot Cournot competition with linear demand, andshow that a firm who has overconfidence about the intercept and/or the slope of
8 These assumptions ensure the concavity of each firm’s payoff function, a downward-sloping

best response curve, and the uniqueness of Nash equilibrium under the correctly specified model.
See, for example, Tirole (1988).

16the inverse demand function earns higher profits than the unbiased rival firm. Intuitively, the overconfident firm is willing to produce more than in the correctly
specified model. Knowing that, the unbiased firm reduces its production level inequilibrium, which yields higher profits to the overconfident firm. This mechanism, which we call the strategic effect, is akin to the commitment to produce
more in the context of quantity competition (e.g., being a first mover in the standard Stackelberg duopoly).9A natural question is whether this strategic effect is beneficial for the overconfident firm in the long run. So long as the game is repeated, the overconfident firm
is persistently “surprised” by a realized price being lower than its anticipation.
This suggests that the overconfident firm’s subjective view about the world wouldchange after a long time, in order to better explain such lower prices. Accordingly,
the overconfident firm changes the behavior, and the rival firm best-responds to it.
In what follows, we will study how this influences the equilibrium outcome.
Suppose that firm 2 incorrectly believes that the true parameter is A > a, where
QA > 0 and QxA ≥ 0 for all x with x1 + x2 > 0. Intuitively, firm 2 is overconfident
about the price level Q and (weakly) overconfident about the slope of the inverse
demand curve Qx . In the special case of linear demand, this assumption reduces tothe one imposed in the literature (e.g., Kyle and Wang (1997)). Firm 1 knows thatthe true parameter is a, and the firms’ first-order beliefs are common knowledge.
We also assume that Qθ > 0 and Qxθ ≥ 0 for all x with x1 + x2 > 0, i.e., the stateθ has positive impacts on the price level and the slope of the inverse demand
function.
Here are two examples which satisfy the assumptions above:10
Q(x1 + x2 , a, θ ) = a − (1 − θ )(x1 + x2 ),(7)

Q(x1 + x2 , a, θ ) = θ − (1 − a)(x1 + x2 ).

(8)

and

9 See also Fershtman and Judd (1987) who investigate the strategic effect induced by managerial

compensation contracts under a correctly specified model.
10 In these examples, as Q is linear in θ , the identifiability condition in Section 5 is satisfied.
Hence, from Proposition 13, the actions and the beliefs converge to a steady state almost surely.

17

In the first example (7), firm 2 is overconfident about the intercept of the inverse
demand function, and learns the slope of the inverse demand function. This happens, for example, when the firm has overconfidence about the preference of therepresentative customers and learns their number.11 In the second example (8),firm 2 is overconfident about the slope of the inverse demand function, and learns
the intercept. This happens, for example, when the firm has overconfidence about
the number of the customers and learns their preference.12
In both these examples, if the firms interact only once, the overconfident firm
earns a higher equilibrium profit than the rival firm (Kyle and Wang, 1997). However, it turns out that these two examples give qualitatively different long-run outcomes; as will be explained, a firm’s overconfidence improves its long-run payoffin the second example, but it is detrimental in the first example.
Recall from Proposition 1 that the impact of firm 2’s overconfidence on its own
2A
action is represented as the base misspecification effect − M
M22 times the multiplier.
For ease of exposition, we assume that misspecification is small (i.e., A is closeto a) so that M22 < 0 and the multiplier is positive. Simple algebra shows that thebase misspecification effect in our Cournot model is written as
indirect effectindirect effect
z
}|
{
z
}|
{
direct effectz}|{ ∂ θ2}|{ ∂ θ2
1 
z

∗
∗
∗
∗
∗
∗
∗
∗
∗
Q
(x
+
x
,
A,
θ
)
+
Q
(x
+
x
,
A,
θ
)
+x
Q
(x
+
x
,
A,
θ
)
+
Q
(x
+
x
,
A,
θ
)
−
 A 1
2
2
2
2  .
θ 1xθ 1
22
2  xA 12
2
M22 |
∂
A
∂
A

{z
}
|
{z
}
direct effecton the price levelon the slope(9)
Q (x∗ +x∗ ,A,θ )By the implicit function theorem, we have ∂∂θA2 = − QA (x1∗ +x2∗ ,A,θ2 ) < 0. Plugging
2
θ 1
2
this into (9), the first two terms in the brackets cancel out. Intuitively, this is aconsequence of the consistency condition (4); in the long run, the overconfidence
about the price level is offset by a pessimistic view about the parameter θ , and thefirm correctly predicts the average price level. Accordingly, the base misspecifi11 Suppose that there are11−θcustomers, and each of them purchases a − p units of products,a−pwhere p denotes a price. Then the total demand is x = 1−θ, which results in the inverse demand
function p = a − (1 − θ )x.
12 Suppose that there are 1 customers, and each of them purchases θ − p units of products.
1−a−pThen the total demand is x = θ1−a, which results in the inverse demand function p = θ − (1 − a)x.

18

cation effect is simplified to
QA (x1∗ + x2∗ , A, θ2 )
x2∗
∗
∗
∗
∗
QxA (x1 + x2 , A, θ2 ) −Qxθ (x1 + x2 , A, θ2 ) .
−M22Qθ (x1∗ + x2∗ , A, θ2 )(10)(10) implies that the overconfident firm’s long-run action crucially relies on
its steady-state belief about the slope of the inverse demand curve. Suppose first
QA
that QxA − Q
Qxθ < 0, i.e., the firm is pessimistic about the demand slope in theθsteady state. In this case, the base misspecification effect is negative, and theoverconfident firm produces less in equilibrium. The rival firm best-responds to it
and produces more, which decreases the overconfident firm’s equilibrium payoff.
Hence, the firm’s overconfidence is detrimental in the long run. It corresponds to
QA
1
the first example (7), where QxA − Q
Qxθ = − x1 +x< 0. Intuitively, after observ2
θ
ing unexpectedly low prices, the overconfident firm becomes pessimistic about thedemand slope and reduces the production level, which hurts own long-run profit.
QA
Next, suppose that QxA − Q
Qxθ > 0, i.e., the overconfident firm is optimisticθ
about the slope in the steady state. In this case, the base misspecification effectis positive, and hence the overconfident firm produces more than in the correctly
specified model. The rival firm best-responds to it and produces less, which improves the overconfident firm’s equilibrium payoff. Hence, the firm’s overconfidence is beneficial even in the long run. It corresponds to the second example (8),QA
where QxA − Q
Qxθ = 1 > 0. In this example, the overconfident firm becomesθ
pessimistic about the demand intercept over time, but its overconfidence about thedemand slope is persistent. Given this optimistic view about the demand slope, inthe steady state, the overconfident firm keeps producing more than the rival firm,and earns a higher long-run profit.
Remark 1. From (9), the indirect learning effect in this Cournot model is


1
∂ θ2
∗
∗ ∂ θ2∗∗∗−Qθ (x1 + x2 , A, θ2 ) + x2
Qxθ (x1 + x2 , A, θ2 ) .
M22 ∂ A∂A
Since we assume Qθ > 0 and Qxθ ≥ 0, this effect is negative. This implies thatlearning mitigates the short-run effect of overconfidence; i.e., in the long run, theoverconfident firm produces less and earns a lower profit than in the corresponding
19

QA
one-shot model. When QxA − Q
Qxθ > 0 or equivalently QQxθθ < QQxAA , this learningθ
effect is relatively small, so overconfidence is still beneficial in the long run. By
contrast, when QQxθθ > QQxAA , the learning effect outweighs the short-run strategic
effect, so overconfidence is detrimental in the long run.

3.2.2

Team production

Consider two players working on a joint project. Each period, each player ichooses an effort level xi , and observes a stochastic output y = Q(x, a, θ ∗ ) + ε
where a is the total capability of the players and θ ∗ is an unknown fundamental. We assume that Q is twice-continuously differentiable and symmetric in that
Q(x1 , x2 , a, θ ) = Q(x2 , x1 , a, θ ). Assume Qxi > 0, Qa > 0, and Qθ > 0. Player
i’s payoff is y − c(xi ), where c(xi ) is the effort cost. Assume c0 > 0. Assume
also that there is a unique Nash equilibrium xcorrect , and it is symmetric in thatx1correct = x2correct .
Suppose that player 2 has first-order misspecification, in that she incorrectlybelieves that the capability is A , a. When A > a, it represents player 2’s overconfidence about her own capability. When A < a, it represents player 2’s underconfidence about her own capability, or prejudice about the opponent’s capability.
Proposition 1 shows that the impact of the misspecification is represented by thebase misspecification effect times the multiplier. In the team production problem,the base misspecification effect is written as


−

1 
 Qx Ai
M22  |{z}+direct effect∂ θ2
Qxi θ 
.
∂A
| {z }(11)

indirect effect

The first term in the brackets is the direct effect of player 2’s bias A on the marginal
productivity, and the second term is the indirect effect through the steady-state
belief θ2 . As is clear from this formula, the sign of the base misspecification
effect crucially depends on the signs of Qxi a and Qxi θ . In what follows, we assume
Qxi a ≤ 0 and Qxi θ > 0, which requires that the marginal return Qxi be negatively
correlated with the capability a, and positively correlated with the fundamental
20

θ .13 The same assumption is imposed in Heidhues, Kőszegi, and Strack (2018), so
by focusing on this case, we can illustrate what is new in our multi-agent setup.14
Assume M22 < 0 as usual. Then, by the assumption Qxi a ≤ 0, the direct effect appearing in (11) is non-positive; in the one-shot game with no learning, theoverconfident player’s effort level is lower than in the correctly specified model.
Intuitively, when Qxi a ≤ 0, the overconfident player thinks that the marginal productivity is lower than the truth, and thus reduces the effort level.
Qa< 0, the indirect learning effect in (11) is
Also, since Qxi θ > 0 and ∂∂θA2 = − Qθnegative. This implies that in the long run, the overconfident player’s effort levelis even lower than in the one-shot game. Intuitively, the overconfident player findsthat the outputs are lower than what she thinks, and becomes pessimistic about θover time, which lowers her effort level. This effect is exactly the self-defeating
property discussed in Heidhues, Kőszegi, and Strack (2018), which states that an
ability to adjust an action depending on an updated belief distorts the action more.
A difference from Heidhues, Kőszegi, and Strack (2018) is that this selfdefeating effect is further amplified by the multiplier 1−BR10 BR0 > 1 in our multi1
2
player setup. Intuitively, if the overconfident player becomes pessimistic about θand reduces the effort level, then the opponent best-respond to it; she increases
the effort if Qx1 x2 < 0, and decreases the effort if Qx1 x2 > 0. This influences theoverconfident player’s optimal action, and in both cases, the overconfident player
13 With this assumption, when A > a, the monotonicity condition stated in Proposition 5 issatisfied, so both the actions and the beliefs indeed converge to a steady state. When A < a, themonotonicity condition fails; but if the identifiability condition stated in Section 5 is satisfied, thenthe actions and the beliefs converge to a steady state. This identifiability condition is satisfied
when Q is linear in θ .
14 It is not difficult to apply the technique here for other cases. For example, suppose that Q < 0
is the damage from drought and agents invest to irrigation which mitigate the damage. Supposethat Q takes a form of


1
1
1
Q=−
+
.
θ x1 + x2 a
In this case, Qxi a ≥ 0 and Qxi θ < 0, so both the direct effect and the indirect effect are positive.
Hence, Corollary 3 holds if we reverse all the inequalities. For example, player 2’s overconfidence
about her capability increases her effort in the one-shot game, and she makes even more effort in
the long run.

21

reduces the effort level further. Due to this mechanism, the impact of misspecification on the steady-state action is larger in our strategic setup than in the singleagent setup.
Our long-run welfare effects are also different from Heidhues, Kőszegi, andStrack (2018). When the players’ actions are strategic substitutes (i.e., Qx1 x2 < 0),
the opponent increases own effort level in response to the overconfident player’s
action, so the overconfident player can free ride. Indeed, if Qx1 x2 < 0, small overconfidence strictly improves own long-run payoff: the above free-riding effectoutweighs the direct cost of misperception.
When A < a, a misspecified player underestimates the total capability of theplayers. For example, the player is underconfident about own capability or has
prejudice about the opponent’s capability. In this case, a player’s small underestimation improves social surplus and the opponent’s long-run payoff. Further, if
Qx1 x2 < 0, it also improves the misspecified player’s long-run payoff. Intuitively,
given such a pessimistic view, the misspecified player becomes optimistic about
the fundamental θ over time. This learning effect outweighs the direct effect ofunderestimation, and the misspecified player contributes more in the long run.
3.2.3

Tournaments

As a third application, we discuss a standard tournament model based on Lazear
and Rosen (1981). Suppose that there are two players. In each period t, eachplayer i chooses an effort level xi and observes a stochastic output y ∈ {w, l}, wherey = w means “player 2 wins” and y = l means “player 1 wins.” The probabilityof y = w (i.e., the probability of player 2 being a winner) is Q(x1 , x2 , a1 , a2 , θ ),where ai denotes player i’s capability and θ is an unknown economic state. We
assume that Qx1 < 0, Qx2 > 0, Qa1 < 0, Qa2 > 0, and Qθ > 0; i.e., player i has abetter chance of winning if she exerts more effort and/or has a better skill. Theseassumptions are satisfied, for example, if
Q(x1 , x2 , a1 , a2 , θ ) = θ22x2 + a2.
x1 + a1 + x2 + a2

(12)

This functional form is commonly used in the literature since Tullock (1980).
The parameter θ represents players’ uncertainty about fairness of the evaluation system: it is a fair contest if θ = 1, but player 1 is favored if θ < 1, andplayer 2 is favored if θ > 1.15 Players’ beliefs about this parameter θ changes
over time, depending on the observed output. A winner receives a payoff W =1, and a loser receives a payoff L = 0. Each agent’s effort cost is c(xi ), andwe assume that c0 > 0. Player 1’s payoff is u1 (x1 , y) = Prob(y = l) − c(x1 ) =[1 − Q(x1 , x2 , a1 , a2 , θ )] − c(x1 ), while player 2’s payoff is u2 (x2 , y) = Prob(y =w) − c(x2 ) = Q(x1 , x2 , a1 , a2 , θ ) − c(x2 ).
Suppose that player 2 has first-order misspecification in that she incorrectlybelieves that her capability is A , a2 . When A > a2 , it represents player 2’s overconfidence about her own capability or prejudice about the opponent’s capability.
When A < a2 , it represents player 2’s underconfidence about her own capability.
This setup is slightly different from the one we have studied so far; we consider the binary signal space Y = {w, l} instead of the continuous signal space.
However, this does not change the steady-state conditions at all, i.e., the conditions (3)-(2) must be satisfied in a steady state in the tournament model withbinary signals. Accordingly, Proposition 1 applies to the tournament model, andthe impact of misspecification is represented by the base misspecification effecttimes the multiplier.
Simple algebra shows that the base misspecification effect in this tournamentmodel is written as


−


1 
Qx A (x1∗ , x2∗ , a1 , A, θ2 ) − ∂ θ2 Qx θ (x1∗ , x2∗ , a1 , A, θ2 ) .
{z} |∂ A 2
M22 | 2{z}direct effect(13)

indirect learning effectThis is exactly the same as the base misspecification effect (11) in the team production, so the results in Section 3.2.2 continue to hold. For example, in Tullock+a1+a1
type tournament (12), we have Qx2 A = −2θ (x +ax1+x< 0 and Qx2 θ = (x +ax1+x
>
+A)3
+A)2
1

1

2

1

1

2

x2 +a215 Another example is Q(x , x , a , a , θ ) = θ +1 2 1 2x1 +a1 +x2 +a2 . In this example, it is a fair contest

if θ = 0, but player 1 is favored if θ < 0, and player 2 is favored if θ > 0. All of the following
discussions hold except that Qxi θ = 0 in this example.

23

0, so assuming M22 < 0, both the direct effect and the indirect effect in the basemisspecification effect are negative. This means that the overconfident player does
not work hard in the one-shot game, and in the long run, her effort level is evenlower than that. Intuitively, the overconfident player incorrectly believes that themarginal return of effort is low (Qx2 A < 0) and does not work hard in the one-shotgame. On top of that, since she wins less frequently than what she thinks, after
a long time, she becomes pessimistic about θ and incorrectly believes that thecontest is unfair. This learning effect further reduces her effort.
However, this base misspecification effect is reduced by the multiplier effect.
Indeed, in this tournament model, the multiplier is
1
1
=
=
0
0
M
21
1 − BR1 BR2 1 − M12 M
M
11

22

1
Q2
1 + M11x1Mx222≤ 1.

Here the inequality follows from the fact that M11 < 0 and M22 < 0 with small misspecification. Note that this inequality is strict whenever Qx1 x2 , 0.16 Intuitively,
when the overconfident player 2 reduces the effort due to the base misspecification
effect, the opponent best-responds to it; she increases the effort if Qx1 x2 < 0, anddecreases the effort if Qx1 x2 > 0. This in turn influences player 2’s optimal action,and in both cases, she increases the effort; this mitigates the base misspecification
effect.
This result is quite different from that in the team production, where the multiplier is greater than one and amplifies the base misspecification effect. A crucial
difference is that players in the tournament have conflicting interests about theoutput y; player 2 prefers y = w while player 1 prefers y = l. Accordingly we havesgn(BR01 ) , sgn(BR02 ), which implies that the multiplier is less than one and strategic interaction weakens the impact of misspecification. By contrast, in the teamproduction, players have a common preference on y, and accordingly we havesgn(BR01 ) = sgn(BR02 ). In this case, the multiplier is larger than one, and strategic interaction strengthens the impact of misspecification. The same argument
16 For example, in the Tullock-type tournament, we have Q

x1 x2 = 0 only when x1 = x2 = 0, so
the multiplier is less than one for all parameter A with which x1first , x2first .

24

applies to a more general setup; in a common interest game, we should expect alarger deviation of long-run actions from a correctly specified model than that in
the single-agent model.

4

Higher-Order Misspecification

This section analyzes higher-order misspecification. Section 4.1 covers secondorder misspecification where a player has an incorrect view about the opponent’sview. Section 4.2 covers double misspecification. It is a combination of firstand second-order misspecification: either one or both players has an incorrect
view about the physical environment and believes that the opponent shares thesame view. Section 4.3 focuses on the case in which a player’s misspecification issmall and derives a simple characterization. Section 4.4 investigates applications
in Cournot duopoly and team production with higher-order misspecification.

4.1

Second-Order Misspecification

In the previous subsection, we consider the situation in which a player has anincorrect view about the physical environment, but correctly understands what
the opponent thinks about the physical environment. However, economic agents
often have various forms of bias about what the opponent thinks. For example,
Madarász (2012) and Gagnon-Bartsch (2016) study the cases in which a player
misperceives the other players’ knowledge/preference, which can be represented
by a in our model. Players may also misperceive other players’ depth of reasoning(see Eyster (2019) for an extensive review).
In this subsection, we consider a long-run impact of a player’s bias about
the opponent’s view about the world. We assume that player 2 has second-ordermisspecification, in that she correctly understands the physical environment, but
has an incorrect view about the opponent’s view about a.17 Formally, we consider
17 As evidence from laboratory experiments, subjects often systematically mispredict other sub-jects’ preferences and actions (Van Boven, Dunning, and Loewenstein, 2000, for example). Lud-25

the following information structure:• Both players believe that for each parameter θ , the signal y is given by
y = Q(x1 , x2 , a, θ ) + ε.
• Player 2 (incorrectly) believes that it is common knowledge that “for eachparameter θ , player 1 believes that the signal y is given by y = Q(x1 , x2 , A, θ )+ε and player 2 believes that the signal y is given by y = Q(x1 , x2 , a, θ ) + ε,”
where A , a.
• Player 1 knows player 2’s information structure above.
Intuitively, this is the case in which player 2 has prejudice, in the sense that she incorrectly believes that she has better information than the opponent does. Player 1
is unbiased, because she knows both the physical environment and the opponent’s
information.
In this setup, player 2 faces inferential naivety. She believes that the opponent
takes an action based on a misspecified model, so she makes an incorrect prediction about the opponent’s play. It turns out that this inferential naivety influences
player 2’s action in two ways. First, player 2 best-responds to this incorrectlypredicted action of the opponent. Second, player 2 interprets an observed signal
conditional on the incorrectly predicted action, which leads to misguided learning.
Assume that players are myopic so that they maximize the expected stagegame payoffs each period. To characterize equilibrium actions when player 2
has second-order misspecification, it is useful to introduce hypothetical player 1
who incorrectly believes that the true parameter is A , a. Player 2 believes thatthe opponent is this hypothetical player 1, so each period, she chooses a Nashequilibrium action against this hypothetical player. The true player 1 correctly
understands player 2’s reasoning, and best responds to player 2’s action.
Formally, let (µ̂1 , x̂t ) denote the action and the belief of the hypothetical player,
and let x = (x1 , x2 , x̂1 ) denote an action profile in the three-player game. The
wig and Nafziger (2011) report that most subjects in their experiments are not aware of or underestimate overconfidence of other subjects.

26

hypothetical player 1’s expected stage-game payoff given θ isÛ1 (x, θ , A) = E[u1 (x̂1 , Q(x̂1 , x2 , A, θ ) + ε)],because she thinks that the parameter is A , a. Player 2’s expected stage-game
payoff isU2 (x, θ ) = E[u2 (x2 , Q(x̂1 , x2 , a, θ ) + ε)],because she thinks that the opponent is the hypothetical player who chooses x̂1 .
Player 1’s subjective expected stage-game payoff is
U1 (x, θ ) = E[u1 (x1 , Q(x1 , x2 , a, θ ) + ε)].
Using these notations, the equilibrium strategy in the infinite-horizon game isdescribed as follows. In period one, all players have the same belief µ11 = µ21 =µ̂11 = µ. So they play a Nash equilibrium (x11 , x21 , x̂11 ), which solves the first-order
)|µ]
)|µ]
)|µ]
= 0, ∂ E[U2∂(x,θ
= 0, and ∂ E[Û1∂(x,θ
= 0. At the end of
conditions ∂ E[U1∂(x,θ
x1
x2
x̂1
1
1
1
∗
period one, players observe a public signal y = Q(x1 , x2 , a, θ ) + ε, and updates
the posterior beliefs using Bayes’ rule. So each player’s belief in period two isµ11 (θ ) f (y − Q(x11 , x21 , a, θ )),
1 (θ̃ ) f (y − Q(x1 , x1 , a, θ̃ ))d θ̃
µ
Θ 1
1 2

µ12 (θ ) = Rµ21 (θ ) f (y − Q(x̂11 , x21 , a, θ )),1
1 1
Θ µ2 (θ̃ ) f (y − Q(x̂1 , x2 , a, θ̃ ))d θ̃µ22 (θ ) = Rµ̂11 (θ ) f (y − Q(x̂11 , x21 , A, θ )).
1
1 1
Θ µ̂1 (θ̃ ) f (y − Q(x̂1 , x2 , A, θ̃ ))d θ̃µ̂12 (θ ) = R

As is clear from this formula, while player 2 correctly knows the parameter a,her posterior µ22 differs from player 1’s posterior µ12 because she uses Bayes’ rule
based on the wrong prediction x̂11 , x11 about player 1’s action. Since actions are
not observable, on the equilibrium path, the beliefs (µ22 , µ̂22 ) are common knowledge between player 2 and the hypothetical player 1. Also player 1 knows the belief profile µ 2 = (µ12 , µ22 , µ̂12 ). So in period two, players play a Nash equilibrium
given this belief profile µ 2 = (µ12 , µ22 , µ̂12 ). Likewise, in any subsequent period27t ≥ 3, players play a Nash equilibrium given the belief profile µ t = (µ1t , µ2t , µ̂1t ),where µ t is computed by Bayes’ rule.
As will be shown in Section 5, under a mild sufficient condition, players’
beliefs and actions almost surely converge to a steady state (x1∗ , x2∗ , x̂1∗ , µ1∗ , µ2∗ , µ̂1∗ )which satisfies the following conditions:x1∗ ∈ arg max U1 (x1 , x2∗ , x̂1∗ , θ ∗ ),(14)

x2∗ ∈ arg max U2 (x2 , x1∗ , x̂1∗ , θ2 ),(15)

x̂1∗ ∈ arg max Û1 (x̂1 , x1∗ , x2∗ , θ̂1 ),(16)µ1∗ = 1θ ∗ ,(17)µ2∗ = 1θ2 s.t. Q(x̂1∗ , x2∗ , a, θ2 ) = Q(x1∗ , x2∗ , a, θ ∗ ),(18)

µ̂1∗ = 1θ̂1 s.t. Q(x̂1∗ , x2∗ , A, θ̂1 ) = Q(x1∗ , x2∗ , a, θ ∗ ).

(19)

x1

x2

x̂1

The first three conditions (14), (15), and (16) are the incentive compatibility conditions, which require that each player maximize her own payoff given some beliefs.
The next three conditions (17), (18), and (19) require that these beliefs satisfy consistency, in that each (actual and hypothetical) player’s belief is concentrated ona state with which each player’s subjective signal distribution coincides with theobjective distribution.
As in the case with first-order misspecification, we assume that for each action
profile x, there is a unique state which solves the consistency condition (18), andwe denote it by θ2 (x, A). This θ2 (x, A) can be interpreted as player 2’s long-run
belief when players choose the same action x each period. Similarly, we assume
that for each x, there is a unique state which solves (19), and we denote it byθ̂1 (x, A). Player 1’s long-run belief is defined as θ1 (x, A) = θ ∗ for all x.
We will characterize how player 2’s misspecification influences the steadystate actions, and to do so, the following notation is useful. For each i, j = 1, 2, 3
(possibly i = j), let
Mi j =∂ 2Ui (x, θ )∂ θi (x, A) ∂ 2Ui (x, θ )+·.
∂ xi ∂ x j θ =θi (x,A)∂xj∂ xi ∂ θ θ =θi (x,A)
28

denote the impact of player j’s action on player i’s marginal utility in the longrun. (Here player 3 refers to the hypothetical player, and x3 , U3 , and θ3 are x̂1 , Û1 ,and θ̂1 , respectively.) The first term is the direct effect, and the second term is theindirect effect through the steady-state belief θi . Then define the slope of player
i’s asymptotic best response curve with respect to player j’s action as
BR0i j = −

Mi j
.
Mii

Intuitively, BR0i j measures how player j’s action influences player i’s optimal
long-run action, while the action of l , i, j being fixed. The slope of player 1’s
asymptotic best response curve, BR012 and BR013 , coincides with that of the standard
best response curve. This is so because she can learn the true state regardless ofthe opponents’ play, and the indirect effects in M11 , M12 , and M13 are zero. In
particular, BR013 = 0, because player 3 is not player 1’s opponent and the direct
effect in M13 is zero. On the other hand, the slopes of the other players’ asymptotic
best response curves are different from those of the standard best response, due
to the indirect effect. For example, BR021 and BR031 need not be zero, even though
players 2 and 3 do not think that player 1 is the opponent. Importantly, the indirect
effects in M21 , M23 , M31 , and M33 do not disappear even in the limit as A → a.
This is so because there is inferential naivety, and θ2 (x, a) and θ̂1 (x, a) can bedifferent from θ ∗ if x1 , x̂1 . This is in a sharp contrast with the case with first-ordermisspecification, where all the indirect effects disappear in the limit as A → a.
Let
M3A :=∂ 2Û1 (x, θ , A)
∂ 2Û1 (x, θ , A)
∂ θ̂1 (x, A)+∂ x̂1 ∂ A
∂ x̂1 ∂ θ∂A
θ =θ̂1 (x,A)θ =θ̂1 (x,A)

denote the impact of the hypothetical player’s bias A on her own marginal utility.
Now we are ready to state the result:Definition 2. A steady state x∗ is regular if the following conditions are satisfied in x∗ : (i) the steady-state action xi∗ is uniquely optimal, (ii) x∗ , θ2 (x∗ , A),and θ̂1 (x∗ , A) are interior points, (iii) BR023 BR032 , 1 and BR012 BR021 + BR023 BR032 +29

BR012 BR023 BR031 , 1, and (iv) Mii < 0 for each i.18
Proposition 2 (Steady State under Second-Order Misspecification). Let x∗ be a
regular steady state for some parameter A∗ .19 Then there is an open neighborhoodof A∗ such that for any value A in this neighborhood, there is a regular steady statex∗ which is continuous with respect to A, and we have



BR023
∂ x2∗
M3A
1
=−
∂A
M33 1 − BR023 BR032
1 − BR012 NE20
∂ x1∗ ∂ x2∗=· BR012∂A∂A
where
NE20 =

BR021 + BR023 BR031.
1 − BR023 BR032(20)

The first equation in this proposition describes how player 2’s second-ordermisspecification influences her own steady-state action x2 , and the second equation states that the rational player 1 simply best-responds to player 2’s play. To
interpret the first equation, recall that the parameter A represents the first-orderbelief (about the physical environment) of the hypothetical player. So when thisparameter A changes, it influences the hypothetical player’s optimal action x̂1 di3A
rectly and indirectly through the steady-state belief. The first term − M
M33 in theequation measures this impact, holding the other players’ actions fixed. Note that18 As in the case with first-order misspecification, the regularity conditions (i) and (ii) ensurethat the steady state is continuous with respect to the parameter A and the first-order condition for
the incentive compatibility is satisfied there. The condition (iii) is needed for the multiplier effectto be well-defined. The condition (iv) ensures that the base misspecification effect and the slopeof the asymptotic best response curve are well-defined. This condition is also useful when weinterpret the base misspecification effect.
0000
19 Under the following additional assumption, we can also show BR23 (BR32 +BR12 BR31 ) < 1. Specif1−BR012 BR021

ically, given x̂1 , let NE(x̂1 ) denote the set of (x1 , x2 ) satisfying (18), (14), and (15) for some θ2 .
Also, given (x1 , x2 ), let BR3 (x1 , x2 ) denote the set of x̂1 satisfying (19) and (16) for some , θ̂1 . If
NE and BR3 are continuous functions (rather than correspondences) and if a steady state is unique,BR0 (BR032 +BR012 BR031 )then 23 1−BR
< 1. A proof is available upon request.
0 BR0
12

21

30

this term is very similar to the base misspecification effect appearing in Proposition 1.
BR0
The second term in the equation, 1−BR0 23BR0 , measures how the hypothetical
23
32
player’s action x̂1 influences player 2’s action, holding player 1’s action being
3A
fixed. When the hypothetical player’s action x̂1 changes by − M
M33 , player 2 bestresponds to it, and her steady-state belief is affected. Accordingly, player 2’s
0
3A
optimal long-run action changes by − M
M33 BR23 . Also, holding player 1’s action
fixed, this effect is amplified by the strategic interaction between player 2 andthe hypothetical player; a change in player 2’s action influences the hypotheticalplayer’s action and belief, which in turn influences player 2’s action and belief,and so on. As in the case with first-order misspecification, this effect is represented by the multiplier 1−BR10 BR0 . So in total, when player 1’s action is fixed,
23
23
player2’s second-ordermisspecification influences her own steady-state action by

BR023
3A
−M
.
M33 1−BR0 BR023

32

The last term in the equation, 1−BR10 NE 0 , measures how player 1’s strategic
12
2play amplifies/reduces the impact of misspecification. To see what it means, it isuseful to define player 2’s asymptotic Nash equilibrium correspondence as
NE2 (x1 ) = {x2 |∃x̂1 satisfying (15), (16), (18), (19)}for each x1 . Intuitively, NE2 (x1 ) denotes player 2’s steady-state action, whenplayer 1 chooses the same action x1 every period while the other players learn
the state and adjust actions. Then the term NE20 appearing in the proposition canbe interpreted as the slope of this Nash equilibrium correspondence NE2 , i.e.,
it measures how a marginal change in player 1’s (constant) action x1 influences
player 2’s steady-state action.20
00

0

20 To see that NE 0 = BR21 +BR23 BR312
1−BR023 BR032

is the slope of NE2 , suppose that the steady-state action(x2 , x̂1 ) is an interior solution for every x1 . Then the following first-order conditions must be
satisfied in any steady state:∂U2= 0,∂ x2 θ2 =θ2 (x,A)

∂ Û1
= 0.
∂ x̂1 θ̂1 =θ̂1 (x,A)

Applying the implicit function theorem to this system of equations (here we regard (x2 , x̂1 ) as a31

With this interpretation in mind, suppose that player 2’s action changes by∆. This influences player 1’s optimal action by BR012 ∆, which in turn influences
player 2’s (and the hypothetical player’s) steady-state beliefs and actions. This
feedback effect on player 2’s action is BR012 NE20 ∆. This process continues infinitely, which results in the multiplier effect 1−BR10 NE 0 .21
122While NE20 is somewhat similar to BR02 appearing in the case of first-ordermisspecification, there are two important differences. First, in NE20 , we consider
the case in which both player 2 and the hypothetical player adjust actions (andplay a Nash equilibrium) every period. In BR02 (and in BR021 ), we consider thecase in which only player 2 adjusts actions. Second, since player 2 does notthink that player 1 is the opponent in the case of second-order misspecification,
BR021 +BR023 BR031
NE20 = 1−BR
involves only the indirect effect; the first term BR021 in the
0 BR0
23
32
numerator represents how player 1’s action influences player 2’s action throughthe steady-state belief, and the second term BR023 BR031 represents how player 1’s
action influences player 2’s action through the hypothetical player’s action. These
effects are amplified by the strategic interaction between player 2 and the hypothetical player, and hence we have 1 − BR023 BR032 in the denominator.

4.2

Double Misspecification

In many economic situations of our interest, a biased player naively thinks that her
own view about the world is absolutely correct and the opponent share the sameview about the world. This happens, for example, when a player is completely
positive about own capability.22 Alternatively, a player may be unaware of orfunction depending on the parameter x1 ), we indeed have ∂∂ xx2 =1

BR021 +BR023 BR031.
1−BR023 BR032

21 In this argument, we implicitly use the fact that player 1’s optimal action is not affected by

the hypothetical player’s action.
22 For example, Camerer and Lovallo (1999) provide lab-experimental evidence of overconfidence in strategic entry settings, and Benoı̂t, Dubra, and Moore (2015) provide lab-experimental
evidence of overconfidence which excludes the possibility of rational Bayesian reasoning. See
Kőszegi (2014) and Grubb (2015) for theoretical studies applying overconfidence in strategic settings.

32underestimate a particular aspect of the environment.23
In our framework, such a situation can be described by considering a doubly
misspecified agent, who has both the first-order misspecification (she incorrectlybelieves that the true parameter is A , a) and the second-order misspecification(she incorrectly believes that the opponent thinks that the true parameter is A). We
first consider the case in which only player 2 is misspecified. Then we consider
the case in which both players are misspecified.
4.2.1

One-Sided Double Misspecification

We consider the case in which only player 2 is misspecified. Specifically, we
assume that:• Player 2 (incorrectly) believes that for each parameter θ , the signal y isgiven by y = Q(x1 , x2 , A, θ ) + ε, where A , a.
• Player 2 (incorrectly) believes that it is common knowledge that “the signal
y is given by y = Q(x1 , x2 , A, θ ) + ε.”• Player 1 knows player 2’s information structure above.
With this information structure, player 2 has an incorrect view about the parametera, and in addition, she has inferential naivety in that she incorrectly believes that
player 1 takes an action based on a misspecified model. Player 1 is unbiased,in the sense that she correctly understands the true parameter a and she knows
player 2’s information structure (which allows her to make a correct prediction
about player 2’s action).
Assume again that the agents are myopic, so that they maximize the expectedstage-game payoff each period. As in the case of second-order misspecification,
we consider a hypothetical player 1 who thinks that it is common knowledge that
23 For example, consumers often systematically ignore or underestimate a specific type of

fee/risk; see Heidhues and Kőszegi (2018) for theoretical applications and evidence.

33

the true parameter is A , a. Let x = (x1 , x2 , x̂1 ) denote an action profile in the threeplayer game, and let Û1 (x, θ , A) denote the hypothetical player’s stage-game payoff, U2 (x, θ , A) denote player 2’s stage-game payoff, and U1 (x, θ ) denote player
1’s stage-game payoff. Note that player 2 and the hypothetical player evaluates
the expected payoff assuming that the signal is given by y = Q(x̂1 , x2 , A, θ ) + ε.
The equilibrium strategy in the infinite-horizon game is very similar to that in thecase of the second-order misspecification; we only need to replace the parametera which appears in player 2’s expected payoff and Bayes’ formula with the biased
parameter A.
In this environment, the following conditions must be satisfied in a steadystate:
x1∗ ∈ arg max U1 (x1 , x2∗ , a, θ ∗ ),(21)

x2∗ ∈ arg max U2 (x̂1∗ , x2 , A, θ2 ),(22)

x̂1∗ ∈ arg max Û1 (x̂1 , x2∗ , A, θ2 ),(23)µ1∗ = 1θ ∗ ,(24)µ2∗ = µ̂1∗ = 1θ2 s.t. Q(x̂1∗ , x2∗ , A, θ2 ) = Q(x1∗ , x2∗ , a, θ ∗ ).

(25)

x1

x2

x̂1

The first three conditions (21), (22), and (23) are incentive compatibility conditions, which require that each player maximizes her payoff given some beliefs.
The next two conditions (24) and (25) assert that these beliefs satisfy consistency,
in that each player’s belief is concentrated on a state under which each player’s
subjective signal distribution coincides with the objective distribution. Note thatthe hypothetical player’s belief is exactly the same as player 2’s belief, as theyboth believe that it is common knowledge that the true parameter is A. We assume
that for each action profile x and parameter A, there is a unique state θ2 (x, A)
which solves (25). Intuitively, this θ2 (x, A) is player 2’s long-run belief whenplayers choose the same action profile x every period. Player 1’s long-run beliefis θ1 (x, A) = θ ∗ .
Define the slope of player i’s asymptotic best response curve with respect to34

player j’s action as
BR0i j := −

Mi j
Mii

where for each i, j = 1, 2, 3 (possibly i = j),
Mi j =∂ 2Ui (x, θ )∂ θi (x, A)∂ 2Ui (x, θ )+.
∂ xi ∂ x j θ =θi (x,A)∂ xi ∂ θ θ =θi (x,A) ∂ x j

measures the impact of player j’s action on player i’s marginal utility in the longrun. Here again, player 3 refers to the hypothetical player, and her action, belief,
and utility are denoted by x3 , θ3 , and U3 rather than x̂1 , θ̂1 , and Û1 .
For each i = 2, 3, let
MiA :=∂ 2Ui (x, θ , A)
∂ 2Ui (x, θ , A)
∂ θi (x, A)+∂ xi ∂ A∂ xi ∂ θ∂Aθ =θi (x,A)θ =θi (x,A)

denote the impact of player i’s first-order misspecification on her marginal utility.
The following proposition characterizes how player 2’s double misspecificationinfluences the steady-state actions.
Definition 3. A steady state x∗ is regular if the following conditions are satisfied
in x∗ : (i) the steady-state action xi∗ is uniquely optimal, (ii) x∗ and θ2 (x∗ , A) =θ̂1 (x∗ , A) are interior points, (iii) BR023 BR032 , 1 and BR012 BR021 + BR023 BR032 +BR012 BR023 BR031 , 1, and (iv) Mii < 0 for each i.
Proposition 3 (Steady State under One-Sided Double Misspecification). Let x∗ bea regular steady state for some parameter A∗ . Then there is an open neighborhoodof A∗ such that for any value A in this neighborhood, there is a regular steady statex∗ which is continuous with respect to A, and we have




∂ x2∗
M2A M3A 0
1
1
=−
+
BR
,
∂A
M22 M33 23
1 − BR023 BR032
1 − BR012 NE20
∂ x1∗ ∂ x2∗=· BR012∂A∂A
where NE20 is defined by (20).
35The first equation in this proposition characterizes how player 2’s double misspecification influences her own steady-state action. This is very similar to thefirst equation in Proposition 2. The only difference is that here player 2 has firstorder misspecification about the parameter a, which influences her optimal action
2A
by the base misspecification effect − M
M22 . All other terms are the same as thosein Proposition 2. The second equation in the proposition simply states that therational player 1 best-responds to player 2’s action.
4.2.2

Two-Sided Double Misspecification

Now consider the case in which both players are misspecified. We assume:• Each player i (incorrectly) believes that for each parameter θ , the signal y
is given by y = Q(x1 , x2 , Ai , θ ) + ε, where Ai , a.
• Each player i (incorrectly) believes that it is common knowledge that “thesignal y is given by y = Q(x1 , x2 , Ai , θ ) + ε.”We allow A1 , A2 , so the dufferent players may have different levels of misspecification. Note that even when Ai = a, player i may be biased, in the sense that she
may not know the opponent’s bias about the technology a. That is, when Ai = a,player i believes that the opponent believes that the technology is a, but in reality,
the opponent may believe A j , a.
For find an equilibrium in this environment, it is useful to consider two hypothetical players. Hypothetical player 1 is player 1 who thinks that it is common
knowledge that the true technology is A2 . Hypothetical player 2 is player 2 who
thinks that it is common knowledge that the true technology is A1 . Note that player
1 and hypothetical player 2 think that they play the game each other, and the same
is for player 2 and hypothetical player 1. Let x̂i , µ̂i , and Ûi denote hypotheticalplayer i’s action and belief.
Then a steady state in this environment satisfies the consistency and incentive-

36

compatibility conditions:xi∗ ∈ arg max Ui (xi , x̂∗j , Ai , θi ) ∀i, j , i(26)x̂i∗ ∈ arg max Ûi (x̂i , x∗j , Ai , θ j ) ∀i, j , i,(27)

µ1∗ = µ̂2∗ = 1θ1 s.t. Q(x1∗ , x̂2∗ , A1 , θ1 ) = Q(x1∗ , x2∗ , a, θ ∗ )(28)

Q(x̂1∗ , x2∗ , A2 , θ2 ) = Q(x1∗ , x2∗ , a, θ ∗ ).

(29)

xi
x̂i

µ2∗ = µ̂1∗ = 1θ2s.t.

We assume that for each action profile x and parameter Ai , there is a unique stateθi (x, Ai ) = θ̂ j (x, Ai ) which solves the consistency condition Q(xi , x̂−i , Ai , θi ) =Q(x1 , x2 , a, θ ∗ ). This is player i’s (and the hypothetical player j’s) long-run belief when the same action profile x is chosen every period.
Define the slope of player i’s asymptotic best response curve with respect toplayer j’s action as
Mi j
BR0i j := −Miiwhere for each i, j = 1, 2, 3, 4 (possibly i = j),
Mi j =∂ 2Ui (x, θ )∂ 2Ui (x, θ )∂ θi (x, A)+.
∂ xi ∂ x j θ =θi (x,A)∂ xi ∂ θ θ =θi (x,A) ∂ x j

measures the impact of player j’s action on player i’s marginal utility in the longrun. Here, players 3 and 4 refer to the hypothetical players 1 and 2, respectively.
Their actions, beliefs, and utilities are denoted by x3 , x4 , θ3 , θ4 , U3 , and U4 , rather
than x̂1 , x̂2 , θ̂1 , θ̂2 , Û1 , and Û2 . Note that M13 = M24 = M34 = M43 = 0, and thus
BR013 = BR024 = BR034 = BR043 = 0. All other Mi j involve indirect effects.
For each i = 2, 3, let
MiA :=∂ 2Ui (x, θ , A)
∂ 2Ui (x, θ , A)
∂ θi (x, A)+∂ xi ∂ A∂ xi ∂ θ∂Aθ =θi (x,A)θ =θi (x,A)

denote the impact of player i’s first-order misspecification on her marginal utility.
The following proposition characterizes how player 2’s double misspecificationinfluences the steady-state actions.
37

Definition 4. A steady state x∗ is regular if the following conditions are satisfied
in x∗ : (i) the steady-state action xi∗ is uniquely optimal, (ii) x∗ and θ2 (x∗ , A) =θ̂1 (x∗ , A) are interior points, (iii) BR014 BR041 , 1, BR023 BR032 , 1, and BR014 BR041 +BR023 BR032 + (BR021 + BR023 BR031 )(BR012 + BR014 BR042 ) , 1, and (iv) Mii < 0 for eachi.
Proposition 4 (Steady State under Two-Sided Double Misspecification). Let x∗
be a regular steady state for some parameter A∗ = (A∗1 , A∗2 ). Then there is an open
neighborhood of A∗2 such that for any value A2 in this neighborhood, there is aregular steady state x∗ which is continuous with respect to A2 , and we have




∂ x2∗
M2A
1
1
0 M3A
=−
+ BR23
,
∂ A2
M22
M33
1 − BR023 BR032
1 − NE10 NE20
∂ x1∗ ∂ x2∗=· NE10∂ A2 ∂ A2
where NE20 is defined by (20) and NE10 is similarly defined as
NE10 =

BR012 + BR014 BR042.
1 − BR014 BR041

The equations in this proposition are very similar to those in Proposition 3,
and the only difference is that the term BR012 in Proposition 3 is replaced with
NE10 . Recall that under one-sided double misspecification, player 1 is rationaland best responds to player 2’s play. The term BR012 in Proposition 3 quantifies
this effect. On the other hand, under two-sided double misspecification, player
1 is also misspecified, and when player 2’s action changes, it influences player
1’s steady-state action in a more complicated way. The term NE10 quantifies thiseffect. More precisely, NE10 can be regarded as the slope of player 1’s asymptotic
Nash equilibrium correspondence defined as
NE1 (x2 ) = {x1 |∃xˆ2 satisfying (26) for i = 1, (27) for i = 2, (28)}.
That is, NE10 considers the case in which player 2 chooses the same action x2 every
period, and measures how a marginal change in x2 influences player 1’s steadystate action. Just like NE20 , this NE10 involves only the indirect effect, because
player 1 does not think that player 2 is the opponent.
38

4.3

Small Misspecification: Limit as A → a

So far we have characterized how a player’s bias influences actions for a general
misspecified parameter A. In this subsection, we will consider the case in which
player 2 has small misspecification, in the sense that the misspecified parameter
A is close to the true value a. We will show that in this special case, the multipliers derived in Propositions 1-4 can be replaced with much simpler forms, whichallows us to make a clean comparison of the impacts of first-order, second-order,
and double misspecification.
Let xfirst , xsecond , xdouble , and xcorrect denote the steady-state action profile with
first-order misspecification, second-order misspecification, one-sided double misspecification, and the correctly specified model, respectively. Also, let xdouble2 denote the steady-state action profile with two-sided double misspecification when
A1 = a and A2 = A; i.e., player 1 knows that the true technology is a, but does notrecognize player 2’s bias A. A game is symmetric if X1 = X2 , u1 (x1 , y) = u2 (x2 , y)for all x1 and x2 with x1 = x2 , and Q(x1 , x2 , a, θ ) = Q(x2 , x1 , a, θ ).
Proposition 5. Suppose that in each misspecification, there is a unique steadystate at A = a and it is regular. Then we have the following result:(i) At A = a, x1first = x1second = x̂1second = x1double = x̂1double = x1double2 = x̂1double2 =x1correct and x2first = x2second = x2double = x2double2 = x̂2double2 = x2correct .
(ii) At A = a,first

∂ xsecond∂ x2double ∂ x2
=
+ 2
.
∂A
∂A
∂A
(iii) Assume in addition that the game is symmetric and that x1correct = x2correct .

39

Then at A = a,first

∂ x2second
Ui j − L ∂ x2
=−
·
,
∂A
Uii − L ∂ A firstfirst∂ x2double ∂ x2
∂ x2second
Ui j − L ∂ x2
=
+
= 1−
,
∂A
∂A
∂A
Uii − L∂A
∂ x2double2 (Uii − L)(Uii +Ui j − L) ∂ x2double=,∂A
Uii (Uii +Ui j − 2L)∂A∂ x1double2
Ui j ∂ x2double2
L
1−
=−
,
∂A
Uii
Uii
∂A
2

correct ,θ ∗ )where Uii = ∂ U1 (x∂ x2∂ 2U1 (xcorrect ,θ ).
∂ x1 ∂ θ

1

2

correct∗

Q, Ui j = ∂ U1∂(xx ∂ x ,θ ) , and L = ∂∂ θ̂x1 ∂ x̂Û1∂ θ = Qxθ1 ·
1

2

1

1

Part (i) shows that if there is no misspecification in that A = a, the steady state
actions are the same for all cases. Part (ii) shows that when misspecification issmall, the impact of one-sided double misspecification can be decomposed intothat of first-order misspecification and that of second-order misspecification. This
result essentially follows from the fact that double misspecification is a combination of first-order misspecification and second-order misspecification.
Part (iii) considers a symmetric game, and compares the impact of various
misspecification, and all these equations follow from simple algebra. The term L
measures the indirect learning effect on the hypothetical player’s marginal utility,
when the real player (secretly) increases the action.

4.4

Higher-Order Misspecification: Applications

To highlight the effects of higher-order misspecification, this subsection investigates applications to Cournot duopoly and team production, with higher-ordermisspecification. We also discuss an application to gender bias, with a focus on
how a teacher’s bias can be endogenously transmitted to a student under two-sided
double misspecification.

40

4.4.1

Cournot duopoly

Consider the Cournot duopoly model introduced in Section 3.2.1. Let xicorrect andπ correct denote firm i’s steady-state action and payoff in the correctly specified
model. The following corollary is an immediate consequence of Proposition 5.
It shows that the long-run impact of misspecification crucially depends on theparameter QQxθθ , which measures the normalized impact of the state θ on the demand slope. We assume that steady state is unique in each misspecification at
a = A1 = A2 .24
Corollary 2. Suppose that in each misspecification, there is a unique steady state
at A = a and it is regular.25 Suppose also that the base misspecification effect (10)is positive at A = a, i.e., QQxθθ < QQxAA in the unique Nash equilibrium in the correctly
specified model with known θ ∗ . Then, the following results hold.
(i) Under first-order misspecification and one-sided double misspecification,2)
2)
we have ∂∂xA1 < 0, ∂∂xA2 > 0, ∂ (x∂1 +x> 0, ∂∂πA1 < 0, ∂∂πA2 > 0, and ∂ (π∂1 +π<0AAat A = a both in the one-shot game with known θ ∗ and in the long-run
steady state.
(ii) Under second-order misspecification, all inequalities in (i) are reversed in
the one-shot game with known θ ∗ . In the long-run steady state, all inequalities in (i) remain true if QQxθθ > QQxxx , but are reversed if QQxθθ < QQxxx .
(iii) Under two-sided double misspecification with A1 = a, there is A2 > A2 suchthat for all A2 ∈ (a, A2 ), we have x1 = x1correct , x2 > x2correct , and πi < πicorrect
for each i in the one-shot game with known θ ∗ . In the long-run steady state,
all inequalities in (i) remain true.
24 In Appendix B, we provide a sufficient condition for the unique steady state in each misspec-

ification.
25 Regularity in the case of second-order misspecification requires M = U − L < 0, which is
ii
33
c00
.
Other
conditions
statedinthedefinitionof regularity
satisfied if and only if QQxθ < QQxxx + 1x − xQxθare satisfied for generic parameters.

41

If the base misspecification effect is negative (i.e., QQxθθ > QQxAA ), all the inequalities
remain true in the one-shot game with known θ ∗ , but are reversed in the long-run
steady state.
Part (ii) of the corollary considers the case with second-order misspecification,in the sense that firm 2 incorrectly believes that firm 1 is overconfident. If thebase misspecification effect is positive, in the long run, firm 2 incorrectly believes
that firm 1 produces more than in the correctly specified model, i.e., ∂∂x̂A1 > 0.26
This incorrect prediction has two countervailing forces on firm 2’s action. First,
there is a direct effect, in that firm 2 best-responds to this incorrect prediction andproduces less. Second, firm 2 observes prices higher than the anticipation andbecomes optimistic about the state θ ; this indirect learning effect increases the
incentive to produce. More formally, simple algebra shows thatUi j − L ∂ x̂1
∂ x2
∂ x̂1
= BR023=−.
∂A∂A
Uii ∂ AIn this Cournot model, Ui j = Qx + x2 Qxx and L = QQθx (Qθ + xi Qxθ ). Plugging this
into the above equation,
indirect effectindirect effect z }| {z }| { effectdirect effectdirectz}|{∂ x2
1  z}|{Qx
Qx
 ∂ x̂1=−Qx −Qθ +x2  Qxx −
Qxθ 
.
∂A
Uii 
Qθ
Qθ
∂A




{z
}
|
{z
} 


|
on the price levelon the slope

Note that the first two terms in the curly brackets cancel out; this happens because
firm 2 correctly predicts the price in the steady state. Hence, we have


∂ x2
x2
Qx
∂ x̂1
=−
Qxx −
Qxθ
.
∂A
Uii
Qθ
∂A
That is, firm 2’s steady-state action is determined by its subjective view about theslope in the steady state. As stated in the corollary, if Qxx − QQθx Qxi θ > 0 so thatfirm 2 is optimistic about the slope, then it produces more than in the correctly
26 Formally, this follows from Proposition 5 (iii), U < 0, U − L < 0, and the base misspecifiii
ii

cation effect being positive.

42

specified model, i.e., ∂∂xA2 > 0. On the other hand, firm 2 produces less if it ispessimistic about the slope.
Part (i) of the corollary considers the case with one-sided double misspecification, and shows that its impact on the steady state is similar to that of first-ordermispecification. That is, when firm 2 is overconfident and the opponent knowsit, firm 2’s higher-order misspecification does not have a substantial impact on
the steady state. To see the intuition, suppose first that Qxx − QQθx Qxi θ > 0. In
this case, both first-order misspecification and second-order misspecification increase firm 2’s production. Since double-misspecification is a combination ofthese two misspecifications (formally, see Proposition 5(ii)), it is immediate thatfirm 2 produces even more under one-sided double misspecification. Suppose
next that Qxx − QQθx Qxi θ < 0. In this case, second-order misspecification has an
opposite impact on firm 2’s production, that is, firm 2 produces less under secondorder misspecification. However, as we show in the proof, this negative impact(of second-order misspecification) is outweighed by the positive impact of firstorder misspecification. So double misspecification increases firm 2’s production,although its impact is smaller than that of first-order misspecification.
Part (iii) of the corollary considers the case with two-sided double misspecification, where firm 1 is not aware of firm 2’s overconfidence. In this case, firm
2’s overconfidence hurts both firms in the short-run. This is because the strategic
effect does not exist in this environment; although the overconfident firm 2 produces more, firm 1 is not aware of it and chooses the Nash equilibrium action in
the correctly-specified model. So firm 2’s overconfidence simply makes her action
suboptimal, just as any bias does in the single-agent problem.
However, in the long run, firm 2’s overconfidence can still improve the equilibrium payoff. A key is the learning effect on firm 1: Since firm 1 is not awareof overproduction by the opponent, it observes a market price lower than the anticipation. Accordingly, firm 1 becomes pessimistic about θ , and in particular
about the demand slope. This means that firm 1 produces less in the long run,which yields a higher profit to the overconfident firm 2 even in the absence ofthe strategic effect. This is an important difference from the case with first-order43

misspecification or one-sided double misspecification, in which the overconfident
firm benefits from the strategic effect. Also, under two-sided double misspecification, firm 2’s overconfidence may improves the total welfare π1 + π2 if thislearning effect is large enough. As stated in parts (i) and (ii) of Corollary 2, thiscannot happen in first-order misspecification and one-sided double misspecification, in which there is no learning effect on firm 1. These results highlight how theawareness about an opponent’s bias makes qualitative differences in multi-player
learning models.
4.4.2

Team production

In Section 3.2.2, we have focused on team production first-order misspecification.
However, various forms of higher-order misspecifications are relevant in practice.
For example, a player may have second-order misspecification and incorrectly believes that the opponent has overconfidence, underconfidence, or prejudice. Or,a player may have double misspecification; she has overconfidence, underconfidence, or prejudice as in the case of first-order misspecification, and on top of
that, she naively thinks that the opponent shares the same view. Using Proposition
5, we can quantify the impact of such misspecification. Again, we assume thatsteady state is unique in each misspecification at a = A1 = A2 .
Corollary 3. Suppose that in each misspecification, there is a unique steady state
Qx Q
at A = a and it is regular. Let L = Qi θxi θ > 0 be as in Proposition 5.
(i) Under first-order misspecification and one-sided double misspecification,(a) If Qx1 x2 > 0, then in the one-shot game with known θ ∗ , ∂∂xA2 ≤ ∂∂xA1 ≤ 0
and ∂∂πA1 ≤ ∂∂πA2 ≤ 0 at A = a with strict inequalities if Qxi a , 0. In thelong-run steady state, all the inequalities are strict for any Qxi a ≤ 0.
(b) If Qx1 x2 < 0, then in the one-shot game with known θ ∗ , ∂∂xA2 ≤ 0 ≤ ∂∂xA1 ,∂ (x1 +x2 )2)≤ 0, ∂∂πA1 ≤ 0 ≤ ∂∂πA2 , and ∂ (π∂1 +π≤ 0 at A = a with strict∂AA
inequalities if Qxi a , 0. In the long-run steady state, all the inequalities
are strict for any Qxi a ≤ 0.
44

(ii) Under second-order misspecification,(a) If Qx1 x2 > L, then in the one-shot game with known θ ∗ , ∂∂xA2 ≤ ∂∂xA1 ≤ 0
and ∂∂πA1 ≤ ∂∂πA2 ≤ 0 at A = a with strict inequalities if Qxi a , 0. In thelong-run steady state, all the inequalities are strict for any Qxi a ≤ 0.
(b) If Qx1 x2 ∈ (0, L), then in the one-shot game with known θ ∗ , ∂∂xA2 ≤ ∂∂xA1 ≤0 and ∂∂πA1 ≤ ∂∂πA2 ≤ 0 at A = a with strict inequalities if Qxi a , 0. In thelong-run steady state, ∂∂xA2 > ∂∂xA1 > 0 and ∂∂πA1 > ∂∂πA2 > 0 at A = a.
(c) If Qx1 x2 < 0, then in the one-shot game with known θ ∗ , ∂∂xA1 ≤ 0 ≤ ∂∂xA2 ,∂ (x1 +x2 )2)≥ 0, ∂∂πA2 ≤ 0 ≤ ∂∂πA1 , and ∂ (π∂1 +π≥ 0 at A = a with strict∂AA
inequalities if Qxi a , 0. In the long-run steady state, all the inequalities
are strict for any Qxi a ≤ 0.
(iii) Under two-sided double misspecification with A1 = a, in the one-shot gamewith known θ ∗ , there is A2 > a such that for any A2 ∈ (a, A2 ), x1 = x1correct ,x2 ≤ x2correct , π1 ≤ π2 ≤ πicorrect with strict inequalities if Qxi a , 0. In thelong-run steady state, ∂∂ xAi < 0 and ∂∂πAi < 0 at A = a.
Part (i) considers the case in which player 2 is overconfident and player 1
knows it. It shows that the overconfident player reduces the effort, regardless ofwhether she knows the opponent’s information or not. This result follows fromthe fact that the base misspecification effect is negative. It also shows that player
1 simply best-responds to this player 2’s action; she increases the effort understrategic substitutes, and reduces the effort under strategic complements. Overconfidence always decreases the total surplus, and the overconfident player 2 always obtains a better payoff than the unbiased player 1.
Part (ii) considers the case in which both players know the true parametera, but player 2 incorrectly believes that the opponent is overconfident. The result
depends on the parameter Qx1 x2 , and notably, when there is weak complementarity,i.e., Qx1 x2 ∈ (0, L), both players work less in the short run, but work harder in thelong run. The intuition is as follows. Due to the bias, player 2 underestimates
player 1’s effort. In the one-shot game, player 2 simply best-responds to it by45

reducing the effort. But in the long run, there is a learning effect; player 2 issurprised by the actual output being better than the anticipation, and over time,
she becomes optimistic about θ and provides an extra incentive to work hard.
When complementarity is weak, this effect dominates the negative impact in the
one-shot game, so player 2 increases the effort.
Part (iii) considers the case in which player 1 is not aware of player 2’s overconfidence. An important difference from part (i) is that player 1 reduces the effort
in the long run, regardless of the parameter Qx1 x2 . This is due to the learning effect. Since player 1 overestimates player 2’s effort, she is disappointed by low
output. Hence, over time, she becomes pessimistic about θ and reduces the effort.
In this case, both players reduce the effort, and both players earn a lower long-run
payoff.
4.4.3

Transmission of Bias

There is a long-standing debate on whether the gender gap in math achievementarises from biological reasons (such as brain functioning) as opposed to culture
and social conditioning. There are recent works which support the latter: For
example, Carlana (2019) finds that the gender gap in performance in math exam
substantially increases when students are assigned to math teachers with stronger
gender stereotypes. She argues that this effect is at least partially driven by lower
self-confidence on math ability of girls exposed to gender-biased teachers. We
will show that our framework is useful to explain such a bias transmission fromteachers to students.
Consider a variant of our team-production model in two-sided double misspecification. Suppose that player 1 is a student and player 2 is a teacher. The student’s
achievement (e.g., math test performance) is given by y = a(x1 +x2 +b)+ε, wherea represents a gender-specific capability and b is the student’s capability. The student (player 1) knows her own capability b, but does not know the gender-specificcapability. So she thinks that the outcome is given by y = θ1 (x1 + x2 + b) + ε
and learns θ1 over time. On the other hand, the teacher has a biased view about
the gender specific capability, and he thinks that the outcome is given by y =46

A2 (x1 + x2 + θ2 ) + ε, where A2 < a represents his bias. He does not know thestudent’s individual capability θ2 , and learns it over time. We assume that eachplayer (incorrectly) thinks that the opponent has the same view about the world,This means that the student is not aware of the teacher’s gender-stereotype A2 < a.
This setup is different from the one presented in Section 4.2.2, in that different
players learn different parameters. However, this does not has a substantial impact
on the property of the steady state. Indeed, it is straightforward to show that theresult similar to Corollary 3 still holds, and in the steady state, both the teacher
and the student work less than in the correctly specified model.
The intuition behind this result is as follows. Since the teacher has a genderstereotype A2 < a, each period, he exerts less effort than in the correctly specified model. Because the student is not aware of it, from the student’s point ofview, the realized outcomes are systematically lower than the anticipation. Accordingly, after a long time, she becomes unrealistically pessimistic about thegender-specific capability θ1 , which weakens the incentive to work. So in thisframework, even though the student initially has an unbiased view about the environment, the teacher’s gender bias is eventually transmitted to the student, due tothe second-order misspecification (i.e., the student’s unawareness of the teacher’sbias).
Recent work by Heidhues, Kőszegi, and Strack (2020b) also argue that a gender bias (more generally, a group discrimination) can endogenously arise as a consequence of misspecified learning. Formally, they develop a single-agent learningmodel, and show that an underconfident (resp. overconfident) agent tends to underestimate (resp. overestimate) the capability of her in-group members. So intheir setup, the source of a group discrimination is one’s misconfidence about
her own capability. Our result complements their work by considering the casein which an agent (a student in our context) does not have underconfidence, ormore generally, any bias about the physical environment. We find that a groupdiscrimination can still arise due to a bias transmission; one’s existing prejudice
may induce other players’ negative self-stereotypes through learning.

47

5

Theorems for ConvergenceThis section generalizes Esponda, Pouzo, and Yamamoto (2019) by allowing multiple players and continuous actions. We show that the asymptotic motion of players’ action frequency is approximated by a differential inclusion, and we use it toderive a sufficient condition for convergence.

5.1

General Setup

For each compact set A ⊂ Rn (or more generally, separable metric space A), let
4A denote the set of probability measures over the set A. We consider the dualbounded-Lipschitz norm on 4A, that is, for each µ ∈ 4A, let
kµk = supZf dµf ∈BL(A) A

where BL(A) is the set of bounded Lipschitz continuous functions f on A withf (y)|≤ 1. This norm has two nice properties. First,
supx∈A | f (x)| + supx,y | f (x)−|x−y|it metrizes the weak topology, that is, the topology induced by the dual boundedLipschitz norm coincides with the weak topology on 4A. Second, with this norm,
4A is a compact subset of a Banach space, i.e., the set of finite signed measures on
A is a Banach space when paired with the dual bounded-Lipschitz norm, and 4A is
a compact subset in it. See Dudley (1966) and Billingsley (1999) for references.
The first property is needed to obtain our Propisition 8. The second property
is crucial in order to use a stochastic approximation technique in the proof of
Proposition 9. The dual bounded-Lipschitz norm is used in Hofbauer, Oechssler,
and Riedel (2009) and Perkins and Leslie (2014), who study learning dynamics ingames with continuous actions.
5.1.1

Objective World

There are two players i = 1, 2 and infinitely many periods t = 1, 2, · · · . In eachperiod t, each player i chooses an action xi from a compact set Xi ⊂ R. These
actions are not observable. Then they observe a noisy public output y ∈ Y which48

is distributed according to a probability measure Q(·|x) ∈ 4Y , where x = (x1 , x2 )denotes the chosen action profile. Each player i’s payoff is ui (xi , y).
In the infinite-horizon game, each player i’s t-period history is hti = (xiτ , yτ )tτ=1 ,where (xti , yt ) is player i’s action and the public outcome in period t. Let Hit denotethe set of all t-period history, and let Hi0 = {0}.
/ Player i’s pure strategy in theS∞infinite-horizon game is a mapping si : t=0 Hit → Xi . Let Si denote the set ofplayer i’s pure strategies. Let hYt = (yτ )tτ=1 denote the t-period public history. A
strategy is public if it depends only on public histories.
5.1.2

Subjective World and Model Hierarchy

We assume that the output distribution Q is not common knowledge among players. Instead, each player i has a set Θi,1 of subjective models, and in each model
θi,1 ∈ Θi,1 , the output distribution given an action profile x is Qθi,1 (·|x). Player
i thinks that the true world is described by one of these models, and her initialprior about the model is µi,1 ∈ 4Θi,1 . Player i’s models are correctly specified if
there is θi,1 such that Q(·|x) = Qθi,1 (·|x) for all x. Otherwise her models are misspecified. Player i also has models about the opponent j’s model, that is, player ibelieves that the opponent j has an initial prior µi,2 over a model set Θi,2 , where
each model θi,2 induces the output distribution Qθi,2 (·|x) for each action profile x.
This triplet Mi,2 = (µi,2 , Θi,2 , (Qθi,2 (·|x))(x,θi,2 ) ) is player i’s second-order model inthat it is her subjective view about player j’s subjective model. More generally, we
assume that each player i has a model hierarchy Mi = (Mi,1 , Mi,2 , · · · ) where each
Mi,k = (µi,k , Θi,k , (Qθi,k (·|x))(x,θi,k ) ) is player i’s kth-order model. That is, player ibelieves that player j believes that player i’s model is Mi,3 , player i believes that
player j believes that player i believes that player j’s model is Mi,4 , and so on.
This framework is flexible and allows us to study a variety of information
structures, including first-order misspecification, second-order misspecification,and double misspecification which we studied in Sections 3 and 4. In what follows, we will maintain the following technical assumptions.
Assumption 1. The following conditions hold:49

(i) Y and Θ are Borel subsets of the Euclidean space, and Θ is compact.
(ii) There is a Borel probability measure ν ∈ 4Y such that Q(·|x) and Qθi,k (·|x)
are absolutely continuous with respect to ν for all x and i, k, and θi,k .
(An implication is that there are densities q(·|x) and qθi,k (·|x) such thatRR
A q(y|x)ν(dy) = Q(A|x) and A qθi,k (y|x)ν(dy) = Qθi,k (A|x) for any A ⊆ Y
Borel.)(iii) q(·|x) and qθi,k (·|x) are continuous in θ and x.
(iv) There is a function g : X ×Y → R such that (a) for each y, g(x, y) is continuous in x, (b) g(x, ·) ∈ L2 (Y, Q(·|x)) for each x, and (c) for all x, x̂ i, k, andθi,k , log q q(·|x)(·|x̂) ≤ g(x, ·) Q(·|x)-a.s..
θi,kThe parts (i)-(iii) are fairly standard. The part (iv) implies that every outcome
y is generated by each player i’s model, which is useful to establish a uniform
version of the law of large numbers. The assumption above is similar to Assumptions 1 and 2 of Esponda, Pouzo, and Yamamoto (2019), but there are twodifferences. First, we allow the action set Xi to be continuous, in which case we
require continuity of q, as described in parts (iii) and (iv-a). Second, we allow
inferential naivety, so when we consider the log-likelihood log q q(·|x)(·|x̂) of the true
θi,koutput probability and the subjective probability, we distinguish the actual action
profile x from the inferred action profile x̂.
We also assume that each player i believes that the models become common
knowledge at higher levels, in the following sense:
Assumption 2. Player i believes that the models (Mi,ki , Mi,ki +1 ) are common knowledge after level ki < ∞, that is, for each i, there is ki < ∞ such that (Mi,ki , Mi,ki +1 ) =(Mi,ki +2n , Mi,ki +1+2n ) for each n = 1, 2, · · · .
To interpret this assumption, suppose that ki = 1, so that
Mi,1 = Mi,3 = Mi,5 = · · · , and Mi,2 = Mi,4 = Mi,6 = · · · .
Then player i believes that...27
27 Here we use the fact that if player i believes E, then she believes that she believes E.

50

(30)

• Her own model is Mi,1 .
• She believes that her own model is Mi,1 .
• The opponent j believes that i’s model is Mi,3 = Mi,1 . And so on.
In this sense, player i believes that her model Mi,1 is common knowledge. Likewise, Mi,2 = Mi,4 = Mi,6 = · · · implies that player i believes that the opponent j’s
model Mi,2 is common knowledge. So (30) indeed implies that player i believes
that the models (Mi,1 , Mi,2 ) are common knowledge. Note that this assumption isabout whether player i thinks that the models are common knowledge, and notabout whether the models are common knowledge in the objective sense Whenki > 1, the condition stated in the assumption above implies that player i believes
that the opponent j believes that · · · that the models (Mi,ki , Mi,ki +1 ) are common
knowledge. We believe that Assumption 2 is satisfied in most applications.28
Pick ki as stated in Assumption 2. Then player i’s problem is strategicallyequivalent to solving the following hypothetical game with ki + 1 agents:• Each period, each agent k = 1, 2, · · · , ki + 1 chooses an action x̂i,k from a set
X̂i,k , where X̂i,k = Xi for odd k, and X̂i,k = X j for even k.
• Agent 1 is player i herself. She has the model Mi,1 , and thinks that her
opponent is agent 2. That is, she thinks that the distribution of the public
outcome is Qθi,1 (x̂i,1 , x̂i,2 ) for some θi,1 , where (x̂i,1 , x̂i,2 ) is the action chosen
by agents 1 and 2.
• Other agents are hypothetical players which describe player i’s reasoning.
Each agent k = 2, 3, · · · , ki − 1 has the model Mi,k , and thinks that her opponent is agent k + 1. That is, she thinks that the distribution of the public
outcome is Qθi,k (x̂i,k , x̂i,k+1 ) for some θi,k
28 This assumption is needed to establish Propositions 8 and 9.

Indeed, if this assumption is
not satisfied, then we need infinite agents to describe player i’s reasoning, so the set X̂ becomesthe product of infinitely many X1 and X2 . This set X̂ is not separable (it is well-known that thel ∞ -space is not separable), so the dual bounded-Lipschitz norm on 4X̂ may not coincide with the
topology of weak convergence.

51• Agents ki and ki + 1 has the models Mi,ki and Mi,ki +1 , respectively, and theyplay the game with each other. That is, they think that the output distribution
is given by Qθi,ki (x̂i,ki , x̂i,ki +1 ) for some θi,ki and Qθi,ki +1 (x̂i,ki , x̂i,ki +1 ) for some
θi,ki +1 , respectively.
• All the information structure above is common knowledge among the agents.
Intuitively, agent 1’s action x̂i,1 in this hypothetical game is player i’s actual action,
agent 2’s action x̂i,2 is player i’s prediction about the opponent j’s action, agent
3’s action x̂i,3 is player i’s prediction about j’s prediction about i’s action, andi +1
in this hypothetical game is essentiallyso on. So the action profile x̂i = (x̂i,k )kk=1
ki +1
player i’s prediction hierarchy. Let X̂i = ×k=1 Xi,k denote the set of all these action
profiles.
In what follows, each agent k in this hypothetical game is labelled as (i, k),because they are agents which describe player i’s reasoning. The opponent j has
a different model hierarchy M j , Mi , and hence her reasoning is represented by adifferent set of agents ( j, k).
Let ŝi,k denote agent (i, k)’s strategy in the infinite-horizon hypothetical game,i +1
denote a strategy profile. This profile ŝi is also interpreted
and let ŝi = (ŝi,k )kk=1
as player i’s prediction hierarchy about strategies in the infinite-horizon game.
That is, ŝi,1 is player i’s actual strategy, ŝi,2 is player i’s prediction about player j’s
strategy, and so on. So ŝi,k ∈ Si for odd k, and ŝi,k ∈ S j for even k. We assume thateach ŝi,k is pure and public.
Given a pure strategy profile ŝi = (ŝi,k ) in the hypothetical game, each agent
t+1
k’s posterior belief µ̂i,k∈ 4Θi,k can be computed using Bayes’ rule, after every
public history hYt . Formally, for each t and k, we havet−1t (θ )Q (yt |ŝ (ht−1 ), ŝ
µ̂i,k
i,k
θi,k
i,k Y
i,k+1 (hY ))t+1µ̂i,k (θi,k ) = R
t−1
t−1
t
t
Θi,k µ̂i,k (θi,k )Qθi,k (y |ŝi,k (hY ), ŝi,k+1 (hY ))dθi,kwhere ŝi,ki +2 = ŝi,ki . Here we use the fact that agent k thinks that the signal yt inperiod t is drawn given the action profile (ŝi,k (hYt−1 ), ŝi,k+1 (hYt−1 )), where ŝi,k (hYt−1 )is her own action, and ŝi,k+1 (hYt−1 ) is the opponent k + 1’s action. The above52

formula is valid only if no one deviates from the profile ŝi ; if some agent k deviates,
then her posterior belief must be computed using a different formula. A strategy
profile ŝi is Markov if each agent’s strategy depends only on the belief hierarchyµ̂it , i.e., for each k and t, ŝi,k (hYt ) depends on hYt only through µ̂it+1 .
Example 1. (Myopically optimal agents) Suppose that the agents are myopic andmaximize their expected stage-game payoffs each period. In such a case, theyplay a one-shot equilibrium given a belief-hierarchy µ̂ t in each period t. Recall
that each agent (i, k) thinks that her opponent is agent (i, k + 1), so her subjective
expected stage-game payoff given a model θi,k isZUθi,k (x̂i,k , x̂i,k+1 ) =Yui,k (x̂i,k , y)Qθi,k (dy|x̂i,k , x̂i,k+1 )where ui,k = u1 when i + k is even, and ui,k = u2 when i + k is odd. So the strategy
profile ŝi must satisfy the following equilibrium condition:ŝi,k (µ̂i ) ∈ arg max

Zx̂i,k ∈X̂i,k Θi,kUθi,k (x̂i,k , ŝi,k+1 (µ̂i ))µ̂i,k (dθi,k ) ∀k∀µ̂i .

(31)

In words, the agents ki and ki + 1 choose actions which are best response to eachother, the agent ki − 1 best responds to the agent ki ’s action, the agent ki − 2 best
responds to that action, and so on. It is obvious that such a strategy profile ŝi is
Markov.
Example 2. (Dynamically optimal agents) Now consider dynamically optimal
agents, who maximize the expectation of the discounted sum of the stage-game∞
payoffs, ∑t=1
δ t−1 ui,k (x̂i,k , y). Many applied papers use Markov perfect equilibria
as a solution concept. In our context, ŝi is a Markov perfect equilibrium if given
any belief hierarchy µ̂i , the continuation strategy profile ŝi |µ̂i satisfies
ŝi,k |µ̂i ∈ arg max
ŝi,k

Z

∞

δ t−1 E[Uθi,k (x̂ti,k , xti,k+1 )|ŝi,k , ŝi,k+1 |µ̂i ]µ̂i,k (dθi,k )
∑
Θi,k
t=1

for each k, where the expectation is taken over (x̂ti,k , xti,k+1 ). Note that this condition reduces to (31) when δ = 0 so that the agents are myopic.
53

∞ denote a sample path (a history in the infinite-horizon game).
Let h = (xt , yt )t=1
Also, let X̂ = X̂1 × X̂2 be the product of the sets of all action profiles of the two
hypothetical games. Given a sample path h and given strategy profiles ŝ = (ŝ1 , ŝ2 )of the two hypothetical games (for players 1 and 2), let σ t (h) ∈ 4X̂ denote theaction frequency up to period t, that is,σ t (h)[(x̂1 , x̂2 )] =1 t
∑ 1{ŝi,k (hYτ−1)=x̂i,k ∀i∀k}.
t τ=1Intuitively, σ t (h)[(x̂1 , x̂2 )] describes how often the action profile x̂i was chosen ineach hypothetical game. (In other words, it describes how often each player i madea prediction hierarchy x̂i .) Note that we cannot directly observe the actions x̂i,k of
the higher-level agents (i, k) with k ≥ 2, as they are hypothetical agents. However,since each agent uses a public strategy ŝi,k , we can back it up from the past public
history; given a history hYτ−1 , the hypothetical agent k’s action in period τ must beŝi,k (hYτ−1 ). This allows us to define the action frequency in the hypothetical gameas a function of the observed history h.

5.2

Posterior Beliefs and Kullback-Leibler Divergence

We first show that after a long time t, the posterior belief is concentrated on themodels which best explain the data. Specifically, we show that the belief is concentrated on the models which minimize the Kullback-Leibler divergence, which
is defined as follows. Let σ ∈ 4X̂ be a probability measure over X̂. For each σ ,the Kullback-Leibler divergence of model θi,k for agent k is defined as
Z Z

Ki,k (θi,k , σ ) =

log
X̂ Y

q(y|x̂1,1 , x̂2,1 )Q(dy|x̂1,1 , x̂2,1 )σ (d x̂).
qθi,k (y|x̂i,k , x̂i,k+1 )Intuitively, Ki,k (θi,k , σ ) measures the distance between the true output distribution
and the subjective distribution induced by agent k’s model θi,k . To see this, think
about the special case in which σ is a degenerate distribution 1x̂1 ,x̂2 . Then the
Kullback-Leibler divergence of model θi,k can be rewritten as
Z

log
Y

q(y|x̂1,1 , x̂2,1 )Q(dy|x̂1,1 , x̂2,1 ).
qθi,k (y|x̂i,k , x̂i,k+1 )54

This measures the distance between the true distribution q(·|x̂1,1 , x̂2,1 ) and the subjective distribution qθi,k (·|x̂i,k , x̂i,k+1 ) induced by the model θi,k . Indeed, this value
is always non-negative, and equals zero if and only if the true and subjective distributions are the same. When σ is not a degenerate distribution, we take a weighted
sum of the Kullback-Leibler divergence over x̂ = (x̂1 , x̂2 ), which leads to the definition of Ki,k (θi,k , σ ) above.
As is clear from this formula, agent k’s subjective signal distribution qθi,k (y|x̂i,k , x̂i,k+1 )is potentially different from the true distribution q(y|x̂1,1 , x̂2,1 ) in two ways. First,
agent k’s model θi,k can be misspecified in that the distribution qθi,k as a functionof the chosen action can be different from the true distribution q. Second, agent k
can have an inferential naivety. That is, while the true distribution is determinedby the actual actions chosen by players 1 and 2 (which is denoted by (x̂1,1 , x̂2,1 ) in
our setup), agent k thinks that the output distribution is determined by the actions
chosen by agents k and k + 1.
For each measure σ ∈ 4X̂, let Θi,k (σ ) denote the minimizers of the KullbackLeibler divergence, that is,
Θi,k (σ ) = arg min Ki,k (θi,k , σ ).
θi,k ∈Θi,kIntuitively, this is the set of models which best explains the data when the past ac∗ (σ ) =tion frequency was σ . The minimized Kullback-Leibler divergence is Ki,kminθi,k ∈Θi,k Ki,k (θi,k , σ ). We first show that these minimizers have useful properties:∗ (σ ) is continuous in (θ , σ ),Lemma 1. For each i and k, (i) Ki,k (θi,k , σ ) − Ki,ki,kand (ii) Θi,k (σ ) is upper hemi-continuous, non-empty, and compact-valued.

The following proposition shows that after a long time t, the posterior is concentrated on the best models Θi,k (σ t ). This extends Theorem 1 of Esponda,
Pouzo, and Yamamoto (2019) to the case with continuous action set Xi and with
∞ . Givenmultiple players. Let H denote the set of all sample paths h = (xt , yt )t=1
strategy profiles ŝ, let Pŝ ∈ 4X denote the probability distribution of the sample
path h. Given a sample path h, let µ̂it (h) denote the belief hierarchy in period t.
55

Proposition 6. Given any i, k, and ŝ, Pŝ -almost surely, we haveZ

limt→∞ Θi,k∗
t+1(Ki,k (θi,k , σ t (h)) − Ki,k(σ t (h)))µ̂i,k(h)[dθi,k ] = 0.

(32)

Let H denote the set of sample paths h which satisfy (32). By Proposition 6,
Pŝ (H ) = 1.

5.3

Asymptotic Motion of Action Frequency

5.3.1

Stochastic Approximation and Differential Inclusion

In this subsection, we show that given any Markov strategy ŝ, the asymptotic
motion of the action frequency σ t is represented by a recursive formula. So picka Markov strategy ŝ, and pick a sample path h ∈ H . Then by the definition, theaction frequency in each period is written ast1
σ t (h) +1 t+1 .
t +1
t + 1 ŝ(µ̂ (h))
That is, the action frequency in period t + 1 is a weighted average of the pastaction frequency σ t and today’s action 1ŝ(µ̂ t+1 (h)) . In what follows, we will showthat this second term 1ŝ(µ̂ t+1 (h)) can be written as a function of σ t , so that σ t+1 isdetermined recursively.
i +1
Pick an arbitrary small ε > 0. Then let Bε : 4X̂ → ∏2i=1 ∏kk=1
4Θi,k be theε-perturbed belief correspondence defined asZ∗Bε (σ ) = µ̂ ∀i∀k
(Ki,k (θi,k , σ ) − Ki,k (σ ))µ̂i,k (dθi,k ) ≤ ε .
σ t+1 (h) =Θi,kRoughly, Bε (σ ) is the set of all belief hierarchies µ̂ such that each µ̂i,k is concentrated on the best models Θi,k (σ ) given the mixture σ in the sense of (32).
Since h ∈ H , there is T such that for all t > T , µ̂ t+1 (h) ∈ Bε (σ t ). This in turnimplies that the action ŝ(µ̂ t+1 ) in period t + 1 must be chosen from the ε-enlarged
policy correspondence Sε (σ t ), which is defined asSε (σ ) = {ŝ(µ̂)|∀µ̂ ∈ Bε (σ )}for each σ . This immediately implies the following result:
56

Proposition 7. Pick a Markov strategy ŝ. Then given any h ∈ H , there is at
∞ with lim
decreasing sequence {ε t }t=1t→∞ ε = 0 such that
σ t+1 (h) ∈1t
σ t (h) +Sε t (σ t (h)).
t +1
t +1

This proposition implies that in a later period t, the action chosen in that period
is selected from the set Sε (σ t ) for small ε. Now we ask how this set looks like inthe limit as ε → 0. Given a Markov strategy ŝ, let
n
o
n
n ∞
n
Ŝ(µ) = x̂ x̂ = lim ŝ(µ̂ ) for some (µ̂ )n=1 with lim (µ̂ ) = µ̂
n→∞

n→∞

for each µ. This Ŝ is an upper hemi-continuous policy correspondence inducedby ŝ. It is obvious that ŝ(µ̂) ∈ Ŝ(µ̂) for each µ̂. Also a standard argument showsthat Ŝ is indeed upper hemi-continuous with respect to µ̂. Note that Ŝ = ŝ if ŝ iscontinuous. Then define
S0 (σ ) = {x̂ ∈ Ŝ(µ̂)|∀µ̂ ∈ B0 (σ )}where
B0 (σ ) = {µ̂|µ̂i,k ∈ Θi,k (σ ) ∀i∀k}.
The following proposition shows that when ε → 0, the set Sε (σ ) which appears
in the previous proposition is approximated by S0 (σ ).
Proposition 8. Sε (σ ) is upper hemi-continuous in (ε, σ ) at ε = 0. So with thedual bounded-Lipschitz norm, 4Sε (σ ) is upper hemi-continuous at ε = 0.
Propositions 7 and 8 suggest that after a long time, the motion of the action
frequency is approximated byσ t+1 (h) ∈t1
σ t (h) +S0 (σ t (h)),
t +1
t +1

which is equivalent toσ t+1 (h) − σ t (h) ∈t(S0 (σ t (h)) − σ t (h))
t +157

That is, the drift of the action frequency, σ t+1 (h) − σ t (h), should be proportional
to the difference between today’s action chosen from S0 (σ t (h)) and the current action frequency σ t (h). The next proposition formalizes this idea using the stochastic approximation technique developed by Benaı̈m, Hofbauer, and Sorin (2005):
It shows that the asymptotic motion of the action frequency is described by thedifferential inclusionσ̇(t) ∈ 4S0 (σ(t)) − σ(t).

(33)

In this differential inclusion, the drift of the action frequency is 4S0 (σ(t)) − σ(t),rather than S0 (σ(t))−σ(t). The reason is as follows. As will be shown in Proposition 9 below, the differential inclusion (33) approximates the motion of the action
frequency in the limit as the period length in the discrete-time model shrinks to
zero. This means that a small time interval [t,t + ε] in the continuous-time model
should be interpreted as a collection of arbitrarily many periods in the discretetime model. Suppose now that players’ beliefs are in a neighborhood of µ during
this time interval [t,t + ε]. In all periods included in this interval, players choose
an action profile from the set S0 (µ), and in particular, if S0 (µ) contains two or
more action profiles, then different action profiles can be chosen in different periods. Accordingly, the action frequency during this interval can take any value in
4S0 (µ), as described by the differential inclusion (33).29
To state the result formally, we use the following terminologies, which arestandard in the literature on stochastic approximation. Let τ0 = 0 and τt = ∑tn=1 1n
for each t = 1, 2, · · · . Then given a sample path h, the continuous-time interpolation of the action frequency σ t is a mapping w(h) : [0, ∞) → 4X̂ such that
τw(h)[τt + s] = σ t (h) +(σ t+1 (h) − σ t (h))τt+1 − τt1for all t = 0, 1, · · · and τ ∈ [0, t+1). Intuitively, w represents the motion of theaction frequency as a piecewise linear path with re-indexed time. A mapping
29 There is also a technical reason: In the proof of Proposition 9, we apply the stochastic ap-

proximation method of Benaı̈m, Hofbauer, and Sorin (2005), which requires that the drift term be
a convex-valued (and upper hemi-continuous) correspondence. So we need to convexify the drift
term by taking 4S0 (σ(t)), rather than S0 (σ(t)).

58σ : [0, ∞) → 4X̂ is a solution to the differential inclusion (33) with an initialvalue σ ∈ 4X̂ if it is absolutely continuous in all compact intervals, σ(0) = σ ,and (33) is satisfied for almost all t. Since 4S0 (σ ) is upper hemi-continuous withclosed convex values, given any initial value σ ∈ 4X̂, the differential inclusion(33) has a solution. (See Theorem 9 of Deimling (1992) on page 117.) Let Z(σ )denote the set of all these solutions.
Proposition 9. Pick a Markov strategy ŝ. Then for any T > 0 and any sample
path h ∈ H ,lim

sup kw(h)[t + τ] − σ(τ)k = 0.

inft→∞ σ∈Z(w(h)[t]) τ∈[0,T ]5.3.2

Steady State and Generalized Berk-Nash Equilibriumσ ∈ 4X̂ is a steady state of the differential inclusion (33) if σ ∈ 4S0 (σ ). The
following proposition shows that if the action frequency σ t converges, then its
limit point must be a steady state. The proof is exactly the same as Proposition 1of EPY, and hence we omit it.
Proposition 10. Pick a Markov strategy s. Then for each sample path h ∈ H , if
the action frequency σ t (h) converges, then its limit point limt→∞ σ t (h) is a steadystate of (33).
In all the examples in this paper, we assume that the agents are myopically
optimal so that the strategy profile ŝ satisfies (31). In this special case, steadystates of our differential inclusion are generalized Berk-Nash equilibria in thefollowing sense:
Definition 5. A probability measure σ ∈ 4X̂ is a generalized Berk-Nash equilibrium (GBNE) if for each pure action profile x̂ = (x̂1 , x̂2 ) in the support of σ , foreach i and for each k, there is a belief µ̂i,k ∈ 4Θi,k (σ ) such thatx̂i,k ∈ arg max
0
x̂i,k

Z
Θi,k

0
Uθi,k (x̂i,k, x̂i,k+1 )µ̂i,k (dθi,k ).

A generalized Berk-Nash equilibrium is degenerate if it is a point mass on some
pure action profile x̂.
59

In words, in a generalized Berk-Nash equilibrium σ , each action profile x̂
which has a positive weight in σ is a one-shot equilibrium for some belief µ̂, andthis belief µ̂ is concentrated on the models Θi,k (σ ) which minimize the KullbackLeibular divergence. In a non-degenerate GBNE which assign positive weights on
multiple action profiles x̂, different action profiles x̂ may be supported by different
beliefs µ̂. We will discuss more on this later.
Proposition 11. Suppose that the strategy profile ŝ satisfies (31). Then any steadystate of our differential inclusion (33) is a generalized Berk-Nash equilibrium. So
for each sample path h ∈ H , if the action frequency σ t (h) converges, then its
limit point limt→∞ σ t (h) is a generalized Berk-Nash equilibrium.
Note that the action frequency may converge to non-degenerate equilibrium
σ , which assigns positive probability to multiple action profiles x̂. An intuition is
as follows. If the action frequency σ t converges to some σ , then from Proposition
6, the posterior belief µ̂ t will be concentrated on 4Θ(σ ) after a long time, thatis, µ̂ t is in a neighborhood of 4Θ(σ ) for large t. If all the beliefs in this neighborhood induce the same equilibrium action x̂ (i.e., ŝ(µ̂) = x̂ for all beliefs µ̂ in aneighborhood of 4Θ(σ )), then the action frequency will eventually converge to apoint mass on x̂. But in general, this need not be the case; different beliefs µ̂ andµ̂ 0 in this neighborhood may induce different equilibrium actions x̂ and x̂0 . In sucha case, both x̂ and x̂0 can be chosen infinitely often on the path, and hence havepositive weights in the limiting action frequency σ .
Note, however, that in many applications, all GBNE are degenerate. Indeed,
if (i) there is a unique equilibrium x̂ for each belief µ̂ and (ii) there is a uniqueminimizer θi,k of the Kullback-Leibular divergence for each action frequency σ ,then obviously any GBNE is degenerate. All our examples in the paper satisfy
these assumptions.
We view GBNE as a natural extension of BNE of Esponda and Pouzo (2016)
to our setup. For comparison, let us think about the information structure considered in Esponda and Pouzo (2016), that is, suppose that the subjective signal
distribution Qθi,k (·|x) is independent of the opponent’s action (i.e., Qθi,k (·|x) =Qθi,k (·|xi ) for each i, k, and x). In this special case, all higher-level agents (i, k)
60

with k ≥ 2 are irrelevant, in the sense that they do not influence the actions andthe beliefs of the actual player (i, 1). This is so because each player i’s subjective expected payoff Uθi,1 (xi ) = ∑y∈Y Qθi,1 (y|xi )ui (xi , y) and her Bayes’ formulaµit+1 (θi,1 ) = Rµit (θi,1 )qθi,1 (yt |xti )tt t
Θi,1 µi (θi,1 )qθi,1 (y |xi )dθi,1

are independent of the opponent j’s action.

Accordingly, GBNE reduces to a probability measure σ ∈ 4(X1 × X2 ) on the set
X1 × X2 such that for each pure action profile x = (x1 , x2 ) in the support of σ andfor each i, there is a belief µi ∈ 4Θi,1 (σ ) such thatxi ∈ arg max
0
xi

Z
Θi,1

Uθi,1 (xi0 )µi (dθi,1 )where Θi,1 (σ ) is the minimizers of the Kullback-Leibular divergence
Z Z

log
X Y

q(y|x)
Q(dy|x)σ (dx).
qθi,1 (y|x)

It is easy to check that GBNE is a weakening of BNE of Esponda and Pouzo(2016), that is, any GBNE σ is a BNE. In particular, degenerate BNE is equivalent to pure-strategy BNE, in that σ is a pure-strategy BNE if and only if it isa degenerate GBNE. However, a non-degenerate GBNE need not be a mixedstrategy BNE, because in a GBNE, (i) different actions xi may be supported bydifferent beliefs µi , and (ii) GBNE distribution σ allows correlation between x1
and x2 . This difference comes from the fact that GBNE is a limit point of theaction frequency, while BNE is a limit point of the action itself. Specifically, in
Esponda and Pouzo (2016), there is an i.i.d. payoff perturbation each period, sothat each player (independently) mix actions each period. A mixed-strategy BNE
σ is regarded as a limit point of this mixed action. In this case, in a steady state,
the mixed strategy σi must be optimal (with a payoff perturbation) given a single
belief, and there is no correlation between actions of different players. In contrast,
in our model, each player chooses a pure action, so there is perfect correlation
between x1 and x2 . Also as noted earlier, different action profiles x and x0 whichappear in non-degenerate GBNE σ are played in different periods t and t 0 ; so they0
are supported by different beliefs µ t and µ t .
61

5.4

Motion of the KL Minimizer

5.4.1

Identifiability and Differential Inclusion

Our Proposition 9 shows that the asymptotic motion of the action frequency σ t is
described by the differential inclusion (33). However, solving the differential inclusion (33) is not easy in general. For example, in many applications (including
the ones in this paper), there are continuous actions, in which case the action frequency σ t is a probability distribution over an infinite-dimensional (continuous)
space, and thus the differential inclusion becomes an infinite-dimensional problem. In this section, we show that this dimensionality problem can be avoided
if we look at the asymptotic motion of the belief, rather than that of the action
frequency.
We will impose the following identifiability assumption, which requires that
there be a unique KL minimizer θi,k (σ ) for each measure σ ∈ 4X̂. This assumption is satisfied in many applications, see Esponda and Pouzo (2016) for moredetailed discussions on this assumption.
Assumption 3. For each i, k, and σ , there is a unique minimizer θi,k (σ ) ∈ Θi,k of
the Kullback-Leibular divergence Ki,k (θi,k , σ ).
Since Θi,k (σ ) is upper hemi-continuous in σ , under the identifiability assumption, each KL minimizer θi,k (σ ) is continuous in σ . The next lemma shows thatθ (σ ) = (θi,k (σ ))i,k is Lipschitz continuous if some additional assumptions hold.
With an abuse of notation, let Ki,k (θi,k , x̂) = Ki,k (θi,k , σ ) for σ = 1x̂ .
Assumption 4. The following conditions hold:∂ K (θ ,x̂)(i) For each i, k, and m, ∂i,kθ i,k < ∞, where θi,k,m denotes the m-th compoi,k,m
nent of θi,k . Also for each x̂, Ki,k (θi,k , x̂) is twice-continuously differentiable∂ 2 K (θ ,x̂)

with respect to θi,k , that is, ∂ θ i,k ∂ θi,k
i,k,m

(ii)

i,k,n

is continuous in θi,k .

∂ Ki,k (θi,k ,x̂)is equi-Lipschitz continuous, that is, there is L > 0 such that
∂ θi,k,m∂ K (θ ,x̂)
∂ K (θ ,x̂0 )| ∂i,kθ i,k − ∂i,kθ i,k | < L|x̂ − x̂0 | for all i, k, m, θi,k , x̂, and x̂0 .
i,k,m
i,k,m

62

(iii) The KL minimizer θ (σ ) satisfies both the first-order and second-order conditions for each σ . (An implication is that the inverse of the Hessian matrix
exists.)Lemma 2. θ (σ ) is Lipschitz continuous in σ . That is, there is L > 0 such that|θ (σ ) − θ (σ̃ )| ≤ Lkσ − σ̃ k.
t ) . Let w denote
Now we consider the motion of the KL minimizer θ t = (θi,ki,kθ

the continuous-time interpolation of θ t . Let ∇Ki,k (θi,k , x) = (∇K(θ , x) = (∂ Ki,k (θi,k ,x)∂ θi,k,m )i,k,m .

∂ Ki,k (θi,k ,x)∂ θi,k,m )m , andAlso let ∇2 Ki,k (θi,k , σ ) denote the Hessian matrixof Ki,k (θi,k , σ ) with respect to θi,k , that is, each component of ∇2 Ki,k (θi,k , σ ) is
∂ 2 Ki,k (θi,k ,σ )2∂ θi,k,m ∂ θi,k,n . Let ∇ K(θ , σ ) denote a block diagonal matrix whose main diagonal
blocks are ∇2 Ki,k (θi,k , σ ), that is,∇2 K1,1 (θ1,1 , σ )0∇2 K1,2 (θ1,2 , σ )∇2 K(θ , σ ) = 

...

0



.

With an abuse of notation, let S0 (θ ) denote S0 (σ ) for σ with θ (σ ) = θ . The
following proposition shows that the asymptotic motion of the KL minimizer is
described by the differential inclusionθ̇(t) ∈[[

−(∇2 K(θ(t), σ ))−1 ∇K(θ(t)), σ 0 ) .

(34)

σ :θ (σ )=θ(t) σ 0 ∈4S0 (θ(t))

Let Zθ (θ(0)) be the set of solutions to the differential inclusion (34) with the initialvalue θ(0).
Proposition 12. Suppose that Assumptions 3 and 4 hold. Then for any T > 0 andany sample path h ∈ H ,
lim

inf

sup |wθ (h)[t + τ] − θ(τ)| = 0.

t→∞ θ∈Zθ (wθ (h)[t]) τ∈[0,T ]To interpret the differential inclusion (34), consider the special case in which
Θi,k ⊂ R, i.e., assume that agent k’s model θi,k is one-dimensional. Then from63

(33), we haveθ̇i,k (t) ∈[[σ :θ (σ )=θ(t) σ 0 ∈4S0 (θ(t))∂ K (θ ,σ )−0 (θ (t), σ 0 )Ki,ki,k00 (θ (t), σ )Ki,ki,k∂ 2 K (θ ,σ )

(35)

i,k
i,k
0 (θ , σ ) =00 (θ , σ ) =for each i and k, where Ki,kand Ki,k
.
∂θ
∂θ2
00
The denominator Ki,k (θi,k (t), σ ) measures the curvature of the Kullback-Leibular
divergence. Note that this term is always positive, because the second-order condition must be satisfied (Assumption 4(iii)). So this term influences the absolutevalue of θ(t), but not the sign of θ̇i,k (t); this in turn implies that this denominatorinfluences the speed of θi,k (t), but not the direction. Intuitively, when the curve
00 is close to zero), all models in a neighborhood of θ(t) almost
is flatter (i.e., Ki,kequally fit the past data. Hence the KL minimizer θ(t) is more sensitive to the newdata generated by today’s action, and it changes quickly.
0 (θ (t), σ 0 ) measures how much an increase in θ
The numerator −Ki,ki,ki,k im0
proves fitness to the new data generated by today’s action σ . This term influences
the sign of θ̇i,k (t), so it determines whether θi,k (t) moves up or down. Intuitively,
when this numerator is positive, (at least in a neighborhood of θ(t)) higher θ better explains the new data generated by today’s action, so θ(t) moves up. On theother hand, when this numerator is negative, lower θ better explains the new data,so θ(t) moves down.
When we consider the dynamic of θ t = θ (σ t ), the drift of θ t cannot be
uniquely determined, for two reaosns. First, the KL minimizer θ t may not uniquelydetermine the agents’ actions today, in the sense that S0 (θ t ) may not be a singleton. (As pointed out by Esponda, Pouzo, and Yamamoto (2019), in the singleagent setup, this happens when the agent is indifferent over multiple actions at amodel θ = θ t .) In our differential inclusion (35), this multiplicity is captured bytaking the union over σ 0 ∈ 4S0 (θ(t)). Note that the same multiplicity problem
appears in the differential inclusion (33).
Second, the KL minimizer θ t may not uniquely determine the past action frequency, in the sense that there may be more than one σ such that θ (σ ) = θ t .
Note that even if two action frequencies σ and σ̃ yield the same KL minimizer64

(i.e., θ (σ ) = θ (σ̃ )), they may yield different curvatures of the KL divergence, sothey influence the speed of θi,k (t) differently. In our differential inclusion, thismultiplicity is captured by taking the union over σ with θ (σ ) = θ(t).
5.4.2

Convergence with Identifiability

Using the differential inclusion (34), now we will derive a sufficient condition for
the agents’ beliefs θ t to converge for each type of misspecification considered in
Sections 3 and 4. First, consider first-order misspecification in Section 3, i.e., set
k1 = k2 = 2, M1,1 = M2,2 , and M2,1 = M1,2 , and assume that agent (1, 1) is correctly
specified. Assume as in Section 3 that players play a static Nash equilibrium
every period and they predict the opponent’s strategy correctly, so that ŝ1,1 = ŝ2,2
and ŝ2,1 = ŝ1,2 . Since k1 = k2 = 2, θ t is a four-dimensional variable, i.e., θ t =t , θ t , θ t , θ t ). However, the differential inclusion (34) is essentially a one(θ1,1
1,2 2,1 2,2dimensional problem. Indeed,• Agent (2, 2) is redundant in that her belief is identical with the belief ofagent (1, 1) every period. (This follows from the fact that player 2 knows
player 1’s posterior belief every period.) Similarly, agent (1, 2) is redundant.
• Agent (1, 1) is correctly specified, so we have θ1,1 (σ ) = θ ∗ for all σ , whicht = θ ∗ for any period t. (This reflects the fact that player 1
implies that θ1,1
eventually learn the true state regardless of players’ play.)
So in order to know the asymptotic motion of θ t , we only need to know the asympt . Similarly, under one-dimensional double
totic motion of player 2’s belief, θ2,1
misspecification considered in Section 4.2.1, the differential inclusion (34) reduces to a one-dimensional problem. This is so because player 1 eventually learns
the true state and the hypothetical player is redundant, in that her belief is exactly
the same as player 2’s belief after every history. Our next proposition shows thatin these cases, the belief converges almost surely, regardless of the initial prior.
To state the result formally, we use the following terminologies. Let I be theset of all agents. An agent (i, k) is correctly specified if there is a correct model

65

∗ ∈Θ
∗ (x) = Q(x) for all x. An agent (i, k) has no inferential
θi,ki,k such that Qθi,knaivety if (i) i + k is even and (ŝi,k (µ̂), ŝi,k+1 (µ̂)) = (ŝ1,1 (µ̂), ŝ2,1 (µ̂)) for all µ̂, or(ii) i + k is odd and (ŝi,k (µ̂), ŝi,k+1 (µ̂)) = (ŝ2,1 (µ̂), ŝ1,1 (µ̂)) for all µ̂. An agent(i, k) is unbiased if she is correctly specified and has no inferential naivety. Let I ∗denote the set of all unbiased agents (i, k).
Given a Markov strategy ŝ, two agents (i, k) and ( j, l) are equivalent if i + k ≡j + l (mod 2), Mi,k = M j,l , and (si,k , si,k+1 ) = (s j,l , s j,l+1 ). Consider a partitionof the set of agents induced by the equivalence classes, and for each equivalenceclass, pick a representative agent (i, k). Let I ∗∗ the set of these representative
agents. (Mathematically, I ∗∗ is a quotient set I/ ∼, where (i, k) ∼ ( j, l) if (i, k)and ( j, l) are eqivalent.) For each (i, k) ∈ I ∗∗ , let I(i, k) denote the set of all agentsequivalent to (i, k). Note that all agents in the set I(i, k) choose the same action
and have the same belief every period.
θ is a steady state if it is a steady state of the differential inclusion (34), i.e.,∇K(θ , σ 0 ) = 0 for some σ 0 ∈ 4S0 (θ ). Let E denote the set of all steady states.

Proposition 13. Suppose that Assumptions 3 and 4 hold. Pick a Markov strategyŝ, and suppose that I ∗∗ \ I ∗ = {(i, k)} and Θi,k ⊂ R. Then for any sample pathh ∈ H , the belief converges to a steady state, i.e., limt→∞ θ t (h) ∈ E. In particular,
if there is a unique steady state, θ t converges there almost surely.
The assumption I ∗∗ \ I ∗ = {(i, k)} implies that there is essentially only one
agent whose belief moves in a non-trivial way; all other agents are redundant orlearn the true state eventually. This assumption is satisfied, for example, in agame with first-order misspecification and one-sided double misspecification.30
The proposition shows that in such a case, the belief converges almost surely
as long as the identifiability condition (Assumption 3) and Assumption 4 hold.
This result is a natural extension of Heidhues, Kőszegi, and Strack (2020a) andEsponda, Pouzo, and Yamamoto (2019), who prove convergence of the belief fora single-agent problem.
30 Note that this proposition applies even when players are patient; if players play a Markov

perfect equilibrium, we have I ∗∗ \ I ∗ = {(i, k)}.

66

Next, consider second-order misspecification studied in Section 4.1. In thiscase, the two beliefs θ2 and θ̂1 evolve in a non-trivial way. Similarly, under twosided double misspecification studied in Section 4.2.2, the two beliefs θ1 and θ2
evolve in a non-trivial way. Our next proposition shows that the belief converges
even in these cases, provided that some additional conditions hold. Consider themodel studied in Section 4.1. Given a misspecified parameter A, let f2 (θ̂1 , A) denote the set of “steady-state belief” of player 2, when the hypothetical player’s belief is fixed at θ̂1 . That is, f2 (θ̂1 , A) is the set of all θ2 such that there is (x1 , x2 , x̂1 )satisfying the consistency condition (19) and the incentive-compatibility condition(14) through (16). Similarly, given a belief θ2 of player 2, let fˆ1 (θ2 , A) be the setof all θ1 such that there is (x1 , x2 , x̂1 ) satisfying the consistency condition (18) andthe incentive-compatibility condition (14) through (16). A steady state studied in
Section 4.1 can be seen as an intersection of the two graphs {(θ2 , fˆ1 (θ2 , A))|∀θ2 }and {( f2 (θ̂1 , A), θ̂1 )|∀θ̂1 }.
Proposition 14. Consider the game with the second-order misspecification in Section 4.1, and suppose that Assumptions 3 and 4 hold. Pick a parameter A suchthat(i) f2 (θ̂1 , A) and fˆ1 (θ2 , A) are singletons for all θ̂1 and θ2 (so that they arefunctions of θ̂1 and θ2 , rather than correspondences),(ii) f2 (θ̂1 , A) and fˆ1 (θ2 , A) are continuously differentiable in θ̂1 and θ2 , respectively, andˆ(iii) maxθ̂1 | ∂ f2∂(θ̂θ̂1 ,A) | maxθ2 | ∂ f1∂(θθ2 ,A) | < 1.
12(iv) The KL divergence is single-peaked, in that for each (i, k), there is θi,k such
0 (θ̃ , σ ) < 0 for all θ̃ < θ and K 0 (θ̃ , σ ) > 0 for all θ̃ > θthat Ki,k
i,k
i,k
i,k
i,k
i,k
i,k i,kThen there is a unique steady state θ ∗ = (θ2∗ , θ̂1∗ ), and the belief converges there
almost surely regardless of the initial prior. The same result holds for two-sided
double misspecification by replacing fˆ1 (θ2 , A) with f1 (θ2 , A), where f1 (θ2 , A) is a“steady-state belief” of player 1 given player 2’s belief θ267This proposition gives a sufficient condition for belief convergence. It requiresthat f2 and fˆ1 are continuously differentiable functions and their derivatives are
not too large (i.e., changing θ̂1 does not influence the steady-state belief of the
opponent much, and vice versa).
In the case of second-order misspecification, if the misspecification is small inthat the parameter A is close to a, there is a simpler sufficient condition for convergence; we can drop assumption (iii) in Proposition 14. To see why, consider thecase in which A = a so that there is no misspecification. We claim that assumption(i) in Proposition 14 automatically implies assumption (iii). Indeed, when A = a,
we always have θ ∗ ∈ fˆ1 (θ2 , a), so that assumption (i) implies fˆ1 (θ2 , a) = θ ∗ forˆ
all θ2 . Then assumption (iii) is satisfied as ∂ f1∂(θθ2 ,A) = 0. By the continuity, the2same is true as long as the parameter A is close to a. The following corollarysummarizes this discussion:
Corollary 4. Consider the game with the second-order misspecification in Section
4.1. Suppose that assumptions (i) and (ii) in Proposition 14 are satisfied at A = a.
Then there is A > a such that for any A ∈ (a, A), there is a unique steady state,and θ t converges there almost surely regardless of the initial prior.
5.4.3

Convergence Without Identifiability

In the previous subsection, we have seen that the motion of the KL minimizer
can be described by a differential inclusion if the identifiability condition holds.
However, there are some economic examples which do not satisfy the identifiability. For example, the model of overconfidence studied in Heidhues, Kőszegi, andStrack (2018) does not satisfy the identifiability in general.
The following proposition shows that even in such a situation, the belief still
converges to a steady state if some additional assumptions on payoffs and information structures are satisfied. For each action frequency σ , let θ i,k (σ ) denotethe minimal KL minimizer, that is, let θ i,k (σ ) = minθi,k ∈Θi,k (σ ) θi,k . Likewise,let θ i,k (σ ) denote the maximal KL minimizer. Also, when the problem is onedimensional (i.e., I ∗∗ \ I ∗ = {(i, k)}), for each model θi,k , let S0 (θi,k ) = S0 (µ)

68where µ is a degenerate belief on θ such that θ j,l = θi,k for all ( j, l) ∈ I(i, k) andθ j,l = θ ∗ for all ( j, l) ∈ I ∗ .
Proposition 15. Pick any Markov strategy ŝ. Assume that(i) The problem is one-dimensional, i.e., I ∗∗ \ I ∗ = {(i, k)} and Θi,k ⊂ R.
(ii) For each pure action profile x, the KL divergence Ki,k (θ , 1x ) is single-peaked,∂ K (θ ,1x )< 0 for θi,k < θi,k (x),i.e., there is a unique KL minimizer θi,k (x), i,k∂θi,k∂ K (θ ,1x )and i,k> 0 for θi,k > θi,k (x).
∂ θi,k∗ }.
(iii) There is a unique steady state σ ∗ , and θi,k (σ ∗ ) = {θi,k(iv) S0 (θ̃i,k ) is a function (rather than a correspondence) of θ̃i,k , and θi,k (S0 (θ̃i,k ))is increasing in θ̃i,k .
(v) For each belief µ̂ whose support is compact, S0 (µ) ⊆S

θi,k ∈co(suppµ̂i,k ) S0 (θi,k ).

Then for each sample path h ∈ H , limt→∞ θ i,k (σ t (h)) = limt→∞ θ i,k (σ t (h)) =
∗ .
θi,k

6

Related Literature

There is a rapidly growing literature on Bayesian learning with model misspecification. Nyarko (1991) presents an example in which the agent’s action does
not converge. Fudenberg, Romanyuk, and Strack (2017) consider a general twostate model and characterize the agent’s asymptotic actions and behavior. Heidhues, Kőszegi, and Strack (2018), Heidhues, Kőszegi, and Strack (2020a), andHe (2019) study a continuous-state setup, and they show that the agent’s action
and belief converge to a Berk-Nash equilibrium of Esponda and Pouzo (2016),
under some assumptions on payoffs and information structure. Esponda, Pouzo,
and Yamamoto (2019) characterize the agent’s asymptotic behavior in a general
single-agent model. Fudenberg, Lanzani, and Strack (2020) discuss robustness ofsteady states. All these papers look at a single-agent problem and focus on firstorder misspecification. More recently, Ba and Gindin (2020) consider two-player
69team production in which both players are overconfident about their own capability. They show that if actions (efforts) are complements and information has
a positive externality, then learning is mutually reinforcing, i.e., one’s strategic
play reduces both players’ efforts and results in a more suboptimal outcome. Our
work strengthens their result, in three ways. First, our Proposition 1 gives a necessary and sufficient condition for mutually-reinforcing learning: Assuming thatthe base misspecification effect is negative, our proposition shows that player 1’s
strategic play reduces both players’ efforts if and only if the two asymptotic best
response curves are upward-sloping (i.e., BR01 > 0 and BR02 > 0).31 Second, our
Proposition 1 can be applied to a more general setup, such as Cournot duopoly
and tournaments, which allows us to study a wide range of applications. Third,
and perhaps most importantly, we develop a model of higher-order misspecification and study how each type of misspecification influences players’ beliefs andactions.
Misspecified learning has also been studied in other settings and applications.
A literature on social learning studies how inferential naivety or model misspecification influences the asymptotic outcomes (e.g.,DeMarzo, Vayanos, and Zwiebel(2003), Eyster and Rabin (2010), Gagnon-Bartsch (2016), Gagnon-Bartsch andRabin (2016), Bohren and Hauser (2020), and Frick, Iijima, and Ishii (2020)).32
Molavi (2020) considers a general equilibrium model in which a representative
agent has a misspecified view about the world. Cho and Kasa (2017) study anasset-pricing model in which an agent incorrectly believes that the environment is
not stationary.
A prominent example of the first-order misspecification is overconfidence on
own capability. Plenty of experimental and empirical papers report that people exhibit overconfidence on own ability in various economic activities, such as strategic entries (Camerer and Lovallo, 1999), corporate investments (Malmendier and
31 The assumptions of Ba and Gindin (2020) (strategic complementarity and positive information

externality) indeed ensure upward-sloping asymptotic best response curves.
32 For experimental evidence on how social-learning outcomes depart from a correctly specified
learning model, see, e.g., Çelen and Kariv (2005), Kübler and Weizsäcker (2004, 2005).

70

Tate, 2005), and merger decisions (Malmendier and Tate, 2008).3334 Furthermore,
recent empirical evidence suggests that overconfidence on a particular aspect ofown capability persists even after a long time and a plenty of feedback (Hoffman and Burks, 2020; Huffman, Raymond, and Shvets, 2019), which calls for the
analysis of long-run behavior under model misspecifications.
Similarly, there is evidence that people sometimes have systematically incorrect views of their opponents’ beliefs and actions, as in our second-order misspecification model. In both laboratory and field studies of disclosure games, departingfrom the theoretical prediction, people often do not disclose unfavorable (but not
the worst) information (Dranove and Jin, 2010; Brown, Camerer, and Lovallo,2012). Jin, Luca, and Martin (forthcoming) conduct more detailed experiments totest potential mechanisms. They find that receivers systematically make incorrect
predictions about their opponents’ actions, and the receivers also choose actions
based on such incorrect views.. In common-value auctions, classical evidence
suggests that bidders often mispredict other bidders’ behavior and suffer from aloss, which is known as the “winner’s curse” (Kagel and Levin, 2002). Avery andKagel (1997) report that such suboptimal bidding behavior systematically persists
even when bidders have received dozens of feedback.
Finally, there is a literature which studies implications of one’s misspecification in a static game theoretic model. As discussed in Section 3.2.1, Kyle andWang (1997) analyze a variant of one-shot Cournot duopoly with first-order misspecification.35 Madarász (2012) considers a player who misspecifies other play33 See Malmendier and Tate (2015) for reviewing managerial overconfidence, Daniel and Hirsh-

leifer (2015) for reviewing overconfidence in financial markets, and Grubb (2015) for reviewing
consumers’ overconfidence.
34 Benoı̂t and Dubra (2011) point out that most of these works may not be sufficient to conclude
that people have overconfidence, because it can also be explained by considering a model in whichpeople do not have overconfidence but face uncertainty about some aspect of the environment.
However, this explanation is not supported by Benoı̂t, Dubra, and Moore (2015); they conduct
a lab experiment which separates out the effect of uncertainty, and find that subjects still exhibit
overconfidence.
35 See Kőszegi (2014), Grubb (2015), and Heidhues and Kőszegi (2018) for reviews of theoretical works which study overconfident agents in contract theory and industrial organization.

71

ers’ knowledge, which can be described by our double misspecification model.
Our paper contributes to this literature in that we characterize how misspecified
players learn about own environment in a general public-monitoring setup, and derive a number of economic implications. Indeed, as discussed in Aghion, Bolton,
Harris, and Jullien (1991), the theory of imperfect competition with complete
information (e.g., its demand function is known) is often defended with the argument that if the economic environment remains fixed over time, then players
eventually learn all relevant parameters from past experience. Since a misspecified player may not be able to correctly learn a true parameter, we believe it isimportant to formally analyze the learning process and the effect of strategic interaction on the long-run outcomes.

72

A

ProofsA.1

Proof of Proposition 1

Pick x∗ and A∗ as stated. Since the steady-state actions (x1∗ , x2∗ ) are interior points,
they must satisfy the first-order conditions∂U1 (x1 , x2∗ , θ ∗ )
= 0,
∂ x1
∂U2 (x1∗ , x2 , θ )
= 0.
∂ x2
θ =θ2 (x∗ ,A)

(36)
(37)

Let M be the Jacobian of this system of the equations. Then each i j-componentof the matrix coincides with Mi j defined in the main text. That is,"#
M11 M12M=
M21 M22Since BR01 BR02 , 1, we have detM , 0, so the implicit function theorem guarantees
that for any parameter A close to A∗ , there is an action profile x∗ which satisfies
the first-order conditions (36) and (37). These action profiles are globally optimal(i.e., maximize the expected payoff given the belief θ1 = θ ∗ and θ2 (x∗ , A)), because of the regularity conditions (i) and (ii). So this x∗ is a steady state given theparameter A. The implicit function theorem also asserts that""## " ∂ x∗ #1
M11 M12
0
∂A
=−
,
∂ x2∗
M21 M22
M
2A∂A
Solving this system of equations,∂ x2∗
M11 M2A
=−
,
∂A
detM
∂ x1∗ M12 M2A
=
.
∂A
detM
Dividing both the numerator and denominator of the first equation by M11 M22∂ x∗
2A
and using detM = M11 M22 − M12 M21 , we have ∂ A2 = − 1−BR10 BR0 M
M22 . Also by∂ x∗
combining the two equations above, we have ∂ A1 = BR1 ∂∂xA2 .

73

1

2

Next, we prove BR01 BR02 < 1 by contradiction. Suppose that BR01 BR02 > 1. Then
we have either (i) BR01 > 0 and BR02 > 0, or (ii) BR01 < 0 and BR02 < 0. Consider
case (i). Then we have BR02 > BR1 0 > 0. This means that if we take x1 on the1horizontal axis and x2 on the vertical axis, then the two asymptotic best response
curves are upward-sloping at the steady state action x∗ , and BR2 is steeper thanBR1 . This and the continuity of BRi immediately imply that BR1 and BR2 must
intersect at some x1 > x1∗ , but this contradicts with the fact that x∗ is a uniquesteady state. The same argument works for case (ii).
Hence we have BR01 BR02 ≤ 1. Also, dividing both sides of detM , 0 by M11 M22 ,we have BR01 BR02 , 1.

A.2

Proof of Proposition 2

Pick A∗ and x∗ as stated. Since x∗ is an interior point, it must satisfy the first-orderconditions∂U1 (x1 , x2∗ , θ ∗ )
= 0,
∂ x1
∂U2 (x̂1∗ , x2 , θ2 )
= 0,
∂ x2
∂ Û1 (x1 , x2∗ , θ̂1 )
= 0.
∂ x1

(38)
(39)
(40)

Let M be the Jacobian of this system of the equations. Then each i j-componentof the matrix coincides with Mi j defined in the main text.
By the regularity condition (iii), detM , 0, so the implicit function theorem
guarantees that for any parameter A close to A∗ , there is an action profile x∗ which
satisfies the first-order conditions (38)-(40). These action profiles are globally
optimal, because of the regularity conditions (i) and (ii). So this x∗ is a steadystate given the parameter A. The implicit function theorem also asserts that  ∂ x∗ 1
M11 M12 M13
0
∂ A∗



 ∂x  M21 M22 M23   ∂ A2  = −  0  ,
∂ x̂1
M3A
M31 M32 M33∂A74

Solving this system of equations,∂ x̂1∗(M11 M22 − M12 M21 )M3A
=−
,
∂A
detM
∂ x2∗ M11 M23 M3A
=
,
∂A
detM
∂ x1∗
M12 M23 M3A
=−
.
∂A
detM
Dividing both the numerator and the denominator of the second equation by M11 M22 M33
and using detM = M11 M22 M33 + M12 M23 M31 − M12 M21 M33 − M11 M32 M23 , we
have
BR023
∂ x2∗
M3A
=−
·
0
0
0
0
0
0
0
∂A
1 − BR12 BR23 BR31 − BR12 BR21 − BR23 BR32 M33
1 − BR023 BR032
BR023
M3A
=−
M33 1 − BR023 BR032
1 − BR012 BR023 BR031 − BR012 BR021 − BR023 BR032




BR023
M3A
1
=−
.
M33 1 − BR023 BR032
1 − BR012 NE20
The second equation in the proposition follows from the second and the third
equations.

A.3

Proof of Proposition 3

Pick A∗ and x∗ as stated. Since x∗ is an interior point, it must satisfy the first-orderconditions∂U1 (x1 , x2∗ , θ ∗ )
= 0,
∂ x1
∂U2 (x̂1∗ , x2 , θ2 )
= 0,
∂ x2
∂ Û1 (x1 , x2∗ , θ2 )
= 0.
∂ x1

(41)
(42)
(43)

Let M be the Jacobian of this system of the equations. Then each i j-componentof the matrix coincides with Mi j defined in the main text.
75

By the regularity condition (iii), detM , 0, so the implicit function theorem
guarantees that for any parameter A close to A∗ , there is an action profile x∗ which
satisfies the first-order conditions (41)-(43). These action profiles are globally
optimal, because of the regularity conditions (i) and (ii). So this x∗ is a steadystate given the parameter A. The implicit function theorem also asserts that  ∂ x∗ 
1
M11 M12 M13
0∂ A∗ ∂x 
 M21 M22 M23   ∂ A2  = −  M2A  ,
∂ x̂1
M3A
M31 M32 M33∂A
Solving this system of equations,∂ x̂1∗(M11 M22 − M12 M21 )M3A − (M11 M32 − M12 M31 )M2A
=−
,
∂A
detM
∂ x2∗
M11 (M33 M2A − M23 M3A )
=−
,
∂A
detM
∂ x1∗ M12 (M33 M2A − M23 M3A )
=
.
∂A
detM
The rest of the proof is very similar to that of Proposition 2, and hence omitted.

A.4

Proof of Proposition 4

Pick A∗ and x∗ as stated. Since x∗ is an interior point, it must satisfy the first-orderconditions∂U1 (x1 , x̂2∗ , θ1 )
= 0,
∂ x1
∂U2 (x̂1∗ , x2 , θ2 )
= 0,
∂ x2
∂ Û1 (x̂1 , x2∗ , θ2 )
= 0,
∂ x̂1
∂ Û2 (x1∗ , x̂2 , θ1 )
= 0.
∂ x̂2

(44)
(45)
(46)
(47)

Let M be the Jacobian of this system of the equations. Then each i j-componentof the matrix coincides with Mi j defined in the main text.
76

By the regularity condition (iii), detM , 0, so the implicit function theorem
guarantees that for any parameter A2 close to A∗2 , there is an action profile x∗ which
satisfies the first-order conditions (44)-(47). These action profiles are globally
optimal, because of the regularity conditions (i) and (ii). So this x∗ is a steadystate given the parameter A. The implicit function theorem also asserts that ∂ x1∗
M11 M12 M13 M14  ∂ A2 0 M
  ∂ x2∗  M  21 M22 M23 M24   ∂ A2 
 2A   ∂ x̂1∗  = − , M31 M32 M33 M34  
 M3A  ∂ A∗2 ∂ x̂2
M41 M42 M43 M44
0∂ A2

Solving this system and using M13 = M24 = M34 = M43 = 0,
∂ x2∗
(M14 M41 M33 − M11 M33 M44 )M2A + (M11 M23 M44 − M23 M14 M41 )M3A
=
∂ A2
detM
∂ x1∗
(M12 M33 M44 − M33 M42 M14 )M2A − (M23 M44 M12 − M23 M42 M14 )M3A
.
=
∂ A2
detM
Dividing both the numerator and the denominator of the first equation by M11 M22 M33 M44 ,


∂ x2∗
M2A
M3A M11 M22 M33 M44= − (1 − BR14 BR41 )+ (BR23 − BR23 BR14 BR41 )
∂ A2
M22
M33
detM


M2A
M3A M11 M22 M33 M44= − (1 − BR14 BR41 ).
+ BR23
M22
M33
detM
Note that
detM =M11 M22 M33 M44 + M14 M21 M33 M42 − M14 M22 M33 M41 − M12 M21 M33 M44− M11 M23 M32 M44 − M14 M23 M31 M42 + M14 M23 M32 M41 + M12 M23 M31 M44 ,so
M11 M22 M33 M44
1
=
0
0
0
0
detM
(1 − BR14 BR41 )(1 − BR23 BR32 ) − (BR012 + BR014 BR042 )(BR021 + BR023 BR031 )



1
1
=
(1 − BR014 BR041 )(1 − BR023 BR032 )1 − NE10 NE20
Plugging this int the equation above, we obtain the first equation in the proposition. The second equation can be derived in a similar way.
77

A.5

Proof of Proposition 5

Proof of Part (i). When A = a, the one-shot Nash equilibrium xcorrect is a steadystate under first-order misspecification, second-order misspecification, and double
misspecification. By the assumption, a steady state is unique, and hence the result
follows.
Proof of Part (ii). Let M first , M second , and M double denote the matrix M defirst , M double , M second , andfined in Sections 3, 4.1, and 4.2, respectively. Define M2A
2A
3A
double in a similar way.
M3A
As shown in the proof of Proposition 3,
double M double − M double M double )M double (M33
∂ x2double
23
2A
3A= − 11.
∂A
detM double
second = M double at A = a,Since M double = M second and M3A
3A
second M second M second
double M double M double
M11
M11∂ x2double
23
33
3A
2A
=−
+
.
doublesecond
∂A
detM
detM

As shown in the proof of Proposition 2,

second M second M second
M11∂ x2second
23
3A
=
, so
∂A
detM second

double M double M double
M11∂ x2double∂ x2second
33
2A
=−
+
.
∂A
∂A
detM double(48)

double = 0, note thatAs M31
double double double
double double double
detM double =M11
M22 M33+ M12
M23 M31
double double double
double double double− M12
M21 M33− M11
M23 M32 .

Since
A = a,

Qx2 (x1∗ ,x2∗ ,a,θ ∗ )−Qx2 (x̂1∗ ,x2∗ ,A,θ2 )∂ θ2double
double = M double at== 0 and hence M1232∂ x2
Qθ2 (x̂1∗ ,x2∗ ,A,θ2 )double double double
detM double =M11
M22 M33
doubledoubledoubledoubledouble double
+ M12
{M23
(M31
− M11
) − M21M33 }.

Since

Qx1 (x1∗ ,x2∗ ,a,θ ∗ )Qx1 (x̂1∗ ,x2∗ ,A,θ2 )∂ θ2double∂ θ2double
double = M double +
=
=
−
and
∗
∗
31
∂ x1
∂ x̂1
Qθ2 (x̂1 ,x2 ,A,θ2 )
Qθ2 (x̂1∗ ,x2∗ ,A,θ2 ) , M11

double at A = a, and hence
M33

doubledouble double
doubledoubledoubledetM double = M33
{M11
M22
− M12
(M23
+ M21
)}.

78

double = M first , M double = M first , M double = M first , and M double +M double =Then since M11
11
22
22
12
12
23
21
first at A = a,M21
doubledetM double = M33
detM first .

(49)

double = M first and M double = M first ,Plugging this into (48) and using M11
11
2A
2A
first
M first M2A∂ xsecond ∂ x2first ∂ x2second∂ x2double= − 11 first
+ 2
=
+
.
∂A
∂A
∂A
∂A
detM

Proof of Part (iii). We first prove the second equation. As shown in the proofof Proposition 2,
second M second M second∂ x2second M11
23
3A
=
.
∂A
detM second
From (49) and M second = M double ,second M second M second∂ x2second M11
23
3A
=
.
second detM first∂A
M33
second = M first . Also M second = M first , so
When the game is symmetric, we have M3A
11
11
2A

M second ∂ x2first∂ x2second= − 23.
second ∂ A∂A
M33
second = U − L and M second = U − L.
Then the result follows from M23
ij
ii
33
The first equation in the proposition follows from this result and Proposition
2. The third equation in the proposition follows from part (ii) of this proposition.
We conclude the proof by showing the remaining two equations in the proposi2Q∂ x2
ition. Let Li = Qxθ1 ∂∂xiU∂ θ . When A1 = A2 = a, the multiplier effect on ∂ A2 appearingin Proposition 4 can be rewritten asU21
1
1 − U12U−L 11 U22 −L2
U21
U21 −L2 U12L2 U12 −L1
L1
L1 U21 −L2
L2
1
1 − U12U−L
1
−
−
−
−
U22 −L2
U22 U11 −L1
U22 −L2 U11
U11
U11 −L1 U22
U22
11

=

U22 (U11 − L1 )(U11U22 −U11 L2 −U21U12 +U21 L1 ).
(U11U22 −U11 L2 −U21U12 +U21 L1 )(U11U22 −U22 L1 −U12U21 +U12 L1 ) − (U12 L1 −U22 L1 )(U21 L1 −U11 L2 )79

When the game is symmetric, this reduces toUii (Uii − L)(Uii −Ui j )(Uii +Ui j − L)
(Uii −Ui j )2 (Uii +Ui j − L)2 − L2 (Uii −Ui j )2
Uii (Uii − L)(Uii +Ui j − L)=
(Uii −Ui j )(Uii2 +Ui2j + 2UiiUi j − 2Uii L − 2Ui j L)
Uii (Uii − L)(Uii +Ui j − L)
(Uii −Ui j )(Uii +Ui j )(Uii +Ui j − 2L)(Uii − L)(Uii +Ui j − L)
U2
= 2 ii 2 ·Uii (Uii +Ui j − 2L)
Uii −Ui j==(Uii − L)(Uii +Ui j − L)
1·00
1 − BR1 BR2
Uii (Uii +Ui j − 2L)

This and Proposition 4 imply the fourth equation in the proposition. Also, the lastUequation follows from BR012 = BR042 = − ULii and BR014 = − Uiiij at A = a.

A.6

Proof of Corollary 2

Let Uii , Ui j , and L be as stated in Proposition 5. That is,∂ 2Ui= 2Qx + xi Qxx − c00 < 0,
∂ 2 xi∂ 2Ui
Ui j == Qx + xi Qxx = Qx + xi Qxx < 0,
∂ xi ∂ x j

Uii =Qxi ∂ 2UiQx
L==
(xi Qxθ + Qθ ) < 0.
Qθ ∂ xi ∂ θ
QθNote thatUii < Ui j < 0.

(50)

Proof of part (i). First, consider the one-shot game with first-order misspecification. A Nash equilibrium action solves the first-order condition∂U1 (x, a, θ ∗ )∂U2 (x, A, θ ∗ )= 0 and
= 0.
∂ x1
∂ x2
80

Md

d =
2A
) where M2A
By the implicit function theorem, we have ∂∂xA2 = 1−BR10 BR0 (− U22
1
2
QA + x2 QxA is the direct effect of misspecification on player 2’s marginal utility,
and BR0i is the slope of the standard best-response function.36 As explained in themain text, the direct effect is positive. Also, (50) implies 1−BR10 BR0 > 1. Hence,1

2

we have ∂∂xA2 > 0. Firm 1 best-responds to this firm 2’s action, so ∂∂xA1 = BR01 ∂∂xA2 .
∂ x1
∂ x1
∂ x2
12
Since (50) implies BR01 = − U
U11 ∈ (−1, 0), we have ∂ A < 0 and ∂ A + ∂ A > 0. For
payoffs, note that at A = a, we have
∂ πi ∂ πi ∂ xi
∂ πi ∂ x−i∂ πi ∂ x−i
∂ x−i
=
+
=
= xi Qx,∂A∂ xi ∂ A ∂ x−i ∂ A∂ x−i ∂ A
∂A
where the second inequality follows from the fact that πi is an equilibrium payoff
so that ∂∂ πxii = 0. This immediately implies ∂∂πA1 < 0, ∂∂πA2 > 0, and ∂∂πA1 + ∂∂πA2 > 0.
Second, consider the infinite-horizon model with first-order misspecification.
Since we assume that the base misspecification effect is positive, from Proposition
1 and 1−BR10 BR0 > 1 at A = a, we have ∂∂xA2 > 0. The remaining inequalities can be
12
shown as in the one-shot game.
Third, consider the one-shot game with one-sided double misspecification.
Player 2 and the hypothetical player 1 plays a Nash equilibrium, which solves thefirst-order condition∂ Û1 (x̂1 , x2 , A, θ ∗ )∂U2 (x̂1 , x2 , A, θ ∗ )= 0 and
= 0.
∂ x̂1
∂ x2
Md

(51)
Md

2A
1A
By the implicit function theorem, we have ∂∂xA2 = 1−BR10 BR0 (− U22− BR02 U11
)=
1

2

d
M2A
1
d = Q + x̂ Q = M d . The remaining(−)(1 + BR02 ) > 0 where M1A
0
0
1 xA
A
2A
U
1−BR1 BR222

inequalities follow as in the case with first-order misspecification.
Finally, consider the infinite-horizon game with one-sided double misspecifiU −L
U −U
cation. From (50) and Uii − L < 0, we have 1 − Uiiij−L = Uiiii −Li j > 0. Then Propo∂ xfirst∂ xdouble

sition 5 and ∂2A > 0 imply ∂2 A > 0. The remaining inequalities can be shownas in the case with first-order misspecification.
36 The asymptotic best response curve coincides with the standard best response curve at A = a,so we use the same notation.

81

Proof of part (ii). Consider the one-shot game with second-order misspecification. Player 2 and the hypothetical player 1 plays a Nash equilibrium, whichsolves the first-order condition∂U2 (x̂1 , x2 , a, θ ∗ )∂ Û1 (x̂1 , x2 , A, θ ∗ )= 0 and
= 0.
∂ x̂1
∂ x2
Md

1A
) < 0. Then
By the implicit function theorem, we have ∂∂xA2 = 1−BR10 BR0 (−BR02 U11
1
2
the remaining inequalities follow as in the case with first-order misspecification.
Next, consider the infinite-horizon game with second-order misspecification.
Simple algebra shows that

Qxθ
Qxx

 < 0 if Qθ < Qx ,Ui j − L == 0 if QQxθθ = QQxxx ,
> 0 if QQxθθ > QQxxx .

∂ xfirst

Then, Proposition 5, Uii − L < 0, and ∂2A > 0 imply the result.
Proof of part (iii). Consider the one-shot game with two-sided double misspecification. Player 1 is not aware of player 2’s bias, so she chooses the Nashequilibrium x1correct regardless of the parameter A2 . Player 2 and the hypotheticalplayer 1 play a Nash equilibrium which solves the first-order condition (51), andas shown in part (i), we have ∂∂xA2 > 0.
Next, consider the infinite-horizon game. Define NEi0 as in Proposition 4.
Note that x1 = x2 = x̂1 = x̂2 = x∗ constitutes the steady state at A1 = A2 = a. A
necessary condition for the unique steady state is |NE10 NE20 | = |NEi0 |2 ≤ 1 at x∗ ;otherwise, as discussed in the proof of Proposition 1, there exist multiple steadystates. Also, det M , 0 implies |NEi0 |2 , 1. So for a steady state to be unique, wemust have |NEi0 |2 < 1 at x∗ , which implies NEi0 ∈ (−1, 1) at x∗ . At x∗ ,NEi0 = −∂ 2Ui ∂ θi
∂ xi ∂ θi ∂ x j

Uii +Ui j +∂ 2Ui∂ xi ∂ θi



∂ θi
∂ θi
∂ xi + ∂ x j

=−

L
.
Uii +Ui j − L

Because Uii − L < 0 and Ui j < 0, the condition NEi0 ∈ (−1, 1) is equivalent toUii +Ui j −L < L < −(Uii +Ui j −L). The first inequality implies Uii +Ui j −2L < 0.
Then, Proposition 5 and∂ x2first∂ A > 0 imply the result.

82

Q.E.D.

A.7

Proof of Corollary 3

Let Uii , Ui j , and L be as stated in Proposition 5. That is,∂ 2Ui
Uii = 2 = Qxi xi − c00 (xi ),
∂ xi
∂ 2Ui
Ui j == Qx1 x2 ,∂ xi ∂ x j
L=Qx
Qxi ∂ 2Ui
= i Qxi θ .
Qθ ∂ xi ∂ θQθSuppose that A = a. From Proposition 1, we have BR01 BR02 < 1 at the uniquesteady state xcorrect . Recall that at A = a, BR0i is simply the slope of the best
response curve in the standard sense. So by symmetry, we have BR01 = BR02 , whichUin turn implies |BR0i | = | Uiiij | < 1. Note also that Uii < 0, because the second-ordercondition for the incentive-compatibility condition must be satisfied according
to the regularity condition imposed in Proposition 1. Taken together, we have
Uii < Ui j .
Proof of part (i). The result for the one-shot game can be shown as in the
proof of Corollary 2. So we will prove only the result for the infinite-horizon
game. Since the base misspecification effect is negative, we have ∂∂xA2 < 0 underfirst-order misspecification. Also, since Uii < 0, L > 0, and Uii < Ui j , we have1−

Ui j − L Uii −Ui j=> 0.
Uii − LUii − L

Then from Proposition 5, we have ∂∂xA2 < 0 even under one-sided double misspecification. The remaining inequalities can be shown as in the proof of Corollary2.
Proof of part (ii). Again, we will prove only the result for the infinite-horizon
game. If Qx1 x2 > L as stated in (a), thenUi j − L< 1.
Uii − L
Here the first inequality follows from Uii < 0, L > 0, and Ui j − L = Qx1 x2 − L > 0,U
and the second inequality follows from Uii < 0, L > 0, and | Uiiij | < 1. Then from
0<−

Proposition 5, ∂∂xA2 < 0 under second-order misspecification.
83

On the other hand, if Qx1 x2 < L as stated in (b) and (c), thenUi j − L< 0.
−
Uii − L
Hence ∂∂xA2 > 0 under second-order misspecification. The remaining inequalities
can be shown as in the proof of Corollary 2.
The proof of part (iii) is also similar to that of corollary 2.
Q.E.D.

A.8

Proof of Lemma 1

For the case in which X is finite, this is exactly the same as Lemma 1 of Esponda,
Pouzo, and Yamamoto (2019). For the case in which X is continuous, we need aminor modification of the proof. We first prove a preliminary lemma:
R

Lemma 3. Assume that X is continuous. Under Assumption 1(iii) and (iv), Y g(x, y)Q(dy|x)
is bounded and continuous in x.
Proof. Take a sequence xn converging to x. ThenZnng(x , y)Q(dy|x ) −Zg(x, y)Q(dy|x)

Y

≤

Y

Z

n

n

g(x , y)Q(dy|x ) −Zg(xn , y)Q(dy|x)

Y

Y

Z

Z

+

g(xn , y)Q(dy|x) −Y

g(x, y)Q(dy|x) .
Y

From Assumption 1(iii), Q(dy|xn ) weakly converges to Q(dy|x), so the first termof the right-hand side converges to zero. Also from Assumption 1(iv-a), g(xn , y)pointwise converges to g(x, y), so the second term converges to zero.
Q.E.D.
As shown in the display in EPY, we havennKi,k (θi,k, σ n ) − Ki (θi,k
,σ) ≤

Z Z

g(x, y)Q(dy|x)σX̂n

XZ Y Z−

X Y

1,1 ×X̂2,1(dx)g(x, y)Q(dy|x)σX̂1,1 ×X̂2,1 (dx)where σX̂1,1 ×X̂2,1 and σX̂n ×X̂ are the marginals of σ and σ n on X̂1,1 × X̂2,1 , re1,1
2,1spectively. From Lemma 3, the right-hand side converges to zero as σ n → σ . The
rest of the proof is exactly the same as in EPY.
84

A.9

Proof of Proposition 6

For the special case in which X is finite, Theorem 1 of Esponda, Pouzo, andYamamoto (2019) proves the same result. We need a minor modification to their
proof, as they use finiteness of X in Step 2 in the proof of Lemma 2.
Pick i, k, θi,k . Then let
q(y|x̂1,1 , x̂2,1 ) q(y|x̂1,1 , x̂2,1 )−fl (x̂) = EQ(·|x̂1,1 ,x̂2,1 )  supq (y|x̂i,k , x̂i,k+1 ) qθi,k0 (y|x̂i,k , x̂i,k+1 )θ 0 ∈O(θ , 1 ) θi,k
i,k

i,k l

where O(θi,k , 1l ) is a 1l -neighborhood of θi,k . Then as explained at the end of thethe first paragraph in EPY’s step 2, liml→∞ fl (x̂) → 0 for each x̂. In what follows,
we will show that this convergence is uniform in x̂; then there is δ (θi,k , ε) withwhich (16) of EPY holds, and the rest of the proof is exactly the same as EPY’s.
Pick an arbitrary ε > 0. For each x̂, let F(x̂) = {l ∈ [0, ∞)| fl (x̂) ≥ ε}. Then
we have the following lemma:
Lemma 4. For each x̂, there is l(x̂) > 0 such that F(x̂) = [0, l(x̂)]. Also F(x̂) is
upper hemi-continuous in x̂.
Proof. The first part follows from the fact that fl (x̂) is continuous and decreasing
in l, and liml→∞ fl (x̂) = 0.
To prove the second part, pick x̂ and an arbitrary small η > 0. Then fl(x̂)+η (x̂) <ε. Since fl (x̂) is continuous in x̂, there is an open neighborhood U of x̂ suchthat fl(x̂)+η (x̂0 ) < ε for all x̂0 ∈ U. This implies that l(x̂0 ) < l(x̂) + η for allx̂0 ∈ U.
Q.E.D.
The above lemma implies that l(x̂) is an upper hemi-continuous function, andfrom the Maximum theorem, l(x̂) is bounded; l(x̂) < l ∗ for some l ∗ . Hence fl (x̂) ≤ε for all x̂ and l ≥ l ∗ , implying uniform convergence.

A.10

Proof of Proposition 8

This is very similar to the first step of the proof of Proposition 2 in EPY. However,
we need a minor modification, as X may not be finite in our setup. We first prove85

upper hemi-continuity of Bε (σ ).
Lemma 5. Bε (σ ) is upper hemi-continuous in (ε, σ ).
i +1Proof. Since ∏2i=1 ∏kk=1
4Θi,k is compact, it is sufficient to show that (ε n , σ n , µ̂ n ) →(ε, σ , µ̂) and µ̂ n ∈ Bε n (σ n ) for each n imply µ̂ ∈ Bε (σ ). Note that
Z

Z
n n
lim
(Ki,k (θi,k , σ )µ̂i,k (dθi,k ) −(Ki,k (θi,k , σ )µ̂i,k (dθi,k )
n→∞
Θi,k
Θi,k
Z

Z
n n
n
(Ki,k (θi,k , σ )µ̂i,k (dθi,k ) −(Ki,k (θi,k , σ )µ̂i,k (dθi,k )
= lim
n→∞
Θi,k
Θi,k

Z
Z
n
(Ki,k (θi,k , σ )µ̂i,k (dθi,k ) −(Ki,k (θi,k , σ )µ̂i,k (dθi,k ) .
+ lim

n→∞

Θi,k

Θi,k

The first term of the right-hand side is zero, because Ki,k (·, σ n ) pointwise converges to Ki,k (·, σ ) (which follows from the fact that σ n weakly converges to σ ).
n weakly converges to
Also the second term of the right-hand side is zero, as µi,k
µi,k .
Zlim

n→∞ Θi,knn(Ki,k (θi,k , σ )µ̂i,k(dθi,k ) =Z(Ki,k (θi,k , σ )µ̂i,k (dθi,k ).
Θi,kSince µ̂ n ∈ Bε n (σ n ),
Z
Θi,k

n
∗
(σ n ))µ̂i,k(dθi,k ) ≤ ε n .
(Ki,k (θi,k , σ n ) − Ki,k∗ (σ ) (which follows from the theory of
Taking n → ∞ and using continuity of Ki,kmaximum),Z∗(Ki,k (θi,k , σ ) − Ki,k(σ ))µ̂i,k (dθi,k ) ≤ ε.
Θi,kHence µ ∈ Bε (σ ), which implies upper hemi-continuity of Bε (σ ).

Q.E.D.

Now we show that Sε (σ ) is upper hemi-continuous at ε = 0. Since X is compact, it suffices to show that (ε n , σ n , xn ) → (0, σ , x) and xn ∈ Sε n (σ n ) for each n,
imply x ∈ Sε (σ ). As noted earlier, we already know that S0 (σ ) is upper hemicontinuous in σ . So without loss of generality, we assume ε n > 0 for all n.
86

Since xn ∈ Sε n (σ n ), there is µ̂ n ∈ Bε n (σ n ) with xn = ŝ(µ̂ n ). The sequence(ε n , σ n , xn , µ̂ n ) is in a compact set, so there is a convergent subsequence, still
denoted by (ε n , σ n , xn , µ̂ n ). Let µ̂ = limn→∞ µ̂ n . Then µ̂ ∈ B0 (σ ), as Bε (σ ) is
upper hemi-continuous and µ̂ n ∈ Bε n (σ n ) for each n. Also, we have x ∈ Ŝ(µ̂),
because Ŝ is upper hemi-continuous and xn ∈ Ŝ(µ̂ n ) for each n. Hence x ∈ S0 (σ ).

A.11

Proof of Proposition 9

The proof is very similar to that of Theorem 2 of EPY. In EPY, the proof consistsof three steps. In the first two steps, they show that w is a perturbed solution of thedifferential inclusion. Then in the last step, they show that a perturbed solution is
an asymptotic pseudotrajectory (i.e., it satisfies (33)).
Our Propositions 7 and 8 imply that w is indeed a perturbed solution in thesense of EPY. We can also show that a perturbed solution is indeed an asymptotic
pseudotrajectory. The proof is omitted because, other than replacing the Euclidean
norm with the dual bounded-Lipschitz norm, it is exactly the same as the last stepof EPY.37

A.12

Proof of Lemma 2

We will show that θ (σ ) is Lipschitz continuous in σ . Under Assumptions 4(i) and(iii), the inverse (∇2 Ki,k (θi,k (σ ), σ ))−1 of the Hessian matrix exists for each σ ,and is continuous in σ . This means that k(∇2 Ki,k (θi,k (σ ), σ ))−1 k is bounded andcontinuous in σ , where kCk = maxi j |ci j | denotes the max norm of a matrix C ={ci j }, . Since 4X̂ is compact, there is L1 such that k(∇2 Ki,k (θi,k (σ ), σ ))−1 k < L1
for all i, k, and σ . Pick such L1 .
37 This parallels Perkins and Leslie (2014), who show that the stochastic approximation tech-nique of Benaı̈m (1999) for the Euclidean space extends to Banach spaces with the same proof.
Our result differs from Perkins and Leslie (2014) in that we consider a differential inclusion, rather
than a differential equation. But this does not cause any technical difficluty, because (i) 4X̂ is acompact subset of a banach space with the dual bounded Lipschitz norm and (ii) Mazur’s lemma,
which is used to establish the result for differential inclusions in Euclidean spaces (Benaı̈m, Hofbauer, and Sorin (2005) and EPY), is valid even in Banach spaces.

87Under Assumption 4(ii), there is L2 > 1 such that
∂ Ki,k (θi,k , x̂) ∂ K(θi,k , x̂0 )−< L2 |x̂ − x̂0 |∂ θi,k,m∂ θi,k,m
for all i, k, m, θi,k , x̂, and x̂0 . Also, under Assumption 4(i), there is L3 > 1 suchthat
∂ Ki,k (θi,k , x̂)< L3∂ θi,k,m
for all i, k, m, θi,k , and x̂. Then for each σ and σ 0 , we have
∂ Ki,k (θi,k , σ ) ∂ Ki,k (θi,k , σ 0 )−∂ θi,k,m∂ θi,k,mZ
Z∂ Ki,k (θi,k , x̂) 0∂ Ki,k (θi,k , x̂)σ (d x̂) −σ (d x̂) ≤ 4L2 L3 kσ − σ 0 k=∂ θi,k,m∂ θi,k,mwhere the inequality follows from the definition of the dual bounded-Lipschitz∂ K (θ ,x̂)
norm and the fact that 4L12 L3 ∂i,kθ i,k ∈ BL(X̂). This in turn implies that ∇Ki,k (θi,k , σ )i,k,mis equi-Lipschitz continuous, that is, there is L4 > 0 such that |∇Ki,k (θi,k , σ ) −∇Ki,k (θi,k , σ 0 )| < L4 kσ − σ 0 k for all i, k, θi,k , σ , and σ 0 .
Let L = L1 L4 . We will show that θ (σ ) is Lipschitz continuous with the constant L. To do so, pick two action frequencies σ and σ 0 , σ arbitrarily. For
each β ∈ [0, 1], let σβ = β σ + (1 − β )σ 0 denote a convex combination of σ and
σ 0 . From Assumption 4(iii), the KL minimizer θi,k (σβ ) must solve the first-ordercondition∇Ki,k (θi,k , σβ ) = 0,
which is equivalent toβ ∇Ki,k (θi,k , σ ) + (1 − β )∇Ki,k (θi,k , σ 0 ) = 0.
Then by the implicit function theorem,dθ (σβ )= −(∇2 Ki,k (θi,k (σβ ), σβ ))−1 ∇Ki,k (θi,k (σβ ), σ ) − ∇Ki,k (θi,k (σβ ), σ 0 ) .
dβ
(52)
88

Using the fundamental theorem of calculus, we haveθ (σ ) − θ (σ 0 )= θ (σ1 ) − θ (σ0 )
Z 1

=−

0


(∇2 Ki,k (θi,k (σβ ), σβ ))−1 ∇Ki,k (θi,k (σβ ), σ ) − ∇Ki,k (θi,k (σβ ), σ 0 ) dβ .

Then by the definition of L1 and L4 ,|θ (σ ) − θ (σ̃ )| ≤Z 1
0

L1 L4 kσ − σ 0 kdβ = Lkσ − σ 0 k,

as desired.

A.13

Proof of Proposition 12

We will first present a preliminary lemma. Pick an arbitrary action frequency
σ(0) ∈ 4X̂ and a solution σ ∈ Z(σ(0)) to the differential inclusion (33) startingfrom this σ(0). Let θ(t) = θ (σ(t)) for each t. The following lemma shows that
{θ(t)}t≥0 solves (34).
Lemma 6. Pick t ≥ 0 such that (33) holds. Then θ̇(t) exists and satisfies (34).
Proof. Pick t as stated, and pick σ ∗ ∈ 4S0 (σ(t)) such that σ̇(t) = σ ∗ − σ(t). Letσβ = β σ ∗ + (1 − β )σ(t) for each β ∈ [0, 1]. Then we have
θ (σ(t + ε)) − θ (σ(t))
=
ε




θ (σε ) − θ (σ0 ) θ (σ(t + ε)) − θ (σε )+.
εεAll we need to show is that the right-hand side has a limit as ε → 0, and the limit(σ(t))
is in the right-hand side of (34). Then θ (σ(t+ε))−θ
also has a limit θ̇(t) andεthis limit value satisfies (34).
(σ0 )exists and is in the right-hand side of (34).
Note first that limε→0 θ (σε )−θε89Indeed, from (52),θ (σε ) − θ (σ0 )ε→0εdθ (σβ )
=
dβ
β =0
lim


= −(∇2 Ki,k (θi,k (σ0 ), σ0 ))−1 ∇Ki,k (θi,k (σ0 ), σ1 ) − ∇Ki,k (θi,k (σ0 ), σ0 )= −(∇2 Ki,k (θi,k (σ(t)), σ(t)))−1 ∇Ki,k (θi,k (σ(t)), σ ∗ )where the second equality follows from the fact that θi,k (σ0 ) solves the first-ordercondition.
(σε )= 0. Since
We conclude the proof by showing that limε→0 θ (σ(t+ε))−θε
θ (σ ) is Lipschitz continuous, there is L > 0 such thatθ (σ(t + ε)) − θ (σε )σ(t + ε)) − σε≤Lεε(σ(t + ε)) − σ(t)) − (σε − σ0 ).
=LεTaking ε → 0,
θ (σ(t + ε)) − θ (σε )σ(t + ε) − σ(t)σε − σ0= L lim
− lim
ε→0
ε→0
ε→0
ε
ε
ε
lim

=L

dσ(t) dσβ−=0dt
dβ β =0
Q.E.D.

Now we prove the proposition. Pick T > 0 and h ∈ H arbitrary. Pick any
small ε > 0. Since θ (σ ) is uniformly continuous in σ (this follows from thecontinuity of θ and the compactness of 4X̂), there is η > 0 such that |θ (σ ) −θ (σ̃ )| < ε for any σ and σ̃ with kσ − σ̃ k < η. From Proposition 9, there is t ∗such that for any t > t ∗ , there is σ ∈ Z(w(h)[t]) such that
kw(h)[t + τ] − σ(τ)k < η90for all τ ∈ [0, T ]. Pick such σ, and consider the corresponding θ, i.e., let θ(t) =θ (σ(t)) for each t. Then by the definition of η, we havekwθ (h)[t + τ] − θ(τ)k < εfor all τ ∈ [0, T ]. Also this θ solves (34).38 This implies the result we want.

A.14

Proof of Proposition 13

Pick an arbitrary sample path h ∈ H . Let (i, k) ∈ I ∗∗ \ I ∗ . Throughout the proof,
for each θi,k , let θ (θi,k ) denote θ such that θ j,l = θi,k for all ( j, l) ∈ I(i, k) and∗ for all ( j, l) ∈ I ∗ .
θ j,l = θ j,lt (h), E ) = 0, where E
We will show that limt→∞ d(θi,ki,ki,k is the set of agent
0 (θ , σ 0 ) = 0(i, k)’s steady state model, i.e., Ei,k is the set of all θi,k such that Ki,ki,k0
for some σ ∈ 4S0 (θ (θi,k )). There are two cases to be considered.
A.14.1t (h) , lim suptCase 1: lim inft→∞ θi,kt→∞ θi,k (h).

t (h), lim supt
In this case, we will show that [lim inft→∞ θi,kt→∞ θi,k (h)] ⊆ Ei,k . Thist (h), E ) = 0.
immediately implies limt→∞ d(θi,ki,kt (h), lim supt
Suppose not, so that there is a model θi,k ∈ [lim inft→∞ θi,kt→∞ θi,k (h)]∗ < E . Then (i) K 0 (θ ∗ , σ 0 ) > 0 for all σ 0 ∈ 4S (θ (θ ∗ )), or (ii)such that θi,k0i,ki,k i,k
i,k
0
∗
0
0
∗
Ki,k (θi,k , σ ) < 0 for all σ ∈ 4S0 (θ (θi,k )). Indeed, if not and there are two mea∗ )) with K 0 (θ ∗ , σ 0 ) > 0 and K 0 (θ ∗ , σ 00 ) < 0, then bysures σ 0 , σ 00 ∈ 4S0 (θ (θi,ki,k i,ki,k i,k000∗∗
0 (θ ∗ , σ ∗ ) = 0,
mixing σ and σ we can construct σ ∈ 4S0 (θ (θi,k )) with Ki,ki,kwhich is a contradiction. In what follows, we will focus on the case (i), where0 (θ ∗ , σ 0 ) > 0 for all σ 0 ∈ 4S (θ (θ ∗ )). The proof for the case (ii) is symmetKi,k
0
i,k
i,k
ric.
0 (θ , σ ) is continuous in (θ , σ ) and 4S (θ (θ )) is upper hemiSince Ki,k
0
i,k
i,k
i,k
0 (θ , σ 0 ) > 0 for any θ with |θ −continuous in θi,k , there is ε > 0 such that Ki,k
i,k
i,k
i,k
38 Note that θ is absolutely continuous because σ is absolutely continuous and θ (σ ) is Lipschitz

continuous. Also from Lemma 6, θ satisfies (34) for almost all t.

91

∗ | ≤ ε and any σ 0 ∈ 4S (θ (θ )). Pick such ε > 0. Then pick T such that
θi,k
0
i,k
∗
θi,k (t) ≥ θi,k
+ε

(53)

∗ − ε.
for any t ≥ T and for any solution θ ∈ Zθ (θ (θi,k )) with any θi,k with θi,k ≥ θi,kFrom Proposition 12, there is t ∗ such that for any t > t ∗ , θ ∈ Zθ (wθ (t)), ands ∈ [0, 2T ],|wθ ,i,k (t + s) − θi,k (s)| <

ε
2

(54)

∗ ≤ lim suptwhere wθ ,i,k denote the (i, k)-component of wθ . Pick such t ∗ . Since θi,kt→∞ θ (h),∗ − ε. Pick such t ∗∗ . Then from (53), we
there is t ∗∗ > t ∗ such that wθ ,i,k (t ∗∗ ) ≥ θi,k
have∗
θi,k (s) ≥ θi,k+ε

for any s ≥ T and for any solution θ ∈ Zθ (wθ (t ∗∗ )). This inequality and (54)impliesε∗wθ ,i,k (t ∗∗ + s) ≥ θi,k+∀s ∈ [T, 2T ].
2∗ + ε , it follows from (53) that
Likewise, since wθ ,i,k (t ∗∗ + T ) ≥ θi,k2∗
+εθi,k (s) ≥ θi,kfor any s ≥ T and for any solution θ ∈ Zθ (wθ (t ∗∗ + T )). This inequality and (54)impliesε∗wθ ,i,k (t ∗∗ + s) ≥ θi,k+∀s ∈ [2T, 3T ].
2
Iterating this argument, we can show that∗wθ ,i,k (t ∗∗ + s) ≥ θi,k+ε2∀s ∈ [T, ∞).

t (h) ≥ θ ∗ + ε , which is a contradiction.
But this means that lim inft→∞ θi,ki,k2

A.14.2t (h) = lim suptCase 2: lim inft→∞ θi,kt→∞ θi,k (h).

t (h) exists. Let θ ∗ = limt
In this case, limt→∞ θi,kt→∞ θi,k (h). We will show thati,k∗ ∈ E.
θi,k92

0 (θ ∗ , σ 0 ) > 0
Suppose not so that θ ∗ < E. Then as in the previous case, (i) Ki,ki,k∗ )), or (ii) K 0 (θ ∗ , σ 0 ) < 0 for all σ 0 ∈ 4S (θ (θ ∗ )). We
for all σ 0 ∈ 4S0 (θ (θi,k0i,k i,ki,kwill focus on the case (i).
0 (θ , σ 0 ) > 0 for any θ
As in the previous case, there is ε > 0 such that Ki,k
i,k
i,k
∗
0
with |θi,k − θi,k | ≤ ε and any σ ∈ 4S0 (θ (θi,k )). Pick such ε > 0. Then pick T
such that (53) holds for any t ≥ T and for any solution θ ∈ Zθ (θ (θi,k )) with any∗ − ε.
θi,k with θi,k ≥ θi,kFrom Proposition 12, there is t ∗ such that (54) holds for any t > t ∗ , θ ∈∗ = limt
Zθ0 (wθ (t)), and s ∈ [0, 2T ]. Pick such t ∗ . Since θi,kt→∞ θ (h), there is∗ − ε. Pick such t ∗∗ . Then as in the previoust ∗∗ > t ∗ such that wθ ,i,k (t ∗∗ ) ≥ θi,kcase, we can show that∗wθ ,i,k (t ∗∗ + s) ≥ θi,k+ε2∀s ∈ [T, ∞).

t (h) ≥ θ ∗ + ε , which is a contradiction.
But this means that limt→∞ θi,k
i,k
2

A.15

Proof of Proposition 14

By assumptions (i)-(iii), it is obvious that the graphs {(θ2 , fˆ1 (θ2 , A))|∀θ2 } and{(θ̂1 , f2 (θ̂1 , A))|∀θ̂1 } intersect once. This immediately implies that there is a
steady state and it is unique. So what remains is to show that the belief converges
to this steady state almost surely.
Recall that under second-order misspecification, there are three players: Player
1, player 2, and the hypothetical player.39 Let σ = (σ1 , σ2 , σ̂1 ) denote an action
frequency of these players. Given σ , let K1 (θ1 , σ ) denote the weighted KL divergence between player 1’s subjective signal distribution and the true distribution, and let θ1 (σ ) denote the minimizer of this KL divergence. Similarly define
K2 (θ2 , σ ), θ2 (σ ), K̂1 (θ̂1 , σ ), and θ̂1 (σ ).
Since player 1 is unbiased, we have θ1 (σ ) = θ ∗ for all σ , which implies that
θ1t is constant over time and θ1t = θ ∗ for all t. On the other hand, θ t = (θ2t , θ̂1t )39 Formally, in the game with second-order misspecification, we have k

1 = 3, k2 = 2, M2,1 =∗∗Agents (1, 2) and (1, 3) are redundant, so I = {(1, 1), (2, 1), (2, 2)}.

M1,2 , and M2,2 = M1,3 .
Agent (1, 1) is player 1, agent (2, 1) is player 2, and agent (2, 2) is the hypothetical player.

93changes as time goes, and our Proposition 12 ensures that the motion of this θ t isapproximated by the differential inclusion (35), which can be rewritten as the twodimensional problem
!


[
[
K20 (θ2 (t), σ 0 ) K̂10 (θ̂1 (t), σ 0 )dθ2 (t) d θ̂1 (t)
,
∈
− 00
,−
dt
dt
K2 (θ2 (t), σ ) K̂100 (θ̂1 (t), σ )σ :θ (σ )=θ(t) σ 0 ∈4S∗ (θ(t))0(55)
where S0∗ (θ ) denote the set of all one-shot equilibrium x = (x1 , x2 , x̂1 ) given the
beliefs θ1 = θ ∗ and θ = (θ2 , θ̂1 ).
In what follows, we will show that regardless of the initial value, any solution
to the differential inclusion (55) converges to the steady state after a long time.
This implies that the steady state is globally attracting in the sense of Esponda,
Pouzo, and Yamamoto (2019), and their Proposition 2 ensures that θ t converges
to the steady state almost surely, as desired.
The following lemma partially characterizes the solution to the differential
inclusion (55): It shows that θ2 (t) moves toward f2 (θ̂1 (t), A) at any time t.
Lemma 7. Pick any initial value θ(0) and any solution θ to the differential inclusion (55). Then for any t ≥ 0 with θ2 (t) > f2 (θ̂1 (t)), we have θ̇2 (t) < 0. Similarly,
for any t ≥ 0 with θ2 (t) < f2 (θ̂1 (t)), we have θ̇2 (t) > 0
Proof. Suppose that θ2 (t) > f2 (θ̂1 (t), A) at some time t. To prove θ̇2 (t) < 0, it
suffices to show that K20 (θ2 (t), σ 0 ) > 0 for any σ 0 ∈ 4S0∗ (θ(t)).
Suppose not and there is σ 0 ∈ 4S0∗ (θ(t)) such that K20 (θ2 (t), σ 0 ) < 0. (Weignore the case with K20 (θ2 (t), σ 0 ) = 0, because in such a case, θ2 (t) ∈ f2 (θ̂1 (t), A),
which contradicts with the uniqueness of f2 (θ̂1 (t), A).) We consider the following
two cases:
Case 1: θ2 (t) = θ . In this case, the KL minimizer given this σ 0 is θ2 (σ 0 ) = θ =θ2 (t) (this follows from the fact that the KL divergence is single-peaked w.r.t. θ2 ).
Hence θ2 (t) = θ is a steady state, i.e., θ2 (t) ∈ f2 (θ̂1 (t), A). But this contradicts
with the uniqueness of f2 (θ̂1 (t), A).
Case 2: θ2 (t) < θ . An argument similar to that in Case 1 shows that at
θ2 = θ , we have K20 (θ , σ 0 ) > 0 for all σ 0 ∈ 4S0∗ (θ , θ̂1 (t)). On the other hand,94

by the assumption, at θ2 = θ2 (t), K20 (θ2 (t), σ 0 ) < 0 for some σ 0 ∈ 4S0∗ (θ(t)).
Then since the set {K20 (θ2 , σ 0 )|∀σ 0 ∈ 4S0∗ (θ2 , θ̂1 (t))} is convex and upper hemicontinuous in θ2 , there must be θ2 ∈ [θ2 (t), θ ] and σ 0 ∈ 4S0∗ (θ2 , θ̂1 (t)) such that
K20 (θ2 (t), σ 0 ) = 0. This implies that θ2 ∈ f2 (θ̂1 , A), but it contradicts with theuniqueness of f2 (θ̂1 , A).
Next, suppose that θ2 (t) < f2 (θ̂1 (t), A) at some time t. Then an argument
similar to the one above shows that K20 (θ2 (t), σ 0 ) < 0 for any σ 0 ∈ 4S0∗ (θ(t)),
which implies θ̇2 (t) > 0.
Q.E.D.
Now we will construct a Lyapunov function V to show that any solution to thedifferential inclusion (55) converges to the steady state. Without loss of generality,
assume that the steady state is θ ∗ = (θ2∗ , θ̂1∗ ) = (0, 0). From assumption (iii), thereis κ > 0 such that maxθ̂1 | f2 (∂θ̂θ̂1 ,A) | < κ <11fˆ (θ ,A)maxθ2 | 1 ∂ θ2

2|

. Pick such κ, and for eachθ = (θ2 , θ̂1 ), let
V (θ ) = max |θ2 |, |κ θ̂1 | .
We will show that given any initial value θ(0) and given any solution θ to thedifferential inclusion (34),V̇ (θ(t)) < 0
for all t with θ(t) , (0, 0). We will consider the following cases separately:
Case 1: |θ2 (t)| > |κ θ̂1 (t)|. Assume first that θ2 (t) > 0. Then by the definitionof κ and f2 (0) = 0, we have f2 (θ̂1 (t)) < |κ θ̂1 (t)| < θ2 (t). Then from Lemma 7
and θ2 (t) > 0, we have V̇ (θ(t)) = θ̇2 (t) < 0.
Assume next that θ2 (t) < 0. By the definition of κ and f2 (0) = 0, we havef2 (θ̂1 (t)) > −|κ θ̂1 (t)| > θ2 (t). Then from Lemma 7 and θ2 (t) < 0, we haveV̇ (θ(t)) = −θ̇2 (t) < 0.
Case 2: |θ2 (t)| < |κ θ̂1 (t)|. An argument similar to those for Case 1 shows thatV̇ (θ(t)) < 0.
Case 3: |θ2 (t)| = |κ θ̂1 (t)|. We will focus on the case with θ2 (t) > 0 andθ̂1 (t) > 0, because a similar argument applies to all other cases. Then as in thefirst half of Case 1, we have θ̇2 (t) < 0. Also, a similar argument shows thatθ̂˙ 1 (t) < 0. Hence we have V̇ (θ(t)) = {θ̇2 (t), κ θ̂˙ 1 (t)} < 0.
95

A.16

Proof of Proposition 15

The result immediately follows from the following lemma:
Lemma 8. Suppose that all the assumptions stated in Proposition 15 are satisfied.
Then for any sample path h ∈ H ,∗ .
(i) lim inft→∞ θ i,k (σ t (h)) ≥ θi,k∗ .
(ii) lim supt→∞ θ i,k (σ t (h)) ≤ θi,k

In what follows, we will prove this lemma. We will focus on part (i), because
the proof of part (ii) is symmetric.
We begin with stating two preliminary lemmas. The first lemma considers thet , andcase in which the current action frequency has a unique KL minimizer θi,kt is lower than the steady state belief
shows that if the current KL minimizer θi,k
∗ , then today’s action S (θ t ) induces a higher KL minimizer. This implies
θi,k
0 i,k
that the KL minimizer tomorrow will be closer to the steady state belief than thecurrent one. Likewise, if the current KL minimizer is higher than the steady state
belief, then today’s action induces a lower KL minimizer.
∗ , and θ (S (θ̃ )) < θ̃ for all
Lemma 9. θi,k (S0 (θ̃i,k )) > θ̃i,k for all θ̃i,k < θi,ki,k 0 i,ki,k∗θ̃i,k > θi,k .

Proof. Note that θi,k (S0 (·)) is a continuous mapping from Θi,k ⊆ R to itself, andits fixed point is a steady state. Since there is a unique steady state, the result
follows from a standard argument.
Q.E.D.
The next lemma considers the case in which the current action frequency neednot have a unique minimizer, and shows that the result similar to the previoust ) is lower than thelemma holds; very roughly, if the smallest KL minimizer θ (θi,ksteady state belief, then it will move up and approaches the steady state belief.
The proof is omitted, as it is very similar to Lemma 4 of Esponda, Pouzo, andYamamoto (2019).

96∗ and any σ such that σ = σ for each ( j, l) ∈ I ∗∗Lemma 10. Pick any θi,k < θi,k
j,l
j˜,l˜
˜ ∈ I( j, l) and such that Ki,k (θi,k , σ ) < Ki,k (θ̃i,k , σ ) for all θ̃i,k < θi,k . Thenand ( j˜, l)for any solution σ ∈ Z(σ ) starting from this σ , we have θ i,k (σ(t)) > θ for allt > 0.
Now we will prove Lemma 8. Suppose not, so that there is a sample path h ∈∗ . Pick such h, and let θ 0 = lim inft
H such that lim inft→∞ θ i,k (σ t (h)) < θi,kt→∞ θ i,k (σ (h)).
i,kLet w : [0, ∞) → 4X denote the continuous-time interpolation of the action fre∞ .
quency (σ t (h))t=1
Pick ε > 0 such that
∂ Ki,k (θi,k , σ )<0∂ θi,k0∀θi,k ≤ θi,k
+ε

(56)

for all σ such that


σ


[
0 −ε
σ̃ :θ i,k (σ̃ )≥θi,k
S0 (σ̃ ) > 1 − 2ε.

To see why such ε exists, note first that from Lemma 9 and Assumption (iv) of
0 for all θ̃ ≥ θ 0 . Since θ (S (·)) is continuProposition 15, θi,k (S0 (θ̃i,k )) > θi,k
i,k
i,k 0
i,k
0
0 − ε. Thenous, for any small ε, we have θi,k (S0 (θ̃i,k )) > θi,k + 2ε for all θ̃i,k ≥ θi,k
0 + 2ε forfrom Assumptions (iv) and (v) of Proposition 15, we have θi,k (x) > θi,k0 − ε and for all x ∈ S (σ ). Then from the singleall σ such that θ i,k (σ ) ≥ θi,k0
peakedness assumption, (56) holds for all σ such that



σ

[
0 −ε
σ̃ :θ i,k (σ̃ )≥θi,k
S0 (σ̃ ) = 1.

This inequality does not change even if σ is perturbed, so ε satisfies the desired
property. (Take ε small, if necessary.)
1
Pick T > 0 such that 1+T< ε. Then pick t ∗ > 0 such that for all t > t ∗ ,
sup

inf

s∈[0,2T ] σ∈Z(w(t))|σ(s) − w(t + s)| < ε.

97

(57)

0 . Pick any
Pick t > t ∗ such that θ i,k (w(t)) is in the ε-neighborhood of θi,ksolution σ ∈ Z(w(t)) to the differential inclusion starting from this w(t). Thenfrom Lemma 10 (we set θi,k = θ i,k (w(t))), we have θ i,k (σ(s)) > θ i,k (w(t)) >θ 0 − ε for all s > 0. So in this solution σ, the share of the set of action profiles
Si,k0 −ε S0 (σ̃ ) increases over time. In particular, by the definition of T ,σ̃ :θ i,k (σ̃ )≥θi,kwe have
σ(s) [0 −ε
σ̃ :θ i,k (σ̃ )≥θi,k
S0 (σ̃ ) > 1 − εThen from (57), we have



[


w(t + s) ∀s ≥ T.

0 −ε
σ̃ :θ i,k (σ̃ )≥θi,k
S0 (σ̃ ) > 1 − 2ε∀s ∈ [T, 2T ].

This and (56) imply∂ Ki,k (θi,k , w(t + s))<0∂ θi,k0∀θi,k ≤ θi,k
+ ε∀s ∈ [T, 2T ].

Now consider a solution σ0 to the differential inclusion starting from w(t0 + T ).
0 +ε), we have θ (σ(s)) > θ 0 +ε >Then again from Lemma 10 (we set θi,k = θi,ki,ki,k0 − ε for all s > 0. Hence
θi,k




σ0 (s) [

0 −ε
σ̃ :θ i,k (σ̃ )≥θi,k
S0 (σ̃ ) > 1 − ε∀s ≥ T,

which impliesw(t + s) 


[

0 −ε
σ̃ :θ i,k (σ̃ )≥θi,kand thusS0 (σ̃ ) > 1 − 2ε∀s ∈ [2T, 3T ]∂ Ki,k (θi,k , w(t + s))0< 0 ∀θi,k ≤ θi,k+ ε∀t ∈ [T, 3T ].
∂ θi,k98

Iterating the same argument, we can show that
∂ Ki,k (θi,k , w(t + s))<0∂ θi,k0∀θi,k ≤ θi,k
+ ε∀s ≥ T.

0 + ε for all s ≥ T , which is a contradiction.
This implies θ i,k (w(t + s)) > θi,kB

Uniqueness of Steady StateWe provide a sufficient condition for the unique steady state in symmetric gamesat A1 = A2 = a in each misspecification.
First-order misspecification. Assume that for any fixed A and θi , Nash equii
librium of the one-shot game is unique. Assume also that ∂U∂ xi > 0 at x1 = x2 = 0i
and that ∂U∂ xi < 0 at some x1 = x2 > 0; this condition ensures that the unique Nashiequilibrium of the one-shot game is interior. By the continuity of ∂U∂ xi , there exi
ists x1 = x2 => 0 such that ∂U∂ xi = 0; that is, the unique Nash equilibrium of the
one-shot game is symmetric.
Note that steady state is an intersection of asymptotic best response correspondences BR1 and BR2 . Hence, the steady state is unique if −1 < BR0i (x j ) < 1 for
all i and x j where j , i.

Second-order and one-sided double misspecification. Assume that for any
fixed A and θi , Nash equilibrium of the one-shot game is unique. Assume also∂Uiithat ∂U∂ xi > 0 at x1 = x̂1 = x2 = 0 and that ∂ xi < 0 at some x1 = x̂1 = x2 > 0; thiscondition ensures that the unique Nash equilibrium of the one-shot game is inte∂Uiirior. By the continuity of ∂U∂ xi , there exists x1 = x̂1 = x2 => 0 such that ∂ xi = 0;
that is, the unique Nash equilibrium of the one-shot game is symmetric.
Note that steady state is an intersection of BR12 and NE2 (x1 ). Hence, thesteady state is unique if −1 < BR012 < 1 for all (x1 ,ˆx2 ) and −1 < NE20 (x1 ) < 1 forall x1 .
The conditions and derivations of the uniqueness of one-sided double misspecification is the same as that of two-sided misspecification.
99

Two-sided double misspecification. Assume that for any fixed Ai and θi , Nashiequilibrium of the one-shot game is unique. Assume also that ∂U∂ xi > 0 at x1 =ix̂2 = x2 = x̂1 = 0 and that ∂U∂ xi < 0 at some x1 = x̂2 = x2 = x̂1 > 0; this condition
ensures that the unique Nash equilibrium of the one-shot game is interior. By the∂Uiicontinuity of ∂U∂ xi , there exists x1 = x̂2 = x2 = x̂1 > 0 such that ∂ xi = 0; that is,
the unique Nash equilibrium of the one-shot game is symmetric. It implies that,∂ Û ji
at any steady state, ∂U∂ xi = ∂ x̂ j = 0 holds only at x1 = x̂2 > 0. Hence, x1 = x̂2 > 0
and x2 = x̂1 > 0 hold at any steady state.
Let NE1 (x2 ) denote the set of steady-state action (x1 , x̂2 ) given x2 . Note that
we omit x̂1 because it does not influence (x1 , x̂2 ). Since x1 = x̂2 , we omit thesecond component, and view NE1 as the set of steady-state action x1 . That is,
NE1 (x2 ) is the set of x1 such that x1 = x̂2 is a steady-state action. Define NE2 (x1 )in a similar way.
Note that steady state is an intersection of NE1 and NE2 , and a sufficient condition for the unique steady state is −1 < NEi0 (x j ) < 1 for all i and x j where j , i.
For xi = NEi (x j ), FOC must be satisfied:∂Ui (xi , x̂ j , Ai , θi (xi , x j , x̂ j ))
= 0.
∂ xi
Using the implicit function theorem with xi = x̂ j = NEi (x j ), the slope of NEi (x j )is2−∂ Ui ∂ θi∂ xi ∂ θi ∂ x j

2
Uii +Ui j + ∂∂xi ∂Uθi i



∂ θi
∂ θi
∂ xi + ∂ x jiHere the numerator measures how much ∂U∂ xi changes when x j changes. Since j
is not the opponent of player i, there is only an indirect effect. The denominatorimeasures how much ∂U∂ xi changes when xi = x̂ j changes. So a sufficient condition
for unique steady state is−∂ 2Ui ∂ θi
∂ xi ∂ θi ∂ x j
2
Uii +Ui j + ∂∂xi ∂Uθi i



∂ θi
∂ θi
∂ xi + ∂ x j

for all i, j , i, x j , and xi ∈ NEi (x j ).
100 ∈ (−1, 1)

CConvergence for Examples in Main Text

In this appendix, we check beliefs in each example covered in the main text converge to a steady state.

Example: Equation 7.
Consider the Cournot model in Section 3.2.1. Suppose that the inverse demand
function is given byQ(x1 + x2 , a, θ ) = a − (1 − θ )(x1 + x2 )and the cost function is linear, i.e., c(xi ) = cxi where c ∈ (0, a). Suppose also that
Θ = [−d, d] where d ∈ (0, 13 ) is a fixed parameter. We will show that for each
misspecification, the belief converges to a steady state as long as misspecificationis small (i.e., A is sufficiently close to a).
First-order misspecification and one-sided double misspecification. Since
the inverse demand function Q is linear in θ , the identifiability condition holds,
and hence Proposition 13 ensures that the belief converges almost surely underfirst-order misspecification and one-sided double misspecification. In particular,when the steady state is unique, the belief converges there almost surely regardlessof the initial prior.
Second-order misspecification. To prove convergence for small misspecification, it suffices to check the conditions stated in Corollary 4.
Given a misspecified parameter A, let f2 (θ̂1 , A) denote the set of “steady-state
belief” of player 2, when the hypothetical player’s belief is fixed at θ̂1 . In such a
steady state, the incentive-compatibility conditions and the consistency condition101

must be satisfied:40
a−c
x2
− ,
∗
2(1 − θ ) 2
a−c
x̂1
x2 =
− ,
2(1 − θ2 ) 2
x2
A−c
− ,
x̂1 =
2(1 − θ̂1 ) 2
x1 =a − (1 − θ2 )(x̂1 + x2 ) = a − (1 − θ ∗ )(x1 + x2 ).
2(a−c)A−c. ComFrom the second and the third equations, we have x2 = 3(1−θ− 3(1−
θ̂1 )2)bining it into the first and the third equations,

a−c
x2
a−c
a−c
A−c
+ =
+
−
,
∗
∗
2(1 − θ ) 2
2(1 − θ ) 3(1 − θ2 ) 6(1 − θ̂1 )
A−c
x2
a−c
A−c
x̂1 + x2 =+ =+.
3(1 − θ2 ) 3(1 − θ̂1 )2(1 − θ̂1 ) 2
x1 + x2 =Plugging them into the last consistency condition,a−(1−θ2 )(

a−c
A−c
a−c
A−c
a−c
+
+
−
)−a+(1−θ ∗ )() = 0,∗
3(1 − θ2 ) 3(1 − θ̂1 )2(1 − θ ) 3(1 − θ2 ) 6(1 − θ̂1 )which is equivalent to−(1 − θ2 )(A − c) (1 − θ ∗ )(a − c) a − c (1 − θ ∗ )(A − c)
+
−
+
= 0.
3(1 − θ2 )6
3(1 − θ̂1 )6(1 − θ̂1 )(58)

40 Here we implicitly assume that steady states actions and beliefs are interior points, and thisassumption is without loss of generality. Indeed, it is straightforward to see that player i’s optimal
action is an interior solution given any belief θi and given any action x−i of the opponent; this
immediately implies that steady state actions are interior points. Regarding beliefs, suppose notand there is f2 (θ̂1 , A) which contains a boundary point, say θ2 = −d, for some θ ∗ , θ̂1 ∈ (−d, d).
a−c
a−c
A−c
Then, from the incentive-compatibility conditions, we have x1 + x2 = 2(1−θ∗ ) + 3(1+d) −6(1−θ̂ )1

a−c
A−c
and x1 + x̂2 = 3(1+d)+ 3(1−. Given these (x1 , x2 , x̂1 ), θ2 = −d does not minimize the KL metric;θ̂1 )because there is θ2 ∈ (−d, d) which makes the KL metric equal to zero (note that such θ2 can befound by solving a − (1 − θ2 )(x̂1 + x2 ) = a − (1 − θ ∗ )(x1 + x2 )). This implies that the boundary
point θ2 = −d is not a steady state belief. The same argument is applied to θ2 = d.
In the rest of this appendix, we will assume that other steady states (such as fˆ1 ) are also interior
points, and an argument similar to the one above shows that this assumption is without loss ofgenerality.

102

A steady-state belief f2 (θ̂1 , A) must be a solution to this equation (58). At A = a,(58) reduces to2(1 − θ ∗ )(1 − θ̂1 ) − (1 − θ2 )(2 − 2θ2 + θ̂1 − θ ∗ ) = 0.

(59)

Note that for any θ2 , θ̂1 , θ ∗ ∈ (−d, d), the left hand side of (59) is strictly
decreasing in θ̂1 and θ ∗ . When θ̂1 = θ ∗ = d, there is a unique solution θ2 = d.
When θ̂1 = θ ∗ = −d, there is a unique solution θ2 = −d. When θ̂1 ∈ (−d, d)or θ ∗ ∈ (−d, d), the left hand side of (59) is strictly increasing in θ2 , negative at
θ2 = −d and positive at θ2 = d. Hence, the intermediate value theorem ensuresthat given any θ̂1 , θ ∗ ∈ (−d, d), (59) has an interior solution. Also this solution
is unique, as the left hand side of (59) is strictly increasing in θ2 . So in sum, foreach θ̂1 , the set f2 (θ̂1 , A) of steady-state beliefs is non-empty and is a singleton.
Similarly, given a misspecified parameter A, let fˆ1 (θ2 , A) denote the set of“steady-state belief” of hypothetical player 1, when player 2’s belief is fixed at θ2 .
In such a steady state, the incentive compatibility condition and the consistency
condition must be satisfied:
x2
a−c
− ,
∗
2(1 − θ ) 2
x̂1
a−c
− ,
x2 =
2(1 − θ2 ) 2
A−c
x2
x̂1 =
− ,
2(1 − θ̂1 ) 2

x1 =

A − (1 − θ̂1 )(x̂1 + x2 ) = a − (1 − θ ∗ )(x1 + x2 ).
The first three equations yield
a−c
a−c
A−c
+
−
,
∗
2(1 − θ ) 3(1 − θ2 ) 6(1 − θ̂1 )a−c
A−cx̂1 + x2 =+.
3(1 − θ2 ) 3(1 − θ̂1 )x1 + x2 =Plugging them into the last equation,
A−(1− θ̂1 )(

A−c
a−c
a−c
A−c
a−c
+
)−a+(1−θ ∗ )(+−) = 0,∗
3(1 − θ2 ) 3(1 − θ̂1 )2(1 − θ ) 3(1 − θ2 ) 6(1 − θ̂1 )103

which is equivalent toa − c A − c (1 − θ̂1 )(a − c) (1 − θ ∗ )(a − c) (1 − θ ∗ )(A − c)
A−a+
−
−
+
−
= 0.
2
3
3(1 − θ2 )3(1 − θ2 )6(1 − θ̂1 )A steady-state belief fˆ1 (θ2 , A) must be a solution to this equation. When A = a,the equation above reduces to(a − c)(θ̂1 − θ ∗ )(1 + θ2 − 2θ̂1 )= 0.
6(1 − θ̂1 )(1 − θ2 )Hence, when Θ = [−d, d] where x ∈ (0, 13 ), θ̂1 = θ ∗ is a unique solution, and theconditions stated in Corollary 4 are satisfied.
Two-sided double misspecification. To prove convergence for small misspecification, it suffices to check the conditions stated in Proposition 14.
Given misspecified parameters A1 , A2 , let f2 (θ1 ) denote the set of “steady-state
belief” of player 2, when player 1’s belief is fixed at θ1 . Note that the incentivecompatibility conditions and the consistency condition are:
A1 − cx̂2− ,2(1 − θ1 ) 2
A2 − cx̂1x2 =− ,2(1 − θ2 ) 2
A2 − cx2x̂1 =− ,2(1 − θ2 ) 2
A1 − cx1x̂2 =− ,2(1 − θ1 ) 2
x1 =A2 − (1 − θ2 )(x̂1 + x2 ) = a − (1 − θ ∗ )(x1 + x2 ).
A2 −c
A1 −c
The first four equations imply x̂1 = x2 = 3(1−θand x1 = x̂2 = 3(1−θ. Plugging
2)
1)
them into the last equation,

A2 − 2(1 − θ2 )A2 − cA1 − cA2 − c− a + (1 − θ ∗ )(+) = 0,
3(1 − θ2 )3(1 − θ1 ) 3(1 − θ2 )104

which is equivalent toθ2 = 1 −(1 − θ1 )(1 − θ ∗ )(A2 − c).
(1 − θ1 )(A2 + 3a − 4c) − (1 − θ ∗ )(A1 − c)

So for any θ1 , f2 (θ1 ) is a singleton. Also, at A1 = A2 = a,
∂ f2 (θ1 )(1 − θ ∗ )2
=−
.
∂ θ1
(3 − 4θ1 + θ ∗ )2Note that this derivative is negative, and is larger than −1 as θ1 , θ ∗ ∈ [− 13 , 13 ].
Hence, if A1 and A2 are sufficiently close to a, then | ∂ f∂2θ(θ1 ) | ∈ (0, 1).
1
Similarly, given misspecified parameters, let f1 (θ2 ) denote the set of “steadystate belief” of player 1, when player 2’s belief is fixed at θ2 . Then we can showthat f1 (θ2 ) is a singleton for all θ2 and | ∂ f∂1θ(θ2 ) | ∈ (0, 1). A proof is similar to that2
for f2 , and hence omitted.

Example: Equation 8.
Suppose the Cournot model with c(xi ) = cxi where c ≥ 0, a < 1, Θ = [k − d, k + d]
where k, d > 0 are fixed parameters which satisfy k − d > c, andQ(x1 + x2 , a, θ ) = θ − (1 − a)(x1 + x2 ).
First-order misspecification and one-sided double misspecification. Note thatthe identifiability condition is satisfied, and hence Proposition 13 implies that the
belief converges almost surely to a steady state.
Second-order misspecification. To prove convergence for small misspecification, it suffices to check the conditions stated in Corollary 4.
Given a misspecified parameter A, let f2 (θ̂1 , A) denote the set of “steady-state
belief” of player 2, when the hypothetical player’s belief is fixed at θ̂1 . Note that105

the incentive-compatibility conditions and the consistency condition are:
θ∗ −c
x2
− ,
2(1 − a) 2
θ2 − cx̂1x2 =− ,2(1 − a) 2θ̂1 − c
x2
x̂1 =
− ,
2(1 − A) 2x1 =

θ2 − (1 − a)(x̂1 + x2 ) = θ ∗ − (1 − a)(x1 + x2 ).
θ2 −cθ̂1 −cFrom the first three equations, we have x̂1 + x2 = 3(1−a)+ 3(1−A)and x1 + x2 =
θ̂1 −cx2
θ2 −c
θ ∗ −cθ ∗ −c2(1−a) + 2 = 2(1−a) + 3(1−a) − 6(1−A) . Plugging them into the last equation andarranging,θ2 =θ ∗ + c (1 − a)(θ̂1 − c)+.
22(1 − A)

Hence, for any θ̂1 , f2 (θ̂1 , A) is a singleton.
Similarly, given a misspecified parameter A, let fˆ1 (θ2 , A) denote the set of“steady-state belief” of hypothetical player 1, when player 2’s belief is fixed at
θ2 . Note that the incentive-compatibility conditions and the consistency condition
are:
θ∗ −c
x2
− ,
2(1 − a) 2x̂1
θ2 − c− ,x2 =2(1 − a) 2θ̂1 − c
x2
x̂1 =
− ,
2(1 − A) 2x1 =

θ̂1 − (1 − A)(x̂1 + x2 ) = θ ∗ − (1 − a)(x1 + x2 ).
∗θ2 −cθ̂1 −cθ −c
The first three equations imply x̂1 + x2 = 3(1−a)+ 3(1−A)and x1 + x2 = 2(1−a)
+
x2
θ2 −c
θ̂1 −cθ ∗ −c2 = 2(1−a) + 3(1−a) − 6(1−A) . Plugging them into the last equation and arranging,

From them, fˆ1 (θ2 , A) is a solution to
6(1 − A) θ ∗ (A − a)c (A − a)(θ2 − c)
θ̂1 =
−
−
.
3 − 4A + a 2
6(1 − A)3(1 − a)106

Hence, for any θ2 , fˆ1 (θ2 , A) is a singleton (also fˆ1 (θ2 , A) = θ ∗ at A = a), and theconditions stated in Corollary 4 are satisfied.
Two-sided double misspecification. To prove convergence for small misspecification, it suffices to check the conditions stated in Proposition 14.
Given misspecified parameters A1 , A2 , let f2 (θ1 ) denote the set of “steady-state
belief” of player 2, when player 1’s belief is fixed at θ1 . Note that the incentivecompatibility conditions and the consistency condition are:x̂2
θ1 − c− ,2(1 − A1 ) 2
θ2 − cx̂1x2 =− ,2(1 − A2 ) 2
θ2 − cx2x̂1 =− ,2(1 − A2 ) 2
θ1 − cx1x̂2 =− ,2(1 − A1 ) 2
x1 =θ2 − (1 − A2 )(x̂1 + x2 ) = θ ∗ − (1 − a)(x1 + x2 ).
θ2 −c
θ1 −c
From the first four equations, x̂1 = x2 = 3(1−A
and x1 = x̂2 = 3(1−A. Plugging
2)
1)
them into the last equation and arranging,
1 − A2(1 − a)c(1 − a)(θ1 − c)
∗
θ2 =
−2c +
+ 3θ −.
2 − a − A2
1 − A2
1 − A1Hence, for any θ1 , f2 (θ1 ) is a singleton. Also,∂ f2 (θ1 )(1 − a)(1 − A2 )
=−
.
∂ θ1
(1 − A1 )(2 − a − A2 )Note that this derivative is negative, and is larger than −1 if A1 and A2 are sufficiently close to a. Hence, | ∂ f∂2θ(θ1 ) | ∈ (0, 1) for any θ1 , θ2 , and θ ∗ .
1
Similarly, given misspecified parameters, let f1 (θ2 ) denote the set of “steadystate belief” of player 1, when player 2’s belief is fixed at θ2 . Then an argument
similar to the one above shows that f1 (θ2 ) is a continuous function and | ∂ f∂1θ(θ2 ) | ∈2(0, 1).
107

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