OSIPP Discussion Paper: DP-2021-E-008

Misspecified Bayesian Learning by Strategic Players:
First-Order Misspecification and Higher-Order
Misspecification
December 2, 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 the world, andinvestigate their long-run behavior when they learn an unknown state from public signals over time. Our
framework is flexible and allows for higher-order misspecification, in that a player may have a bias
about the physical environment, a bias about the opponent's bias about the physical environment, and soon. We provide a condition under which players' beliefs and actions converge to a steady state, and thencharacterize how one's misspecification influences the long-run (steady-state) outcome.

We apply theseresults to various economic examples such as Cournot competition, team production, and discrimination.
We find that higher-order misspecification can have a significant impact on the equilibrium outcome:One's overconfidence can have opposite effects on the equilibrium outcome, depending on whether the
opponent is aware of this bias or not.
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

Misspecified Bayesian Learning by Strategic
Players: First-Order Misspecification andHigher-Order Misspecification*
Takeshi Murooka
Osaka University

Yuichi Yamamoto
Hitotsubashi University

December 2, 2021
Abstract
We consider strategic players who may have a misspecified view about
the world, and investigate their long-run behavior when they learn an unknown state from public signals over time. Our framework is flexible andallows for higher-order misspecification, in that a player may have a bias
about the physical environment, a bias about the opponent’s bias about thephysical environment, and so on. 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) outcome.
We apply these results to various economic examples such as Cournot competition, team production, and discrimination. We find that higher-ordermisspecification can have a significant impact on the equilibrium outcome:One’s overconfidence can have opposite effects on the equilibrium outcome,depending on whether the opponent is aware of this bias or not.
JEL Codes: C73, D83, D90, D91
Keywords: model misspecification, learning,convergence, overconfidence,
bias transmission* We thank Matthias Fahn, Drew Fudenberg, Kevin He, Botond Kőszegi, Shintaro Miura, Yoko

Okuyama, Klaus Schmidt, and 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 misspecified view about the world:
A worker may be overconfident about his own capability, a firm may incorrectly
assume that the demand function is linear in prices (in reality, the demand is
non-linear), an investor may incorrectly believe that the economy is driven by
fewer variables, and so on.1 Recent literature on model misspecification studies
how such a bias influences the agents’ behavior and payoffs, assuming either asingle-agent setup or a multi-agent setup in which the agents’ misspecifications
are common knowledge (e.g., Esponda and Pouzo, 2016; Heidhues, Kőszegi, andStrack, 2018; Ba and Gindin, 2021). However, this common knowledge assumption leaves out many potential applications, as it does not allow players’ higherorder misspecification. For example, when a worker is overconfident about his
own capability, his colleague may not be aware of it; in this case, this colleague
has a misspecified view about the opponent’s view about the world. This paper
proposes a general model which allows for such higher-order misspecification,and studies its economic consequences.
Specifically, we consider an infinite-horizon game in which players take actions each period and learn an unknown state from public signals over time. Players are Bayesian and maximize the expected payoffs just as in the standard gametheoretic model, but they evaluate information using misspecified models. We
assume myopic players in order to rule out the folk-theorem type result.2 Actions
are unobservable. Our goal is to understand how a misspecified player behavesdifferently than the unbiased player in the long run, and how it influences the
behavior of other players.
Why should we be interested in the long-run behavior of misspecified players,
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). See Daniel and Hirshleifer (2015), Malmendier and Tate(2015), and Grubb (2015) for reviews of the literature.
2 Our results are valid even for forward-looking players by assuming that they play a Markov
perfect equilibrium.

2rather than an equilibrium in a one-shot model? When players have a misspecified
view about the world, they observe outcomes which are systematically different
from the anticipation. Accordingly, it is likely that they eventually change the belief about some economic variable. For example, if a firm is persistently overconfident about some aspect of the demand function (e.g., the intercept of the inverse
demand curve), on average, actual prices are lower than the firm’s anticipation.3
After a long time, this firm becomes (unrealistically) pessimistic about other aspect of the demand (e.g., the slope of the inverse demand curve). Similarly, in
tournaments, if an agent is persistently overconfident about her own capability,
after a series of unexpected losses, she may 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.
In Section 2, we consider a benchmark case in which there is no higher-ordermisspecification. Specifically, we assume that players have first-order misspecification only, in that they may have misspecified views about the physical environment and these first-order beliefs about the environment are common knowledge.
This setup covers a wide range of applications, such as Cournot duopoly with misspecified demand and team production with overconfidence/prejudice. We showthat players’ beliefs and actions converge to a steady state under some condition, and then characterize how one’s misspecification influences the steady-stateoutcomes. A novelty here is that we quantify the impact of misspecification on
the steady-state outcomes, which allows us to discuss how each parameter of themodel influences the equilibrium outcome, and how strategic interaction amplifies/reduces the impact of misspecification. For example, our result implies thatin any symmetric game, both strategic substitutes and strategic complements am3 Recent evidence suggests that overconfidence can be persistent: Hoffman and Burks (2020)

find that workers are persistently overconfident about their own productivity, and Huffman, Raymond, and Shvets (2019) find that managers are persistently overconfident about future performance. In a laboratory experiment, Grossman and Owens (2012) report that subjects’ responses
are consistent with Bayesian updating and overconfident prior beliefs, but overconfidence about
their ability is persistent in the face of repeated feedback.

3

plify the impact of first-order misspecification.
Then in Section 3, we consider a model with higher-order misspecification.
There are many types of higher-order misspecification we can think of, and here
we focus on a particular one which seems economically relevant.4 Specifically,
we assume that each player may have a misspecified view about the physical environment as in the case of first-order misspecification, and on top of that, eachplayer naively thinks that the opponent has the same view. In this setup, players
are not aware of the opponent having a different view about the world; they thinkthat their view about the world is absolutely correct. This describes, for example, a worker who is unaware of a bias of his colleague. We find that even withsuch higher-order misspecification, players’ beliefs and actions still converge toa steady state under an additional assumption. We then quantify how one’s misspecification influences this steady-state outcome.
In Section 4, we apply these results to more specific examples. We find thatthe presence of higher-order misspecification (i.e., unawareness of the opponent’sbias) can have a significant impact on the equilibrium outcome. For example,
when Alice and Bob work on a joint project, it is possible that Bob’s overconfidence (about his own capability) improves his equilibrium payoff if his overconfidence is common knowledge, but reduces his payoff if Alice is not aware of theoverconfidence. A point is that when Alice is unaware of Bob’s overconfidence,
she faces inferential naivety in that she makes an incorrect prediction about Bob’s
action. Accordingly, she may take an action different from the one she would takeif Bob’s overconfidence was common knowledge, which may result in a qualitative difference in equilibrium payoffs.
We also find that our framework of higher-order misspecification is useful toexplain bias transmission. In Section 4.3, we consider a teacher who has a bias
against a particular type of students (e.g., female students). We show that theteacher’s bias can endogenously induce these students’ negative self-stereotypes,
4 We propose a general model of higher-order misspecification in Appendix A.

In Appendix
C, we present the analysis of different types of higher-order misspecification, as well as otherapplications such as a tournament.

4

if the students are not aware of the teacher’s bias.
Section 5 summarizes the related literature and concludes. Appendix A presents
a general model and characterize the asymptotic behavior of players’ actions andbeliefs, which is useful to prove the convergence theorems in the main text. Appendix B provides proofs. Appendix C presents other types of misspecification,
additional convergence results, and other applications. Appendix D checks convergence of long-run beliefs in each example covered in the main text.

2

First-Order Misspecification

2.1

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 theother player 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) 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 differentiable.
We assume that one of the players (player 2) has a biased view about theparameter a, while the other player is unbiased and knows the parameter a. We
call it first-order misspecification, because player 2 has an incorrect first-orderbelief about the parameter a. 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.
5• The above beliefs are common knowledge (i.e., player 2 believes that player
1 believes that y = Q(x1 , x2 , a, θ ) + ε, and the like).
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)]
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 play a static Nash equilibrium every period. Thisessentially means that in our model, (i) players are myopic, and (ii) they predict
the opponent’s play correctly and best-respond to it. Condition (i) shuts down therepeated-game effect, so that a result similar to the folk theorem (which is not of
our interest) does not arise.5 Condition (ii) implies that players recognize that the
opponent also learns the state and changes the action as time goes. This setup isdifferent from the one in the literature on learning in games (e.g., Fudenberg andKreps, 1993; Esponda and Pouzo, 2016), which asks when and why players play
equilibria; they assume that players do not know the opponent’s strategy and learn
it from experience. In our model, players know the opponent’s strategy, and learn
only the unknown economic state θ .6
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
5 Another way to avoid the repeated-game effect is to use a Markov-perfect equilibrium (where

the state is players’ beliefs about θ ) as a solution concept. With an additional assumption, Appendix A shows that players’ long-run behavior is exactly the same as that of myopic players
studied in this section. In this sense, our result remains true even for forward-looking players.
6 Condition (ii) is inessential if the game is dominance solvable (e.g., Cournot duopoly withlinear demand in Section 4.1).

6

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 ). Because the players’ information structure about the parametera is common knowledge, these posteriors are common knowledge among players.
So in period two, players play a Nash equilibrium given the belief profile µ 2 =∂ E[Ui (x,θ )|µi2 ](µ12 , µ22 ), which solves= 0 for each i. Likewise, in any subsequent∂ xi
period t, players play a Nash equilibrium given the belief profile µ t = (µ1t , µ2t ),where µ t is computed by Bayes’ rule.

2.2

Steady-State Analysis

In this subsection, we will assume that the actions and the beliefs converge to asteady state in the long run, and characterize how one’s misspecification influences
this long-run (steady-state) outcome. In the next subsection, we will show that theactions and the beliefs indeed converge to this steady state under some conditions.
As will be seen, these conditions are satisfied in many economic examples suchas Cournot competition and team production.
A steady state in this model is a pair (x1∗ , x2∗ , µ1∗ , µ2∗ ) of an action profile and a7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, which requires that eachplayer maximizes her payoff given some beliefs. The next two conditions requirethat 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 isconcentrated on a state θ2 with which her subjective signal distribution coincides
with the true distribution. This condition must be satisfied in a steady state; 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 ∗ .
We assume that for each (x, A), there is a unique state θ2 which solves theconsistency condition Q(x, a, θ ∗ ) = Q(x, A, θ ), and we denote it by θ2 (x, A). Intuitively, θ2 (x, A) is player 2’s long-run belief given an action profile x; if players
choose the same action profile x every period, then almost surely, player 2’s belief
will be concentrated on the state θ2 (x, A) after a long time (Berk, 1966). Player 1’s
long-run belief is defined as θ1 (x, A) = θ ∗ for all x and A, because she is unbiased
and can learn the true state θ ∗ .
Our goal is to quantify how player 2’s misspecification A influences the steadystate action defined above. We first describe how one’s action influence the opponent’s steady-state action. Consider player i’s asymptotic best response correspondence, which is defined 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 steady-state action in a single-agent learning problem where player i learns the state while the opponent simply chooses the8same action x−i every period. Player 2’s asymptotic best response BR2 is different from the standard best response, as it describes the optimal action in the longrun where the belief θ2 (x, A) is endogenously determined. On the other hand,player 1’s asymptotic best response BR1 coincides with the standard best response
correspondence given the state θ ∗ , because her long-run belief is constant, i.e.,θ1 (x, A) = θ ∗ for all x. By a fixed-point theorem, BRi (x−i ) is non-empty for all
x−i . A standard argument shows that BRi is upper hemi-continuous in x−i .
When the asymptotic best response is a function (rather than a correspondence), its slope BR0i can be computed by
BR0i = −

Mi j
,
Mii

where for each i and j (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 how player j’s action influences player i’s marginal utility in the longrun. To be precise, suppose that players choose the same action every period,
and that player j increases the action x j at bit. This influences player i’s marginalutility directly and indirectly through the belief θi (x, A) in the long run. The firstterm of Mi j represents this direct effect, and the second term represents the indirect
effect. For i = 1, the indirect effect is zero, because player 1’s long-run belief isconstant and does not depend on the actions (θ1 (x, A) = θ ∗ for all x). Hence BR01 =∂ 2U1 /∂ x1 ∂ x2
, which is precisely the slope of the standard best-response curve. Also,∂ 2U1 /∂ 2 x1
for i = 2, the indirect effect disappears in the limit as A → a, because θ2 (x, a) = θ ∗for all x. So when misspecification is small (i.e., A is close to a), each Mi j is2
approximated by ∂∂xi ∂Uxi j , which means that BR0i is approximately the same as theslope of the standard best-response function.
Let
M2A : =

∂ 2U2 (x, A, θ )∂ 2U2 (x, A, θ )∂ θ2 (x, A)+∂ x2 ∂ A∂ x2 ∂ θ∂Aθ =θ2 (x,A)θ =θ2 (x,A)9(6)

denote how player 2’s bias A influences her marginal utility in the long run. Again,22
2 ∂ θ2the first term ∂∂x U∂2A measures the direct effect, while the second term ∂∂x U
2
2 ∂ θ2 ∂ Ameasures the indirect effect through the belief. Our first proposition quantifies theimpact of player 2’s first-order misspecification on the steady-state action.
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.
Proposition 1 (Steady State under First-Order Misspecification). Let x∗ be a regular steady state for some parameter A∗ . Then there is an open neighborhood ofA∗ 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
Note that the regularity conditions (i) and (ii) are standard, and the condition(iii) is satisfied for generic parameters. The condition (iv) reduces to player i’s
second-order condition for incentive compatibility when misspecification is small(i.e., A is close to a).8
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.
8 Heidhues, Kőszegi, and Strack (2018) impose a similar assumption: They consider a singleagent learning problem and assume a unique steady state, which requires Mii ≤ 0 in the steadystate.

10

Under this regularity condition, Proposition 1 shows that the impact of firstorder misspecification on the steady-state action is represented as the base mis1
2A
specification effect − M
M22 times the multiplier effect 1−BR01 BR02 . The base misspecification effect measures how player 2’s bias influences her steady-state action x2∗
in the absence of strategic interaction. To see what it means, suppose that player 1chooses the same fixed action each period, so player 2 faces a single-agent problem. Suppose that player 2’s bias A increases a bit. This influences player 2’s
marginal utility by M2A (recall that this includes the indirect effect through thebelief in the long run), and hence her optimal long-run action changes. The base
2A
misspecification effect − M
M22 measures this change.
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 allthese feedback effects, player 2’s action changes by∞1∑ (BR01BR02)k ∆ = 1 − BR0 BR0 ∆.
1

k=0

2

So the multiplier 1−BR10 BR0 can be seen as a result of the infinite adjustment process12
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.
11

(ii) If sgn(BR01 ) = sgn(BR02 ), then the multiplier 1−BR10 BR0 is greater than one.
1
2
So both strategic substitutes and strategic complements amplify the impactof misspecification.
(iii) If sgn(BR01 ) , sgn(BR02 ), then the multiplier 1−BR10 BR0 is less than one. So a1
2
strategic interaction reduces the impact of misspecification.
This corollary characterizes when a strategic interaction amplifies/weakens
the impact of misspecification. An interesting special case is symmetric games.
When A = a, we have sgn(BR01 ) = sgn(BR02 ) in any symmetric equilibrium of asymmetric game. So part (ii) of the corollary implies that a strategic interaction
always amplifies the impact of misspecification in these games, if misspecificationis small. On the other hand, part (iii) shows that a strategic interaction reduces theimpact of misspecification if sgn(BR01 ) , sgn(BR02 ). This condition is satisfied, forexample, in a tournament model.9
Remark 1. So far we have assumed that player 1 knows the true parameter a,but the result similar to Proposition 1 still holds even when both players havefirst-order misspecification. Suppose that player 1 believes that the true parameter
is A1 , a, and player 2 believes that the true parameter is A , a. Suppose alsothat these first-order beliefs are common knowledge. Then the impact of player
2’s misspecification A on the steady-state actions is still described by the formula
presented in Proposition 1, with a minor modification on the definition of BR01 ;now it must involve an indirect learning effect, in order to take into account theendogeneity of her long-run belief θ1 .

2.3

Sufficient Condition for Convergence

In a single-agent finite-action setup, Esponda, Pouzo, and Yamamoto (2021) provide a fairly general condition for convergence; they show that the agent’s belief
9 Consider a tournament model by Lazear and Rosen (1981) in which player i’s payoff is P (e −i ie j )w − c(ei ) where w > 0 is the prize for a winner, c(·) is an increasing and convex cost, and Pi (·)
is i’s probability of winning which satisfies P1 (·) = 1 − P2 (·). Then, so long as Pi00 (0) , 0, thestandard best response curves have opposite signs.

12

converges to a steady state almost surely if an “identifiability condition” holds.
In this subsection, we will show that the same result holds in our two-player
continuous-action model. Having two players does not cause a serious difficulty,
because our problem is essentially a single-agent problem; since player 1 is unbiased and learns the true state, we only need to take care of player 2’s belief
evolution.
On the other hand, having continuous actions causes a technical complication.
When actions are finite, an agent’s action frequency is represented as a finitedimensional vector. Esponda, Pouzo, and Yamamoto (2021) show that the motionof this action frequency is approximated by a differential inclusion, and use thisresult to prove convergence. In our continuous-action model, the action frequency
is an infinite-dimensional vector, and it moves in a Banach space, rather than a
Euclidean space. We can still show that (with an appropriate choice of a norm)
the motion of the action frequency is approximated by a differential inclusion justas Esponda, Pouzo, and Yamamoto (2021), but this is a differential inclusion in a
Banach space, which is difficult to solve. So we do not work on this differential inclusion directly, and instead, we look at players’ beliefs. As we show in Appendix
A, the asymptotic motion of the beliefs is approximated by finite-dimensional differential inclusion, which is much more tractable than that in a Banach space. We
use this result to prove convergence.
Let us define the identifiability condition in our environment. For each action
profile x, define player 2’s surprise function as
K2 (θ , x) =(Q(x, θ , A) − Q(x, θ ∗ , a))2
2

Intuitively, this surprise function measures how player 2’s subjective expectation
Q(x, θ , A) about the output is different from the truth, when she believes that thestate is θ .10 Then for each probability measure σ ∈ 4X on the set of action
10 This surprise function is exactly the Kullback-Leibular divergence between the true output

distribution and the subjective distribution. See Appendix A for the general definition of the
Kullback-Leiblar divergence.

13

profiles, define a weighted surprise function asZ

K2 (θ , σ ) =K(θ , x)σ (dx).
X

Intuitively, this function measures how player 2’s subjective expectation is different from the truth on average, when players take different actions in different
periods. The identifiability requires that for each σ , the weighted surprise function
K2 (θ , σ ) has a unique minimizer θ2 (σ ) and it is an interior point. The following
proposition shows that this identifiability condition ensures convergence.
Proposition 2. Suppose that there is a unique steady state (x1∗ , x2∗ , θ1 , θ2 ) and it
is regular.11 Suppose also that the identifiability condition holds. Then almost
surely, player 2’s belief converges to the steady state belief, i.e., limt→∞ µ2t = 1θ2 .
Remark 2. Identifiability is sufficient for convergence, but not necessary. For
example, Proposition 9 of Esponda, Pouzo, and Yamamoto (2021) show that in asingle-agent problem, the agent’s action converges if payoffs and information are“monotone.” We can show that a similar result holds for general Q in the teamproduction problem studied in Section 4.2. For more details, see C.

3

Higher-Order Misspecification

In Section 2, we have studied the case in which players correctly understand what
the opponent thinks about the environment. However, economic agents often havehigher-order misspecification, in that they may have a biased view about the opponent’s view about the environment (second-order misspecification), a biased view
about the opponent’s second-order misspecification, and so on.12
11 Assuming a unique steady state is not essential, but it simplifies the statement. In the proof,

we actually show that the belief converges even when there are multiple steady states.
12 As evidence from laboratory experiments, subjects often systematically mispredict other subjects’ preferences and actions (e.g., Van Boven, Dunning, and Loewenstein, 2000). Ludwig andNafziger (2011) report that most subjects in their experiments are not aware of or underestimate
overconfidence of other subjects.

14

In this section, we will focus on a special form of higher-order misspecification: We will assume that each player has a biased view about the environment,
and on top of that, she naively thinks that the opponent shares the same view
about the world (in reality, the opponent has her own view about the world). We
call it double misspecification, because players have a biased view about the world(first-order misspecification) and a biased view about the opponent’s view about
the world (second-order misspecification). Our goal in this section is to characterize how such misspecification influences players’ long-run behavior. Of course,
we can think of various other forms of higher-order misspecification. In Appendix
A, we will present a more general model of higher-order misspecification.

3.1

Setup: Double Misspecification

In the economic environment described in Section 2.1, we will consider the following information structure:• 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 different players may have different levels of misspecification.
This setup is substantially different from the first-order misspecification in theprevious section, because now players have inferential naivety and make incorrect
predictions about the opponent’s play. Indeed, while player i believes that the opponent (player j) maximizes the payoff conditional on the parameter Ai , in reality,
the opponent maximizes the payoff conditional on the parameter A j . Accordingly,
player i’s prediction about the opponent’s action does not match the opponent’s
actual action in general.
To analyze players’ behavior in the presence of such inferential naivety, it isuseful to consider two hypothetical players. Hypothetical player 1 is player 1
who thinks that it is common knowledge that the true technology is A2 . Intuitively, player 2 thinks that this hypothetical player is her opponent, and hence
each period, player 2 chooses a Nash equilibrium action against this hypothetical
15

player. Similarly, hypothetical player 2 is player 2 who thinks that it is common
knowledge that the true technology is A1 . Each period, player 1 chooses a Nashequilibrium against this hypothetical player.
Let x̂i and µ̂i denote hypothetical player i’s action and belief, and let x =(x1 , x2 , x̂1 , x̂2 ) denote an action profile in the four-player game. Player i’s expectedstage-game payoff is defined as
Ui (x, θ , Ai ) = E[ui (xi , Q(xi , x̂−i , Ai , θ ) + ε)],because she thinks that the parameter is Ai and the opponent is a hypotheticalplayer. Similarly, hypothetical player i’s expected stage-game payoff given θ isÛi (x, θ , A−i ) = E[ui (x̂i , Q(x̂i , x−i , A−i , θ ) + ε)].
Using these notations, the equilibrium strategy in the infinite-horizon game is described as follows. In period one, all players have the same belief µi1 = µ̂i1 = µ.
So they play a Nash equilibrium (x11 , x21 , x̂11 , x̂21 ), which solves the first-order con)|µ])|µ]= 0 and ∂ E[Ûi∂(x,θ
= 0. At the end of period one, players
ditions ∂ E[Ui∂(x,θ
xi
x̂i
1
1
1
observe a public signal y = Q(x1 , x2 , a, θ ∗ ) + ε, and update the posterior beliefs
using Bayes’ rule. Their beliefs in period two are given by
1 , A , θ ))µi1 (θ ) f (y − Q(xi1 , x̂−i
i
,
1
1
1
Θ µi (θ̃ ) f (y − Q(xi , x̂−i , Ai , θ̃ ))d θ̃µi2 (θ ) = R

1 , A , θ ))µ̂i1 (θ ) f (y − Q(x̂i1 , x−i
−i
.
1
1
1
µ̂
(
θ̃
)
f
(y
−
Q(
x̂
,
x
,
A
,
θ̃
))d
θ̃
−i
Θ i
i −i

µ̂i2 (θ ) = R

As is clear from this formula, player i’s posterior belief is biased in two ways:
She updates the belief conditional on the wrong parameter Ai , and on the wrong
1 about the opponent’s play. Then in period two, players play a Nashprediction x̂−iequilibrium given this belief profile µ 2 = (µ12 , µ22 , µ̂12 , µ̂22 ).13 Likewise, in any
subsequent period t, players play a Nash equilibrium given the posterior beliefs
computed by Bayes’ rule.
13 Since y is public, player 1 correctly predicts hypothetical player 2’s posterior belief µ̂ 2 , and2similarly, hypothetical player 2 correctly predicts player 1’s posterior belief µ12 . So they will

indeed play a Nash equilibrium given these beliefs.

16

3.2

Steady-State Analysis

As in the case of first-order misspecification, we assume that players’ actions andbeliefs converge to a steady state, and study how one’s misspecification influences
this steady-state outcome. We will provide a sufficient condition for convergencein Section 3.3.
Given an action profile x = (x1 , x2 , x̂1 , x̂2 ), let θi (x, Ai ) denote player i’s longrun belief when the same action x is chosen every period. That is, let θi (x, Ai ) bea state θ which solves
Q(xi , x̂ j , Ai , θ ) = Q(x1 , x2 , , a, θ ∗ ),so that player i’s subjective model (the left-hand side) explains the actual output(the right-hand side). A critical difference from the case of first-order misspecification is that player i has inferential naivety and uses x̂ j (rather than x j ) whenevaluating the average output. In what follows, we will assume that θ2 (x, A) is
unique for each x and Ai .
With this notation, a steady state under double misspecification is defined as∗(x1 , x2∗ , x̂1∗ , x̂2∗ , µ1∗ , µ2∗ , µ̂1∗ , µ̂2∗ ) which satisfies∗xi∗ ∈ arg max Ui (xi , x̂−i
, Ai , θi ) ∀i,(7)∗x̂i∗ ∈ arg max Ûi (x̂i , x−i, A−i , θ−i ) ∀i,(8)µ1∗ = µ̂2∗ = 1θ1 (x,A1 ) ,(9)µ2∗ = µ̂1∗ = 1θ2 (x,A2 ) .

(10)

xi

x̂i

The first two conditions are the incentive-compatibility conditions, which requirethat each player maximize her own payoff given some beliefs. The next two conditions require that these beliefs satisfy consistency, in that each (actual and hypothetical) player’s belief is concentrated on a state with which her subjective signal
distribution coincides with the objective distribution.
As in the case of first-order misspecification, we will quantify how one’s misspecification influences the steady-state outcome, using the slope of asymptotic
17

best response curve. For notational convenience, let player 3 refer to hypotheticalplayer 1, and player 4 refer to hypothetical player 2. Then define the slope ofplayer i’s asymptotic best response curve with respect to player j’s action as
BR0i j = −

Mi j
Mii

where 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,Ai )∂ xi ∂ θ θ =θi (x,Ai )∂xjdenotes the impact of player j’s action on player i’s marginal utility in the longrun. Here the first term is the direct effect, and the second term is the indirect
effect through the steady-state belief θi .
Intuitively, BR0i j measures how player j’s action influences player i’s optimal
long-run action, when other players’ actions are fixed. The mathematical definition of BR0i j is exactly the same as that for first-order misspecification, but there
are two important remarks. First, each Mi j here involves the indirect effect caused
by inferential naivety, and thus BR0i j may not be zero even when player i does
M12
not think that player j is the opponent.14 For example, consider BR012 = − M.
11Since player 1 does not think that player 2 is the opponent, she does not respondto player 2’s action, which means that the direct effect in M12 is zero. However, achange in player 2’s action influences player 1’s steady-state belief θ1 ; it leads toincorrect learning, because player 1 is not aware of a change in player 2’s action.
Hence the indirect effect in M12 is non-zero, and so is BR012 .
Second, this indirect effect from inferential naivety does not vanish even when
Ai approaches a. So even in the limit case with Ai = a, the slope BR0i j is not
approximated by the slope of the standard best response curve. This is in a sharp
contrast with the case of first-order misspecification, where all indirect effects
vanish in the limit as A → a.
14 More precisely, we always have BR0 = BR0 = BR0 = BR0 = 0, but other slopes BR0 are
ij
13
24
34
43

non-zero in general.

18For each i = 2, 3, let
MiA :=∂ 2Ui (x, θ , A2 )∂ θi (x, A2 )∂ 2Ui (x, θ , A2 )+∂ xi ∂ A2∂ xi ∂ θ∂ A2
θ =θi (x,A2 )θ =θi (x,A2 )denote the impact of player i’s first-order misspecification on her marginal utility. The following proposition characterizes how player 2’s misspecification influences the steady-state actions.
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∗ and θi (x∗ , Ai ) 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 each i.15
Proposition 3 (Steady State under Double Misspecification). Let x∗ be a regularsteady 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 a regularsteady state x∗ which is continuous with respect to A2 , and we have




∂ x2∗
1
M2A
1
0 M3A
,
= −
− BR23∂ A2
M22
M33
1 − BR023 BR032
1 − NE10 NE20
∂ x1∗ ∂ x2∗=· NE10 ,∂ A2 ∂ A2
where
NE10 =

BR012 + BR014 BR042
1 − BR014 BR041

and

NE20 =

BR021 + BR023 BR031.
1 − BR023 BR032

0 M3A
2A
The term − M
M22 − BR23 M33 in the first equation is the base misspecification
effect of double misspecification. Suppose that the parameter A2 increases a bit.
15 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 Ai and the first-order condition
for the incentive compatibility is satisfied there. The condition (iii) is needed for the multipliereffect to be well-defined. The condition (iv) ensures that the base misspecification effect and theslope of the asymptotic best response curve are well-defined. This condition is also useful when
we interpret the base misspecification effect.

19

This influences player 2’s steady-state action in two ways: First, it increases player
2’s bias about the physical environment a, which influences her optimal action di2A
rectly and indirectly through the belief. This effect is measured by − M
M22 , justas in the case of first-order misspecification. Second, when A2 increases, hypothetical player 1’s bias about the physical environment a increases. Hence thishypothetical player modifies the action, and player 2 best-responds to it. This ef3A
fect is measured by −BR023 M
M33 . This second effect is a consequence of player 2’s
second-order misspecification (inferential naivety).
Proposition 3 shows that this base misspecification effect is further amplified by the two multipliers, 1−BR10 BR0 and 1−NE10 NE 0 . The first multiplier ef23

32

1

2

fect, 1−BR10 BR0 , is similar to the multiplier effect appearing in Proposition 1,
23
32
and it represents how the strategic interaction between player 2 and hypothetical player 1 (which happens in player 2’s mind) amplifies the base misspecifi2A
cation effect, holding other players’ actions being fixed. So the term (− M
M22 −
1
3A
BR023 M
M33 )( 1−BR023 BR032 ) in the equation measures how player 2’s misspecificationinfluences her own action, when player 1’s action is fixed.
The second multiplier effect, 1−NE10 NE 0 , captures what happens when player
12
1’s action is not fixed and she changes her action over time. The economic interpretation of this multiplier effect is very different from the first one, in that it
is not about the impact of strategic interaction; rather, it measures how incorrect
learning due to inferential naivety amplifies the impact of misspecification.
To see what it means, suppose that player 2’s action changes by ∆ due tomisspecification. This does not influence player 1’s action immediately, because
player 1 is not aware of player 2’s misspecification, and thus does not best-respondto this change. However, in the long run, it causes incorrect learning and influences player 1’s steady-state belief µ1 , which in turn influences player 1’s action
directly and indirectly through hypothetical player 2’s action. (Note that the beliefµ1 influences hypothetical player 2’s optimal action, and player 1 best-responds
to it.) This effect is BR12 ∆ + BR14 BR42 ∆. Since this effect is further amplified
by 1−BR114 BR41 due to the strategic interaction between player 1 and hypothetical player 2, in total, player 1’s action changes by NE10 ∆. Then for the same20

reason, this change in player 1’s action influences player 2’s action, which influences player 1’s action, and so on. This infinite process leads to the multiplier1.161−NE 0 NE 0
1

2

Remark 3. In this section, we have assumed two-sided misspecification, in thatboth players 1 and 2 are misspecified. For some applications, it is also importantto think about one-sided misspecification; e.g., one can think of a seller-buyer
problem where only a buyer is misspecified while a seller is fully rational. It turns
out that even with such one-sided misspecification, the result similar to Proposition 3 still holds. Specifically, the equations in Proposition 3 are still valid, if we
replace NE10 with the slope of the best-response function BR01 . This is because
when player 1 is fully rational, then she correctly learns the state and simply bestresponds to player 2’s action.

3.3

Sufficient Condition for Convergence

Now we will think about whether players’ actions and beliefs indeed converge toa steady state. Under double misspecification, both player 1’s belief µ1t and player
2’s belief µ2t evolve in a non-trivial way. This makes our analysis significantly
more complicated than that of first-order misspecification, where there is only
one belief which moves in a non-trivial way. Nonetheless, we find that the belief
converges if the identifiability condition and some additional assumption hold.
Recall that in the case with first-order misspecification, identifiability requires
each (weighted) surprise function to have a unique minimizer. Under double mis16 NE 0 can be also seen as the slope of player 1’s asymptotic Nash equilibrium correspondence
1

defined as
NE1 (x2 ) = {x1 |∃xˆ2 satisfying (7) for i = 1, (8) for i = 2, (9)}.
In words, NE1 (x2 ) denotes player 1’s steady-state action, when player 2 chooses the same action
x2 every period while the other players learn the state and adjust actions. Note that player 2 is notplayer 1’s opponent, but nonetheless her action x2 influences player 1’s steady-state action due tothe incorrect learning: If player 2 changes the action and player 1 is not aware of it, player 1’s
long-run belief is affected, and so is her long-run optimal action. So its slope, NE1 , measures how
a marginal change in player 2’s (constant) action x2 influences player 1’s steady-state action.

21

specification, player i’s surprise function is defined as(Q(xi , x̂−i , θ , Ai ) − Q(xi , x−i , θ , a))22
for each action profile x = (x1 , x2 , x̂1 , x̂2 ), and her weighted surprise function isdefined asZ
Ki (θ , σ ) = Ki (θ , x)σ (dx)Ki (θ , x) =X

for each probability measure σ ∈ 4(X1 × X2 × X1 × X2 ). These surprise functions
are a bit different from those under first-order misspecification because of inferential naivety; player i thinks that players play (xi , x̂−i ), but the actual actions are(xi , x−i ). Identifiability requires that each of the above surprise functions has aunique minimizer.
Under double misspecification, we need an additional assumption for convergence. To state our condition formally, consider the single-agent learning problem
in which player 1 (and hypothetical player 2) learns the state over time, while thebelief of player 2 (and of hypothetical player 1) is fixed at some value θ2 . Let
f1 (θ2 ) denote the set of steady-state beliefs of player 1 in this problem, that is,
f1 (θ2 ) is the set of all θ1 such that there is (x1 , x2 , x̂1 , x̂2 ) satisfying the consistency condition (9) and the incentive-compatibility condition (7) and (8) givenµ2 = µ̂1 = 1θ2 .
Likewise, consider the single-agent learning problem in which player 2 (andhypothetical player 1) learns the state over time, while the belief of player 1 (andof hypothetical player 2) is fixed at some value θ1 . Then let f2 (θ1 ) denote theset of steady-state belief of player 2, that is, f2 (θ1 ) is the set of all θ2 such that
there is (x1 , x2 , x̂1 , x̂2 ) satisfying the consistency condition (10) and the incentivecompatibility condition (7) and (8) given µ1 = µ̂2 = 1θ1 .
The following proposition present a sufficient condition for convergence under
double misspecification.
Proposition 4. Suppose that there is a unique steady state (x1∗ , x2∗ , θ1 , θ2 ) and it
is regular. Suppose also that for each i and σ , the weighted surprise function
Ki (θ , σ ) has a unique minimizer θ2 (σ ) and it is an interior point. In addition,
assume that22

(i) For each i, fi (θ−i ) is a function (rather than a correspondence), and iscontinuously differentiable in θ−i .
(ii) maxθ1 | ∂ f∂2θ(θ1 ) | maxθ2 | ∂ f∂1θ(θ2 ) | < 1.
1

2

Then players’ beliefs converge to the steady state almost surely, regardless of theinitial prior.
Assumption (i) in this proposition is identifiability, and it ensures that the belief converges in every one-dimensional problem. Assumption (ii) requires thateach player’s steady-state belief fi is not too sensitive to the opponent’s belief;
this means that one’s learning is not influenced by the the opponent’s learning by
much, at least asymptotically. Proposition 4 shows that the beliefs indeed converge under these conditions.

4

Applications

4.1

Cournot duopoly

Consider a symmetric Cournot duopoly. Each firm i = 1, 2 simultaneously chooses
its quantity xi , and then they observe a market price y = Q(x1 +x2 , a, θ )+ε, wherea is a parameter which influences the demand and θ is an unknown economic
state. Firm i’s payoff is ui (xi , y) = xi y − c(xi ), where xi y is firm i’s revenue andc(xi ) is firm i’s production cost. We assume that the inverse demand function Q isstrictly decreasing and weakly concave in the first element, and the cost function
c is strictly increasing and weakly convex.17
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 moderate level of) overconfidence about the market demand is beneficial, in that an overconfident firm earns higher equilibrium payoffs than theunbiased rival firm. Intuitively, the overconfident firm is willing to produce more
17 These assumptions imply a concave payoff function, a downward-sloping best response curve,

and a unique Nash equilibrium under the correctly specified model. See, e.g., Tirole (1988).

23than in the correctly specified model. Knowing that, the unbiased firm reduces
its production level in equilibrium, which yields higher profits to the overconfident firm. This mechanism is similar to the commitment effect in the Stackelberg
duopoly.
Their result relies on two implicit assumptions. First, they assume that thegame is one-shot. It is not a priori clear if their result persists in the long run,because when the game is repeatedly played, the overconfident firm is “surprised”
by a realized price being lower than its anticipation, and modifies the (subjective)
view about the demand function. Second, they assume that a firm’s overconfidence is common knowledge, while in reality the opponent may not be aware of
the firm’s overconfidence. In this section, we will relax these assumptions andinvestigate how it changes the result.
First-order misspecification. To begin with, we will relax the first assumption
only, and consider a dynamic model in which one’s overconfidence is common
knowledge. Specifically, we consider the model of first-order misspecification inwhich firm 2 incorrectly believes that the true parameter is A > a. We assume
that QA > 0 and QxA ≥ 0 for all x with x1 + x2 > 0, which means that firm 2 isoverconfident about the price level Q and (weakly) overconfident about the slopeof the inverse demand curve Qx . Firm 1 knows that the true parameter is a, and thefirms’ first-order beliefs are common knowledge. Note that similar assumptions
are imposed in Kyle and Wang (1997). We also assume that Qθ > 0 and Qxθ ≥ 0
for all x with x1 + x2 > 0, i.e., the state θ has positive impacts on the price level
and the slope of the inverse demand function.
Here are two examples which satisfy the assumptions above:18
Q(x1 + x2 , a, θ ) = a − (1 − θ )(x1 + x2 ),(11)

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

(12)

18 These examples satisfy the regularity condition for first-order misspecification with A = a,and for double misspecification with A1 = A2 = a. They also satisfy the conditions stated in
Propositions 2 and 4, so the firms’ beliefs and actions converge to the steady state. The proof is in
Appendix D.

24

In the first example (11), firm 2 is overconfident about the intercept of the demand
function and learns its slope.19 Conversely, in the second example (12), firm 2 isoverconfident about the slope and learns the intercept.20
We will consider how firm 2’s overconfidence influences the long-run steadystate outcome. Recall from Proposition 1 that the impact of firm 2’s overconfidence on its own steady-state action is represented as the base misspecification
2A
effect − M
M22 times the multiplier. For ease of exposition, we assume that misspecification is small (i.e., A is close to a) so that M22 < 0 and the multiplier is positive.
Simple algebra shows that the base misspecification effect in our Cournot modelis written as

z
}|
{
z
}|
{
direct effectdirect effectz}|{ ∂ θ2}|{ ∂ θ2
1 

z
∗
∗
∗
∗
∗
∗
∗
∗
∗
Qθ (x1 + x2 , A, θ2 ) +x2 QxA (x1 + x2 , A, θ2 ) +Qxθ (x1 + x2 , A, θ2 ) .
−QA (x1 + x2 , A, θ2 ) +M22 |
∂
A
∂
A

{z
}
|
{z
}


indirect effecton the price level

indirect effecton the slope(13)

This expression shows that the long-run behavior of the overconfident firm is governed by two countervailing forces. The first one is the direct effect, QA + x2 QxA ,which measures how firm 2’s misspecification directly influences its marginal utility. Since we assume QA > 0 and QxA ≥ 0, this effect is positive, and hence increases the firm’s incentive to produce. This is exactly the effect studied in Kyle
and Wang (1997). The second one is the indirect effect, ∂∂θA2 (Qθ + x2 Qxθ ), whichmeasures how firm 2’s learning (about θ ) influences the marginal utility in thelong run. Since ∂∂θA2 < 0, Qθ > 0, and Qxθ ≥ 0, this effect is negative, and hence
weakens the firm’s incentive to produce.
If the direct effect is larger than the indirect effect, the overconfident firm
19 This happens, for example, when the firm is overconfident about the preference of the cus1
tomers and learns their number. Suppose that there are 1−θcustomers, and each of them purchasesa−p
a − p units of products, where p is a price. Then, the total demand is x = 1−θ, which results in the
inverse demand function p = a − (1 − θ )x.
20 This happens, for example, when the firm is overconfident about the number of the customers
1and learns their preference. Suppose that there are 1−acustomers, and each of them purchases
θ −p
θ − p units of products. Then, the total demand is x = 1−a , which results in the inverse demand
function p = θ − (1 − a)x.

25

is willing to produce more even in the long run. This means that the result ofKyle and Wang (1997) persists, i.e., in the long-run steady state, the rival firm
best-responds by producing less, which yields a higher profit to the overconfident
firm.21On the other hand, if the indirect effect outweighs the direct effect, the overconfident firm is willing to produce less in the long run. Then the commitment
effect works towards the opposite direction, i.e., the rival firm best-responds by
producing more, which harms the overconfident firm’s profit. So in this case,the result of Kyle and Wang (1997) is overturned and a firm’s overconfidence isdetrimental in the long run.
There is another interpretation of the base misspecification effect (13). Notethat the first two terms in the brackets cancel out, because the overconfident firmcorrectly predicts the price level in the steady state.22 Accordingly, the base misspecification effect is rewritten asx2∗ QA QxA Qxθ
x2∗∂ θ2
Qxθ = −
.
QxA +
−
−
M22
∂A
M22 QAQθ
This expression implies that the long-run behavior of the overconfident firm isdetermined by its steady-state belief about the demand slope: If QQxAA − QQxθθ > 0
so that the firm is optimistic about the demand slope, then it produces more thanin the correctly-specified model, and obtains a higher profit. This happens inexample (12) where the firm is persistently overconfident about the demand slope.
On the other hand, if QQxAA − QQxθθ < 0 so that the firm is pessimistic about thedemand slope, then it produces less and earns a lower profit. This happens inexample (11) where the overconfident firm becomes pessimistic about the demandslope through learning. This discussion leads to the following corollary:
Corollary 2. Consider the model of first-order misspecification. Suppose that
there is a unique steady state at A = a and it is regular.23 Then at A = a, we have
21 However, due to the indirect effect, the overconfident firm’s profit is less than that in the

one-shot game: By the incorrect learning, the commitment effect is weakened in the long run.
22 By the implicit function theorem, ∂ θ2 = − QA < 0, and hence we indeed have Q + ∂ θ2 Q = 0.
A
Qθ
∂A
∂A θ
23 Because a steady state is an intersection of asymptotic best response correspondences BR
1

26







∂ x∗
∂ x∗
∂ π∗
∂ x∗
∂ x∗
∂ π∗
sgn ∂ A2 = sgn − ∂ A1 = sgn ∂ A2 = sgn − ∂ A1 = sgn QQxAA − QQxθθ , | ∂ A2 | > | ∂ A1 |,
∂ π∗

∂ π∗

and | ∂ A1 | > | ∂ A2 |.
Double misspecification. Now we will relax the common knowledge assumption, and consider the case in which firm 1 is unaware of the rival firm’s overconfidence. Specifically, consider double misspecification with A1 = a and A2 > a, sothat the firm’s beliefs about the parameter a are the same as before, but each firm(incorrectly) believes that the other firm shares the same view about the parametera.
How does the overconfident firm 2 behave in such a situation? Proposition 3
shows that the impact of one’s overconfidence on its own steady-state action is the
0 M3A
2Abase misspecification effect, − M
M22 − BR23 M33 , times the multipliers. As shownin Lemma 4 in Appendix B.7, when the game is symmetric, the sign of this basemisspecification effect is the same as that of first-order misspecification.24 So under double misspecification, firm 2 produces more than in the correctly-specified
model if it does so in the case of first-order misspecification, and produces less if
it does so in the case of first-order misspecification.
How about the behavior of the rival firm? A critical difference from the caseof first-order misspecification is that the commitment effect does not exist in thisenvironment; although firm 2’s overconfidence influences its behavior, the opponent is not aware of it and does not best-respond to it. This in particular implies
that if we look at the one-shot game, firm 2’s overconfidence does not influence
the opponent’s behavior, and hence never be beneficial.
However, in our dynamic model, firm 2’s overconfidence can still improve theequilibrium payoff. Indeed, we have the following result:Corollary 3. Consider the model of double misspecification with A1 = a. Supposethat there is a unique steady state at A2 = a and it is regular. Then at A2 = a, all
the results stated in Corollary 2 still hold.
and BR2 , the steady state is unique if BR0i (x j ) ∈ (−1, 1) for all i and x j where j , i.
24 Intuitively, this happens because the impact |BR0 M3A | of second-order misspecification is23 M332A
smaller than the impact | M
M22 | of first-order misspecification in symmetric games.

27

So in the long-run, first-order misspecification and double misspecification
lead to similar steady-state outcomes, i.e., firm 1’s unawareness about firm 2’s
overconfidence does not have a significant impact on their long-run behavior. In
particular, when QQxAA − QQxθθ > 0, firm 2’s overconfidence improves its long-run
equilibrium payoff, regardless of whether the opponent is aware of it.
Why do we have such a result, even though the commitment effect does notexist under double misspecification? A key is that firm 1’s long-run behavior is
influenced by incorrect learning. To illustrate the idea, suppose that the base misspecification effect is positive, so that firm 2 produces more than in the correctlyspecified model. Under double misspecification, firm 1 is not aware of it, andhence observes prices which are systematically lower than the anticipation. Accordingly, firm 1 becomes pessimistic about θ and produces less; this yields ahigher profit to the overconfident firm 2 even in the absence of the commitment
effect.

4.2

Team production

In the Cournot model, we have seen that one’s unawareness about the opponent’s
overconfidence does not have a significant impact on the equilibrium outcome.
However, this result does not hold in general; there are many economic examples
in which first-order misspecification and double misspecification lead to qualitatively different outcomes. In this subsection, we will present one of such examples: a team production problem.
Consider two players working on a joint project. Each period, player i = 1, 2chooses an effort level xi , and observes a stochastic output y = Q(x, a, θ ∗ ) + ε
where a is a fixed parameter (e.g., one’s capability) and θ ∗ is an unknown fundamental. We assume that Q is twice-continuously differentiable, Qxi > 0, Qa > 0,
and Qθ > 0. Player i’s payoff is y − c(xi ), where c(xi ) is the effort cost satisfyingc0 > 0 and c00 > 0. Assume also that there is a unique Nash equilibrium in the
one-shot game. This setup is fairly general, and includes the following examples
as special cases.25
25 Again, these examples satisfy the regularity condition for first-order misspecification with A =28

Example 1. Let ai denote player i’s capability, and let a = a1 + a2 denote thetotal capability. Let ci (xi ) = xi2 andQ = θ (x1 + x2 + kx1 x2 + a),
where k ∈ (− θ2∗ , θ2∗ ) is a fixed parameter.26 Note that efforts are complements
if k > 0, and substitutes if k < 0. Each player i may have a bias and incorrectlybelieve that the total capability is Ai , a. When Ai > a, it represents one’s overconfidence. When Ai < a, it represents one’s underconfidence or prejudice about
the opponent’s capability. Players learn the profitability θ of the business overtime. This setup corresponds to a multi-player version of Example 2 of Heidhues,Kőszegi, and Strack (2018).
Example 2. Let a denote player 1’s capability. Let ci (xi ) = xi2 andQ = θ (ax1 + x2 + kx1 x2 + 2),
where k ∈ (− θ2∗ , θ2∗ ) is a fixed parameter. A difference from Example 1 is that
player 1’s capability a influences her marginal productivity, which makes the function Q asymmetric, in that Q(x1 , x2 , a, θ ) , Q(x2 , x1 , a, θ ) for a , 1. As will be explained, this property has a qualitative impact on the steady-state outcome under
double misspecification.
First-Order Misspecification. Again, we start with the benchmark case in which
player 2 has first-order misspecification, in that she incorrectly believes A , a.
Simple algebra shows that the base misspecification effect of first-order misspecification is
indirect effectz }| {
direct effect
z}|{

1 
 Q x A + ∂ θ2 Q x θ  .
−
(14)
i
i

M22 |{z ∂ A}on marginal productivity

a and for double misspecification with A1 = A2 = a, and the sufficient conditions for convergencestated in Propositions 2 and 4. The proof is in Appendix D.
26 This assumption ensures that the equilibrium is an interior point.

29

So the base misspecification effect is determined by the biased player 2’s subjective view about the marginal productivity in the steady state; the term Qxi A
measures how player 2’s misspecification influences her view about the marginal
productivity, and the term ∂∂θA2 Qxi θ measures how the incorrect learning modifies
it.
Following Heidhues, Kőszegi, and Strack (2018), we will assume Qx2 A ≤ 0
and Qx2 θ > 0, i.e., the marginal return Qxi of the misspecified player is negatively
correlated with the capability, and positively correlated with the fundamental. It iseasy to check that this assumption is satisfied in Examples 1 and 2 above. Under
this assumption, both the direct effect and the indirect effect in (14) are negative.
Thus the overconfident player 2 works less in the static model, and even less in the∂ x∗long run (so we have ∂ A2 < 0). Intuitively, when a player is overconfident about
her own capability, she observes outputs systematically lower than the anticipation. Accordingly, as time goes, she becomes pessimistic about the state θ andreduces the effort.27
How about the equilibrium payoffs? There are two cases to be considered.
First, suppose that Qx1 x2 > 0 so that efforts are complements. (In the above examples, this corresponds to k > 0.) Then the rational player 1 reduces the effort
as a response to the lower effort of the overconfident player, which reduces theoverconfident player’s payoff. So one’s overconfidence is detrimental in this case.
Next, suppose that Qx1 x2 < 0 so that efforts are substitutes. (In the above examples, this corresponds to k < 0.) In this case, one’s overconfidence is beneficial;indeed, the rational player 1 increases the effort as a response, which improves
27 Of course, the argument here can be extended to a more general setup.

The bottom-line isthat the short-run effect of player 2’s overconfidence on her own action is determined by Qx2 A , andQA
the long-run effect is determined by Qx2 A + ∂∂θA2 Qx2 θ = Qx2 A − Q
Qx2 θ . For example, suppose thatθQ < 0 is the damage from drought and agents invest to irrigation which mitigate the damage, andit 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.
This means that 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.

30

the overconfident player’s payoff. The mechanism here is exactly the same as
the commitment effect in the Cournot model, i.e., one’s misspecification may influence the opponent’s behavior, which may improve the misspecified player’s
payoff. This result is in a sharp contrast with the single-agent case studied by Heidhues, Kőszegi, and Strack (2018), where one’s overconfidence is always detrimental. In their model, the overconfident player reduces the effort just as in ourmodel, but it never improves the overconfident player’s payoff due to the lack of
the commitment effect.
Double Misspecification. Now we consider the case of double misspecification, where player 1 is not aware of player 2’s overconfidence. We will focus on
Examples 1 and 2 above, and show that first-order misspecification and double
misspecification can have opposite effects on players’ actions and payoffs.
First, consider Example 1. This game is symmetric, in that Q(x1 , x2 , a, θ ) =Q(x2 , x1 , a, θ ) and u1 (x1 , y) = u2 (x1 , y). As discussed in the Cournot model, in
0 M3A
2A
such a case, the sign of the base misspecification effect − M
M22 −BR23 M33 of double
misspecification coincides with that of first-order misspecification. So the over∂ x∗
confident player 2 reduces the effort (i.e., ∂ A2 < 0) just as in the case of first-order2
misspecification.
How does it influence player 1’s action? Under double misspecification, player
1’s long-run behavior is influenced by her incorrect learning: Player 1 is not awareof the overconfident player reducing the effort, and thus observes outputs lower
than the anticipation on average. This makes her pessimistic about the state overtime, and she reduces the effort in the end. So in this example, one’s overconfidence lowers both players’ efforts and payoffs.
Note that this result does not rely on whether efforts are complements or substitutes. This is in a sharp contrast with the case of first-order misspecification,
where an agent’s overconfidence improves her profit when the efforts are substitutes. That is, first-order misspecification and double misspecification have opposite effects on the overconfident player’s payoff (and player 1’s action) whenk < 0.

31

Next, consider Example 2. Here the difference between first-order misspecification and double misspecification is even more striking, in that the base misspecification effects of these two misspecifications can have opposite signs. For
example, let Θ = [0.1, 0.3], θ ∗ = 0.2, a = A1 = 1, and k = 4. Then at A2 = a, thebase misspecification effect of double misspecification is positive, and thus theoverconfident player 2 increases effort.28 Intuitively, in the case of double misspecification, player 2 (incorrectly) believes that player 1 is overconfident about aand makes higher effort than in the reality. Because efforts are complements (recall k = 4), with this perception, player 2 makes more effort. Given the specified
parameters, this effect dominates all the other effects coming from the misspecification, so the overconfident player 2 makes higher effort in the steady state.

4.3

Bias Transmission and Self-fulfilling Prophecies

Recent evidence suggests that the gender gap in math achievement arises from
culture and social conditioning rather than from biological reasons (such as brain
functioning). For example, Lavy and Sand (2018) and Carlana (2019) find that thegender gap in performance in math exam substantially increases when students are
assigned to math teachers with stronger gender stereotypes. In particular, Carlana(2019) argues that this effect is at least partially driven by lower self-confidence
on math ability of female students who are exposed to gender-biased teachers.29
We will show that our framework is useful to explain such a bias transmission
from teachers to students.30
Suppose that player 1 (she) is a student and player 2 (he) is a teacher. The stu28 In this case, M = − 1 , M = M = − 47 , BR0 = 63 , and M = 39 . Hence, the base
22
33
2A
3A
23
44
22
235
220

misspecification effect of double misspecification is about 0.012.
29 Relatedly, Gong, Lu, and Song (2018) report that a male math teacher in their survey ismore likely to question and praise male students than female students compared with a female
math teacher, and that having such a teacher lowers female students’ beliefs about gender-specificcapability: They find that female students who have a male math teacher are more likely to agreewith a question “boys are more talented in learning math than girls” than those who have a female
math teacher.
30 See Giuliano (2020) for a survey of the transmissions of gender-biased norms and beliefs.

32

dent’s achievement (e.g., math test performance) is given by y = a(x1 +x2 +b)+ε,
where a > 0 represents a gender-specific capability and b > 0 is the student’s
own capability. The student knows her own capability b, but does not know
the gender-specific capability θ1 . So she thinks that the outcome is given byy = θ1 (x1 + x2 + b) + ε and learns θ1 over time. On the other hand, the teacher hasa biased view about the gender specific capability, and he thinks that the outcome
is given by y = A2 (x1 + x2 + θ2 ) + ε, where A2 < a represents his bias. He does notknow the student’s individual capability θ2 , and learns it over time. We assume
that each player (incorrectly) thinks that the opponent has the same view about theworld. 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 3, in that different
players learn different parameters. However, this does not has a substantial impact
on the property of the steady state. Similarly to the analysis in Section 4.2, itis straightforward to show that both the teacher and the student exert less effort
than in the correctly specified model in the steady state, and the student becomesunderestimating the gender-specific capability θ1 .
A notable feature in this framework is that the student initially has an unbiased
view about the environment, but nonetheless, the teacher’s gender bias is eventually transmitted to the student.31 A key driving force is the student’s inferential
naivety; a biased teacher secretly reduces the effort, and the student is not awareof it. Then on average, the realized outcomes are lower than the student’s expectation, which makes the student unrealistically pessimistic about the gender-specificcapability θ1 .
31 Heidhues, Kőszegi, and Strack (2020) also argue that a gender bias (more generally, a groupdiscrimination) can endogenously arise as a consequence of misspecified learning. Formally, they
develop a single-agent learning model, 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 case in which an agent does not have underconfidence, or more generally, any bias about the physical environment. Our analysis showsthat one’s existing prejudice may induce other players’ negative self-stereotypes through learning.

33

It seems that bias transmissions and self-fulfilling prophecies are as prevalent
in workplaces as they are in school classrooms. Livingston (1969) finds that amanager’s high expectation of the subordinates improves the subordinates’ job
performance, and it is confirmed by many subsequent papers.32 This phenomenon
is known as the Pygmalion effect in management, and Eden (1984) and Davidson
and Eden (2000) argue that this effect stems from bias transmission: A manager’s
higher expectation raises workers’ beliefs about their own self-efficacy, which leadthem to greater motivation and achievement. 33 Our framework here is useful tobetter understand why such a bias transmission occurs. Roughly speaking, if an
optimistic manager makes an extra effort and a worker is not (fully) aware of it,then on average, the worker observes an output better than his expectation. This
makes the worker more confident of his own capability, which in turn improves
the outcome further.
If a manager is negatively biased, the mechanism above can work in the opposite direction. For example, Hoobler, Wayne, and Lemmon (2009) report thatin many industries, managers tend to think that female workers are unfit for promotion compared to male workers. Our analysis suggests that such a bias canbe transmitted to female workers, and they become underconfident about theirown capabilities. This is consistent with the recent work by Born, Ranehill, andSandberg (forthcoming) who find that women are less confident than men in theirrelative ability as being a leader position. This mechanism may help understand
why a “glass ceiling,” an invisible barrier that discourages women and minorities,
persists in various institutions.
32 Kierein and Gold (2000) and McNatt (2000) provide meta-analysis results of the Pygmalion

effect in management. See Bertrand and Duflo (2017) for a survey of the self-fulfilling prophecies
in economics.
33 Of course, there are other mechanisms which explain the correlation between a manager’s bias
and a worker’s performance. For example, Glover, Pallais, and Pariente (2017) find that minority
workers in French grocery stores tend to perform worse when they work with biased managers,
while working with biased managers does not activate self-stereotyping of the minority workers.
Glover, Pallais, and Pariente (2017) argue that this result can be explained by the fact that biased
managers are less comfortable around minorities: Such managers do not monitor minority workers
frequently and do not ask them to stay after the end of their shifts.

34

5

Related Literature and Conclusion

There is a rapidly growing literature on Bayesian learning with model misspecification. Nyarko (1991) presents a model in which the agent’s action does notconverge. Fudenberg, Romanyuk, and Strack (2017) consider a general two-statemodel and characterize the agent’s asymptotic actions and behavior. Heidhues,Kőszegi, and Strack (2018, 2021) and He (2021) 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 andinformation structure. Esponda, Pouzo, and Yamamoto (2021) characterize theagent’s asymptotic behavior in a general single-agent model. Fudenberg, Lanzani, and Strack (2021) discuss robustness of steady states. All these papers lookat a single-agent problem and focus on first-order misspecification.
More recently, Ba and Gindin (2021) consider two-player team productionin which both players are overconfident about their own capability. They showthat if efforts are complements and information has a positive externality, thenlearning is mutually reinforcing, i.e., one’s strategic play reduces both players’
efforts and results in a worse outcome. Our work strengthens their result, inthree ways. First, our Proposition 1 gives a necessary and sufficient condition
for mutually-reinforcing learning: When the 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).34 Second, our Proposition 1 does not impose any
assumptions on payoffs and information structure, and allows us to study a wide
range of applications such as Cournot duopoly and tournaments. Third, and mostimportantly, we develop a model of higher-order misspecification and study howeach type of misspecification influences players’ beliefs and actions.
Misspecified learning has also been studied in other settings. In the literature
on social learning, many papers study how inferential naivety or model misspecification influences the asymptotic outcomes (e.g., DeMarzo, Vayanos, and Zwiebel,
34 Strategic complementarity and positive information externality assumed by Ba and Gindin(2021) imply upward-sloping asymptotic best response curves.

35

2003; Eyster and Rabin, 2010; Gagnon-Bartsch and Rabin, 2016; Bohren andHauser, 2021; Frick, Iijima, and Ishii, 2020). Molavi (2020) considers a general equilibrium model in which a representative agent has a misspecified view
about the world. Cho and Kasa (2017) study an asset-pricing model in which anagent incorrectly believes that the environment is not stationary. Our model would
help investigate other applications, such as firm-consumer or principal-agent relationships under higher-order misspecification.

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40

AppendicesA

Asymptotic Behavior of Misspecified Players

In this appendix, we will provide a general model which encompasses both firstorder misspecification and double misspecification as special cases, and show thatthe motion of players’ action frequency is asymptotically approximated by a solution to a differential inclusion. This result can be seen as a generalization of themain theorem of Esponda, Pouzo, and Yamamoto (2021) to the case with multiple
players and continuous actions. Then we show that the motion of players’ beliefs
is also approximated by a solution to a differential inclusion. This result is new,and we use it to derive a sufficient condition for belief convergence.

A.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 Proposition 7. The second property
is crucial in order to use a stochastic approximation technique in the proof of
Proposition 8. 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.

41

A.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 which
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 denote
0 = {0}.
the set of all t-period history, and let HiS/ Player i’s pure strategy in the∞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.
A.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. For example, we obtain the model of first-order misspecification studied in Section 2 when M1,1 = M2,2 = M1,3 = M2,4 = M1,5 = · · · , M2,1 = M1,2 =M2,3 = M1,4 = M2,5 = · · · , and M1,1 is correctly specified; here the first condition implies that player 1’s model M1,1 is common knowledge, and the second condition implies that player 2’s model M2,1 is common knowledge. Similarly, we obtain the model of double misspecification studied in Section 3 when
Mi,1 = Mi,2 = Mi,3 = · · · for each i.
42

In what follows, we will maintain the following technical assumptions.
Assumption 1. The following conditions hold:(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 thereRare densities q(·|x) and qθi,k (·|x) such thatRq(y|x)ν(dy)= Q(A|x) and A qθi,k (y|x)ν(dy) = Qθi,k (A|x) for any A ⊆ Y
ABorel.)(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 (2021), 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.
Recall that in the cases of first-order misspecification and double misspecification, each player i believes that (i) her view Mi,1 about the world is common
knowledge (i.e., Mi,1 = Mi,3 = Mi,5 = · · · ) and that (ii) her view Mi,2 about the opponent’s view about the world is common knowledge (i.e., Mi,2 = Mi,4 = Mi,6 =· · · ). This ensures that player i’s decision making problem is equivalent to solving
a game played by this player i and a hypothetical player.35 In the general model
here, we will impose a (similar but) weaker assumption:
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, · · · .
35 In the case of first-order misspecification, this hypothetical player is redundant in that her

action coincides with the actual player’s action. So such a hypothetical player does not appear in
our analysis in Section 2.

43

For the special case in which ki = 1, this assumption implies that player ibelieves that the models (Mi,1 , Mi,2 ) are common knowledge, just as in the case of
first-order misspecification and double misspecification. The assumption above ismore general than that, because it allows ki > 1; in such a case, the assumption
implies that player i believes that models are common knowledge at higher levels,
i.e., she believes that the opponent believes that · · · that the models (Mi,ki , Mi,ki +1 )are common knowledge. Note that this assumption is about whether player ithinks that the models are common knowledge, and not about whether the models
are common knowledge in the objective sense. We believe that Assumption 2 issatisfied in most applications.36
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 appearing in 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 . Here, agent ki + 2 refers to agent
ki , so agents ki and ki + 1 play the game with each other.
• 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
so on. So the action profile x̂i = (x̂i,k )kk=1
in this hypothetical game is essentiallyki +1
player i’s prediction hierarchy. Let X̂i = ×k=1 Xi,k denote the set of all these action
profiles.
36 This assumption is needed to establish Propositions 7 and 8.

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.

44

In what follows, each agent k in this hypothetical game is labelled as (i, k),because these agents describe player i’s reasoning. The opponent j has a different
model hierarchy M j , Mi , and hence her reasoning is represented by a different
set of agents labelled as ( j, k).
Let ŝi,k denote agent (i, k)’s strategy in the infinite-horizon hypothetical game,i +1
and let ŝi = (ŝi,k )kk=1
denote a strategy profile. This profile ŝi is also interpreted
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∈ 4Θi,k can be computed using Bayes’ rule, after every
k’s posterior belief µ̂i,ktpublic history hY . Formally, for each t and k, we havet−1t (θ )Q (yt |ŝ (ht−1 ), ŝµ̂i,ki,k+1 (hY ))
i,k Y
i,k
θi,k
t+1
µ̂i,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 above
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 .

It is obvious that this strategy profile ŝi is Markov.
45

(15)

Example 2. (Dynamically optimal agents) Now consider dynamically optimal
agents, who maximize the expectation of the discounted sum of the stage-game∞
δ t−1 ui,k (x̂i,k , y). Many applied papers use Markov perfect equilibria
payoffs, ∑t=1
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

Z

ŝi,k

∞

∑ δ t−1E[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 ).
∞ 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,

1 t
σ (h)[(x̂1 , x̂2 )] = ∑ 1{ŝi,k (hτ−1 )=x̂i,k ∀i∀k} .
Y
t τ=1
t

Intuitively, 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.

A.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 )46

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 )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 (2021) 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.
47

Proposition 5. 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.

(16)

Let H denote the set of sample paths h which satisfy (16). By Proposition 5,

Pŝ (H ) = 1.

A.3

Asymptotic Motion of Action Frequency

A.3.1

Stochastic Approximation and Differential Inclusion

Now we will show that given any Markov strategy ŝ, the asymptotic motion of theaction frequency σ t is approximated by a solution to a differential inclusion. Picka Markov strategy ŝ, and pick a sample path h ∈ H . By the definition, the action
frequency in each period is written as
σ t+1 (h) =t1
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 .
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 ) in the sense of (16), given the mixture σ .
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:

48

Proposition 6. 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) +S t (σ t (h)).
t +1
t +1 εThis 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, letnonŜ(µ) = x̂ x̂ = lim ŝ(µ̂ n ) for some (µ̂ n )∞
with
lim
(
µ̂
)
=
µ̂
n=1
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 ∈ 4Θi,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 7. 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 6 and 7 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 +1That 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):
49

It shows that the asymptotic motion of the action frequency is described by thedifferential inclusionσ̇(t) ∈ 4S0 (σ(t)) − σ(t).

(17)

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 8 below, the differential inclusion (17) 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 (17).37
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
σ : [0, ∞) → 4X̂ is a solution to the differential inclusion (17) with an initialvalue σ ∈ 4X̂ if it is absolutely continuous in all compact intervals, σ(0) = σ ,and (17) 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(17) has a solution. (See Theorem 9 of Deimling (1992) on page 117.) Let Z(σ )denote the set of all these solutions.

Proposition 8. Pick a Markov strategy ŝ. Then for any T > 0 and any sample
path h ∈ H ,
lim

inf

sup kw(h)[t + τ] − σ(τ)k = 0.

t→∞ σ∈Z(w(h)[t]) τ∈[0,T ]37 There is also a technical reason: In the proof of Proposition 8, 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)).

50

A.3.2

Steady State and Generalized Berk-Nash Equilibriumσ ∈ 4X̂ is a steady state of the differential inclusion (17) 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 9. 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 (17).
In all the examples in this paper, we assume that the agents are myopic sothat the strategy profile ŝ satisfies (15). In this special case, steady states of ourdifferential inclusion are generalized Berk-Nash equilibria in the following sense:
Definition 3. 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.
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.
Proposition 10. Suppose that the strategy profile ŝ satisfies (15). Then any steadystate of our differential inclusion (17) 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
5, the posterior belief µ̂ t will be concentrated on 4Θ(σ ) after a long time, that51

is, 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) identifiability holds inthat there is a unique minimizer θi,k of the Kullback-Leibular divergence for eachaction frequency σ , then obviously any GBNE is degenerate. All our examples in
the paper satisfy these assumptions.
Proposition 10 above implies that when agents are myopic, a limiting action
frequency must be a GBNE. It turns out that the same result holds for dynamically optimal agents, provided that identifiability holds and agents play a Markov
perfect equilibrium. This follows from the fact that under identifiability, the differential inclusion (17) for myopic agents is exactly the same as that for dynamically
optimal agents who play a Markov perfect equilibrium. So all the results presented i the main text of the paper area valid for dynamically optimal agents, aslong as identifiability holds.

A.4

Motion of the KL Minimizer

A.4.1

Identifiability and Differential Inclusion

Our Proposition 8 shows that the asymptotic motion of the action frequency σ t is
described by the differential inclusion (17). However, solving the differential inclusion (17) 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.

52

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 ,x̂0 )∂ K (θ ,x̂)| ∂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

(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

53



.

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 inclusion[[θ̇(t) ∈−(∇2 K(θ(t), σ ))−1 ∇K(θ(t)), σ 0 ) .
(18)
σ :θ (σ )=θ(t) σ 0 ∈4S0 (θ(t))

Let Zθ (θ(0)) be the set of solutions to the differential inclusion (18) with the initialvalue θ(0).
Proposition 11. Suppose that Assumptions 3 and 4 hold. Then for any T > 0 andany sample path h ∈ H ,lim

sup |wθ (h)[t + τ] − θ(τ)| = 0.

inft→∞ θ∈Zθ (wθ (h)[t]) τ∈[0,T ]To interpret the differential inclusion (18), consider the special case in which
Θi,k ⊂ R, i.e., assume that agent k’s model θi,k is one-dimensional. Then from(17), 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 )

(19)

i,k
i,k
0 (θ , σ ) =00 (θ , σ ) =for each i and k, where Ki,k.
and Ki,k∂θ∂θ200
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.

54

When we consider the dynamic of θ t = θ (σ t ), the drift of θ t cannot be
uniquely determined, for two reasons. 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 (2021), in the singleagent setup, this happens when the agent is indifferent over multiple actions at amodel θ = θ t .) In our differential inclusion (19), this multiplicity is captured bytaking the union over σ 0 ∈ 4S0 (θ(t)). Note that the same multiplicity problem
appears in the differential inclusion (17).
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 minimizer(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).

B

ProofsB.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)

(20)
(21)

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 (20) and (21). 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 the55parameter 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
1

2

∂ x∗
∂ x∗
combining the two equations above, we have ∂ A1 = BR1 ∂ A2 .
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 imply that BR1 and BR2 must intersect at somex1 > x1∗ , but this contradicts with the fact that x∗ is a unique steady state. The same
argument works for case (ii). Hence, we have BR01 BR02 ≤ 1. Also, dividing bothsides of detM , 0 by M11 M22 , we have BR01 BR02 , 1.
Q.E.D.

B.2

Proof of Corollary 1

Immediate from Proposition 1.

B.3

Q.E.D.

Proof of Proposition 2∞
In this proof, we will use the tools developed in Section A. Let h = (xt , yt )t=1
denote a sample path of the infinite horizon game. Given a sample path h, let
σ t (h) ∈ 4X denote the action frequency up to period t, i.e.,σ t (h)[x] =|{τ ≤ t|xt = x}|t56

for each action profile x. Proposition 5 shows that almost surely, each player i’s
belief in a later period t will be concentrated on the minimizer of the KL divergence (the surprise function) with weight σ t−1 . More formally, there is a set H
of sample paths such that a sample path h must be in this set H with probability
one, and such that for any sample path h ∈ H , each player i’s belief in period t isapproximately 1θi (σ t−1 (h)) for large t. This result immediately implies that player 1
correctly learns the true state θ ∗ , as her KL minimizer is constant and θ1 (σ ) = θ ∗for any frequency σ ∈ 4X.
We will show that player 2’s belief also converges to the steady-state belief
almost surely. For this, it suffices to show that for every sample path h ∈ H , her
KL minimizer θ2 (σ t (h)) converges to the steady state. In what follows, we will
prove a bit stronger result; we allow multiple steady states, and show that for eachsample path h ∈ H , limt→∞ d(σ t (h), E2 ) = 0 where E2 is the set of all steady-state
beliefs θ of player 2. This implies that player 2’s belief converges even when thesteady state is not unique.
So pick an arbitrary sample path h ∈ H . To think about a dynamic of the KL
minimizer θ2t (h) = θ2 (σ t (h)), Proposition 11 is useful; it shows that the motionof θ2t is asymptotically approximated by the differential inclusion (19), which
reduces to the one-dimensional differential inclusionθ̇2 (t) ∈[σ :θ (σ )=θ(t)−K20 (θ2 (t), s(1θ ∗ , 1θ2 (t) ))K200 (θ2 (t), σ )

(22)

2

where K20 = ∂∂Kθ2 and K200 = ∂∂ θK22 . Here we ignore the dynamic of player 1’s KL
minimizer θ1 , as it is constant and θ1 (σ ) = θ ∗ for all σ . With an abuse of notation,
let Zθ (θ ) denote the set of solutions to the differential inclusion above with an
initial value θ ∈ Θ.
We will consider the following two cases separately.
B.3.1

Case 1: lim inft→∞ θ2t (h) , lim supt→∞ θ2t (h).

We will show that [lim inft→∞ θ2t (h), lim supt→∞ θ2t (h)] ⊆ E2 .
Suppose not, so that there is a model θ 0 ∈ [lim inft→∞ θ2t (h), lim supt→∞ θ2t (h)]such that θ 0 < E2 . Then K20 (θ 0 , s(1θ ∗ , 1θ 0 )) , 0, meaning that (i) K20 (θ 0 , s(1θ ∗ , 1θ 0 )) >0 or (ii) K20 (θ 0 , s(1θ ∗ , 1θ 0 )) < 0. In what follows, we will focus on the case (i). The
proof for the case (ii) is symmetric.
Since K20 (θ , σ ) is continuous in (θ , σ ) and s(1θ ∗ , 1θ ) is continuous in θ , thereis ε > 0 such that K20 (θ , s(1θ ∗ , 1θ )) > 0 for any θ with |θ − θ 0 | ≤ ε. Pick such ε >57

0. Then the right-hand side of (22) is positive for any θ(t) in the ε-neighborhoodof θ 0 , which means that θ(t) increases as time goes in this neighborhood.38 Hence
there is T > 0 such that
θ2 (t) ≥ θ 0 + ε(23)

for any t ≥ T and for any solution θ2 ∈ Zθ (θ ) to the differential inclusion withany initial value θ with θ ≥ θ20 − ε. Pick such T .
With an abuse of notation, let wθ (t) denote the continuous-time interpolation∞ . From Proposition 11, there is t ∗ such that for anyof the KL minimizer (θ2t (h))t=1∗
t > t , θ2 ∈ Zθ (wθ (t)), and s ∈ [0, 2T ],ε|wθ (t + s) − θ2 (s)| < .
2(24)

Pick such t ∗ . Since θ 0 ≤ lim supt→∞ θ2t (h), there is t ∗∗ > t ∗ such that wθ (t ∗∗ ) ≥θ 0 − ε. Pick such t ∗∗ . Then from (23), we have
θ2 (s) ≥ θ 0 + ε
for any s ≥ T and for any solution θ ∈ Zθ (wθ (t ∗∗ )). This inequality and (24)impliesεwθ (t ∗∗ + s) ≥ θ 0 +∀s ∈ [T, 2T ].
2
Likewise, since wθ (t ∗∗ + T ) ≥ θ 0 + ε2 , it follows from (23) that
θ2 (s) ≥ θ 0 + ε
for any s ≥ T and for any solution θ2 ∈ Zθ (wθ (t ∗∗ + T )). This inequality and (24)impliesεwθ (t ∗∗ + s) ≥ θ 0 +∀s ∈ [2T, 3T ].
2
Iterating this argument, we can show thatwθ (t ∗∗ + s) ≥ θ 0 +ε
2∀s ∈ [T, ∞).

But this means that lim inft→∞ θ2t (h) ≥ θ 0 + ε2 , which is a contradiction.
38 Note that K 00 < 0 because K is convex.

58

B.3.2tt (h) = lim supCase 2: lim inft→∞ θi,kt→∞ θi,k (h).

tt (h) exists. Let θ ∗ = lim
In this case, limt→∞ θi,kt→∞ θi,k (h). We will show thati,k∗θi,k ∈ E.
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,ki,ki,k∗ | ≤ ε and any σ 0 ∈ 4S (θ (θ )). Pick such ε > 0. Then pick Twith |θi,k − θi,k0
i,ksuch that (23) 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 11, there is t ∗ such that (24) 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,kε2∀s ∈ [T, ∞).

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

B.4

Q.E.D.

Proof of Proposition 3

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

(25)
(26)
(27)
(28)

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 A2 close to A∗2 , there is an action profile x∗ which59

satisfies the first-order conditions (25)-(28). 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
  ∂ x2∗ 
 M
 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


M11 M22 M33 M44 M2A
M3A+ BR23= − (1 − BR14 BR41 )
.
detM
M22
M33
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 ,so1
M11 M22 M33 M44
=
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 into the equation above, we obtain the first equation in the proposition. The second equation can be derived in a similar way.
Q.E.D.

60

B.5

Proof of Proposition 4

We use the tools developed in Section A. Recall that under double misspecification, there are two real players and two hypothetical players. Let x = (x1 , x2 , x̂1 , x̂2 )∞ ,denote the action profile of these players, and given a sample path h = (xt , yt )t=1
t
let σ (h) ∈ 4(X1 × X2 × X1 × X2 ) denote the action frequency up to period t. Notethat this σ t (h) contains information about the past actions of the real players andthe hypothetical players.
Proposition 5 shows that after a long time, each player i’s posterior belief
will be concentrated on the KL minimizer θit = θi (σ t (h)). Also Proposition 11
shows that the motion of these KL minimizers, (θ1t , θ2t ), is approximated by thedifferential inclusion (19), which can be rewritten as the two dimensional problemdθ1 (t) dθ2 (t)
,
dt
dt


∈

[
σ :θ (σ )=θ(t)K 0 (θ1 (t), s(θ1 (t), θ2 (t))) K20 (θ2 (t), s(θ1 (t), θ2 (t))),−− 1
K100 (θ2 (t), σ )K200 (θ̂1 (t), σ )!

(29)where s(θ1 , θ2 ) denotes a static equilibrium x = (x1 , x2 , x̂1 , x̂2 ) given the beliefs(θ1 , θ2 , θ̂1 , θ̂2 ) with θ̂1 = θ2 and θ̂2 = θ1 .
In what follows, we will show that regardless of the initial value, any solution
to the differential inclusion (29) 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 (2021), and their Proposition 2 ensures that θ t converges
there almost surely, as desired.
The following lemma partially characterizes the solution to the differential
inclusion (29): It shows that θ2 (t) moves toward f2 (θ1 (t)) at any time t.
Lemma 3. Pick any initial value θ(0) = (θ1 (0), θ2 (0)) and any solution θ =(θ1 , θ2 ) to the differential inclusion (29). Then for any t ≥ 0 with θ2 (t) > f2 (θ1 (t)),
we have θ̇2 (t) < 0. Similarly, for any t ≥ 0 with θ2 (t) < f2 (θ1 (t)), we haveθ̇2 (t) > 0
Proof. We will prove only the first part of the lemma, because the proof of thesecond part is symmetric. Suppose that θ2 (t) > f2 (θ̂1 (t)) at some time t. To proveθ̇2 (t) < 0, it suffices to show that K20 (θ2 (t), s(t)) > 0, where s(t) denotes the static
equilibrium s(θ1 (t), θ2 (t)) in time t.
Suppose not and K20 (θ2 (t), s(t)) < 0. (We ignore the case with K20 (θ2 (t), s(θ1 (t), θ2 (t))) =0, because in such a case, θ2 (t) ∈ f2 (θ1 (t)), which contradicts with the uniquenessof f2 (θ1 (t)).) We consider the following two cases:
61

Case 1: θ2 (t) = θ . In this case, the KL minimizer given the equilibrium s(t)is θ2 (s(t)) = θ = θ2 (t) (this follows from the fact that the KL divergence K2 issingle-peaked w.r.t. θ2 ). Hence θ2 (t) = θ is a steady state, i.e., θ2 (t) ∈ f2 (θ1 (t)).
But this contradicts with the uniqueness of f2 (θ1 (t)).
Case 2: θ2 (t) < θ . An argument similar to that in Case 1 shows that at
θ2 = θ , we have K20 (θ , s(θ1 (t), θ )) > 0. On the other hand, by the assumption,
K20 (θ2 (t), s(θ1 (t), θ2 (t))) < 0. Then since K20 (θ , s(θ1 (t), θ )) is continuous in θ ,there must be θ ∈ (θ2 (t), θ ) such that K20 (θ , s(θ1 (t), θ )) = 0. This implies thatθ ∈ f2 (θ1 ), but it contradicts with the uniqueness of f2 (θ1 ).
Q.E.D.
Now we will construct a Lyapunov function V to show that any solution to thedifferential inclusion (29) converges to the steady state. Without loss of generality,
assume that the steady state is (θ1∗ , θ2∗ ) = (0, 0). From assumption (iii), there1is κ > 0 such that maxθ1 | f2∂(θθ 1 ) | < κ <f1 (θ2 ) . Pick such κ, and for each1

maxθ2 | ∂ θ2|

θ = (θ1 , θ2 ), 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 (18),V̇ (θ(t)) < 0
for all t with θ(t) , (0, 0). We will consider the following cases separately:
Case 1: |θ2 (t)| > |κθ1 (t)|. Assume first that θ2 (t) > 0. Then by the definitionof κ and f2 (0) = 0, we have f2 (θ1 (t)) < |κθ1 (t)| < θ2 (t). Then from Lemma 3
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 3 and θ2 (t) < 0, we haveV̇ (θ(t)) = −θ̇2 (t) < 0.
Case 2: |θ2 (t)| < |κθ1 (t)|. An argument similar to those for Case 1 shows thatV̇ (θ(t)) < 0.
Case 3: |θ2 (t)| = |κθ1 (t)|. We will focus on the case with θ2 (t) > 0 andθ1 (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 thatθ̇1 (t) < 0. Hence we have V̇ (θ(t)) = {θ̇2 (t), κ θ̇1 (t)} < 0.
Q.E.D.

62

B.6

Proof of Corollary 2

Consider the infinite-horizon model with first-order misspecification. Let Mii , Mi jbe the ones in Proposition 1. Then we have
Mii = 2Qx + xi Qxx − c00 < 0,
Mi j = Qx + xi Qxx < 0,x2∗ QA QxA Qxθ
M2A
=−
−
−
.
M22
M22 QAQθ
M12∈
The first two inequalities imply Mii < Mi j < 0, and hence we have BR01 = − M11∂ x∗(−1, 0) and 1−BR10 BR0 > 1. Thus it follows from Proposition 1 that sgn ∂ A2 =
1 2



∂ x∗
∂ x∗
∂ x∗
sgn − ∂ A1 = sgn QQxAA − QQxθθ and | ∂ A2 | > | ∂ A1 |.
For payoffs, note that at A = a, we have∗∗
∂ x∗
∂ πi∗ ∂ πi∗ ∂ xi∗ ∂ πi∗ ∂ x−i∂ πi∗ ∂ x−i=+== xi∗ Qx −i ,∂A
∂ xi ∂ A ∂ x−i ∂ A∂ x−i ∂ A∂A
∂ π∗

where the second inequality follows from ∂ xii = 0. Since x1∗ = x2∗ at A = a and


∂ x∗
∂ π∗
∂ π∗
∂ π∗
∂ π∗
Qx < 0, we have sgn ∂ A2 = sgn ∂ A2 = sgn − ∂ A1 and | ∂ A1 | > | ∂ A2 |.
Q.E.D.

B.7

Proof of Corollary 3

We first prove a lemma which is useful to analyze a symmetric game, where
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 ). Let xicorrect and π correct denote firm i’s steady-state action and payoff in the correctly specified model. Let xfirst denote the steady-state action profile
for first-order misspecification, where player 1 believes that the true parameter is
A1 = a and player 2 believes that the true parameter is A2 , a. Likewise, let xdouble
denote the steady-state action profile for double misspecification with A1 = a andA2 , a. The following lemma relates these two steady states when player 2’s
misspecification is small.
Lemma 4. Consider a symmetric game with x1correct = x2correct . Suppose that in
the case of first-order misspecification with A = a, there is a unique steady stateand it is regular. Suppose also that in the case of double misspecification with63

A1 = A2 = a, there is a unique steady state and it is regular. Then in the case of
double misspecification with A1 = A2 = a, we have NEi0 ∈ (−1, 1), and∂ x2double (Uii − L)(Uii +Ui j − L)
Ui j − L ∂ x2first
=
1−
,
∂ A2
Uii (Uii +Ui j − 2L)
Uii − L∂A
∂ x2double∂ x1double
L
=−
,
∂ A2
Uii +Ui j − L ∂ A2
2

correct ,θ ∗ )where Uii = ∂ U1 (x∂ x2
12

correct∗Q

2

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

2

1

1

1

∂ xdouble∂ xfirst∂ xdouble∂ xdouble
We also have sgn ∂2A = sgn ∂2A and sgn ∂1A = sgn ∂2A L.
2
2
2

Proof. We first prove NEi0 ∈ (−1, 1). Note that x1 = x2 = x̂1 = x̂2 = xicorrect constitutes a steady state at A1 = A2 = a. Then we must have |NE10 NE20 | ≤ 1 at
xdouble = (x1correct , x2correct ); the proof is very similar to that of BR01 BR02 ≤ 1 in the
proof of Proposition 1, and hence omitted. (We only need to replace BR0i in the
proof if Proposition 1) with NEi0 .) Also, the regularity condition detM , 0 implies |NE10 NE20 | , 1. Accordingly, we have |NE10 NE20 | = |NEi0 | < 1, which implies
NEi0 ∈ (−1, 1).
Q

2i
Let Li = Qxθ1 ∂∂xiU∂ θ . When A1 = A2 = a, the multiplier effect on∂ x2double∂ A2 appear-

22 M33 M44, can be rewritten
ing in the proof of Proposition 3, (1 − BR14 BR41 ) M11 MdetMas

1 − BR14 BR41
(1 − BR014 BR041 )(1 − BR023 BR032 ) − (BR012 + BR014 BR042 )(BR021 + BR023 BR031 )U21
1
1 − U12U−L
U22 −L2
11







=
U21
U21 −L2 U12L1U12 −L1
L2
L2
U21 −L2
L1
1
1 − U12U−L
1
−
−
−
+
−
+
U22 −L2
U22 U11 −L1
U11
U11 U22 −L2
U22
U22 U11 −L1
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 )64

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
·
.
0
0
1 − BR1 BR2
Uii (Uii +Ui j − 2L)∂ xdouble

Similarly, when the game is symmetric, the base misspecification effect on ∂2A
2


M3A
M2A
appearing in the proof of Proposition 3, M22 + BR23 M33 , can be rewritten as
Ui j − L M2A.
1−
Uii − L M22

These results and Proposition 3 imply the first equation in the proposition. Also,
the second equation follows from
Ui j −L L
L−L(Uii − L) + L(Ui j − L)
BR012 + BR014 BR042 − Uii + Uii Uii −L
0
=
=
NE1 =
0
0
U
−L
U
i
j
i
j
1 − BR14 BR41
Uii (Uii − L) −Ui j (Ui j − L)
1 − U U −L
ii

ii

L(Uii −Ui j )L==.
(Uii +Ui j − L)(Uii −Ui j ) Uii +Ui j − L∂ xdouble

Next, we will show sgn ∂2A2∂ xfirst= sgn ∂2A . Recall that
∂ x2double (Uii − L)(Uii +Ui j − L)
Ui j − L ∂ x2first
=
1−
.
∂ A2
Uii (Uii +Ui j − 2L)
Uii − L∂A
Note that Uii = M11 < 0 and Uii − L = M33 < 0 under the regularity condition,

65

Also Uii +Ui j − L < 0 because if not and Uii +Ui j − L > 0,
L
<1
Uii +Ui j − L⇔ −(Uii +Ui j − L) < L < Uii +Ui j − L⇒ Uii +Ui j > 0,

NEi0 ∈ (−1, 1) ⇔ −1 <Uwhich contradicts with Uii < 0 and |BR0i | = | Uiiij | < 1. Similarly, Uii +Ui j − 2L < 0becauseL<1
Uii +Ui j − L⇔ Uii +Ui j − L < L < −(Uii +Ui j − L)
⇒ Uii +Ui j − 2L < 0,

NEi0 ∈ (−1, 1) ⇔ −1 <(Uii −L)(Uii +Ui j −L)
Uii (Uii +Ui j −2L) appearing in the above display is positive. Similarly,
U
Ui j −L
the term 1 − Uii −L is positive, because Uii < 0 and |BR0i | = | Uiiij | < 1 imply

So the term

Uii −Ui j < 0 ⇔ (Uii − L) − (Ui j − L) < 0 ⇔ 1 −Ui j − L>0
Uii − L∂ xdouble

where the last inequality uses Uii − L < 0. Hence we have sgn ∂2A2∂ xfirst= sgn ∂2A as∂ xdouble∂ xdouble
desired. Finally, sgn ∂1A = sgn ∂2 A L directly follows from NEi0 = Uii +ULi j −L
2and Uii +Ui j − L < 0.

Q.E.D.

Let Uii , Ui j , and L be as stated in Lemma 4. That is,∂ 2Ui= 2Qx + xi Qxx − c00 < 0,
∂ 2 xi∂ 2Ui
Ui j == Qx + xi Qxx = Qx + xi Qxx < 0,
∂ xi ∂ x j

Uii =

L=

Qxi ∂ 2UiQx=
(xi Qxθ + Qθ ) < 0.
Qθ ∂ xi ∂ θ
QθThen, Lemma 4 and the argument similar to the proof of Corollary 2 imply the
result.
Q.E.D.
66

B.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 (2021). For the case in which X is continuous, we need aminor modification of the proof. We first prove a preliminary lemma:
R

Lemma 5. 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

g(xn , y)Q(dy|xn ) −

Z

Y

Y

Z

Z

+

g(xn , y)Q(dy|x) −Yg(xn , y)Q(dy|x)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 5, the right-hand side converges to zero as σ n → σ . The
rest of the proof is exactly the same as in EPY.
Q.E.D.

B.9

Proof of Proposition 5

For the special case in which X is finite, Theorem 1 of Esponda, Pouzo, andYamamoto (2021) 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

67

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 6. 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.
Q.E.D.

B.10

Proof of Proposition 7

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 prove
upper hemi-continuity of Bε (σ ).
Lemma 7. 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
= 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
+ lim
(Ki,k (θi,k , σ )µ̂i,k (dθi,k ) −(Ki,k (θi,k , σ )µ̂i,k (dθi,k ) .

n→∞

Θi,k

Θi,k

68

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
(Ki,k (θi,k , σ n ) − Ki,k(σ n ))µ̂i,k(dθi,k ) ≤ ε n .

∗ (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.
Since xn ∈ Sε n (σ n ), there is µ̂ n ∈ Bε n (σ n ) with xn = ŝ(µ̂ n ). The sequencen(ε , 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 ).
Q.E.D.

B.11

Proof of Proposition 8

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 (17)).
Our Propositions 6 and 7 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
69

norm with the dual bounded-Lipschitz norm, it is exactly the same as the last stepof EPY.39
Q.E.D.

B.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 .
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 .
39 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 difficulty, 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 Esponda, Pouzo, and Yamamoto (2021)), is valid even in Banach
spaces.

70

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(30)
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

B.13

L1 L4 kσ − σ 0 kdβ = Lkσ − σ 0 k. Q.E.D.

Proof of Proposition 11

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 (17) startingfrom this σ(0). Let θ(t) = θ (σ(t)) for each t. The following lemma shows that
{θ(t)}t≥0 solves (18).
Lemma 8. Pick t ≥ 0 such that (17) holds. Then θ̇(t) exists and satisfies (18).
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))θ (σε ) − θ (σ0 ) θ (σ(t + ε)) − θ (σε )
=
+
.
ε
ε
ε
71

All 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 (18). Then θ (σ(t+ε))−θ
also has a limit θ̇(t) andεthis limit value satisfies (18).
(σ0 )Note first that limε→0 θ (σε )−θexists and is in the right-hand side of (18).
εIndeed, from (30),dθ (σβ )θ (σε ) − θ (σ0 )
=
ε→0
ε
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 )=L.
εTaking 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 8, 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 < η72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 (18).40 This implies the result we want.
Q.E.D.

40 Note that θ is absolutely continuous because σ is absolutely continuous and θ (σ ) is Lipschitz

continuous. Also from Lemma 8, θ satisfies (18) for almost all t.

73

C

Other Types of Misspecification and Applications

This section presents the analysis which was not covered in the main text: otherapplications, different types of higher-order misspecification, and an additional
convergence result. Section C.1 discusses tournaments. Section C.2 covers secondorder misspecification, where a player has an incorrect view about the opponent’sview, but she correctly understands the physical environment. Section C.3 covers
one-sided double misspecification, where only one player has double misspecification. Section C.4 presents a convergence result without identifiability.

C.1

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 ) = θ

x2 + a2.
x1 + a1 + x2 + a2

(31)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 > 1.41 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 overx2 +a2
41 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.

74

confidence 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 ) .
22{z
} |∂ A
M22 |{z}direct effect(32)

indirect learning effectThis is exactly the same as the base misspecification effect (14) in the team production, so the results in Section 4.2 continue to hold. For example, in Tullock+a1
x1 +a1
type tournament (31), we have Qx2 A = −2θ (x +ax1+x
3 < 0 and Qx2 θ = (x +a +x +A)2 >
1
1
2 +A)
1
1
2
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.

1 + M11x1Mx222

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 , 0.42 Intuitively,
42 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 .

75

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
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.

C.2

Second-Order Misspecification

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-order misspecification, in that she correctly understands the physical environment, but has
an incorrect view about the opponent’s view about a.43 Formally, we consider thefollowing 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.
43 As evidence from laboratory experiments, subjects often systematically mispredict other sub-jects’ preferences and actions (Van Boven, Dunning, and Loewenstein, 2000, for example). Ludwig and Nafziger (2011) report that most subjects in their experiments are not aware of or underestimate overconfidence of other subjects.

76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
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]
conditions ∂ E[U1∂(x,θ
= 0, ∂ E[U2∂(x,θ
= 0, and ∂ E[Û1∂(x,θ
= 0. At the end of
x1
x2
x̂1
1
1
1
∗
period one, players observe a public signal y = Q(x1 , x2 , a, θ ) + ε, and updates

77the posterior beliefs using Bayes’ rule. So each player’s belief in period two isµ11 (θ ) f (y − Q(x11 , x21 , a, θ )),1
1 1
Θ µ1 (θ̃ ) f (y − Q(x1 , x2 , a, θ̃ ))d θ̃

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 periodt ≥ 3, players play a Nash equilibrium given the belief profile µ t = (µ1t , µ2t , µ̂1t ),where µ t is computed by Bayes’ rule.
As in Section 3.3, 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 ∗ ),(33)

x2∗ ∈ arg max U2 (x2 , x1∗ , x̂1∗ , θ2 ),(34)

x̂1∗ ∈ arg max Û1 (x̂1 , x1∗ , x2∗ , θ̂1 ),(35)µ1∗ = 1θ ∗ ,(36)µ2∗ = 1θ2 s.t. Q(x̂1∗ , x2∗ , a, θ2 ) = Q(x1∗ , x2∗ , a, θ ∗ ),(37)

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

(38)

x1

x2

x̂1

The first three conditions (33), (34), and (35) are the incentive compatibility conditions, which require that each player maximize her own payoff given some beliefs.
The next three conditions (36), (37), and (38) 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.
78

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 (37), 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 (38), 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)

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
∂ 2Û1 (x, θ , A)
∂ 2Û1 (x, θ , A)
∂ θ̂1 (x, A)+
M3A :=∂ x̂1 ∂ A
∂ x̂1 ∂ θ∂A
θ =θ̂1 (x,A)θ =θ̂1 (x,A)
79

denote the impact of the hypothetical player’s bias A on her own marginal utility.
Now we are ready to state the result: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∗ , θ2 (x∗ , A),and 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.44
Proposition 12 (Steady State under Second-Order Misspecification). Let x∗ be a
regular steady state for some parameter A∗ .45 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(39)

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 the
44 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.
0

0

0

0

45 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 (37), (33), and (34) for some θ2 .
Also, given (x1 , x2 ), let BR3 (x1 , x2 ) denote the set of x̂1 satisfying (38) and (35) 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

80

equation measures this impact, holding the other players’ actions fixed. Note that
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 (34), (35), (37), (38)}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.46
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 influences000
46 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 afunction depending on the parameter x1 ), we indeed have ∂∂ xx2 =1

81

BR021 +BR023 BR031.
1−BR023 BR032

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 .47
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.
Proof of Proposition 12
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

(40)
(41)
(42)

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 (40)-(42). These action profiles are globally
optimal, because of the regularity conditions (i) and (ii). So this x∗ is a steady
47 In this argument, we implicitly use the fact that player 1’s optimal action is not affected by

the hypothetical player’s action.

82

state given the parameter A. The implicit function theorem also asserts that  ∂ x∗ 1
M11 M12 M130  ∂∂xA2∗ 



M
M
M
 21
22
23   ∂ A  = −  0  ,∂ x̂1
M31 M32 M33
M3A
∂A


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.
Q.E.D.

C.3

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.”83• 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 thatthe 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 ∗ ),(43)

x2∗ ∈ arg max U2 (x̂1∗ , x2 , A, θ2 ),(44)

x̂1∗ ∈ arg max Û1 (x̂1 , x2∗ , A, θ2 ),(45)µ1∗ = 1θ ∗ ,(46)µ2∗ = µ̂1∗ = 1θ2 s.t. Q(x̂1∗ , x2∗ , A, θ2 ) = Q(x1∗ , x2∗ , a, θ ∗ ).

(47)

x1

x2

x̂1

The first three conditions (43), (44), and (45) are incentive compatibility conditions, which require that each player maximizes her payoff given some beliefs.
The next two conditions (46) and (47) 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 assume84

that for each action profile x and parameter A, there is a unique state θ2 (x, A)
which solves (47). 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 toplayer j’s action as
Mi j
BR0i j := −Miiwhere for each i, j = 1, 2, 3 (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 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)
∂ θi (x, A)∂ 2Ui (x, θ , 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 5. 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 13 (Steady State under One-Sided Double Misspecification). Let x∗
be a regular steady state for some parameter A∗ . Then there is an open neighborhood of A∗ such that for any value A in this neighborhood, there is a regularsteady state x∗ which is continuous with respect to A, and we have




∂ x2∗
1
M2A M3A 0
1
=−
+
BR
,
∂A
M22 M33 23
1 − BR023 BR032
1 − BR012 NE20
∂ x1∗ ∂ x2∗=· BR012∂A∂A
where NE20 is defined by (39).
85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 12. 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 12. The second equation in the proposition simply states that therational player 1 best-responds to player 2’s action.
Proof of Proposition 13
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

(48)
(49)
(50)

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 (48)-(50). 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
0
M11 M12 M13  ∂∂xA∗  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
86

The rest of the proof is very similar to that of Proposition 12, and hence omitted.
Q.E.D.

C.4

Convergence Without Identifiability

There are some economic examples which do not satisfy the identifiability presented in the main text. For example, the model of overconfidence studied in
Heidhues, Kőszegi, and Strack (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)
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 14. 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,
i.e., there is a unique KL minimizer θi,k (x),and

∂ Ki,k (θ ,1x )< 0 for θi,k < θi,k (x),∂ θi,k∂ Ki,k (θ ,1x )> 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

87

Proof of Proposition 14
The result immediately follows from the following lemma:
Lemma 9. Suppose that all the assumptions stated in Proposition 14 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 10. 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 (2021).
∗ and any σ such that σ = σ for each ( j, l) ∈ I ∗∗Lemma 11. Pick any θi,k < θi,k
j,l
j˜,l˜
˜
˜
and ( j, l) ∈ I( j, l) and such that Ki,k (θi,k , σ ) < Ki,k (θ̃i,k , σ ) for all θ̃i,k < θi,k . Thenfor any solution σ ∈ Z(σ ) starting from this σ , we have θ i,k (σ(t)) > θ for allt > 0.

88

Now we will prove Lemma 9. 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

(51)

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 10 and Assumption (iv) of
0 for all θ̃ ≥ θ 0 . Since θ (S (·)) is continuProposition 14, θi,k (S0 (θ̃i,k )) > θi,k
i,k
i,k 0
i,k
0 + 2ε for all θ̃ ≥ θ 0 − ε. Thenous, for any small ε, we have θi,k (S0 (θ̃i,k )) > θi,ki,ki,k0 + 2ε forfrom Assumptions (iv) and (v) of Proposition 14, 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, (51) 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.

(52)

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 11 (we set θi,k = θ i,k (w(t))), we have θ i,k (σ(s)) > θ i,k (w(t)) >0 − ε for all s > 0. So in this solution σ, the share of the set of action profiles
θi,k

89

S

0 −ε S0 (σ̃ ) increases over time.
σ̃ :θ i,k (σ̃ )≥θi,kwe have

In particular, by the definition of T ,σ(s) [0 −ε
σ̃ :θ i,k (σ̃ )≥θi,k
S0 (σ̃ ) > 1 − εThen from (52), 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 (51) 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 11 (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,k
S0 (σ̃ ) > 1 − 2ε∀s ∈ [2T, 3T ]and thus∂ Ki,k (θi,k , w(t + s))0< 0 ∀θi,k ≤ θi,k+ ε∀t ∈ [T, 3T ].
∂ θi,kIterating 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,k
Q.E.D.

90

D

Convergence for Examples in Main Text

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

Cournot Duopoly: Equation 11.
Consider the Cournot model in Section 4.1. 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. Since the inverse demand function Q is linear inθ , the identifiability condition holds, and hence Proposition 2 ensures that the
belief converges almost surely under first-order misspecification. In particular,when the steady state is unique, the belief converges there almost surely regardlessof the initial prior.
Double misspecification. To prove convergence for small misspecification, it
suffices to check the conditions stated in Proposition 4.
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 ) 2x2
A2 − cx̂1 =− ,2(1 − θ2 ) 2
A1 − cx1x̂2 =− ,2(1 − θ1 ) 2
x1 =A2 − (1 − θ2 )(x̂1 + x2 ) = a − (1 − θ ∗ )(x1 + x2 ).
91

A1 −cA2 −cand x1 = x̂2 = 3(1−θ. Plugging
The first four equations imply x̂1 = x2 = 3(1−θ2)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 )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,
(1 − θ ∗ )2
∂ f2 (θ1 )
=−
.
∂ θ1
(3 − 4θ1 + θ ∗ )2This derivative is negative and larger than −1 if θ1 , θ ∗ ∈ [− 13 , 13 ]. Hence, if A1and 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 is omitted.

Cournot Duopoly: Equation 12.
Suppose the Cournot model with c(xi ) = cxi where c ≥ 0, a < 1, Θ = [θ , θ ] wherec < θ < θ , andQ(x1 + x2 , a, θ ) = θ − (1 − a)(x1 + x2 ).
First-order misspecification. Note that the identifiability condition is satisfied,and hence Proposition 2 implies that the belief converges almost surely to a steadystate.
Double misspecification. To prove convergence for small misspecification, it
suffices to check the conditions stated in Proposition 4.

92

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:
θ1 − cx̂2− ,2(1 − A1 ) 2x̂1
θ2 − c− ,x2 =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 − a)c(1 − a)(θ1 − c)
1 − A2∗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 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).

Team Production: Example 1.
Consider the team production model in Section 4.2. Suppose that Q is given byQ(x1 , x2 , a, θ ) = θ (x1 + x2 + kx1 x2 + a)
and the cost function is c(xi ) = 2c xi2 where c > 0. We will show that for each
misspecification, the belief converges to a steady state when c = A1 = A2 = a = 2,k ∈ [−4, 4], and Θ = [0.1, 0.3].
93

First-order misspecification. Since the inverse demand function Q is linear inθ , the identifiability condition holds, and hence Proposition 2 ensures that the
belief converges almost surely under first-order misspecification.
Double misspecification. To prove convergence for small misspecification, it
suffices to check the conditions stated in Proposition 4.
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:
θ1 (1 + kx̂2 ),c
θ2 (1 + kx̂1 )x2 =,c
θ2 (1 + kx2 )
x̂1 =
,
c
θ1 (1 + kx1 )
x̂2 =
,
c
θ2 (x̂1 + x2 + kx̂1 x2 + A2 ) = θ ∗ (x1 + x2 + kx1 x2 + a).
x1 =

θ2
θ1
The first four equations imply x̂1 = x2 = c−θand x1 = x̂2 = c−θ. Plugging
2k
1k
them into the last equation,




θ22
θ1
2θ2
θ2
θ1
θ2
∗
θ2
+k
+
+k
+ a = 0,+ A2 −θc − θ2 k(c − θ2 k)2c − θ1 k c − θ2 kc − θ1 k c − θ2 kwhich is equivalent to
G := 2θ22 +kθ23θ ∗ θ1 c− θ ∗ θ2 + (c − θ2 k)(A2 θ2 − aθ ∗ ) −= 0.
c − θ2 kc − θ1 k

Note that G is strictly decreasing in θ1 , and is also strictly increasing in θ2∂G
if ∂ θ > 0, and this holds when θ is sufficiently close to zero (see ∂∂θG in the22
following). If it holds, then for any θ1 , f2 (θ1 ) is a singleton. Also,∗ 2θ c− (c−θk)2∂G

d f2 (θ1 )∂θ= − ∂ G1 = −
3kθ22
dθ1
4θ +
+
∂θ
2

2

c−θ2 k1

k2 θ23− θ ∗ − 2θ2 A2 k + A2 c + akθ ∗(c−θ2 k)2

94.

Because c = A1 = A2 = a = 2, ∂∂θG = 4+4θ2 (1−k)+θ ∗ (2k −1)+22kθ22 (3−kθ2 )>(2−kθ2 )2

2 (θ1 )
0 for any k ∈ [−4, 4]. Also, because d fdθ
is increasing in θ ∗ and hence is maxi1

2 (θ1 )
mized at θ ∗ = θ̄ , given Θ = [0.1, 0.3], we have | d fdθ
| < 1 for any k ∈ [−4, 4].
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.

Team Production: Example 2.
Consider the team production model in Section 4.2. Suppose that Q is given byQ(x1 , x2 , a, θ ) = θ (ax1 + x2 + kx1 x2 + s)
and the cost function is c(xi ) = 2c xi2 where c > 0. We will show that for each
misspecification, the belief converges to a steady state when A1 = A2 = a = 1,c = s = 2, k ∈ [−4, 4], and Θ = [0.1, 0.3].
First-order misspecification. Since the inverse demand function Q is linear inθ , the identifiability condition holds, and hence Proposition 2 ensures that the
belief converges almost surely under first-order misspecification.
Double misspecification. To prove convergence for small misspecification, it
suffices to check the conditions stated in Proposition 4.
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:
θ1 (A1 + kx̂2 ),c
θ2 (1 + kx̂1 )x2 =,c
θ2 (A2 + kx2 )
x̂1 =
,
c
θ1 (1 + kx1 )
x̂2 =
,
c
θ2 (A2 x̂1 + x2 + kx̂1 x2 + s) = θ ∗ (ax1 + x2 + kx1 x2 + s).
x1 =

95

θ1 (c+θ1 kA1 )1 c+θ1 k)
2 c+θ2 k)
The first four equations imply x1 = θ1c(A2 −θ, x̂1 = θ2c(A2 −θ
2 k2 , x̂2 =2 k2 , x2 =c2 −θ 2 k2
1

θ2 (c+θ2 kA2 ). Plugging them into the last equation,c2 −θ22 k2

12θ2 (c + A22 c + 2A2 kθ2 )θ22 (A2 c + θ2 k)(c + θ2 kA2 )G :=θ2+k+sc2 − θ22 k2(c2 − θ22 k2 )2
θ1 (A1 c + θ1 k) θ2 (c + θ2 kA2 )θ1 (A1 c + θ1 k) θ2 (c + θ2 kA2 )∗++k 2+ s = 0.
−θ a 2c − θ12 k2
c2 − θ22 k2
c − θ12 k2
c2 − θ22 k2


If A1 = A2 = a = 1, then it becomes




θ22
2θ2
θ2
θ1
θ2
θ1
∗
θ2
+k
+
+k
+ s = 0,+ s −θ
c − θ2 k(c − θ2 k)2c − θ1 k c − θ2 kc − θ1 k c − θ2 kwhich is reduced to
G := 2θ22 +kθ23θ ∗ θ1 c− θ ∗ θ2 + s(c − θ2 k)(θ2 − θ ∗ ) −= 0.
c − θ2 kc − θ1 k

As in the previous example, when s = c = 2 and Θ = [0.1, 0.3], both ∂∂θG > 0 and
2

2 (θ1 )
| < 1 hold for any k ∈ [−4, 4].
| d fdθ1

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.

96