OSIPP Discussion Paper: DP-2023-E-002-Rev

Higher-Order Misspecification and Equilibrium Stability
September 4, 2023
Takeshi Murooka*
Osaka School of International Public Policy (OSIPP), Osaka University

Yuichi Yamamoto†
Institute of Economic Research, Hitotsubashi University

Abstract: This paper considers a Bayesian learning problem where strategic players jointly learn an
unknown economic state, and show that one's higher-order misspecification (i.e., one's misspecification
about the opponent's misspecification) can have a significant impact on the equilibrium outcome. We
consider a simple environmental problem where players' production, as well as an unknown state, affects
the quality of the environment. Crucially, we assume that one of the players is unrealistically optimistic
about the quality of the environment. When this optimism is common knowledge, the equilibriumoutcome is continuous in the amount of optimism, and hence small optimism leads to approximately
correct learning of the state. In contrast, when the optimism is not common knowledge and each player
is unaware of the opponent having a different view about the world, the equilibrium outcome is
discontinuous, and even vanishingly small optimism leads to completely incorrect learning. We thenanalyze a general Bayesian learning model and discuss when such discontinuity arises.

JEL Codes: C73, D83, D90, D91
Keywords: model misspecification, learning, unawareness, convergence, stability, inferential naivety,overconfidence*

Email: murooka@osipp.osaka-u.ac.jp†

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

Higher-Order Misspeciﬁcation and Equilibrium Stability*
Takeshi Murooka

Yuichi Yamamoto

Osaka University

Hitotsubashi University

September 4, 2023

Abstract
This paper considers a Bayesian learning problem where strategic players jointly learn
an unknown economic state θ , and show that one’s higher-order misspeciﬁcation (i.e., one’s
misspeciﬁcation about the opponent’s misspeciﬁcation) can have a signiﬁcant impact on theequilibrium outcome. We consider a simple environmental problem where players’ production,
as well as an unknown state θ , affects the quality of the environment. Crucially, we assume
that one of the players is unrealistically optimistic about the quality of the environment. When
this optimism is common knowledge, the equilibrium outcome is continuous in the amount of
optimism, and hence small optimism leads to approximately correct learning of the state θ .
In contrast, when the optimism is not common knowledge and each player is unaware of the
opponent having a different view about the world, the equilibrium outcome is discontinuous,and even vanishingly small optimism leads to completely incorrect learning. We then analyze
a general Bayesian learning model and discuss when such discontinuity arises.
JEL CODE: C73, D83, D90, D91
Keywords: model misspeciﬁcation, learning, unawareness, convergence, stability, inferential
naivety, overconﬁdence* We thank Matthias Fahn, Drew Fudenberg, Kevin He, Paul Heidhues, 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, JP22K13365). 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 thatthe 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 a single-agent
setup or a multi-agent setup in which the agents’ misspeciﬁcations are common knowledge (e.g.,
Esponda and Pouzo, 2016; Heidhues, Kőszegi, and Strack, 2018; Ba and Gindin, 2023). However,this common knowledge assumption leaves out many potential applications, as it does not allow
players’ higher-order 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 misspeciﬁedview about the opponent’s view about the world.
This paper shows that such higher-order misspeciﬁcation has a signiﬁcant impact on players’
play. In particular, we ﬁnd that even a negligible amount of misspeciﬁcation can drastically change
the equilibrium outcome. To illustrate this, we consider a simple model of an environmental problem with two players. There are inﬁnitely many periods, and the players’ production, as well as
an unknown state θ , inﬂuences the quality of the environment each period. We assume that one ofthe players (say, player 2) is misspeciﬁed and is unrealistically optimistic about the quality of theenvironment. Players are myopic, and actions are unobservable.
In Section 2, we consider a benchmark case in which the players do not have higher-ordermisspeciﬁcation, and each player correctly understands the opponent’s view about the world. Thatis, player 1 knows that she is more pessimistic than player 2, while player 2 knows that she ismore optimistic than player 1. (Also, these beliefs are common knowledge.) As we show in
our companion paper (Murooka and Yamamoto, 2023), in this case, the players’ beliefs about
the unknown state θ eventually converge to a steady state almost surely, regardless of the initialcommon prior. This steady-state outcome is continuous with respect to the level of player 2’s bias
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

(i.e., optimism). So when player 2’s bias is small, its impact on the long-run outcome is small, i.e.,
after a long time, the players have approximately correct beliefs about θ .
Then Section 3 considers the model with higher-order misspeciﬁcation, where the players areunaware of the opponent having a different view about the world. Speciﬁcally, player 2 is optimistic about the environment, and on top of that, she naively thinks that the opponent shares thesame view with her. (In reality, player 1 isn’t optimistic.) Similarly, player 1 has the unbiased view
about the environment, and naively thinks that player 2 also has the unbiased view.
As in the benchmark case, when player 2’s optimism is small, there is a steady state in which
the players approximately learn the true state θ ∗ . However, it turns out that this steady state is
unstable, and the players’ beliefs converge there with zero probability; indeed, the players’ beliefs
tend to be polarized over time and converge to a boundary point almost surely. This result showsthat the players’ higher-order misspeciﬁcation has a signiﬁcant impact on the long-run equilibriumoutcome. With small optimism of player 2, the players approximately learn the true state if the
optimism is common knowledge, while their beliefs converge to boundary points if the players areunaware of the opponent having a different view about the world.
Our result also shows discontinuity of the equilibrium outcome with respect to the information
structure. Indeed, the players’ long-run beliefs are concentrated on the true state in the case of no
misspeciﬁcation, but these beliefs jump to boundary points once player 2 has (even vanishingly
small) optimism. One may think that this discontinuity contradicts with various continuity resultsin the literature on incomplete information; e.g., Chen, Di Tillio, Faingold, and Xiong (2017) showthat a small change in information structure can have only a small impact on equilibrium in any
normal-form game. We will explain how to reconcile this in Section 3.3.
In our model, instability of the steady state is closely related to the inferential naivety arisingfrom the players’ higher-order misspeciﬁcation. Since player 1 is unaware of player 2’s optimism,
she believes that player 2 will maximize payoffs and update the belief given the unbiased view
about the environment (but in reality, player 2 does so given the optimistic view about the world). It
turns out that this inferential naivety (about the opponent’s action and belief) is reinforced throughlearning in our model; a small gap between one’s belief about the opponent’s belief (about the3state) and the opponent’s actual belief can become arbitrarily large after a long time. As we will
explain in Section 3.3, this is the source of the instability of the steady state.
Recent work by Frick, Iijima, and Ishii (2020) shows that a small misspeciﬁcation can lead to
a complete breakdown of correct learning, in the context of social learning. In their model, agentsobserve the opponents’ actions every period and learn a payoff-relevant unknown state from it.
The agents are misspeciﬁed in that they have incorrect views about how the opponents interpret
information (and hence they have incorrect views about the opponents’ behavior). They show thata steady state is discontinuous in the amount of misspeciﬁcation, and in particular, even with avanishingly small amount of misspeciﬁcation, in the unique steady state, the agents have a pointmass belief on a state which is far away from the true state.
Frick, Iijima, and Ishii (2020) also argue that their result relies on the assumption that the agents
have only a limited amount of information about the state, in that the agents observe a noisy signal
about the state only once. (So the agents learn mostly from the opponents’ actions, and in this sense
it is a model of social learning.) Indeed, they show that if the agents observe signals in every periodof the inﬁnite-horizon model, then the result is overturned and steady states are continuous in theamount of misspeciﬁcation. So one may naturally expect that a small misspeciﬁcation can destroy
correct learning in social learning models, but not in models where agents receive feedbacks (i.e.,
signals) repeatedly. Note that repeated feedbacks are common in many economic applications;e.g., if agents observe their own payoffs every period, then it is a model of repeated feedbacks, as
payoffs are informative about the state in general.2
Our result shows that such a conjecture is not true, and a small misspeciﬁcation can still have ahuge impact on the learning outcome even in a model of repeated feedbacks. In our model, when
the agents become slightly misspeciﬁed, the probability of the belief converging to the steady state
suddenly drops from one to zero. So even though a small misspeciﬁcation has only a negligible impact on the steady state, it leads to a complete breakdown of correct learning. This is a
new mechanism which causes discontinuity of the learning outcome, and in this sense our work
complements Frick, Iijima, and Ishii (2020).

2

Frick, Iijima, and Ishii (2020) assume that the agents do not observe payoffs.

4

In Section 3.4, we extend this non-convergence result to a general setup, and study when asmall amount of misspeciﬁcation leads to a complete breakdown of correct learning. We ﬁndthat learning tends to be fragile when the state θ and one’s belief about the state θ have opposite
impacts on the outcome. As we will explain, this condition can be satisﬁed in a wide range of
economic applications, such as team production and Cournot competition.
To prove the fragility of correct learning, we extend the non-convergence result of Pemantle(1990), which shows that if a steady state of a stochastic process is unstable in some sense, then theprocess converges there with zero probability. His theorem does not apply to our setup directly, forthree reasons. First, we assume that players observe public signals and update their beliefs, so thestochastic shocks on these beliefs are perfectly correlated. Assumption (iii) in Theorem 1 of Pemantle does not allow such a correlation. Second, we consider a Gaussian noise, which violates thebounded support assumption of Pemantle. Third, the drift term of our stochastic process involves a
perturbation term, which is not considered by Pemantle.3 We show that these features do not cause
a serious problem, and the result of Pemantle remains valid in our environment. We believe that
this result can be useful for future works which consider a stochastic process similar to ours (inparticular, problems in which stochastic shocks are Gaussian or shocks on multiple variables are
perfectly correlated).

2 First-Order Misspeciﬁcation
2.1 Setup
As a benchmark, we ﬁrst consider the case in which higher-order misspeciﬁcation does not exist,
i.e., each player correctly understands the opponent’s view about the world. There are two players
i = 1, 2 and inﬁnitely many periods t = 1, 2, · · · . At the beginning of the game, an unobservableeconomic state θ ∗ is drawn from a closed interval Θ = [θ , θ ], according to a common prior distri3

Benaı̈m (1999) also prove a similar non-convergence theorem, but his result does not apply to our model forthe same reason. Benaı̈m and Faure (2012) prove a non-convergence result which allows a Gaussian noise, but they
assume that the process is cooperative. Furthermore, they make various assumptions on the noise term, which are notsatisﬁed in our model (e.g., i.i.d. noise, positive-deﬁnite assumption which rules out perfect correlation of a noise).

5bution µ ∈ △Θ. We assume that µ has a continuous density µ ′ with full support. In each period t,each player i has a belief µit ∈ △Θ about the state θ , and chooses an action xi from a closed interval
Xi = [0, xi ]. Player i’s action xi is not observable by the opponent j , i. Given an action proﬁlex = (x1 , x2 ), the players observe a noisy public signal y = Q(x1 , x2 , a, θ ∗ ) + ε , where a ∈ R is aﬁxed parameter and ε is a random noise which follows the standard normal distribution N(0, 1).
Player i’s stage-game payoff is ui (xi , y). We assume that both Q and ui are twice continuously
differentiable.
Crucially, we assume that one of the players (player 2) incorrectly believes that the true parameter is A , a, while the other player is unbiased and knows the parameter a. These ﬁrst-orderbeliefs (about the parameter a) are common knowledge, e.g., player 1 knows that player 2 believes
that the true parameter is A , a. We call it ﬁrst-order misspeciﬁcation, because player 2 has anincorrect ﬁrst-order belief about the parameter a.
Player 1’s subjective expected stage-game payoff given an action proﬁle x and a 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 ) insteadof U2 (x, A, θ ) when it does not cause a confusion.
We assume that players play a static Nash equilibrium every period. This essentially meansthat in our model, (i) players are myopic, and (ii) they predict the opponent’s play correctly andbest-respond to it. Condition (i) shuts down the repeated-game effect, so that a result similar to
the folk theorem (which is not of our interest) does not arise.4 Condition (ii) implies that players
recognize that the opponent also learns the state and changes the action as time goes. This setup
4 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, and with an additional assumption, we can show 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 forforward-looking players.

6

is different from the one in the literature on learning in games (e.g., Fudenberg and Kreps, 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 .5
In period one, players play a Nash equilibrium (x11 , x21 ). Assuming an interior solution, it isθ )|µ ]
= 0 for each i. At the end ofan action proﬁle which solves the ﬁrst-order condition ∂ E[Ui∂(x,xi

period one, players observe a public signal y1 , and update the posterior beliefs using Bayes’ rule.
Assuming that no one has deviated in period one, each player i’s posterior belief µ12 , µ22 in period
two isµ11 (θ ) f (y − Q(x1 , a, θ ))11
Θ µ1 (θ̃ ) f (y − Q(x , a, θ̃ ))d θ̃

µ12 (θ ) = Randµ21 (θ ) f (y − Q(x1 , A, θ )),1
1
Θ µ2 (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 function of the noise
term ε . Note that player 2’s posterior µ22 differs from player 1’s posterior µ12 , as she incorrectlybelieves that the mean output is Q(x1 , A, θ ) rather than Q(x1 , a, θ ). Because the players’ information structure about the parameter a is common knowledge, these posteriors are common knowledge among players. So in period two, players play a Nash equilibrium given the belief proﬁleµ 2 = (µ12 , µ22 ), which solves∂ E[Ui (x,θ )|µi2 ]= 0 for each i.
∂ xi

Likewise, in any subsequent period t,

players play a Nash equilibrium given the belief proﬁle µ t = (µ1t , µ2t ), where µ t is computed by
Bayes’ rule.
A steady state in this dynamic learning model is a pair (x1∗ , x2∗ , µ1∗ , µ2∗ ) of an action proﬁle anda 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)

x1

x2

µ2∗ = 1θ2 s.t. θ2 ∈ arg min |Q(x∗ , A, θ ) − Q(x∗ , a, θ ∗ )|.
θ ∈Θ

(4)

5 Condition (ii) is inessential if the game is dominance solvable. Note that all the examples studied in this paper

are actually dominance solvable.

7

The ﬁrst two conditions (1) and (2) are incentive compatibility, which requires that each player
maximizes her payoff given some beliefs. The other two conditions require that these beliefs
satisfy consistency: (3) asserts that the unbiased player 1 correctly learns the true state θ ∗ in a
steady state. (4) requires that player 2’s belief is concentrated on a state θ2 which best explains thedata, in that with this state θ2 , player 2’s subjective view about the mean output is closest to theactual mean. This condition must be satisﬁed in a steady state; otherwise, player 2 is “surprised”
by observed signals and her belief will move to the state which better explains the data. In many
economic applications (including the environmental problem example in the next subsection), theconsistency condition (4) reduces toQ(x∗ , A, θ2 ) = Q(x∗ , a, θ ∗ ),(5)i.e., the subjective mean output exactly matches the true mean.
As we show in our companion paper (Murooka and Yamamoto, 2023), if there is a uniquesteady state and if a mild condition called identiﬁability is satisﬁed, then players’ actions andbeliefs converge to this steady state almost surely.6 So the steady state can be thought of as a“long-run outcome” of the dynamic learning model.
Note that in most economic applications (including all the examples studied in this paper), thesteady-state outcome is continuous with respect to the parameter A. So a small misspeciﬁcation
leads to approximately correct learning of the state θ , i.e., player 2’s steady-state belief θ2 is close
to the true state θ ∗ when the parameter A is close to a.

2.2 Application: Optimism in Environmental Problems
To see how the steady state looks like, we will consider a model of environmental problems,
which includes air pollution, deforestation, and ﬁshery as special cases.7 Every period, each player
i = 1, 2 chooses a production level xi ∈ [0, 1], which has a negative impact on the quality of the
The identiﬁability condition requires that given any action frequency σ ∈ △X, there is a unique state θ which
minimizes the Kullback-Leibler divergence between the true signal distribution and the subjective signal distribution.
See Murooka and Yamamoto (2023).
7 More generally, this is a model of production with negative externalities.
68

environment. As in Chapter 24 of Varian (1992), we assume that the quality of the environment isgiven by the formulay = Q(x, a, θ ) + ε = a − θ (x1 + x2 ) + ε ,(6)where a ∈ R is a ﬁxed parameter, θ ∈ Θ = [0.7, 0.9] is an unknown fundamental, and ε is a noise
term which follows the standard normal distribution N(0, 1). Player i’s payoff is y + xi − c(xi ),where xi is a private beneﬁt from production and c(xi ) = 12 x2 is a production cost. Since weassume θ ∈ (0, 1), regardless of players’ beliefs, the Nash equilibrium in the one-shot game is aninterior point. In what follows, we will assume that the true state is θ ∗ = 0.8.
We assume that one of the players is unrealistically optimistic about the quality of the environment. Formally, we assume that player 2 incorrectly believes that the true parameter is A > a.
Such optimism is commonly observed in various environmental problems, see Dechezleprêtre etal. (2022) and references therein.
This example satisﬁes the identiﬁability condition of Murooka and Yamamoto (2023), andhence regardless of the initial prior, the players’ actions and beliefs almost surely converge to thesteady state, which is characterized by (1)-(4). We will consider how player 2’s bias inﬂuences thissteady-state outcome.
Let Qz denote the derivative of Q with respect to a variable z. Since Qa > 0 and Qθ < 0,Condition (5) implies that player 2’s steady-state belief is θ2 > θ ∗ , i.e., the optimistic player overestimates the state in the long run. Intuitively, player 2 is disappointed by observed environmental
quality being worse than the anticipation, and becomes pessimistic about the state θ as time goes.
This in turn implies that player 2 overestimates the marginal social cost Qxi of the production.8
Thus, her steady-state action x2 is lower than in the correctly-speciﬁed case. On the other hand,the unbiased player 1’s production is exactly the same as in the correctly-speciﬁed case, because
player 1’s optimal production is independent of the opponent’s action. Accordingly, player 2’s
payoff is lower than in the correctly-speciﬁed case, while player 1’s payoff is higher than that. So
player 2’s bias is detrimental for herself, but improves the opponent’s payoff. Also, simple algebra
8 Attari et al. (2010) report that people overestimate certain energy-saving activities (e.g., driving less to savegasoline), but their answers also have substantial variation.

9

shows that the latter effect is larger than the former effect, so player 2’s bias improves the social
surplus.

3

Higher-Order Misspeciﬁcation

The benchmark model in the previous section assumed that players correctly understand the opponent’s view about the world. Now we consider the case in which players have higher-order misspeciﬁcation, in that they have a biased view about the opponent’s view about the world (secondorder misspeciﬁcation), a biased view about the opponent’s second-order misspeciﬁcation, and so
on.9
In what follows, we will focus on a special form of higher-order misspeciﬁcation: We willassume that each player has a biased view about the world, and on top of that, she naively thinksthat 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 theworld (ﬁrst-order misspeciﬁcation) and a biased view about the opponent’s view about the world(second-order misspeciﬁcation). Of course, we can think of various other forms of higher-ordermisspeciﬁcation. In our companion paper Murooka and Yamamoto (2023), we present a moregeneral model of higher-order misspeciﬁcation.10

3.1 Setup: Double Misspeciﬁcation
Our model is the same as the one studied in Section 2.1, except the information structure; now wewill assume that each player i (incorrectly) believes that it is common knowledge that the signal 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.

9

As evidence from laboratory experiments, subjects often systematically mispredict other subjects’ preferences
and actions (e.g., Van Boven, Dunning, and Loewenstein, 2000). Ludwig and Nafziger (2011) report that most subjects
in their experiments are not aware of or underestimate overconﬁdence of other subjects.
10 In static games with strategic complementarity/substitutability, recent work by McGee (2023) analyzes how
certain higher-order misspeciﬁcation affects equilibrium outcomes.

10

A critical difference from the ﬁrst-order misspeciﬁcation is that players have inferential naivety
and make incorrect predictions about the opponent’s play.11 Indeed, while player i believes that the
opponent (player j) maximizes the payoff conditional on the parameter Ai , the opponent maximizes
the payoff conditional on the parameter A j in reality. 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 is useful to consider
two hypothetical players i = 1, 2. Hypothetical player i is player j who thinks that it is common
knowledge that the true technology is A j . Intuitively, player j thinks that hypothetical player i is her
opponent, and hence each period, player j chooses a Nash equilibrium action against hypotheticalplayer i.
Let x̂i and µ̂i denote hypothetical player i’s action and belief, and let x = (x1 , x2 , x̂1 , x̂2 ) denotean action proﬁle in the four-player game. Player i’s expected stage-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 hypothetical player. 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 equilibriumθ )|µ ]=
(x11 , x21 , x̂11 , x̂21 ), which (assuming interior solutions) satisﬁes the ﬁrst-order conditions ∂ E[Ui∂(x,xiθ )|µ ]
0 and ∂ E[Ûi∂(x,= 0. At the end of period one, players observe a public signal y1 = Q(x11 , x21 , a, θ ∗ )+x̂i

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
µ
θ̃
)
f
(y
−
Q(x
,
x̂
,
A
,
θ̃
))d
θ̃
(
i
Θ i
i −i

µi2 (θ ) = R

1 , A , θ ))µ̂i1 (θ ) f (y − Q(x̂i1 , x−i−i2µ̂i (θ ) = R 1.
11
Θ µ̂i (θ̃ ) f (y − Q(x̂i , x−i , A−i , θ̃ ))d θ̃

11 See Eyster (2019) for a review of the literature.

11

As is clear from this formula, player i’s posterior belief is biased in two ways: She updates the
1 about the opbelief conditional on the wrong parameter Ai , and on the wrong prediction x̂−iponent’s play. Then in period two, players play a Nash equilibrium given this belief proﬁleµ 2 = (µ12 , µ22 , µ̂12 , µ̂22 ).12 Likewise, in any subsequent period t, players play a Nash equilibrium
given the posterior beliefs computed by Bayes’ rule.
Given an action proﬁle x = (x1 , x2 , x̂1 , x̂2 ), let θi (x, Ai ) denote player i’s long-run belief when
the same action x is chosen every period. That is, let θi (x, Ai ) be a state θ which solves
min |Q(xi , x̂ j , Ai , θ ) − Q(x1 , x2 , a, θ ∗ )|.
θ ∈ΘNote that the term Q(xi , x̂ j , Ai , θ ) is player i’s subjective mean output given θ , while the termQ(x1 , x2 , a, θ ∗ ) is the true mean. So θi (x, Ai ) deﬁned above is the state which best explains thedata, given an action proﬁle x. 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)

xi

∗
x̂i∗ ∈ arg max Ûi (x̂i , x−i, A−i , θ−i )
x̂i

∀i,

(8)

µ1∗ = µ̂2∗ = 1θ1 (x∗ ,A1 ) ,(9)µ2∗ = µ̂1∗ = 1θ2 (x∗ ,A2 ) .

(10)

The ﬁrst two conditions are the incentive-compatibility conditions, which require that 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 statewith which best explains the data.

12 Since y is public, player 1 correctly predicts hypothetical player 2’s posterior belief µ̂ 2 , and similarly, hypothetical2
player 2 correctly predicts player 1’s posterior belief µ12 . So they will indeed play a Nash equilibrium given these

beliefs.

12

3.2 Environmental Problem under Double Misspeciﬁcation
Consider the environmental problem discussed in Section 2.2, but assume now that players are notaware of the fact that the opponent has a different view about the world. Speciﬁcally, we consider
double misspeciﬁcation with parameters A1 = a and A2 ≥ a. Note that in this setup, player 1 isalso misspeciﬁed; she has a correct view about the parameter a, but she is unaware of player 2’s
optimism and naively thinks that player 2 also knows a. To simplify the exposition, assume thatthe initial prior is a uniform distribution on Θ = [0.7, 0.9].
The steady state in this setup is characterized by the conditions (7)-(10). For the special casein which A1 = A2 = a (i.e., the case with no misspeciﬁcation), there are three steady states: Oneof the steady state is an interior point, in which both players learn the true state (θ1 = θ2 = θ ∗ ),and choose the Nash equilibrium for this state θ ∗ . The remaining two steady states are boundary
points. In these steady states, players’ beliefs are (θ1 , θ2 ) = (θ , θ ) or (θ1 , θ2 ) = (θ , θ ), and theychoose a Nash equilibrium given these beliefs.13 These boundary steady states are not essential, inthat they do not arise as a long-run outcome. Indeed, because there is no misspeciﬁcation, startingfrom a common prior µ , players learn the true state with probability one, i.e., the beliefs converge
to the interior steady state almost surely when A1 = A2 = a.
Now, consider the case in which player 2 is slightly optimistic (i.e., A2 is a bit larger than a).
By the continuity, there is an interior steady state in which players’ beliefs are close to (θ ∗ , θ ∗ ).
Let m∗ = (m∗1 , m∗2 ) denote this steady-state belief. Also, the boundary points (θ1 , θ2 ) = (θ , θ ) and(θ1 , θ2 ) = (θ , θ ) are still steady states in this case.
One may expect that the boundary states are still inessential as in the case of no misspeciﬁcation, and that the the beliefs converge to the interior steady state m∗ , just as in the case of no
misspeciﬁcation. Proposition 1 shows that such a conjecture is incorrect, and the beliefs converge
to the boundary points when A2 > a.

To see that (θ1 , θ2 ) = (θ , θ ) is a steady-state belief, note that ∂∂xi ∂Qθ < 0, so we have x1 > x̂1 in this steady state.
This means that player 2 underestimates the opponent’s production, and thus ﬁnds that the quality of the environment
is worse than the anticipation. This makes player 2 more pessimistic, but her current belief θ already hits the upperbound of the set Θ, so her belief stays there. Similarly, player 1’s belief stays at θ , which imply that (θ1 , θ2 ) = (θ , θ )is indeed a steady-state belief. For the same reason, (θ1 , θ2 ) = (θ , θ ) is a steady-state belief.
2

13

13

Proposition 1. (i) Suppose that A2 = a. Then almost surely, players eventually learn the true stateθ ∗ , i.e.,
Pr lim (µ1t , µ2t ) = (1θ ∗ , 1θ ∗ ) = 1.
t→∞(ii) There is A2 > a such that for any A2 ∈ (a, A2 ), players’ posterior beliefs µ t = (µ1t , µ2t )converge to the interior steady state (1m∗1 , 1m∗2 ) with zero probability. Indeed, almost surely, thebeliefs converge to the boundary points, i.e.,

Pr lim µ ∈ {(1θ , 1θ ), (1θ , 1θ )} = 1.
tt→∞

Proposition 1 shows that unawareness about the opponent’s bias can have a huge impact on
the equilibrium outcome. Recall that in the case of ﬁrst-order misspeciﬁcation, small optimism ofplayer 2 has only a marginal impact on the long-run outcome, and players approximately learn thetrue state. In contrast, when players are unaware of the opponent having a different view about
the world, the long-run outcome becomes discontinuous at A2 = a, and even vanishingly small
optimism completely changes the learning outcome.
In the literature of incomplete-information games, it is well-known that an equilibrium ina normal-form game is continuous with respect to the information structure; Chen, Di Tillio,
Faingold, and Xiong (2017) show that a small perturbation of one’s belief hierarchy (a belief
about an economic state, a belief about the opponent’s belief about the state, and so on) has only
a marginal impact on the equilibrium. Our Proposition 1 above does not contradict with this result. Indeed, in our model, the equilibrium strategy in the inﬁnite-horizon game, which maps one’s
belief µi to an action, is continuous in the parameter A2 , so a small perturbation of one’s belief
hierarchy has a negligible impact on the equilibrium strategy.14 In this sense, the main result ofChen, Di Tillio, Faingold, and Xiong (2017) still holds in our model. However, this need not imply
that the resulting equilibrium outcome is continuous in the parameter A2 , and Proposition 1 showsthat our model is one of the cases in which such discontinuity arises.

In our model, the belief hierarchies induced by A2 and A′2 , A2 are close in the uniform-weak topology of Chen,
Di Tillio, Faingold, and Xiong (2017) if A2 and A′2 are close.
14

14

3.3 Proof Sketch of Proposition 1
Now we will describe an outline of the proof of Proposition 1 (ii). Given an action proﬁle x =(x1 , x2 , x̂1 , x̂2 ), player i believes that the signal y is generated by the formulay = Ai − θ (xi + x̂−i ) + ε
which can be rewritten asεAi − y.
=xi + x̂−i xi + x̂−iHence, if she observes a signal y, then the likelihood of the state θ ∈ Θ is the truncated normal dis2 15
1tribution on the set Θ, induced by a normal distribution with mean xAi +i −yandvariancex̂−ixi +x̂−i .
θ−For shorthand notation, let Ii (x) denote the inverse of this variance, i.e.,
Ii (x) = (xi + x̂−i )2 .
Intuitively, this Ii (x) measures the informativeness of the signal y for player i given an action proﬁlex; high Ii implies low variance, meaning that the signal is more informative.
Since the initial prior is uniform and the likelihood induced by signals is truncated normal, eachplayer’s posterior belief is also a truncated normal distribution. Let Ñ(m, σ 2 ) denote the truncatednormal distribution induced by the normal distribution N(m, σ 2 ). Then player i’s posterior at the
beginning of period t + 1 is the truncated normal distribution Ñ(mt+1i ,mt+1
and ξit+1 are given by
i
mt+1
=
i

ξit+1 =

Ai −yτ
τxiτ +x̂−iτ )∑tτ =1 Ii (xiτ , x̂−iτ )∑tτ =1 Ii (xiτ , x̂−i1), where the parameterst ξit+1


,

(11)

1 t
τ
Ii (xiτ , x̂−i
).
t τ∑
=1

(12)
τ

In words, the parameter mt+1
is the weighted average of player i’s estimate xAτ i+−yx̂τ each period,ii−iwhere the weight is the informativeness Ii . The parameter ξit+1 is simply the average of the infor1
mativeness Ii of the past signals. Our goal is to show that this posterior belief Ñ(mt+1
i , t ξ t+1 ) doesi
not converge to the interior steady state.
15 The truncated normal distribution is derived from a normally distributed random variable by bounding the randomvariable from either below or above (or both). For example, when a random variable X follows N(µ , ξ1 ), the truncatednormal distribution on Θ = [θ , θ ] is obtained by conditioning X on θ ≤ X ≤ θ .

15

Step 1: Difference Equation We ﬁrst show that the motion of the parameters (mti , ξit ) can be
described by a system of difference equations. Given an action proﬁle x, let θi (x) be a solution toQ(xi , x̂−i , Ai , θ ) = Q(x1 , x2 , a, θ ∗ ), i.e.,θi (x) =Ai − a + θ ∗ (x1 + x2 ).
xi + x̂−iIntuitively, this θi (x) can be thought of as player i’s estimate of θ when the noise is zero (i.e.,ε = 0).16 This suggests that player i’s actual estimate xAi +i −yx̂−i appearing in (11) can be represented
as θi (x) plus a noise, and we indeed have17
Ai − yε= θi (x) − p.
xi + x̂−i
Ii (xi , x̂−i )Plugging the above equation into (11) and arranging it and (12), we obtain the following recursive equations which completely describe the evolution of (mt , ξ t ).



t


ε
t
t
t
t


 Ii (xi , x̂−i ) θi (x ) − mi − √Ii (xt ,x̂t ) 
1
i −i
t
,
mt+1
−
m
=
i
i
1
t−1 t
t , x̂t )

t
ξ
+
I
(x
i


i −i
t i
t



ξit+1 − ξit =
1
Ii (xti , x̂t−i ) − ξit .
t

(13)

(14)

In words, (13) implies that player i updates the mean belief mti depending on how her estimateτθi (xt ) − √ ε t t

Ii (xi ,x̂−i )based on the new information today differs from her current mean belief mti . If

the new estimate coincides with the current mean belief, she does not update it. Otherwise, theτmean belief moves toward the new estimate θi (xt ) − √ ε t t , and its magnitude is inﬂuenced
Ii (xi ,x̂−i )1tt t
by the informativeness of the signals; if Ii (xti , x̂t−i ) is relatively larger than t−1
t ξi + t Ii (xi , x̂−i ), itmeans that the signal today is more informative relative to the past signals, and hence inﬂuences
the posterior more.
The second equation (14) has a similar interpretation, and ξit is updated depending on how
the informativeness Ii (xti , x̂t−i ) of the signal today differs from the informativeness ξit of the pastsignals.
For some x, θi (x) deﬁned above may not be in the state space Θ. So θi (x) should be regarded as a parameter
which best explains the data, when the choice of θ is not restricted on the state space Θ. In contrast, θi (x, Ai ) deﬁned
in the previous subsection must be chosen from the state space Θ.
17 This equation follows from the deﬁnition of θ (x) and y = a − θ ∗ (x + x ) + ε .
i
1
2
16

16

Note that players’ actions (xti , x̂t−i ) in period t is a one-shot Nash equilibrium given the posterior
1), so all terms in the right-hand sides of the above equations are the functions ofbelief Ñ(mti , (t−1)ξti(mti , ξit ), except the noise term ε t . Hence the above difference equations are indeed recursive, i.e.,t+1
once we ﬁx the current value (mti , ξit ) and the noise term ε t , the next value (mt+1) is uniquelyi , ξidetermined.
Step 2: Stochastic Approximation Since the difference equations derived in Step 1 involves astochastic noise term ε , ﬁnding its exact solution is a hard problem. So instead, we borrow theidea of stochastic approximation and use the fact that the solution to the difference equations canbe approximated by much simpler differential equations.
1
Recall that player i’s posterior belief in period t is the truncated normal distribution Ñ(mti , (t−1)
),
ξt
i

1
where mti is the mean and (t−1)
is the variance of the (untruncated) normal distribution. Whenξit
1
1
t is sufﬁciently large, this variance (t−1)
approaches zero, and hence the belief Ñ(mti , (t−1)
) is
ξt
ξt
i

i

approximately a degenerate belief. Given mi , let Ñ(m, 0) denote this limiting belief, that is,
1 if m ∈ [θ , θ ] mÑ(m, 0) = lim Ñ(m, σ 2 ) =1θ if m < θσ 2 →0
 1 if m > θθwhere θ = 0.7 and θ = 0.9 denote the boundary points of the state space Θ = [0.7, 0.9].
Then players’ actions (xti , x̂t−i ) in period t can be approximated by the one-shot Nash equilibrium given this limiting belief Ñ(mti , 0). Let θi (mt1 , mt2 ) and Ii (mti ) denote player i’s estimate andthe signal informativeness when players play this Nash equilibrium. That is, θi (m1 , m2 ) = θi (x)and Ii (mi ) = Ii (xi , x̂−i ), where (x̂ j , x̂− j ) is the Nash equilibrium for the belief Ñ(m j , 0) for each j.
This in turn implies that the drift terms of the difference equations (13) and (14) are approxiI (mti )(θi (mt )−mti )and Ii (mti ) − ξit , in the sense that there is K > 0 suchξitmated by much simpler terms ithat for any t and for α = 0.5,

Ii (xti , x̂t−i )(θi (xt ) − mti )
t−1 t
1
t t
t ξi + t Ii (xi , x̂−i )−Ii (mti )(θi (mt ) − mti )
K
< α,
t
ξi
t


K
Ii (xti , x̂t−i ) − ξit − (Ii (mti ) − ξit ) < α .
t
17

(15)
(16)

Then it follows from the theory of stochastic approximation (e.g., Theorem 2.1 of Kushner and Yin(2003)) that the asymptotic behavior of the process (13) and (14) is approximated by the ordinaldifferential equations (ODE)
dmi (t) Ii (mi (t))(θi (m(t)) − mi (t))=,
dt
ξi (t)d ξi (t)= Ii (mi (t)) − ξi (t),
dt

(17)
(18)

which do not involve a noise term.
Step 3: Instability of the Interior Steady State

Figure 1 is the phase portrait which describes

the solution to the ODE (17), when there is no misspeciﬁcation (i.e., A2 = a) and the variable ξ (t)is ﬁxed at ξ1 (t) = ξ2 (t) = 1 for all t.18 The origin is the point in which both players learn the true
state (i.e., (m1 , m2 ) = (θ ∗ , θ ∗ ) where θ ∗ = 0.8), which is the interior steady state in this specialcase. There are only two paths converging to this steady state m∗ , one from the top-right and theone from bottom-left. These paths are the basin of attraction of the steady state. If the initial value
is not on this basin of attraction, the solution to the ODE does not converge to m∗ , and it moves
toward the boundary points. (This is so even if the initial value is in a neighborhood of the origin.)
In this sense, the origin is an unstable steady state.
Figure 2 is the phase portrait when player 2 is optimistic (precisely, when A2 − a = 0.03). Dueto the misspeciﬁcation, the origin is not a steady state; now the steady state m∗ moves toward the
top-right corner. Other than that, the motion of the mean belief m(t) is very similar to that for thecase with no misspeciﬁcation. In particular, the steady state m∗ is still unstable, in that there are
only two paths converging to this point.
The instability of the steady state m∗ here is deeply related to the inferential naivety. To see
this, note ﬁrst that in the steady state m∗ , each player’s subjective output exactly matches the objective output. Suppose now that the mean belief is perturbed toward the bottom-right direction,
and we have m(t) = (m1 (t), m2 (t)) = (m∗1 + η , m∗2 − η ) for small η > 0. That is, we consider the
In reality, the parameter ξi is not ﬁxed, and evolves according to (18). However, this does not inﬂuence themotion of the mean belief m(t) much, because the variable ξ does not inﬂuence the sign of dmdti (t) in the ODE (17), i.e.,
it does not inﬂuence whether the mean belief mi (t) increases or decreases at the next instant. Accordingly, the motionof m(t) is very similar to the one described in Figure 1 even for the case in which ξ (t) changes over time.
18

18

m2

m2
0.90

0.90

0.85

0.85

0.80

0.80

0.75

0.75

0.75

0.80

0.85

0.90

m1

0.75

Figure 1: Motion of m(t) when A2 = a

0.80

0.85

0.90

m1

Figure 2: Motion of m(t) when A2 > acase in which player 1 is more pessimistic about the state θ than in the steady state, while player 2
is more optimistic about the state. With this belief proﬁle, player 1 reduces the production than in
the steady state, while player 2 increases the production. However, due to the inferential naivety,player 1 misestimates the opponent’s production; player 1 incorrectly believes that the opponent
is similarly pessimistic and reduces the production (in reality, the opponent increases the production). This means that player 1 observes the environmental quality worse than her anticipation,and becomes even more pessimistic about the state θ . Similarly, player 2 becomes even more optimistic. Hence the gap between players’ beliefs become larger, and the belief proﬁle m(t) moves
toward the bottom-right corner, rather than moving back to the origin. This “amplifying effect”
continues over and over; so even if the initial gap η is small, it becomes arbitrarily large, and thebelief converges to the boundary point.
Remark 1. For the argument in the last paragraph of this step to work, it is critical that thestate θ and one’s belief about θ have conﬂicting effects on the outcome y, in the sense that Qθ
(which measures the effect of θ ) and Qxi ∂∂mxii (which measures the effect of player i’s belief aboutθ through her action) have opposite signs. To see this, suppose that the current mean belief ism(t) = (m1 (t), m2 (t)) = (m∗1 + η , m∗2 − η ) as in the discussion above, and suppose now that Qθ > 0

19

and Qxi ∂∂mxii > 0.19 The latter inequality implies that player 1’s inferential naivety leads her to overestimate the output, i.e., she ﬁnds that the actual output is worse than her anticipation. Thenbecause we have Qθ > 0, player 1’s mean belief m1 goes down and approaches to the steady state
belief m∗1 . In Section 3.4, we will consider a general setup, and show that this property is indeed
necessary for the instability. See Proposition 3.
Step 4: Non-Convergence to Unstable Steady StateIn the previous step, we have seen that the

motion of the solution to the ODE is very sensitive to the initial value: If the initial value is on thebasin of attraction of the interior steady state, the solution eventually converges to there. However,once the initial value is perturbed, the solution moves toward a different direction and converges
to the boundary points. It turns out that this property is the key to obtain our discontinuity result.
To begin with, consider the case with A2 > a, and suppose that the current mean belief mt is at
the steady state (or in its neighborhood). The key observation is that due to the stochastic shock ε ,the mean belief mt cannot stay at the basin of attraction of the steady state forever. Indeed, as canbe seen from (13), the shock ε pushes mt toward the direction
√
√


I1
I2
b=
,
,
(t − 1)ξ1 + I1 (t − 1)ξ2 + I2
which is simpliﬁed to b = ( t √1I , t √1I ) in the steady state.20 This vector b is represented by the
12

black thick line in Figure 2, and it does not coincide with the basin of attraction of the steady state.
Hence at some point, the mean belief mt will be “kicked out” from the basin due to the shock, andthen the mean belief should move toward the boundary points, rather than reverting to the steadystate.21
In contrast, when there is no misspeciﬁcation (i.e., A2 = a), the mean belief mt remains on thebasin of attraction, even after the shock ε . Indeed, if the current mean belief is at the steady state,
19 Note that Q∂ xi
θ < 0 and Qxi ∂ mi > 0 in the above example.

Here we use the fact that ξi = Ii at the steady state.
More precisely, in the proof of the non-convergence theorem, we show that the mean belief leaves √c t neighborhood of the basin of attraction inﬁnitely often, due to the shock ε . Once the mean belief leaves this neighborhood, the drift term (which represents the “amplifying effect” we explained in the last step) is bounded away fromzero and dominates the impact of the stochastic shock ε , and hence the mean belief moves toward the boundary points
as described in Figure 2.
20

21

20

the shock pushes the mean belief toward the direction


1
1
√ , √
b=
,
t I1 t I2
which is proportional to (1, 1) in this special case. This means that the mean belief mt remains
on the 45-degree line, even after the shock. As can be seen from Figure 1, this 45-degree line
is precisely the basin of attraction, and hence the solution to the ODE starting from the current
mean belief reverts to the steady state. This suggests that the beliefs converge to the interior steadystate in this case. Intuitively, in the case of no misspeciﬁcation, player 1’s posterior belief about

θ coincides with player 2’s belief after every history, and hence the amplifying effect is nevertriggered; accordingly the mean belief does not move toward the boundary points.
In the proof of Proposition 1, we formalize the idea above by borrowing a technique developedby Pemantle (1990), who establish a non-convergence theorem for a class of stochastic processes.
Pemantle’s theorem does not apply to our model directly, as our stochastic process does not satisfy
some technical assumptions of Pemantle. So we extend Pemantle’s result and show that the same
result applies to our setup; see Appendix B for details.
Remark 2. Heidhues, Kőszegi, and Strack (2021) consider a single-agent learning problem andshow that the agent’s belief does not converge to an unstable steady state. Unfortunately, their
proof has a ﬂaw: In the proof, they show that the mean belief visits the basin of attraction of stablesteady states inﬁnitely often, but this need not imply non-convergence to unstable equilibria. (Asshown in Theorem 6.10 of Benaı̈m (1999), we need to show that the mean belief visits a compact
subset of the basin of stable steady states inﬁnitely often.) However, their theorem is correct as is,and it is an immediate corollary of our non-convergence theorem in Appendix B. (So this is one of
the examples in which our non-convergence theorem is valuable.)

3.4 Non-Convergence in a General Setup
We have seen that in the environmental problem under double misspeciﬁcation, players’ beliefs
do not converge to the interior steady state. Now we will consider a general model and provide a
condition under which discontinuity similar to that in the environmental problem occurs.
21Consider a general model of double misspeciﬁcation with a compact state space Θ = [θ , θ ].
Consider an initial prior µ with a continuous density. Assume that for any state θ , a Nash equilibrium (xi , x̂−i ) given (Ai , θ ) is unique.
Assume that the output function Q is linear in θ , in that Q = R(x, a)θ + S(x, a), and deﬁne
Ii (xi , x̂−i ) = (R(xi , x̂−i , Ai ))2 . This linearity assumption is a bit restrictive, but it ensures thatthe likelihood induced by any signal sequence (y1 , · · · , yt ) is the truncated normal distribution1), where the parameters mti and ξit are determined by (11) and (12) with the termÑ(mti , (t−1)
ξt
i

τ

τ

τ

y −S(xi ,x̂−i ,Ai )Ai −yτ
τ in (11) being replaced byxiτ +x̂−i
R(xi ,x̂−i ,Ai ) . We assume that R(xi , x̂−i , Ai ) , 0 for all on-path1
actions (xi , x̂−i ); this implies that ξit > 0, and hence the distribution Ñ(mti , (t−1)) is well-deﬁned.
ξitTo keep our notation as simple as possible, in what follows, we strengthen this assumption andfocus on the case in which R(xi , x̂−i , Ai ) < 0 for all on-path actions (xi , x̂−i ).22 Then just as in theenvironmental problem, the evolution of the parameters mti and ξit is governed by the difference
equations (13) and (14), where θi (x) is a solution to Q(xi , x̂−i , Ai , θ ) = Q(x1 , x2 , a, θ ∗ ).
Assume that the drift terms of these difference equations are approximated by the drift termsof the ODE. That is, assume that there is K > 0 and α > 0 such that (15) and (16) hold. Then it
follows from the theory of stochastic approximation that the evolution of the parameters (mt , ξ t )is asymptotically approximated by the ODE (17) and (18). Here, the functions Ii (mti ) and θi (mt )in these equations are deﬁned as in the environmental problem (i.e., these are the informativenessIi (xi , x̂−i ) and the estimate θi (x) when players have degenerate beliefs and play a Nash equilibrium
given these beliefs).
A steady state of the ODE is a point p = (m1 , m2 , ξ1 , ξ2 ) where dmdti (t) = d ξdti (t) = 0. A steady statep is linearly unstable if the Jacobian J of the ODE at the point p has at least one eigenvalue withpositive real part. When p is linearly unstable, the basin of attraction of p is locally approximatedby p + H, where H is the space spanned by the eigenvectors associated with eigenvalues withnegative real parts. So if the initial value is not in this set p + H, the solution to the ODE eventuallyleaves a neighborhood of p. The interior steady state of the environmental problem in Section 3.2
q1) in (13) should be replaced
When R(xi , x̂−i , Ai ) > 0 for some actions (xi , x̂−i ), the term −ε t / Ii (mti , (t−1)ξitq1) for such actions, and it may inﬂuence the speciﬁcation of the vector b. All the remaining
with +ε t / Ii (mti , (t−1)
ξt
22

i

arguments are not affected.

22

is an example of linearly unstable steady states; as described in Figure 1, when A1 = A2 = a, thesolution to the ODE leaves a neighborhood of the origin unless the initial value is on the 45-degreeline.
In our general model, the set H, which determines the basin of attraction of the stead state,
can be computed as follows: Note that θi (m) − mi = 0 and ξi = Ii in any steady state. Hence theJacobian J of the ODE at a steady state p can be written as


∂ θ1
∂ θ1
−1
0
0
∂ m2
 ∂ m1 ∂ θ2

∂ θ2
 ∂m
−1 0
0 
∂
m
1
2
.
J=


 ∂∂mI10−1 0 
1


∂ I2
0 −1
0
∂m
2

Obviously this Jacobian J has an eigenvalue λ = −1 (multiplicity 2), and the corresponding
eigenspace is{(0, 0, ξ1 , ξ2 )|∀ ξ1 , ξ2 ∈ R}.

(19)

The remaining two eigenvalues of J are the ones for the submatrix


∂ θ1
∂ θ1
−1
∂ m2
.
J ′ =  ∂ m1
∂ θ2
∂ θ2
∂m
∂m − 1
1

2

So a steady state p is linearly unstable if and only if this submatrix J ′ has an eigenvalue withpositive real part.
If the two eigenvalues of J ′ have positive real part, then the set H is simply the set described by(19). So the solution to the ODE leaves a neighborhood of p, unless the mean belief (m1 , m2 ) of
the initial value exactly matches that of the steady-state belief. If the matrix J ′ has one eigenvalue
with positive real part and one eigenvalue with negative real part, then the set H is{(ch1 , ch2 , ξ1 , ξ2 )|∀c, ξ1 , ξ2 ∈ R}
where h = (h1 , h2 ) is an eigenvector associated with the eigenvalue with negative real part. In thiscase, if the initial value of the mean belief (m1 , m2 ) is perturbed toward a direction other than h,then it leaves the basin p + H, so that the solution to the ODE must leave a neighborhood of p.
23

The following proposition shows that the process does not converge to a linearly unstablesteady state, if the noise kicks out the mean belief from its basin of attraction in the above sense.
This result is a direct consequence of the general non-convergence theorem in Appendix B (Proposition 5), and hence we omit the proof. Let b denote the coefﬁcient vector on the noise term ε inthe difference equations (13) and (14) at the steady state p, i.e.,
!
√
√


1
1
I1
I2
1
√ , √ , 0, 0 .
,, 0, 0 =b=t ξ1 + I1 t ξ2 + I2
t +1
I1
I2
Proposition 2. Pick the parameters (A1 , A2 ) arbitrarily, and pick an initial prior µ with a continuous density. Let p be a linearly unstable steady state of the ODE (17) and (18). Assume that thefollowing properties hold.
(i) For each i and θ , a Nash equilibrium (xi , x̂−i ) given (Ai , θ ) is unique.
(ii) The noise term ε follows the standard normal distribution N(0, 1),(iii) Q = R(x, a)θ + S(x, a) and R < 0 for all on-path actions.
(iv) Given any on-path action proﬁle x, there is a unique θ ∈ R which solves Q(xi , x̂−i , Ai , θ ) =Q(x1 , x2 , a, θ ∗ ). (Hence θi (x) is well-deﬁned.)(v) The functions Ii (mi ) and θi (m) are Lipschitz-continuous.
(vi) There is K > 0 such that (15) and (16) hold for α = 1 at the steady state p.
(vii) b < H.
Then the probability of limt→∞ (mt , ξ t ) = p is zero.
The critical assumption in Proposition 2 is (vii), which implies that the process (mt , ξ t ) cannot
stay in the basin of attraction of p due to the stochastic shock ε . Proposition 2 asserts that if thisassumption (as well as other standard assumptions (i)-(vi)) holds, then the process converges to theunstable steady state p with zero probability.
We view assumption (vii) as a mild restriction, because it is satisﬁed for generic choice ofparameters. For example, in the environmental problem studied in the previous subsections, the24

assumption (vii) is satisﬁed for any value A2 , a in a neighborhood of a. In this sense, linear
instability of p “almost always” implies non-convergence to p.
So it is important to understand when an interior steady state p is linearly unstable, and thenext proposition characterizes it. We use the following notation. For each i and θ−i , let fi∗ (θ−i )denote the set of all θi such that θi = θi (x, Ai ) for some (x1 , x2 , x̂1 , x̂2 ) such that (xi , x̂−i ) is a Nashequilibrium given (Ai , θi ) for each i. Intuitively, this fi∗ (θ−i ) is the set of steady states in player
i’s single-agent learning problem, where the opponent −i’s belief is ﬁxed at θ−i (and hence shechooses the same Nash equilibrium action for θ−i every period), while player i incorrectly believes
that the opponent −i’s belief changes over time.
Figure 3 describes the graph of fi∗ for the environmental problem. The blue line is the graph off1∗ (θ2 ). It shows that when θ2 is ﬁxed at a low value, we have f1∗ (θ2 ) = {θ 1 }, i.e., the highest stateθ 1 is the unique steady state for player 1’s learning problem. Indeed, as can be seen from Figure
3, if θ2 is low, the arrow points toward the east, so θ1 goes up. Similarly, when θ2 is ﬁxed at a highvalue, we have f1∗ (θ2 ) = {θ 1 }. When θ2 takes an intermediate value, both boundary points θ 1 andθ 1 are still steady states, and on top of that, there is an interior steady state which is described by
the upward-sloping blue curve in the ﬁgure. Note that at any point on this blue curve, the solutionof the ODE moves toward the vertical direction only; i.e., player 1’s belief m1 does not change at
the next instant. This means that these are indeed steady states for player 1’s learning problem.
The orange line in the ﬁgure is the graph of f2∗ (θ1 ), and it can be interpreted in the same way. Notethat a steady state of the joint learning problem is the intersections of the blue and orange lines.
Proposition 3. Let p = (m∗1 , m∗2 , ξ1∗ , ξ2∗ ) be a steady state such that Q(xi , x̂−i , Ai , m∗i ) = Q(x1 , x2 , a, θ ∗ )where (x1 , x2 , x̂1 , x̂2 ) denotes steady-state actions. Suppose also that for each i, there is an open interval Ui containing m∗−i such that there is a unique continuous function fi : Ui → R with fi (m∗−i ) =m∗i and fi (θ−i ) ∈ fi∗ (θ−i ) for all θ−i ∈ Ui . Assume that fi is differentiable. Then the following properties hold:(i) Suppose that ∂ θ∂im(m)− 1 > 0 at p for each i. Then p is linearly unstable.
i− 1 < 0 at p for each i. Then p is linearly unstable if f1′ (θ2 ) f2′ (θ1 ) > 1(ii) Suppose that ∂ θ∂im(m)
i
at p.
25

m2

m2

0.60

0.90

0.58

0.85

0.56

0.80

0.54

0.75
0.52

0.75

0.80

0.85

m1

0.90

0.52

0.54

0.56

0.58

0.60

m1

−1 < 0Figure 4: Example of ∂ θ∂im(m)iFigure 3: Example of ∂ θ∂im(m)−1 > 0i

The function fi deﬁned in Proposition 3 maps the opponent’s belief θ−i to player i’s interior
steady-state belief. For example, in Figure 3, f1 is the blue ﬂatter upward-sloping curve, and f2 is
the orange steeper upward-sloping curve.
Proposition 3 (i) assumes ∂ θ∂im(m)− 1 > 0. To interpret this assumption, consider player i’sisingle-agent learning problem in which the opponent’s belief is ﬁxed at m∗−i . In this problem,
player i’s posterior belief is represented by the parameters (mti , ξit ) just as in the original problem,
and the motion of (mti , ξit ) is approximated by the ODE (17) and (18). The Jacobian of this systemof the ODE at the steady-state belief is∂ θi∂ mi − 1

Ji =
∂ Ii
∂ mi


0
−1

,

− 1 > 0 ensures that the latterso its eigenvalues are λ = −1, ∂∂mθii − 1. The assumption ∂ θ∂im(m)ieigenvalue is positive, which means that the steady state is linearly unstable in this single-agent
problem. Proposition 3 (i) shows that in such a case, the steady state is similarly unstable evenwhen players jointly learn the state.
Proposition 3 (ii) assumes ∂ θ∂im(m)− 1 < 0, in which case the Jacobian Ji has two negative eigeni
values, λ = −1, ∂∂mθii − 1. This means that in the single-agent learning problem, the steady state p
is asymptotically stable; i.e., if the opponent’s belief is ﬁxed at the steady-state value, any solution
26

to the ODE starting from a neighborhood of p converges to p. Proposition 3 (ii) shows even insuch a case, the steady state p can be unstable when players jointly learn the state. Speciﬁcally, if
players’ beliefs have strong complementarity/substitutability in that f1′ f2′ > 1, then the steady stateis unstable. On the other hand, if f1′ f2′ < 1, then it is not difﬁcult to show that the steady state is
asymptotically stable, in that all eigenvalues of the Jacobian have negative real part. In this case, if
the initial value is in a neighborhood of p, any solution to the ODE (17) and (18) converges to p.
The environmental problem studied in the previous subsections is one of the examples which
satisﬁes the assumption stated in Proposition 3 (i). Indeed, as described in Figure 3, when m2 (t)is ﬁxed at the steady-state value and m(t) moves on the horizontal axis only, the solution leaves aneighborhood of p. Hence the steady state p is unstable in the single-agent learning problem.
We can also construct an example which satisﬁes the assumption stated in Proposition 3 (ii), by
replacing the parameters of the environmental problem with Θ = [0.5, 0.6] and θ ∗ = 0.55. Figure
4 describes the solution of the ODE for this modiﬁed environmental problem with small optimism(speciﬁcally, A2 − a = 0.003). Now the interior steady state is asymptotically stable in the singleagent learning problem; indeed, when m2 (t) is ﬁxed at the steady-state value and m(t) moves on
the horizontal axis only, the solution converges to p. Nonetheless this steady state is unstable in
the joint learning problem, because the slope of the curve fi is steep at the origin and f1′ f2′ > 1.
Proposition 3 is applicable to other economic applications as well. Here are two examples:
1. Team production. Suppose that two players work on a joint project. Each period, each player
i chooses an effort level xi ∈ [0, 1] and observes an output


1
1
+ ε,
−
y = a−θ
x1 + x2 2(20)

where a is the capability of the team, θ is an unknown state, and ε is a noise term which
follows the standard normal distribution. Assume that the true state is θ ∗ = 0.5. Player i’s
payoff is y − c(xi ), where c(xi ) = 18 x2 is a production cost. Consider the double misspeciﬁcation model with A1 = a and A2 > a, where player 2 is overconﬁdent about the capability
and player 1 is unaware of it. When A2 = a, simple algebra shows that f1′ f2′ ≈ 17.7 at theinterior steady state θ ∗ = 0.5. By the continuity, this implies that f1′ f2′ > 1 for any A2 close27

to a. So from Proposition 3, the interior steady state is linearly unstable whenever player 2
has small overconﬁdence.
2. Cournot duopoly with linear demand. Suppose that each period, each ﬁrm i = 1, 2 chooses
its quantity xi ∈ [0, x], and a publicly observable market price is given by
y = a − θ (x1 + x2 ) + ε ,where θ is an unknown state and ε is a noise term which follows the standard normal distribution. Firm i’s payoff is yxi − c(xi ), where yxi is ﬁrm i’s revenue and c(xi ) is ﬁrm i’s
production cost. Consider the double misspeciﬁcation model with A1 = a and A2 > a, whereﬁrm 2 is overconﬁdent about the demand and ﬁrm 1 is unaware of it. When A2 = a, simple
algebra shows that f1′ f2′ > 1 at the interior steady state θ ∗ = 0.5 if and only if c′′ (xi ) < 0. By
the continuity, the same is true for any A2 > a close to a. This means that if the cost function
is strictly concave at the steady-state action, then player 2’s small overconﬁdence leads toinstability of the steady state.
As noted in Remark 1, for the steady state to be unstable in the environmental problem, it iscritical that the state θ and one’s belief about θ have conﬂicting effects on the output y. Proposition 3 implies that the same is true for a general setup, i.e., for a steady state to be unstable, it is∂ xi
necessary that ∂∂ Q
θ and Qxi ∂ mi have opposite signs, at least for symmetric games with small mis-speciﬁcation. Indeed, Proposition 3 asserts that in symmetric games with A1 = A2 = a, a steadystate is unstable if ∂∂mθii > 1 or | fi′ (θ−i )| > 1.23

23

Although it is not stated in Proposition 3, it is straightforward to show that these conditions are necessary andsufﬁcient for instability.

28

By the implicit function theorem, we have24∂ xi

Qxi ∂ mi
∂ θi
=−
∂ mi
Qθ

fi′ (θ−i ) =andQxi ∂∂mxii

Qxi ∂∂mxii + Qθ.

Hence, the condition for instability can be rewritten as−

Qxi ∂∂mxii
Qθ

>1

Qxi ∂∂mxii

orQxi ∂∂mxii + Qθ

> 1.

(21)

It is obvious that this condition requires Qθ and Qxi ∂∂mxii to have different signs, as claimed above.

3.5 Convergence under Double Misspeciﬁcation
So far we have seen that one’s unawareness about the opponent’s misspeciﬁcation can inﬂuencea learning dynamic and cause non-convergence to an interior steady state in some economic examples. However, this does not imply that one’s unawareness always cause non-convergence, andindeed, there are many cases in which the beliefs do converge to an interior steady state under
double misspeciﬁcation.
For example, consider the team production problem discussed in the last subsection, but assume
now that the output function (20) is replaced with
y = a + (1 − θ )(x1 + x2 ) + ε .

(22)

The formal derivation is as follows.
Note that θi (mi ) is a solution to Q(xi (mi ), x−i , θ ∗ , a) =Q(xi (mi ), x̂−i (mi ), θi , A). Let Q denote the left-hand side (the true mean) and Q̂ denote the right-hand side (the subjective mean). Then by the implicit function theorem, we have
24

∂ xi

∂ x̂−i∂ xi∂ xi

Q̂xi ∂ mi + Q̂x−i ∂ mi − Qxi ∂ mi
Qxi ∂ mi
∂ θi
=−
=−
.
∂ mi
Qθ
Q̂θ
Here the second equality follows from the fact that we assume A1 = A2 = a (and hence Q = Q̂) and symmetry.
Similarly, fi (θ−i ) is a solution to Q(xi ( fi ), x−i (θ−i ), θ ∗ , a) = Q(xi ( fi ), x̂−i ( fi ), fi , A). With an abuse of notation, let Q
denote the left-hand side and Q̂ denote the right-hand side. Thenfi′ (θ−i ) =Qx−i ∂∂ θx−i−iQ̂xi ∂∂mxii + Q̂x−i ∂∂x̂m−ii + Q̂θ − Qxi ∂∂mxii

Again, the second equality uses A1 = A2 = a and symmetry.

29=

Qxi ∂∂mxii

Qxi ∂∂mxii + Qθ.

This new output function is similar to the previous one in that the output y is increasing in thecapability a and the total effort x1 + x2 , and is decreasing in the state θ . However, there is one
critical difference: with this new output function, the state θ has a negative impact on the marginal
productivity (i.e., ∂∂xi ∂Qθ < 0), and and hence has a negative impact on the effort level (i.e., ∂∂mxii < 0).
2

Accordingly, Qθ and Qxi ∂∂mxii have the same sign, which means that the condition for instability (21)is not satisﬁed.
Figure 5 is the phase portrait of the solution to the ODE (17) for this team production problem,with A1 = A2 = a, θ ∈ Θ = [0.3, 0.7], θ ∗ = 0.5, and ﬁxed ξ1 = ξ2 .25 As can be seen, the interior
steady state (the origin) is globally attracting in that for any initial value, the solution reaches aneighborhood of the steady state in ﬁnite time.
From this observation, one may expect that players’ beliefs converge to this interior steadystate almost surely in this example. The following proposition, which directly follows from a wellknown result in the literature of stochastic approximation (e.g., Theorem 6.9 of Benaı̈m (1999)),
conﬁrms that this is indeed the case. That is, players’ actions and beliefs converge to the interior
steady state almost surely, regardless of the initial prior. Given a vector x ∈ R4 and a compact set
A ⊂ R4 , let d(x, A) = miny∈A |x − y| denote the distance from x to the set A.
Deﬁnition 1. A set A ⊂ R4 is attracting if there is a set W such that A ⊆ intW and such that forany ε > 0, there is T > 0 such that d((m(t), ξ (t)), A) < ε for any initial value (m, ξ ) ∈ W and forany t > T . A set A is globally attracting if it is attracting and the stochastic process (mt , ξ t ) moves
within the basin W almost surely.
Proposition 4. Suppose that Assumption (i)-(v) in Proposition 2 hold, that lim inft→∞ mti > −∞ andlim supt→∞ mti < ∞ for each i with probability one, and that there are K > 0 and α > 0 such that (15)and (16) hold for all t and (mt , ξ t ). If a set A is globally attracting, then the process approaches
this set A almost surely, i.e., the probability of limt→∞ d((mt , ξ t ), A) = 0 is one regardless of theinitial prior. In particular, if A is a singleton, then the process converges to this point almost surely.
Here we ﬁx the parameter ξi , but the result does not change (i.e., the interior steady state is globally attracting)even when ξi changes over time. Also the same is true when the parameter A2 is perturbed so that there is a small
amount of misspeciﬁcation.
25

30

m2

m2

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.4

0.5

0.6

0.7

m1

0.4

Figure 5: Motion of m = (m1 , m2 )

0.5

0.6

0.7

m1

Figure 6: Graphs of f1 and f2

Of course, the condition stated in the above proposition is just a sufﬁcient condition for convergence, and there are many cases in which this condition does not hold but nonetheless players’
beliefs converge to the steady state. See our companion paper (Murooka and Yamamoto, 2023) for
more results and discussions on convergence.

4 Related Literature and Concluding Remarks
There is a rapidly growing literature on Bayesian learning with model misspeciﬁcation. Nyarko(1991) presents a model in which the agent’s action does not converge. Fudenberg, Romanyuk, andStrack (2017) consider a general two-state model and characterize the agent’s asymptotic actions
and behavior. Ba and Gindin (2023), He (2022), and Heidhues, Kőszegi, and Strack (2018, 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 the agent’s asymptotic
behavior in a general single-agent model. Fudenberg, Lanzani, and Strack (2021) discuss stabilityof steady states. All these papers look at a single-agent problem or a multi-agent setup in which
each player’s bias (ﬁrst-order misspeciﬁcation) is common knowledge.
Higher-order misspeciﬁcation has been studied in the literature on social learning (e.g., De31

Marzo, Vayanos, and Zwiebel, 2003; Eyster and Rabin, 2010; Gagnon-Bartsch and Rabin, 2016;
Bohren and Hauser, 2021). Most of these papers do not discuss discontinuity of the equilibriumoutcome, and indeed, one of the main result of Bohren and Hauser (2021) is that the long-run outcome is robust to a small perturbation of the information structure. An exception is Frick, Iijima,and Ishii (2020), who show that the equilibrium outcome is discontinuous in the information structure in a model of information aggregation. As explained in Introduction, a key assumption is thatthe agents observe a noise signal about the state only once, which leads to discontinuity of thesteady state. In contrast, in our model, the agents have repeated feedbacks about the state, andaccordingly the steady states are continuous in the information structure. Nonetheless the equilibrium outcome is discontinuous, because a small misspeciﬁcation inﬂuences the entire learning
dynamics and the convergence probability suddenly drops to zero.
In this paper, we have focused on a particular form of higher-order misspeciﬁcation, where
both players are unaware of the opponent having a different view about the world. Of course, there
are many other forms of higher-order misspeciﬁcation which are prevalent in the real world. For
example, in some markets, a fraction of consumers is overconﬁdent; cellular phone customers tend
to underestimate their usage next month, gym members tend to overestimate how often they willvisit the gym, and so on.26 In these cases, misspeciﬁcation can be on one side, i.e., the consumers
are misspeciﬁed while the companies are rational and understand the consumers’ overconﬁdence.
It turns out that this type of one-sided double misspeciﬁcation also leads to the discontinuity of theequilibrium outcome as in our model, i.e., small overconﬁdence leads to a complete breakdown ofcorrect learning.27 This shows that our model is just an example in which the equilibrium outcome
is sensitive to small misspeciﬁcation. For future research, it may be interesting to study whether
other forms of misspeciﬁcation lead to this kind of sensitivity.
26 See Grubb (2015) for a review of the literature.
27 A rough idea is as follows. Suppose that player 2 is doubly misspeciﬁed (i.e., she believes that the true parameter

is A2 > a and she believes that the opponent also believes A2 ), while player 1 is fully rational in that she correctly
understands the parameter a and player 2’s belief hierarchy. When A2 = a, there is no misspeciﬁcation, so bothplayers learn the true state θ ∗ . When A2 > a, rational player 1 learns the true state, but player 2 does not. In particular,
the evolution of player 2’s belief is asymptotically approximated by the belief evolution in the single-agent learning
problem appearing in Section 3.4, where player 1’s belief is ﬁxed at θ ∗ . So if the interior steady state of this singleagent learning problem is unstable (Proposition 3 (i) gives the precise condition under which the steady state is indeed
unstable), then player 2’s belief converges there with zero probability under one-sided double misspeciﬁcation.

32

Appendix A: Proof of Proposition 3
Let (xi (θi ), x̂−i (θi )) denote the one-shot Nash equilibrium (xi , x̂−i ) given a state θi and a parameter
Ai . By the assumption, for the steady-state belief θ2 = m∗2 , f1 (θ2 ) solves Q(x1 (θ1 ), x̂2 (θ1 ), θ1 ) =Q(x1 (θ1 ), x2 (θ2 ), θ ∗ ). So by the implicit function theorem,∂ Q∗ ∂ x2 (θ2 )∂ f1∂ x2 ∂ θ2=,∗∂ Q ∂ x1 (θ1 )∂ Q ∂ x̂2 (θ1 )∂ θ2++ ∂ Q − ∂ Q ∂ x1 (θ1 )
∂ x1

∂ θ1

∂ x̂2

∂ θ1

∂ θ1

∂ x1

∂ θ1

where Q = Q(x1 (θ1 ), x̂2 (θ1 ), θ1 ) and Q∗ = Q(x1 (θ1 ), x2 (θ2 ), θ ∗ ).
On the other hand, given (m1 , m2 ), the KL minimizer θ1 (m1 , m2 ) solves Q(x1 (m1 ), x̂2 (m1 ), θ1 ) =Q(x1 (m1 ), x2 (m2 ), θ ∗ ). So by the implicit function theorem,∂ Q ∂ x1 (θ1 )∂ Q ∂ x̂2 (θ1 )∂ Q∗ ∂ x1 (θ1 )∂ θ1 + ∂ x̂2 ∂ θ1 − ∂ x1 ∂ θ1
.
∂Q
∂ θ1

∂ θ1
∂x
=− 1
∂ m1
Similarly,∂ Q∗ ∂ x2 (θ2 )

∂ θ1
∂x
∂θ
= 2 ∂Q 2 .
∂ m2
∂ θ1

Then simple algebra shows that
1
∂ f1
∂ θ1
=
∂ θ 2 ∂ m2 1 − ∂ θ 1

(23)

∂ m1

To interpret this equation, suppose that player 2’s belief θ2 increases. Then her optimal action
x2 changes, which inﬂuences player 1’s belief (KL minimizer) by ∂∂mθ1 . Since player 1’s belief
2changes, her optimal action changes, which inﬂuences her own belief (KL minimizer) by ∂∂mθ1 ∂∂mθ1 .
2

1Then again player 1’s optimal action changes, which inﬂuences her own belief by ∂∂mθ1 ( ∂∂mθ1 )2 , and2
1
so on. The total effect of this process is
(
)


∂ θ1
∂ θ1
∂ θ1 21+++··· ,
∂ m2
∂ m1
∂ m1
which equals the right-hand side of the above equation.
Now, recall that the eigenvalues of the matrix J ′ solves



∂ θ1
∂ θ2
∂ θ1 ∂ θ2−1−λ−1−λ −
= 0,
∂ m1
∂ m2
∂ m2 ∂ m1
33

which is equivalent to





∂ θ1 ∂ θ2∂ θ1∂ θ2∂ θ1 ∂ θ22λ −+−2 λ +−1−1 −= 0.
∂ m1 ∂ m2∂ m1∂ m2∂ m2 ∂ m1(24)

Part (i): ∂∂mθii − 1 > 0 for each i.
Suppose that ∂∂mθii − 1 > 0 for each i. Then ∂∂mθ1 + ∂∂mθ2 − 2 > 0, Hence if (24) have real solutions,1

2

at least one of them must be positive, implying linear instability. If (24) have imaginary solutions,then the real part of these solutions is 21 ( ∂∂mθ1 + ∂∂mθ2 − 2) > 0, which again implies linear instability.
1

2Part (ii): ∂∂mθii − 1 < 0 for each i.
Assume ﬁrst that f1′ f2′ > 1. Plugging (23) into f1′ f2′ > 1, we have

∂ θ1
1
∂ θ2
1
> 1.
∂ m2 1 − ∂ θ 1 ∂ m1 1 − ∂ θ 2
∂ m1

∂ m2

Since we assume ∂∂mθii − 1 < 0, this inequality is equivalent to∂ θ1 ∂ θ2
∂ θ1
∂ θ2
> 1−
1−
∂ m2 ∂ m1∂ m1∂ m2
This implies that the y-intercept of the quadratic curve appearing in (24) is negative, which in turnimplies that (24) has one positive solution and one negative solution. Hence the steady state islinearly unstable.
Next, consider the case with f1′ f2′ < 1. Algebra similar to the one above yields∂ θ1 ∂ θ2
∂ θ1
∂ θ2
< 1−
1−
,
∂ m2 ∂ m1∂ m1∂ m2
which means that the y-intercept of the curve appearing in (24) is positive. Since we assume
∂ θ1
∂ θ2
∂ θi
∂ mi < 1, we have ∂ m1 + ∂ m2 − 2 < 0.

Hence if (24) have real solutions, they must be negative,

implying asymptotic stability. If (24) have imaginary solutions, then the real part of these solutions
is 12 ( ∂∂mθ1 + ∂∂mθ2 − 2) < 0, which again implies asymptotic stability.
1

2

34

Q.E.D.

References
Attari, S.Z., M.L. DeKay, C.I. Davidson, and W.B. de Bruin (2010): “Public Perceptions of Energy
Consumption and Savings” Proceedings of the National Academy of Sciences, 107 (37), 1605416059.
Ba, C. and A. Gindin (2023): “A Multi-Agent Model of Misspeciﬁed Learning with Overconﬁdence,” Working Paper.
Benaı̈m, M. (1999): “Dynamics of Stochastic Approximation Algorithms,” Séminaire de Probabilités XXXIII, Lecture Notes in Math. 1709, Springer.
Benaı̈m, M. and M. Faure (2012): “Stochastic Approximation, Cooperative Dynamics and Supermodular Games,” Annals of Applied Probability, 22, 2133-2164.
Benaı̈m, M., J. Hofbauer, and S. Sorin (2005): “Stochastic Approximations and Differential Inclusions,” SIAM Journal on Control and Optimization, 44, 328-348.
Billingsley, P. (1999): Convergence of Probability Measures, Wiley.
Bohren, J.A. and D.N. Hauser (2021): “Learning with Heterogeneous Misspeciﬁed Models: Characterization and Robustness,” Econometrica, 89 (6), 3025-3077.
Camerer, C. and D. Lovallo (1999): “Overconﬁdence and Excess Entry: An Experimental Approach,” American Economic Review, 89 (1), 306-318.
Chen, Y-C., A. Di Tillio, E. Faingold, and S. Xiong (2017): “Characterizing the Strategic Impactof Misspeciﬁed Beliefs,” Review of Economic Studies, 84, 1424-1471.
Daniel, K. and D. Hirshleifer (2015): “Overconﬁdent Investors, Predictable Returns, and Excessive
Trading,” Journal of Economic Perspectives, 29 (4), 61-88.
Dechezleprêtre, A., A. Fabre, T. Kruse, B. Planterose, A.S. Chico, and S. Stantcheva (2022):“Fighting Climate Change: International Attitudes Toward Climate Policies,” Working Paper.
Deimling, K. (1992): Multivalued Differential Equations, Walter de Gruyter.
DeMarzo, P.M., D. Vayanos, and J. Zwiebel (2015): “Persuasion Bias, Social Inﬂuence, and Unidimensional Opinions,” Quarterly Journal of Economics, 118 (3), 909-968.
Dudley, R. (1966): “Convergence of Baire Measures,” Studia Mathematica, 27, 251-268.
Esponda, I. and D. Pouzo (2016): “Berk-Nash Equilibrium: A Framework for Modeling Agentswith Misspeciﬁed Models,” Econometrica, 84, 1093-1130.
Esponda, I., D. Pouzo, and Y. Yamamoto (2021): “Asymptotic Behavior of Bayesian Learners with
Misspeciﬁed Models,” Journal of Economic Theory, 195, 105260.

35

Esponda, I., D. Pouzo, and Y. Yamamoto (2022): “Corrigendum to “Asymptotic Behavior ofBayesian Learners with Misspeciﬁed Models” [J. Econ. Theory 195 (2021) 105260],” Journalof Economic Theory, 204, 105513.
Eyster, E. (2019): “Errors in Strategic Reasoning,” In D.B. Bernheim, S. DellaVigna & D. Laibson,(Eds.), Handbook of Behavioral Economics: Foundations and Applications 2, 187-259, North
Holland.
Eyster, E. and M. Rabin (2010): “Naı̈ve Herding in Rich-Information Settings,” American Economic Journal: Microeconomics, 2 (4), 221-43.
Frick, M., R. Iijima, and Y. Ishii (2020): “Misinterpreting Others and the Fragility of Social Learning,” Econometrica, 88, 2281-2328.
Fudenberg, D. and D. Kreps (1993): “Learning Mixed Equilibria,” Games and Economic Behavior,
5, 320-367.
Fudenberg, D., G. Lanzani, and P. Strack (2021): “Limit Points of Endogenous Misspeciﬁed Learning,” Econometrica, 89, 1065-1098.
Fudenberg, D., G. Romanyuk, and P. Strack (2017): “Active Learning with a Misspeciﬁed Prior,”
Theorectical Economics, 12, 1155-1189.
Gagnon-Bartsch, T. and M. Rabin (2016): “Naive Social Learning, Mislearning, and Unlearning,”
Working Paper.
Grubb, M. (2015): “Overconﬁdent Consumers in the Marketplace,” Journal of Economic Perspectives, 29 (4), 9-36.
He, K. (2022): “Mislearning from Censored Data: The Gambler’s Fallacy and Other Correlational
Mistakes in Optimal-Stopping Problems,” Theoretical Economics, 17, 1269-1312.
Heidhues, P., B. Kőszegi, and P. Strack (2018): “Unrealistic Expectations and Misguided Learning,” Econometrica, 86, 1159-1214.
Heidhues, P., B. Kőszegi, and P. Strack (2021): “Convergence in Models of Misspeciﬁed Learning,” Theoretical Economics, 16, 73-99.
Hofbauer, J., J. Oechssler, and F. Riedel (2009): “Brown-von Neumann-Nash Dynamics: the Continuous Strategy Case,” Games and Economic Behavior, 65, 406-429.
Kushner, H.J. and G.G. Yin (2003): Stochastic Approximation and Recursive Algorithms and Applications, Springer.
Ludwig, S. and J. Nafziger (2011): “ Beliefs about Overconﬁdence,” Theory and Decision, 70,
475-500.
Malmendier, U. and G. Tate (2005): “CEO Overconﬁdence and Corporate Investment,” Journal of
Finance, 60 (6), 2661-2700.
36

Malmendier, U. and G. Tate (2008): “Who Makes Acquisitions? CEO Overconﬁdence and theMarket’s Reaction,” Journal of Financial Economics, 89 (1), 20-43.
Malmendier, U. and G. Tate (2015): “Behavioral CEOs: The Role of Managerial Overconﬁdence,”
Journal of Economic Perspectives, 29 (4), 37-60.
McGee, D. (2023): “Stereotypes and Strategic Discrimination,” Working Paper.
Murooka, T. and Y. Yamamoto (2023): “Convergence and Steady-State Analysis under Higher-Order Misspeciﬁcation,” mimeo. https://drive.google.com/file/d/
1tUNo98mLPCoabLbJctVKfJJCIc8QL8IE/view
Nyarko, Y. (1991): “Learning in Mis-Speciﬁed Models and the Possibility of Cycles,” Journal of
Economic Theory, 55, 416-427.
Pemantle, R. (1990): “Nonconvergence to Unstable Points in Urn Models and Stochastic Approximations,” Annals of Probability, 18, 698-712.
Perkins, S. and D.S. Leslie (2014): “Stochastic Fictitious Play with Continuous Action Sets,”
Journal of Economic Theory, 152, 179-213.
Van Boven, L., D. Dunning, and G. Loewenstein (2000): “Egocentric empathy gaps between
owners and buyers: Misperceptions of the endowment effect,” Journal of Personality and Social
Psychology, 79 (1), 66-76.
Varian, H.R. (1992): Microeconomic Analysis (3rd ed.), Norton.

37

Online Appendix
B

Non-Convergence Theorem for a General Stochastic Process

In this appendix, we will extend the non-convergence theorem of Pemantle (1990) and show thatthe same non-convergence result holds in a more general environment which includes our model
as a special case. This result is used in the proofs of the various non-convergence results in themain text.
Consider a stochastic difference equationv(t + 1) − v(t) =1(F(v(t)) + b(t, v(t))ε )
t +1

(25)

where v(t) ∈ Rn , F : Rn → Rn , b(t, v(t)) ∈ Rn , and ε ∼ N(0, 1). We assume that F is Lipschitzcontinuous, and that there is b such that |bi (t, v)| < b for all i, t, and v ∈ Rn , where bi (t, v) is the ith
component of the vector b(t, v). This second assumption essentially means that the variance of thenoise is bounded.
∞ is a perturbed solution to (25) if it solves
A stochastic process {v(t)}t=1v(t + 1) − v(t) =1F̃(t, v(t)) + b(t, v(t))ε
t +1

for some F̃ such that there is K > 0 and α > 0 such that for all t and v,
K|F(v) − F̃(t, v)| < α .
tIt follows from the theory of stochastic approximation (e.g, Theorem 2.1 of Kushner andYin (2003)) that if a stochastic process {v(t)} is a perturbed solution to (25), and if this process∞ is bounded with probability one, i.e.,{v(t)}t=1
Pr lim sup |v(t)| < ∞ = 1.
t→∞then the asymptotic motion {v(t)} is approximated by the ODE
dw(t)= F(v(t)).
dt

(26)

A point p ∈ Rn is a steady state of the ODE if F(p) = 0. A steady state p is linearly unstable
if the Jacobian of F at p has at least one eigenvalue with a positive real part. Pemantle (1990)
1

shows that there is zero probability of the stochastic process converging to linearly unstable steadystates, i.e., Pr(limt→∞ v(t) = p) = 0 for any linearly unstable steady state p, under a few technicalassumptions. His result does not apply to our setup, because (i) we consider a perturbed solution
to (25), (ii) the noise term ε has an unbounded support, and (iii) the noise term ε is common for
all variables, F1 , · · · , Fn . An important consequence of the second assumption (ii) is that the stepc̃
size v(t + 1) − v(t) is bounded by t+1, which is frequently used in Pemantle’s proof.

The following proposition shows that these features are not essential and Pemantle’s result stillholds in our model.
Proposition 5. Let p be a linearly unstable steady state of the ODE (26). Let H be the afﬁne spacespanned by the eigenvectors associated with the eigenvalues with negative real parts, and let H ∗ be
the set of all unit vectors orthogonal to H. Assume that there is κ > 0, t ∗ > 0, and a neighborhood
U of p such that |b(t, v) · h| ≥ κ for all h ∈ H ∗ , t ≥ t ∗ , and v ∈ U. If there is K and a neighborhood
U ′ of p such that |F(v) − F̃(t, v)| < Kt for all v ∈ U ′ and t, then Pr(limt→∞ v(t) = p) = 0.

C

ProofsC.1

Proof of Proposition 1

For part (i), note that when A2 = a(= A1 ), players have the same view about the world and hence
have the same posterior belief µ1t = µ2t every period. Hence the problem reduces to the classical individual learning problem with no misspeciﬁcation, and the result follows from a standard
argument (a version of the law of large numbers).
So in what follows, we will prove part (ii) of the proposition. We will ﬁrst show that theprocess converges to the interior steady state with zero probability. Then we will show that theprocess converges to the boundary steady states.
Part 1: Non-convergence to p First, we will show that there is zero probability of the process
converging to the interior steady state p. For this, it sufﬁces to show that assumptions (i)-(vii)stated in Proposition 2 hold and that the interior steady state p is linearly unstable.

2

From the ﬁrst-order condition, a Nash equilibrium (xi , x̂−i ) given θi is unique and xi = x̂−i =1 − θi . Hence assumption (i) holds.
Assumptions (ii)-(v) are obviously satisﬁed. To check assumptions (vi), let m̃(m, ξ ) denote themean of the truncated normal Ñ(m, ξ1 ). Since each player’s payoffs are linear in θ , given a posterior
1belief µit = Ñ mti , (t−1), player i and hypothetical player j chooses a Nash equilibrium for aξtistate θ = m̃(mti , (t − 1)ξit ). So from the Lipschitz-continuity of θi and Ii , assumption (vi) follows
from part (iv) of the next lemma. (Parts (i)-(iii) of this lemma are not used here, but we will use
them when we prove convergence to the boundary steady states.)
Lemma 1. There is k > 0 and t > 0 such that for all t > t and all ξ which arises on the equilibrium
path,(i) |m̃(m,t ξ ) − m| < √k t for all m ∈ Θ,(ii) |m̃(m,t ξ ) − θ | < √k t for all m < θ ,(iii) |m̃(m,t ξ ) − θ | < √k t for all m > θ .
Also, for any interior point θ ∗ ∈ Θ, there is a neighborhood U of θ ∗ , k > 0, and t ′ > 0 such thatfor all t > t ′ and m ∈ U,(iv) |m̃(m,t ξ ) − m| < kt .
Proof. Let ξ be the minimum of Ii (xi , x̂−i ) over all actions (xi , x̂−i ) which can be chosen on theequilibrium path, and we will show that (i)-(iv) hold for this particular ξ . Then it is straightforward
to see that (i)-(iv) holds for all other ξ .
Let ϕ denote the pdf of the standard normal N(0, 1), and let Φ denote its cdf. Pick sometruncated normal distribution Ñ(m, t1ξ ). It is well-known that that the mean of this truncated normal
distribution ispp
ϕ ( t ξ (θ − m)) − ϕ ( t ξ (θ − m))
1ppm̃ (m,t ξ ) = m + p ·t ξ Φ( t ξ (θ − m)) − Φ( t ξ (θ − m))

Since ϕ (x) = √12π exp(−0.5x2 ) ≤ √12π ,
|ϕ (

p
p
1
t ξ (θ − m)) − ϕ ( t ξ (θ − m))| < √ .
2π
3

(27)

Also there is t > 0 such that for all m ∈ Θ and t > t,
Φ(

p
p
1
t ξ (θ − m)) − Φ( t ξ (θ − m)) > .
3(28)

Plugging these into (27), we have13|m̃(m,t ξ ) − m| < p · √2πtξ
for all m ∈ Θ and t > t, which implies (i).
Next, we will prove (iv). Note thatpp1
|ϕ ( t ξ (θ − m)) − ϕ ( t ξ (θ − m))| = √
2π

1

!t ξ

√
2
( e)(θ −m)1−√ (θ −m)2( e)

Pick θ ∗ ∈ (θ , θ ). Then there is a neighborhood U of θ ∗ such that we have′1
√ (θ −m)2 < 1 for all m ∈ U. Then there is t such that
( e)

|ϕ (

t ξ
.

1
√
2 < 1 and( e)(θ −m)pp1t ξ (θ − m)) − ϕ ( t ξ (θ − m))| <t

for all m ∈ U and t > t ′ . Plugging this and (28) into (27), we have (iv).
Finally, we will prove (ii) and (iii). Let ϕ̃ (m, ξ1 ) denote the pdf of the truncated normal Ñ(m, ξ1 ).
Then for any x > 0 and m < θ ,




p
p
1
1
ϕ̃ θ , ξ [θ + x] ϕ ( ξ x) ϕ ( ξ (θ − m + x)) ϕ̃ m, ξ [θ + x]




p
=
=
.
>
ϕ (0)
ϕ ( ξ (θ − m))ϕ̃ θ , ξ1 [θ ]ϕ̃ m, ξ1 [θ ]This means that the truncated normal Ñ(θ , ξ1 ) ﬁrst-order stochastically dominates Ñ(m, ξ1 ) for allm < θ . Henceθ < m̃(m, ξ ) < m̃(θ , ξ )for all m < θ . Together with part (i) of the lemma, we obtain (ii). The proof of (iii) is similar andhence omitted.

Q.E.D.

Next, we will check (vii). We need to show that ( √1I , √1I ) is not an eigenvector of J ′ , i.e.,
1
2





1
1
∂ θ1
∂ θ1
√
√
−1
 I (p) ∂ m2   I1 (p)  ∂ m1 1  , λ ·  11 
∂ θ2
∂ θ2
√
√
∂m
∂m − 11I2 (p)

2

4

I2 (p)

for any λ ∈ R. This is equivalent to show that
p
p




∂ θ1
∂ θ1 I1 (p) ∂ θ2 I2 (p)
∂ θ2
p
p
−1 +
−1 .
,
+
∂ m1
∂ m2 I2 (p) ∂ m1 I1 (p)
∂ m2

(29)

Note that θi (m) solves Q(xi (mi ), x̂−i (mi ), Ai , θi ) = Q(xi (mi ), x−i (m−i ), a, θ ∗ ). By the implicit
function theorem, we have
∗

∂ x̂

∗

∂ Qi ∂ xi∂ Qi∂ Q ∂ xi−i
2 ∂∂Qxii ∂∂mxii − ∂∂Qxi ∂∂mxii∂ θi∂ xi ∂ mi + ∂ x̂−i ∂ mi − ∂ xi ∂ mi
√
=−
,
=−
∂ Qi
∂ mi
Ii

∂ θi
∂ Q∗ ∂ x j∂ Q∗ ∂ x j∂ θi∂xj ∂mj∂xj ∂mj= ∂Q = √ ,
i
∂mj
Ii
∂ θi

where Qi = Q(xi (mi ), x̂−i (mi ), Ai , θi ) denotes player i’s subjective expectation of the output andQ∗ = Q(xi (mi ), x−i (m−i ), a, θ ∗ ) is the true mean. Combining these two equalities, we have∗√
∂ Q∗ ∂ x−i−2 ∂∂Qxii ∂∂mxii + ∂∂Qxi ∂∂mxii∂ θi
Ii ∂ θi∂ x−i ∂ m−i
√
−1+ p
=
−1+ √
.
∂ mi
I−i
Ii
Ij ∂ mj

Using this equation, (29) can be rewritten as−2 ∂∂Qx 1 ∂∂mx1−2 ∂∂Qx 2 ∂∂mx2
1
1√√2 2 ,,
I1
I2
which is further simpliﬁed to
−

m2
m1
,−
1 − m1
1 − m2

because xi = x̂−1 = 1 − θi . This inequality indeed holds (and hence assumption (vii) is satisﬁed),
mibecause 1−m
is increasing in mi on the set Θ, and the consistency condition implies that m∗1 , m∗2iin any interior steady state with A2 , A1 = a.
To conclude the proof, we will show that the interior steady state p is linearly unstable. From
> 1 for each i. For the special case with A1 = A2 = a,
Proposition 3 (i), it sufﬁces to show that ∂ θ∂im(m)i= 2. Then by the continuity, for any A2 close to a, we still have ∂ θ∂im(m)> 1.
we have ∂ θ∂im(m)iiPart 2: Convergence to boundary beliefs We will ﬁrst show that the stochastic process (mt , ξ t )is bounded with probability one. Recall that regardless of the parameter Ai , a Nash equilibrium

5

given a state θi is xi = x̂−i = 1 − θi . Hence on the equilibrium path, each player’s production is atleast x = 1 − θ but does not exceed x = 1 − θ .
Let mi be such that
Ai − mi (x + x) = a − θ ∗ (x + x).
In words, mi ∈ R denotes a state with which player i’s subjective expectation about the output
matches the true mean, when player i thinks that the opponent chooses the maximal effort x but inreality she chooses the minimal effort x. Note that this mi is the minimum of θi (m) over all m, andthat mi need not be in the state space Θ. Similarly, let mi be such that
Ai − mi (x + x) = a − θ ∗ (x + x).
This is a state with which player i’s subjective expectation about the output matches the true mean,
when player i thinks that the opponent chooses the minimal effort x but in reality she chooses the
maximal effort x (which yields the most optimistic belief mi ). Note that this mi is the maximum of

θi (m) over all m.
The following lemma shows that almost surely, mti is in a neighborhood of [mi , mi ] after a longtime. This immediately implies that the process (mt , ξ t ) is bounded almost surely; indeed, Ii (xτ )has the minimal value I = Ii (x, x) and the maximal value I = Ii (x, x), so it is obvious that ξit isalways in the bounded interval [I, I].
Lemma 2. Given any A2 , almost surely, mi ≤ lim inft→∞ mti ≤ lim supt→∞ mti ≤ mi for each i.
From Lemma 1, it is obvious that there is K > 0 such that (15) and (16) hold for α = 0.5.
Then since the process is bounded with probability one, Theorem 2.1 of Kushner and Yin (2003)
implies the following lemma: Given a realized inﬁnite-horizon outcome (mt , ξ t )t=1∞ , deﬁne thecontinuous-time interpolation as a mapping w : [0, ∞) → R4 such thatw[τt + s] = (mt , ξ t ) +τ((mt+1 , ξ t+1 ) − (mt , ξ t ))
τt+1 − τt1). This w is an asymptotic pseudotrajectory of the ODE if for any
for all t = 0, 1, · · · and τ ∈ [0, t+1

T > 0,
lim sup |w(t + τ ) − s(w(t))[τ ]| = 0

t→∞ τ ∈[0,T ](30)

where s(m, ξ ) : R+ → R4 is a solution to the ODE (17) and (18) given the initial value (m, ξ ).
6

Lemma 3. With probability one, w is an asymptotic pseudotrajectory of the ODE (17) and (18).
This lemma implies that after a long time, the path w of the stochastic process is approximatedby the solution s to the ODE (17) and (18). So in order to know the long-run outcome of thestochastic process, it sufﬁces to investigate the ODE.
The next lemma characterizes the behavior of the solution to the ODE when player 2’s overconﬁdence is small.
Lemma 4. Pick a arbitrarily. There is A > a such that for any A2 ∈ [a, A) and i, there are values′ and θ ′′ with θ < θ ′ < θ ′′ < θ and differentiable functions f : [θ ′ , θ ′′ ] → Θ, f˜ : [θ , θ ′′ ] →
θ−i
i
i
−i
−i
−i
−i −i
−i
′ , θ ] → [m , θ ] such that the following properties hold:[θ , mi ], and fˆi : [θ−ii′ ) = θ , f (θ ′′ ) = θ , f˜′ (m ) < 0 for all m , f˜ (θ ) = m ,(i) fi′ (m−i ) > 1 for all m−i , fi (θ−i
i −i
−i
−i
i
i
i
′′ ) = θ , fˆ′ (m ) < 0 for all m , fˆ (θ ′ ) = θ , fˆ (θ ) = m ,f˜i (θ−i−i−i i −iiii(ii) For any m−i < θ , θi (m) − mi is positive if mi < mi , is zero if mi = mi , and is negative if
mi > mi .
′ ), θ (m) − m is positive if m < f˜ (m ), is zero if m = f˜ (m ), and is(iii) For any m−i ∈ [θ , θ−i
i
i
i
i
−i
i
i
−i

negative if mi > f˜i (m−i ),′ , θ ′′ ], θ (m)−m is positive if m < fˆ (m ), is zero if m = fˆ (m ), is neg(iv) For any m−i ∈ [θ−i
i
i
i
i
−i
i
i
−i
−i

ative if mi ∈ ( fˆi (m−i ), fi (m−i )), is zero if mi = fi (m−i ), is positive if mi ∈ ( fi (m−i ), f˜i (m−i )),is zero if mi = f˜i (m−i ), and is negative if mi > f˜i (m−i ).
′′ , θ ], θ (m) − m is positive if m < fˆ (m ), is zero if m = fˆ (m ), andv) For any m−i ∈ (θ−i
i
i
i
i
−i
i
i
−i

and is negative if mi > fˆi (m−i ).
(vi) For any m−i > θ , θi (m) − mi is positive if mi < mi , is zero if mi = mi , and is negative if
mi > mi .
′ , θ ′′ , f , f˜ , and fˆ . Let θ ′ be a state θ which solves
Proof. We will ﬁrst explain how to choose θ−ii−i i i−i

Ai − θ (xi (θ ) + x−i (θ )) = a − θ ∗ (xi (θ ) + x−i (θ )).

7

When Ai = a, the right-hand side (a − θ ∗ (2 − θ − θ )) is less than the left-hand side (a − θ (2 − 2θ ))at θ = θ , and is greater than that at θ = θ ∗ . Also the right-hand side is increasing in θ . Hence′ which solves the equality above is unique and θ < θ ′ < θ ∗ . Then by the continuity, the same
θ−i−iresult holds as long as A2 is close to a.
′′ be a state θ which solves
Similarly, let θ−iAi − θ (xi (θ ) + x−i (θ )) = a − θ ∗ (xi (θ ) + x−i (θ )).
′′ is uniquely determined and θ ∗ < θ ′′ < θ . Hence we have
Then again, for Ai close to a, θ−i−i′ < θ ′′ < θ as stated in the lemma.
θ < θ−i−i′′ ], deﬁne f˜ (m ) as a value m which solves
Then for each m−i ∈ [θ , θ−ii−iiAi − mi (xi (θ ) + x−i (θ )) = a − θ ∗ (xi (θ ) + x−i (m−i )),i.e., with this belief mi , player i’s subjective expectation about the output matches the true mean
when she believes that the Nash equilibrium for θ will be chosen but in reality the opponent
chooses the Nash equilibrium action for m−i . Note that the above equation is linear in mi , and′′ ) = θ . Also by thehence indeed has a unique solution. By the deﬁnition, f˜i (θ ) = mi and f˜i (θ−iimplicit function theorem, f˜i (m−i ) is decreasing in m−i , as stated in the lemma.
′ , θ ], deﬁne fˆ (m ) as a value m which solves
Similarly, for each m−i ∈ [θ−ii−iiAi − mi (xi (θ ) + x−i (θ )) = a − θ ∗ (xi (θ ) + x−i (m−i )).
Again this equation is linear in mi , and hence has a unique solution. Also it is easy to check that′ ) = θ , fˆ (θ ) = m , and fˆ (m ) is decreasing in m .
fˆi (θ−i
i
i
−i
−i
i
′ , θ ′′ ], deﬁne f (m ) as a value m ∈ Θ which solves
Also for each m−i ∈ [θ−ii−iiAi − mi (xi (mi ) + x−i (mi )) = a − θ ∗ (xi (mi ) + x−i (m−i )).
To see that this equation has a solution, letg(mi , m−i ) = Ai − mi (xi (mi ) + x−i (mi )) − a + θ ∗ (xi (mi ) + x−i (m−i )).
′′ , g(θ , θ ′′ ) = 0. Then since g is decreasing in m , we have g(θ , m ) ≥ 0
By the deﬁnition of θ−i−i−i−i′ , θ ′′ ]. Likewise, since g(θ , θ ′ ) = 0. we have g(θ , m ) ≤ 0 for all m ∈ [θ ′ , θ ′′ ].
for all m−i ∈ [θ−i
−i
−i
−i
−i

8

′ , θ ′′ ], we have g(θ , m ) ≤ 0 ≤ g(θ , m ), so there is at least
Taken together, given any m−i ∈ [θ−i−i−ione mi ∈ Θ which solves g(mi , m−i ) = 0. Also this solution is unique, because given any m−i ∈′ , θ ′′ ], g is strictly increasing in m when m ∈ Θ. (Note that g is a quadratic function of m .)
[θ−i
i
i
i
′ and θ ′′ , we have f (θ ′ ) = θ and f (θ ′′ ) = θ . Also, by the implicit
By the deﬁnition of θ−i
i −i
i −i
−i

function theorem,∂g∂mfi′ (m−i ) = − ∂ g−i =
∂ mi

θ∗
.
−2 + 4mi − θ ∗We have fi′ (m−i ) = 2 at mi = m−i = θ ∗ = 0.8 and fi′ (m−i ) > 1 for any mi , m−i ∈ Θ. So all theproperties stated in part (i) holds.
′ , θ ′′ ) arbitrarily. By the deﬁnition of fˆ , we
Next, we will prove part (iv). Pick m−i ∈ (θ−ii−ihave θi (m) = fˆi (m−i ) for any mi ≤ θ . Hence θi (m) − mi is positive for mi < fˆi (m−i ), is zero for
mi = fˆi (m−i ), and is negative for mi ∈ ( fˆi (m−i ), θ ], as stated in the lemma.
For mi ∈ (θ , fi (m−i ), we claim that θi (m) − mi is negative. Suppose not so that θi (m) − mi ≥ 0.
If θi (m) − mi = 0, then by the deﬁnition of fi , we must have mi = fi (m−i ), which contradicts withmi < fi (m−i ). If θi (m) − mi > 0, then there must be m′i ∈ (θ , mi ) such that θi (m′i , m−i ) − m′i =0. (This is so because θi (θ , m−i ) − θ < 0.) But then we must have m′i = fi (m−i ), which is acontradiction. Hence θi (m) − mi is negative in this case.
By the symmetry, for mi > fi (m−i ), all the properties stated in part (iv) of the lemma aresatisﬁed. Also, by the deﬁnition of fi , we have θi (m) − mi = 0 for mi = fi (m−i ). Hence part (iv)follows.
The proofs of the other parts of the lemma are very similar, and hence omitted.

Q.E.D.

Figure 7 highlights what is shown in the lemma above. Here the the horizontal axis represents
m−i and the vertical axis represents mi . The origin is the interior steady-state belief. The large
dotted square is ×i=1,2 [mi , mi ], and recall that after a long time, (mt1 , mt2 ) is in a neighborhood of
this square almost surely. The small dotted square is the state space ×i=1,2 Θ. The thick polygonalline is the set of points at which d θdti (t) = θi (m(t)) − mi (t) = 0; the downward-sloping line at the
top is the graph of the function f˜i (m−i ) deﬁned in the lemma above, the upward-sloping line in the
middle is the graph of fi , and the downward-sloping line at the bottom is the graph of fˆi . On theleft side of this thick line, d θdti (t) = θi (m(t)) − mi (t) > 0, which means that the solution θi (t) to the9ODE increases over time. In contrast, on the right side of the line, d θdti (t) = θi (m(t)) − mi (t) < 0,and hence θi (t) decreases over time. See the thick arrows in the ﬁgure.
Figure 8 describes how the solution to the ODE behaves when both m1 (t) and m2 (t) changeover time. The horizontal axis represents m1 and the vertical axis represents m2 . The two thickpolygonal lines are the set of points at which dmdti (t) = 0. If the current value m(t) is on the polygonalline with dmdt1 (t) = 0, only m2 (t) changes at the next instant, so m(t) moves vertically, as shown by
the arrows in the ﬁgure. Similarly, If the current value is on the polygonal line with dmdt2 (t) = 0,
only m1 (t) changes at the next instant, so m(t) moves horizontally. For all other points, both m1
and m2 move simultaneously. We cannot pin down the exact motion of m(t) in this case (hence
we have fork arrows in the picture) because it depends on the current value of ξ (t), which is not
speciﬁed here; in general, when ξ1 is relatively larger than ξ2 , m1 moves faster than m2 , and hence
the arrow becomes ﬂatter.
m2
mi

θ
m−i

θ

′
θ−i

′′
θ−i

θ

m1

m−i

θ
mi

ṁ2 (t) = 0ṁ1 (t) = 0
Figure 8: Motion of m(t)Figure 7: Motion of mi (t) for Fixed m−i

As can be seen from the ﬁgure, the polygonal lines intersect three times, and these are thesteady states of the ODE. That is, the ODE have one interior steady state (the origin) and two
boundary steady states ((m1 , m2 ) and (m1 , m2 )). From the ﬁgure, it is easy to check that given any
initial value (m, ξ ), the solution to the ODE eventually converges to one of these steady states.
However, this does not imply that the set of steady states is globally attracting; a problem is thatin a neighborhood of the origin (the interior steady state), ( dmdt1 (t) , dmdt2 (t) ) is approximately (0, 0),10

meaning that the motion of m(t) can be very slow. Accordingly, for some initial value, it takes
arbitrarily long time for the solution to reach a neighborhood of the boundary steady state, so wecannot ﬁnd a uniform bound T appearing in the deﬁnition of attracting sets.
Nonetheless, we can show that mt converge to the boundary steady states. This implies theresult we want, as in such a case the actual belief Ñ(mti , t ξ1 t ) converges to 1θ or 1θ .
iFormally, our goal is to prove the following lemma. Let B = {(m1 , m2 , I, I), (m1 , m2 , I, I)}
denote the set of the boundary steady states. Also, let M = (×i=1,2 [mi , mi ]) × [I, I]2 .
Lemma 5. Pick a particular path w : R → R4 such that (i) w is an asymptotic pseudotrajectory of
the ODE, (ii) limt→∞ d(w(t), M) = 0, and (iii) limt→∞ w(t) , p. (Note that these properties holdwith probability one, as shown by the earlier lemmas.) Then limt→∞ d(w, B) = 0.
Proof. Pick w as stated. Since limt→∞ w(t) , p, there is ε > 0 such that for any T > 0, there ist > T such that w(t) < (×i=1,2 [m∗i − ε , m∗i + ε ]) × [I, I]2 . Pick such ε .
Now, note that the inverse function fi−1 is increasing and fi−1 (m∗i ) = m∗−i . Hence we havefi−1 (m∗i − ε ) < fi−1 (m∗i ) < fi−1 (m∗i + ε ). Then there is η > 0 such thatfi−1 (m∗i − ε ) + 2η < fi−1 (m∗i ) < fi−1 (m∗i + ε ) − 2η(31)

for all i. Pick such η > 0. Then let A ⊂ R4 be such that
A = {(m, ξ ) ∈ M| min{m1 − m∗1 , m2 − m∗2 } ≤ η } ∩ {(m, ξ )| max{m1 − m∗1 , m2 − m∗2 } ≥ −η }.
See Figure 9.
From Figure 8, given any initial value chosen from the ε -neighborhood of M, the solution to
the ODE converges to this set A. Also, the solution does not enter a neighborhood of the origin on
the way to a neighborhood of A; this means that the solution reaches a neighborhood of A by sometime T , which is independent of the initial value. Thus the set A is attracting, and its basin is theε -neighborhood of M.
Theorem 6.10 of Benaı̈m (1999) asserts that if a path w visits the basin W of an attracting
set A inﬁnitely often and if W is compact, then w converges to the set A. Since we assume that
limt→∞ d(w(t), M) = 0, our path w indeed visits the ε -neighborhood of M inﬁnitely often (actuallyw stays there forever, after a long time). Also ε -neighborhood of M is compact. Hence w converges
11

m2

m2

m1

m1

Figure 9: The projection of the set A.

Figure 10: The projection of the set A′ .

to the set A, i.e., limt→∞ d(w(t), A) = 0. This in particular implies that there is T > 0 such that forany t > T , w(t) stays in the η -neighborhood of the set A.
At the same time, by the assumption w leaves the set (×i=1,2 [m∗i − ε , m∗i + ε ]) × [I, I]2 inﬁnitely
often. This means that w visits the set
A′ = {(m, ξ )|d((m, ξ ), A) ≤ η and m < ×i=1,2 (m∗i − ε , m∗i + ε )}inﬁnitely often. See Figure 10.
Note that this set A′ is compact and is a basin of the set B of the boundary steady states.28
Hence again from Theorem 6.10 of Benaı̈m (1999), w converges to B, as desired.

Q.E.D.

To see that A′ is a basin of B, pick any point (m, ξ ) ∈ A′ . If (m, ξ ) is in the fourth quadrant, we have dmdt1 (0) > 0
dm2 (0)and dt < 0, i.e., the solution m(t) to the ODE move toward the south-east direction, and eventually converge to the28boundary point (m1 , m2 ). See Figure 8. Also the solution does not enter the ε -neighborhood of the origin, so it reachesa neighborhood of the boundary point by some time T which is independent of the initial value. Next, consider thecase in which (m, ξ ) is in the ﬁrst quadrant. In this case we have either m1 < m∗1 + 2η or m2 < m∗2 + 2η , and without
loss of generality, we will focus on the case with m2 < m∗2 + 2η . Then from (31), the point (m, ξ ) is below the graph of
f1 (the ﬂatter upward-sloping line in Figure 8). Then again we have dmdt1 (0) > 0 and dmdt2 (0) < 0, so that the solution m(t)moves toward the south-east direction and eventually converges to the boundary point (m1 , m2 ). A similar argument
applies when (m, ξ ) is in the second or the third quadrant. Hence A′ is indeed a basin of B.

12

C.2 Proof of Proposition 5
Let N be a neighborhood of p, and choose a function η : N → R+ as in Section 3 of Pemantle(1990), given the ODE (26). Roughly, η (v) measures the distance between a point v and (the setof) the paths pointing to the steady state p. For example, any point v with η (v) = 0 is on such apath, so starting from this point v, a solution to the ODE (26) converges to p.
On the other hand, any point v with η (v) > 0 is not on such a path. So the solution to the
ODE does not converge to p. Indeed, as shown by Proposition 3(v) of Pemantle (1990), we have
Dv (η )(F(v)) > 0 for any v with η (v) > 0. So a solution to the ODE moves away from the paths
converging to p. (Here, the notation for multidimensional derivatives uses Dv (η ) for the differential of η at a point v.)
Let St = η (v(t)) and Xt = St − St−1 . Lemma 1 of Pemantle (1990) shows that after every∗c(i.e., v(t) leaves a neighborhood of the paths
history Ft , the stochastic process {Sk } can exceed √tconverging to p) at some point in the future with probability at least 0.5. The following lemmashows that the same result holds in our setup. The proof can be found in Appendix C.2.1
Lemma 6. There is a constant c∗ > 0 and t ∗ such that for any t > t ∗ and Ft ,c∗Pr sup Sk > √ or v(k) < N for some k > t Ft > 0.5.
t
k>t
∗

c
Lemma 2 of Pemantle (1990) shows that once the process {v(t)} leaves a √-neighborhood oftp as stated in the lemma above, then it fails to return to p with positive probability. The proof canbe found in Appendix C.2.2
Lemma 7. Let c∗ > 0 be as in Lemma 6. Then there is a > 0 such that


c∗
c∗
Pr inf Sk > √ or v(k) < N for some k ≥ t Ft , St ≥ √ ≥ a.
k>t
2 t
t
The rest of the proof is exactly the same as the argument in the full paragraph on page 711
of Pemantle (1990): Suppose that Pr(v(t) → p) > 0. Then there is some history Ft after which
the probability that v(M) converges to p and never leaves the neighborhood N is at least 1 − a2 .
However, Lemmas 6 and 7 imply that the probability that v(M) fails to converge to p or leaves N
is at least a2 conditional on any history Ft . This is a contradiction.
13C.2.1 Proof of Lemma 6
Without loss of generality, assume that N (the domain of the “distance function” η ) is a closed
ball surrounding p. (This is so because given a neighborhood U of the point p, we can always ﬁnd a closed ball N ⊆ U containing p.) Then enlarge the domain of η by letting η (v) =η (arg maxṽ∈N d(ṽ, N )) for each v < N . Here d(v, N ) measures the Euclidean distance betweenv and the ball N . This function η is well-deﬁned because N is a closed ball. Since η is Lipschitz
in N , it is so in the entire space Rn .
∗c
Pick a sufﬁciently large t, and deﬁne a stopping time τ = {M ≥ t|SM > √}. We will show thattPr(τ = ∞|Ft ) < 0.5.
Step 1: Inequalities (12) and (14) of Pemantle (1990).
In the proof Pemantle (1990), he shows that there is k2 > 0 such that for any M > t with∗c,
SM ≤ √
t

 
2
k2 SM
SM
E[2XM+1 SM |FM ] ≥
+O
,
M+1
M2
const.
2
E[XM+1|FM ] is at least
.
M2

(32)
(33)

See (12) and (14) of Pemantle (1990). His proof relies on the assumption that the noise term has a
1). We will show that the same result holds
bounded support (and hence the step size is of order t+1

in our setup where the noise is Gaussian.
Note that for any v, ṽ ∈ Rn and sufﬁciently large M, b(M, ṽ)εE η v+
FM
M+1=E[η (v + zb(M, ṽ)ε )],1
where z = M+1∂ E[η (v + zb(M, ṽ)ε )]z + O(z2 )∂zz=0n∂ η (v)=η (v) + ∑bi (M, ṽ)E[ε ]z + O(z2 )
∂
v
i
i=1
 
1
2
.
=η (v) + O(z ) = η (v) + O
M2=η (v) +14

To obtain the second equation, we regard the whole term as a function of z and apply Taylor
expansion at z = 0. Intuitively, this shows that the impact of the noise ε in period M on the
expected value of η (v(M + 1)) is of order O( M12 ). Then we have
E[SM+1 |FM ]
 



1
=E η v(M) +F̃(t, v(M)) + b(M, v(M))ε
FM
M+1


 


 
F̃(t, v(M))
1
F(v(M))1=η v(M) += η v(M) +
+O
+O
2
M+1
M
M+1
M2
 
k2 SM
1
≥
,
+O
M+1
M2
which immediately implies (32). Here the third equation follows from the Lipschitz continuityKof η , and |F(v) − F̃(M, v)| < M. The last inequality follows from Proposition 3(iv) of Pemantle(1990).
To obtain (33), note that2+
E[XM+1|FM ] =(E[XM+1|FM ])2+≥(Pr(|ε (M)| < 1|FM )E[XM+1|FM , |ε (M)| < 1])2 .
1. Hence as in the ﬁrst
Conditional on |ε (M)| < 1, the step size v(M + 1) − v(M) is of order M+1

display on page 709 of Pemantle (1990), we have+E[XM+1|FM , |ε (M)| < 1]
"
#

+

F̃(M, v(M)) + b(M, v(M))ε
≥E
Dv(M) (η )+ O(|v(M + 1) − v(M)|2 )FM , |ε (M)| < 1
M+1
"
#


 +
F(v(M)) + b(M, v(M))ε
1
FM , |ε (M)| < 1=E
Dv(M) (η )
+O
M+1
M2
"
#


 +
b(M, v(M))ε1≥E
Dv(M) (η )+O
FM , |ε (M)| < 1
M+1
M2
 
const.
1
≥
+O
.
M+1
M2
K
Here the equality follows from linearity of Dv (η ), |F(v) − F̃(M, v)| < M, and the fact that the step1
. The second to the last inequality follows from Proposition
size v(M + 1) − v(M) is of order M+1

15

3(v) of Pemantle (1990). The last inequality uses the fact that the gradient of η at p is c′ h for somec′ > 0 and h ∈ H ∗ , which implies Dv (η )(b(M, v(M))) ≥ c′ κ for any v(M) in a neighborhood of p.
Substituting this inequality to the previous one, we obtain (33).
Step 2: Main Proof.
As argued in the full paragraph on page 709 of Pemantle (1990), combining (32) and (33)yields2
E[2XM+1 SM + XM+1
|FM ] ≥

const.
,
M2

which in turn implies
2
E[Sτ2∧(M+1) |Ft ] − E[Sτ2∧M |Ft ] = E[1τ >M (2XM+1 SM + XM+1)|Ft ]
2= E[E[1τ >M (2XM+1 SM + XM+1)|FM ]|Ft ]const.
≥
E[1τ >M |Ft ]M2
const.
≥
Pr(τ = ∞|Ft )M2

for M > t. Pemantle (1990) applies this inequality iteratively and obtains
M−1

1
2
i=t i


1 1
−
.
≥ const. · Pr(τ = ∞|Ft )t M

E[Sτ2∧M |Ft ] ≥ St2 + const. · Pr(τ = ∞|Ft ) ∑(34)

Then in the ﬁrst paragraph on page 710, Pemantle (1990) shows that4(c∗ )2
2≥ E(SM∧τ |Ft ),
t(35)

using the assumption that the noise has a bounded support (which ensures that the step size is of1). We can show that the same result holds in our setup, the proof can be found at the end.
order M+1

Then the rest of the proof is the same as Pemantle (1990): Combining (34) and (35),
1 14(c∗ )2≥ const. · Pr(τ = ∞|Ft )
−
.
t
t M
This inequality holds for all M, and when M → ∞, it reduces to4(c∗ )2 ≥ const. · Pr(τ = ∞|Ft ).
By taking c∗ small enough, we have Pr(τ = ∞|Ft ) ≤ 0.5, as desired.
16

Step 3: Proof of (35).
We will show that (35) holds in our setup: Let L > 0 be the Lipschitz constant of η , and ĉ > 0
be such that |F̃(M, v) − v| < ĉ for all M > t and for all v in a neighborhood of p. Then
|SM∧τ − S(M∧τ )−1 )| < L|v(M ∧ τ ) − v((M ∧ τ ) − 1)|ĉL + nb|ε |LM∧τĉL + nb|ε |L
≤
.
t
≤

(36)
√

whenever v(M) is in the neighborhood of p. Since the mean of the half-normal distribution is √π2
and its variance is 1 − π2 , we haveconst.
,
tconst.
E[(SM∧τ − S(M∧τ )−1 )2 |Ft ] < 2 .
t
E[|SM∧τ − S(M∧τ )−1 ||Ft ] <Then we have
2
2
E[SM∧
τ |Ft ] = E[{S(M∧τ )−1 + (SM∧τ − S(M∧τ )−1 )} |Ft ] ∗ 2c∗ const. const.
c
√
√
+2
+ 2
≤
t
t
t t
∗

c
(which follows from the deﬁnition of τ ), the Lipschitzwhere the inequality uses S(M∧τ )−1 ≤ √t∗ 2continuity of η , and the previous inequalities. When t is large, the last line is less than 4(ct ) , andhence (35) follows.
C.2.2 Proof of Lemma 7
The proof is almost the same as that of Pemantle (1990). However, at some places, his proof uses
the assumption that the noise has a bounded support (which ensures that the step size of the process1
is of order t+1). In what follows, we will explain how to extend his argument to our setup with

Gaussian noise.
Enlarge the domain of η as in the proof of Lemma 6. Pick t large enough, and assume that∗∗c. Let τ = inf{k ≥ t|Sk ≤ 2c√t }. Recall that Xk = Sk − Sk−1 is a difference sequence. Let
St ≥ √t17kµk = E[Xk |Fk−1 ]. Consider a martingale {Zk }∞k=t deﬁned as Zk = St + ∑ j=t+1 Y j , where Yk = 0 forτ > k and Yk = Xk − µk for τ ≤ k.
In the seventh to the last line on page 710, Pemantle (1990) argues that if the step size is oforder 1k , then {Zk } is L2 -bounded (and hence the martingale convergence theorem applies).
In our setup, we can still prove that {Zk } is L2 -bounded. It is well-known that a martingale{Zk } is L2 -bounded if and only if∑ E[(Zk − Zk−1)2] < ∞.
k

We will show that this inequality holds in our model. Using the argument similar to (36), we have
|Sk − Sk−1 | <ĉL + nb|ε |L
,
k

(37)

and hence
E[(Sk − Sk−1 )2 |Fk−1 ] <const.
,
k2

E[|Sk − Sk−1 ||Fk−1 ] <const.
.
k

These inequalities imply
E[(Zk − Zk−1 )2 |Fk−1 ] ≤ E[(Xk − µk )2 |Fk−1 ]= E[(Sk − Sk−1 )2 − 2µk (Sk − Sk−1 ) + µk2 |Fk−1 ]= E[(Sk − Sk−1 )2 − 2E[Sk − Sk−1 |Fk−1 ](Sk − Sk−1 ) + (E[Sk − Sk−1 |Fk−1 ])2 |Fk−1 ]const.
<.
k2

for every k. Then obviously ∑k E[(Zk − Zk−1 )2 ] < ∞, asHence we have E[(Zk − Zk−1 )2 ] < const.
k2
desired.
Also in the last line on page 710, Pemantle (1990) shows that if the step size is of order 1k , then!
τ∞
const.
Var ∑ Yk ≤ ∑
,
2
k
k=t+1
k=t+1
In our model, we can still prove the same inequality. It is well-known that the covariance of themartingale difference (Yi ,Y j ) is zero, and hence
!
τ

Var

∑ Yk

k=t+1

= Var

∞

∑ Yk

k=t+1

!

∞

=

∑ Var(Yk ).

k=t+1

Note that E[(Zk −Zk−1 )2 ] < const.
. Because Var(Yk ) = E[(Zk −Zk−1 )2 ], the desired inequality holds.
k218