text:
\section{Introduction}

In many economic settings, people draw \emph{misspecified inferences}
about the world: that is, they start with a prior belief that dogmatically
precludes the true data-generating process. For instance, behavioral
economics documents a number of prevalent statistical biases. When
people reason about economic fundamentals under the spell of one of
these biases, they engage in misspecified learning. Following \citet{esponda2016berk},
a growing literature has focused on the implications of Bayesian learning
under different misspecifications, taking the errors as exogenously
given.

Why and when might misspecified inferences persist in a society? We
examine this question using an evolutionary framework.
Our framework evaluates competing (mis)specifications based on their
 objective expected payoffs, so that one specification is more fit than another if
the adherents of the first specification receive higher payoffs than those of the second specification, on average.  In our
setting, an individual may come to hold different beliefs about parameters of the underlying stage game, and thus adopt different best-reply functions, when
facing game outcomes generated from different strategy profiles. This endogeneity of the equilibrium best-reply function through mislearning is a defining characteristic
of our exercise, and our contribution is to show its implications
on evolutionary stability.

Our main message is that  a framework where agents can learn and draw (possibly wrong) inferences from data
strictly expands the scope for mistaken worldviews to invade
 rational societies. Mistakes might \emph{only} persist when inference is feasible. That is, the rational society
can be stable against invasion from any mutants who do not learn
from data, yet be vulnerable to some mutants who undertake \emph{strategically
beneficial misinferences}. We provide general conditions for when  such phenomenon happens,
without restrictions on the admissible misspecifications of the invaders. We also study applications where the same happens
for natural misspecifications encoded in economically meaningful biases.

In the course of doing so, we additionally provide simple intuition for how the persistence of particular biases  depends jointly on the social interaction structure, the possibility of learning, and the stage game's payoff structure. All three factors  influence the selection of biases, so studying only one factor in isolation may provide an incomplete understanding. For example, the mapping between different matching assortativities and the selected biases may be reversed for mutants who do not learn from data relative to mutants who do.

\subsection{Correlational Mistakes in a Linear-Quadratic-Normal Game}

As an example of how inference may matter for the selection of
biases, we informally describe one application of our framework. Fix
a linear-quadratic-normal game of incomplete information from \citet{vives1988aggregation}
as the stage game, which can be interpreted as an incomplete-information version of Cournot duopoly. There is a population of players (firms) who match in pairs
every period to play the stage game. The intercept of the demand curve
is drawn i.i.d. across games, and every pair of matched players receive
correlated information about this intercept in their game (as in \citet{bergemann2013robust}).
Firms privately observe a signal about today's demand shock before
choosing how much to produce.

The population initially consists of a homogeneous group of firms
who have correct beliefs about all game parameters, including the
correlation between rival players' private signals. But now a small
fraction of new firms enter the society. The entrants hold a dogmatically wrong
belief about the signal correlation. Can biased beliefs about signal correlation persist over the correct
belief?

Results from past work provide the intuition that commitments to
different kinds of strategies benefit the entrants in environments with different
matching assortativities. If entrants are only paired with each other (perfectly assortative matching),
then they can improve their payoffs by committing to more cooperative strategies. If entrants tend to be
paired with the rational incumbents (uniform matching), then committing to more aggressive strategies
can help them obtain more favorable outcomes compared to what the incumbents get when they play
each other. Our innovation is to show that whether a certain biased belief about
signal correlation leads to more cooperative or more aggressive play, and hence whether it will be selected
for a given matching assortativity, depends on whether the learning channel is present.

Suppose that the entrants are uncertain about the elasticity of market price with respect to total supply and learn this fundamental from
the realized prices in their games across different periods. The entrants
therefore engage in misspecified learning: after seeing their own
signals, they hold wrong beliefs about rivals' signals (due to misperceived signal correlation) and hence rivals'
production, so they misinterpret the market price and mislearn price elasticity. Such misinferences about elasticity distort perceived best replies. With uniform matching, entrants who believe in excessively correlated information (i.e.,
projection bias)  obtain objectively higher profits than the correctly
specified firms. Entrants who believe in excessively independent information
(correlation neglect), on the other hand, do objectively worse. With perfectly assortative matching, the conclusion is reversed.

But, this mapping from assortativity to the selected misperception of signal correlation changes when the
learning channel is absent. If the entrants who misperceive signal correlation  instead start off knowing
the correct price elasticity, the conclusions from the previous paragraph
would be reversed. Intuitively, the direct effect of the correlation misspecification
on the entrant's strategy goes in the opposite direction as its indirect effect through the
channel of misspecified inference about elasticity, but the latter channel dominates when it is present.
Thus, a  correlational mistake that is favorable when inference
is present may be unfavorable when inference is absent.

The presence of the inference channel has an even more stark effect on the selection of errors when the underlying elasticity parameter can take on multiple possible values.  In such cases, a fixed belief about elasticity can  be beneficial for some  realizations of the true elasticity parameter, but harmful for others. We use this insight to show that generally, there is some amount of uncertainty under which \emph{no} entrant with a fixed misperception about correlation and elasticity can invade a rational society, but some entrant with a biased belief over signal correlation who  makes flexible inferences about elasticity from data can do strictly better than the incumbents.

\subsection{A Framework of Competing Specifications}

This example illustrates  two main points of our paper.  First,  learning and the endogeneity of equilibrium
beliefs can strictly expand the possibility for  errors to invade a rational society.
Second,  environmental factors like the social interaction structure and the presence of learning
affect which errors get selected. To generalize beyond this example, we develop a  framework to formalize
how inference interacts with evolutionary stability. We provide a
characterization of when inference is necessary for the correct worldview
to be susceptible to invasion, and additionally describe new stability
phenomena which only emerge when learning is feasible.

Our framework amounts to an application of the \emph{indirect evolutionary
approach} to the selection of (mis)specifications.\footnote{As we discuss in the literature review, under the indirect evolutionary
approach, evolution acts on a parameter that determines best replies
rather than directly on strategies. Many of our modelling choices
are made in order to stay as close to this approach as possible, accommodating
misspecified inference in a transparent way without unnecessary complicating
features.} We posit that specifications are encoded in \emph{theories} that
delineate feasible beliefs and serve as the basic unit of cultural
transmission. Each theory represents a set of (possibly all incorrect)
relationships between environmental parameters and observables, which we call \emph{models}. The
theory's adherents estimate the best-fitting model and use it to play the stage game.
Theories rise and fall in prominence
based on the objective welfare of their adherents, as higher payoffs
confer greater evolutionary success.

When we allow for inference in the example above, the incumbents and the entrants differ in their
theories about the signal correlation structure in the stage game. Every firm learns about a parameter of the environment
(price elasticity) through the lens of its theory. Firms that believe
in different correlations interpret the same observation differently
when inferring price elasticity, as they make different estimates
about rival firm's production based on their own demand signal. We
imagine that, for instance, future entrepreneurs flock to the theory
yielding objectively higher firm profits.

The fitness of a theory is determined by its average payoff in stage
games, and this average depends on the distribution of opponents.
We introduce the concept of a \emph{zeitgeist} to capture the relevant
social interaction structure in the society --- the sizes of the
subpopulations with different theories, and the matchmaking technology
that pairs up opponents to play the game. In equilibrium, each agent
forms a Bayesian belief about her environment using data from all
of her interactions, and subjectively best responds to this belief.
We define the \emph{evolutionary stability} of theory A against theory
B based on whether theory A has a weakly higher equilibrium fitness
than theory B when the population share of theory A is close to 1.
The example above, for instance, says the correct specification about
the information structure  is evolutionarily fragile against
either projection bias or correlation neglect, depending on the matching
assortativity. Using this and other examples, we articulate how details
of the evolutionary environment (e.g., social interaction structure,
the stage game) affect which misspecifications persist.

Adherents of a misspecified theory may come to different conclusions
about the economic fundamentals in different zeitgeists, with these
different beliefs translating into different subjective best-response
functions in the stage game. Our framework assumes that the agents
might face one of a variety of games---which we refer to as \emph{situations}---and
therefore richer specifications can in principle help as they allow agents to adapt their behavior to the situation.

Our first general contribution is to show that the ability to draw inferences within a theory (as opposed to committing to a fixed model) may be necessary for misspecifications to defeat rationality. In Section \ref{sec:TheoryOverPref}, we  characterize when the correctly specified theory is not evolutionarily fragile against any theory that does not allow for inferences, while being  evolutionarily fragile against some richer misspecified theory.  Our argument identifies a particular kind of misspecified theory, resembling an ``illusion of control'' bias, where agents think the outcomes they get in a game only depend on their own strategy and not on the opponent's strategy. The theory is constructed in such a way that its adherents end up adopting
the optimal commitment against a correctly specified opponent situation-by-situation, thus misinference becomes a channel to tailor commitment to the true situation. The correctly specified theory is  evolutionarily fragile   against this misspecified theory with uniform matching, unless the former already gets the Stackelberg payoff in every situation.

Our next general results identify some new evolutionary phenomena which emerge in our framework due to the presence of learning.  First, we demonstrate the possibility of a strong form
of multiplicity in the stability comparison between two theories:
\emph{stability reversals}. Two theories exhibit stability reversal
if (i) theory A's adherents strictly outperform theory B's adherents
not only on average, but even conditional on opponent's type, whenever
theory A is dominant; and (ii) theory B's adherents strictly outperform
theory A's adherents, whenever theory B is dominant. Second, we show
that the relative stability of one theory over another may be non-monotonic
in matching assortativity. One theory may be evolutionarily stable
against another when assortativity is either high or low, but not
when it is intermediate. Both of these stability phenomena rely on
the idea that a fixed misspecification can lead to different equilibrium
beliefs and hence different behavior in different settings. That is,
they never arise in a world where the basic unit of cultural transmission
is a single belief about the economic environment instead of a theory
(i.e., a collection of feasible beliefs).

The rest of this section reviews related literature. Section \ref{sec:Environment-and-Stability}
introduces the environment and the evolutionary framework for assessing
the stability of specifications. Section \ref{sec:new_stability_phenomena}
discusses how the learning channel enables novel stability phenomena.
Sections \ref{sec:LQN} contains application to misspecified information
structures in linear-quadratic-normal games. Section \ref{sec:ABEE}
contains a second application that shows our framework can endogenize
analogy classes, a solution concept that \citet{jehiel2005analogy}
introduced to capture simplified strategic thinking in complex environments.
It also illustrates how payoff structure of the stage game can alter
the prevalence of analogy-based thinking in the society. Section \ref{sec:Concluding-Discussion}
concludes. Appendix \ref{sec:Proofs} and Online Appendix \ref{sec:omitted_proofs}
contain proofs of results from the main text. Online Appendix \ref{sec:Existence-and-Continuity}
presents sufficient conditions for the existence and upper hemicontinuity
of equilibrium zeitgeists,  Online Appendix \ref{sec:Learning-Foundation}
provides a learning foundation for equilibrium zeitgeists, and Online
Appendix \ref{OA:SingleAgent} discusses the single-agent case.

\subsection{\label{subsec:literature}Related Literature}

Our paper contributes to the literature on misspecified Bayesian learning
by proposing a framework to assess which specifications are more likely
to persist based on their objective performance. Most prior work on
misspecified Bayesian learning study implications of particular errors
in specific active-learning environments (i.e., when actions affect
observations), including both single-agent decision problems \citep*{nyarko1991learning,fudenberg2017,heidhues2018unrealistic,he_gambler}
and multi-agent games \citep*{bohren2016informational,bohren2017bounded,jehiel2018investment,molavi2019macro,dasaratha2020network,ba2020overconfidence,frick2019misinterpreting,murooka2021multi}.
A number of papers establish general convergence properties of misspecified
learning \citep*{esponda2016berk,esponda2019asymptotic,frick2019stability,FLS_general_conv}.
All of the above papers take misspecifications as exogenously given.
By contrast, we propose endogenizing misspecifications using ideas
from evolutionary game theory. This also lets us ask how details of
the evolutionary process (e.g., the matching assortativity) shape
the stability of misspecifications.

Some past work has shared
our central focus on selecting between multiple specifications for
Bayesian learning. Papers in this literature have focused on different
criteria, including performance in financial markets \citep{sandroni2000markets,massari2020under},
subjective expectations of payoffs \citep*{olea2019competing,Levyetal2020,eliaz2020model,gagnon2018channeled},
and goodness-of-fit tests \citep{cho2015learning,cho2017gresham,ba2020,schwartzstein2020using}.
We instead consider the objective expected payoffs of agents with
different specifications who coexist in the same society and interact
strategically. This difference is substantial, since in the absence of strategic responses, mistakes will typically lead to objectively lower payoffs. Thus, our results on how mistakes can invade with inference (and sometimes only with inference) do not follow from the existing literature.  We are implicitly motivated by a story of cultural
transmission where agents play ``games'' with random opponents and
derive welfare based on these interactions, and those with higher
welfare are more likely to pass down their theories to future agents.

This paper is closest to two independent and contemporaneous papers,
\citet*{FL_mutation} and \citet*{FII_welfare_based}, who consider
welfare-based criteria for selecting among misspecifications in single-agent
decision problems. \citet*{FL_mutation} study a framework where a
continuum of agents with heterogeneous misspecifications arrive each
period and learn from their predecessors' data. When the population
shares of different misspecifications change according to their objective
performance, \citet*{FL_mutation} ask which Berk-Nash equilibria
under one misspecification are robust to invasion by a small fraction
of mutants with a different misspecification. \citet*{FII_welfare_based}
assign a \emph{learning efficiency index} to every misspecified signal
structure and conduct a robust comparison of welfare under different
misspecifications.

In Online Appendix \ref{OA:SingleAgent}, we point out that correctly
specified agents always perform weakly better than misspecified agents
in the long run in single-agent problems (except when there are non-identifiability
issues, see Proposition \ref{prop:decision_problem}), so welfare-based
criteria do not provide a strict advantage in equilibrium to misspecified
individuals compared to the correctly specified ones in the same society.\footnote{This conclusion relates to the market-selection hypothesis that dates
back to at least \citet{friedman1953essays}. \citet{blume2002optimality,blume2006if}
come to the same conclusion in market equilibria where agents act
as price-takers, provided the market environment leads to Pareto efficient
outcomes. Our environment where agents pair off into two-player games
and interact as local oligopolies is a natural setting where outcomes
are not Pareto efficient.} \citet*{FII_welfare_based} also find that correctly specified agents
who know the data-generating process converge to the truth faster
than misspecified agents.\footnote{But, correctly specified agents who are uncertain about the data-generating
process may converge more slowly than misspecified agents.} By contrast, we focus on a theory of welfare-based selection of misspecifications
in games, where strategic concerns may imply that learning under a
misspecification confers a strict evolutionary advantage relative
to learning under the correct specification and relative to dogmatically
knowing the true data-generating process.\footnote{Some papers studying misspecified learning in games also point out
that misspecifications can improve an agent's welfare in particular
situations (e.g., \citet{jehiel2005analogy} and \citet{ba2020overconfidence}).
We contribute by introducing a general framework that can be applied
broadly.}

Our framework of competition between different specifications for
Bayesian learning is inspired by the evolutionary game theory literature.
While to the best of our knowledge this literature has not yet spoken
to applications featuring misspecified Bayesian learning, it also
uses objective payoffs as the selection criterion, and studies the
evolution of subjective preferences in games and decision problems
(e.g., \citet*{Dekeletal2007}, see also the surveys \citet*{robson2011evolutionary}
and \citet*{Alger2019survey}) and the evolution of constrained strategy
spaces \citep*{heller2015three,heller2016rule}. Perhaps closest to
our exercise is \citet*{GuthNapel2006}, who allow for heterogeneity
in the underlying stage game with the same preferences. Specifically,
they assume assume players either play a dictator game or an ultimatum
game, and study the ability to discriminate between these games.

By contrast, our work provides a foundation for the exogenously given
misspecifications in the recent literature on misspecified Bayesian
learning, and our results depend crucially on the mislearning channel
(as highlighted in Section \ref{sec:new_stability_phenomena} and
Section \ref{subsec:learningkey}). In settings where agents entertain
fundamental uncertainty about payoff parameters in the stage game,
we can think of our framework as applying evolutionary forces to \emph{sets
of} preferences, viewing every misspecification (i.e., a set of feasible
stage-game parameters) as a set of preferences over strategy profiles.
Doing so allows us to ask when (mis)inference is necessary to defeat
a correctly specified worldview, as discussed in Section \ref{subsec:When-Is-the}. A few papers in this literature study
the evolution of different belief-formation processes \citep{heller2020biased,berman2020naive},
but they take a reduced-form (and possibly non-Bayesian) approach
and consider arbitrary inference rules. We require agents to be Bayesians
who only differ in the support of their Bayesian prior (i.e., their
specification), given our emphasis on better understanding the persistence
of errors in models of misspecified Bayesian learning.

\section{Environment and Stability Concept \label{sec:Environment-and-Stability}}

In this section, we introduce the general environment and stability
concept. We begin with the objective stage game and subjective theories
that encode specifications. We define the notion of an \emph{equilibrium
zeitgeist}, which describes the steady-state behavior and beliefs
in a society populated by agents with heterogeneous specifications.
We then present the stability concept, based on objective welfare
in equilibrium zeitgeists when one theory is sufficiently prevalent.
For the ease of exposition, in this section we focus on a solution
concept where agents correctly know others' strategies in equilibrium.
We extend this and consider the notion of an \emph{equilibrium zeitgeist
with strategic uncertainty} in Section \ref{sec:ABEE}. Online Appendix
\ref{sec:Learning-Foundation} provides a combined learning foundation
of equilibrium zeitgeists and equilibrium zeitgeists with strategic
uncertainty.

\subsection{\label{subsec:Objective-Primitives}Objective Primitives}

We first set up the objective primitives of the general environment.
The stage game is a symmetric two-player game with a common strategy
space $\mathbb{A}$, assumed to be metrizable. There is a set of possible
states of nature $G\in\mathcal{G}$, called \emph{situations}. The
strategy choices $a_{i},a_{-i}\in\mathbb{A}$ of $i$ and $-i$, together
with the situation, stochastically generate consequences $y_{i},y_{-i}\in\mathbb{Y}$
from a metrizable space $\mathbb{Y}$. Each $i$'s consequence $y_{i}$
determines her utility, according to a common utility function $\pi:\mathbb{Y}\to\mathbb{R}$.
The objective distribution over consequences is $F^{\bullet}(a_{i},a_{-i},G)\in\Delta(\mathbb{Y}).$
We denote the density or probability mass function associated with
this distribution by $f^{\bullet}(a_{i},a_{-i},G),$ where $f^{\bullet}(a_{i},a_{-i},G)(y)\in\mathbb{R}_{+}$
for each $y\in\mathbb{Y}$. We suppress $G$ from $f^{\bullet}$ and
$F^{\bullet}$ in cases where $|\mathcal{G}|=1$.

This general setup can allow for mixed strategies (if $\mathbb{A}$
is the set of mixtures over some pure actions) and incomplete-information
games (if $S$ is a space of private signals, $A$ a space of actions,
and $\mathbb{A}=A^{S}$ is the set of signal-contingent actions).
It can also describe asymmetric games. Suppose there is a game with
action sets $A_{1},A_{2}$ for player roles P1 and P2, and that the
consequences of P1 and P2 under the action profile $(a_{1},a_{2})\in A_{1}\times A_{2}$
are generated according to the distributions $F_{1}^{\bullet}(a_{1},a_{2},G)$
and $F_{2}^{\bullet}(a_{2},a_{1},G)$ over $\mathbb{Y}$, where we
assume the consequence also fully reveals the agent's role. We may
construct a symmetric stage game by letting $\mathbb{A}=A_{1}\times A_{2}$,
so the strategy of each agent spells out what action she would take
if she were assigned into each of the player roles. The agents are
then placed into the player roles uniformly at random and play according
to their strategies. That is, the objective distribution over $i$'s
consequence when playing $(a_{i1},a_{i2})\in\mathbb{A}$ against $(a_{-i1},a_{-i2})\in\mathbb{A}$
is given by the 50-50 mixture over $F_{1}^{\bullet}(a_{i1},a_{-i2},G)$
and $F_{2}^{\bullet}(a_{i2},a_{-i1},G)$.

Allowing for multiple possible situations $G$ captures the idea that
there may be uncertainty about the underlying stage game. A given
action (e.g., making an investment) may be cooperative in some situations
but competitive in others. Agents do not observe the true situation
and may not even understand the set $\mathcal{G}$. Instead, they
draw inferences about the environment by using their data to find
the best-fitting model within their theory of the world. We now describe
how this inference works.

\subsection{Models and Theories}

Throughout this paper, we will take the strategy space $\mathbb{A},$
the set of consequences $\mathbb{Y},$ and the utility function over
consequences $\pi$ to be common knowledge among the agents. But,
agents are unsure about how play in the stage game translates into
consequences: that is, they have \emph{fundamental uncertainty} about
the function $(a_{i},a_{-i})\mapsto F^{\bullet}(a_{i},a_{-i},G).$
For example, the agents may be uncertain about some parameters of
the stage game, such as the market price elasticity in a quantity-competition
game.

We will consider a society with two observably distinguishable groups
of agents, A and B, who may behave differently in the stage game (due
to each group having a different belief about the economic fundamentals,
for example). All agents entertain different \emph{models} of the
world as possible resolutions of their fundamental uncertainty. A
model $F:\mathbb{A}^{2}\to\Delta(\mathbb{Y})$ is a conjecture about
how strategy profiles translate into consequences for the agent. Assume
each $F$ is given by a density or probability mass function $f(a_{i},a_{-i}):\mathbb{Y}\to\mathbb{R}_{+}$
for every $(a_{i},a_{-i})\in\mathbb{A}^{2}$.

A \emph{theory} $\Theta$ is a collection of models: that is, a subset
of $(\Delta(\mathbb{Y}))^{\mathbb{A}^{2}}$. We assume $\Theta$ is
metrizable. Each agent enters society with a persistent theory, which
depends entirely on whether they are from group A or group B. We think
of this exogenously endowed theory as coming from education or cultural
background, and each agent dogmatically believes that her theory contains
the correct model of the world. The\emph{ correctly specified }theory
is $\Theta=\{F^{\bullet}(\cdot,\cdot,G):G\in\mathcal{G}\}.$

In general, a theory may exclude the true $F^{\bullet}(\cdot,\cdot,G)$
that produces consequences for at least some situation $G$. Such
\emph{misspecified} theories can represent a scientific paradigm about
the economy based on a false premise, a religious belief system with
dogmas that contradict facts about the world, or heuristic thinking
stemming from a psychological bias that deems the true environment
as implausible. Each agent plays the stage game with a random opponent
in every period, and uses her personal experience in these matches
to calibrate the most accurate model within her theory in a way that
we will make precise in Section \ref{subsec:EZDef}.

The main innovation of this paper is our focus on the evolutionary
selection of \emph{theories}. We apply what is known as the ``indirect
evolutionary approach'' (see \citet{robson2011evolutionary}) to
ask which theories will be selected on the basis of their fitness.
There is an extensive literature that uses the same approach to study
the selection of \emph{preferences}, and we can accommodate these
works as a special case when inference is absent. If $\Theta$ is
a singleton theory that only contains one model $F$, then we may
view $\Theta$ as a preference in the stage game given by the subjective
utility function $(a_{i},a_{-i})\mapsto\mathbb{E}_{y\sim F(a_{i},a_{-i})}[\pi(y)]$.
Theories are more general in that they can contain multiple models,
so agents may adapt their beliefs and hence their subjective preferences
to their observations. We show in Section \ref{sec:new_stability_phenomena}
that allowing misinference strictly expands the possibility for mistaken
agents to invade a rational society and generates new stability phenomena.

An agent endowed with a theory is called an \emph{adherent} of the
theory. As alluded to above, the two observable groups A and B in
the society correspond to the adherents of the two theories. This
assumes that agents can identify which group their matched opponent
belongs to, though we do not assume that agents know the models contained
in other theories or that they are capable of making inferences using
other theories. As an example, consider hedge funds that regularly
trade against each other. Funds differ in their philosophies about
market rationality, with some focusing on fundamental analysis and
others on technical analysis.\footnote{In practice, each fund's theory about the financial market is well
known to other market participants, as it is always prominently marketed
to their clients. For instance, the founders of Dimensional Advisors
base their trading strategies on the efficient-market hypothesis,
attributing this to their education at the University of Chicago.
On the other hand, Paul Tudor Jones's Tudor Investment Corporation
focuses heavily on technical analysis. Jones is known for his view
on price movements that ``There's typically no logic to it; irrationality
reigns supreme, and no class can teach what to do'' \citep{quotes2008}.} But, simply observing another fund's actions would not lead a technical
analyst to embrace efficient markets, or vice versa. Part of the obstacle
is that both fundamental analysis and technical analysis are complex
forecasting systems that involve calibrating sophisticated models
and take many years of training and experience to master. That is,
agents can identify an opponent's theory without knowing the models
contained in said theory.

\subsection{\label{subsec:ZeitDef}Zeitgeists}

To study competition between two theories, we must describe the social
composition and interaction structure in the society where learning
takes place. We introduce the concepts of zeitgeists and equilibrium
zeitgeists to capture these details. We define a zeitgeist as a \emph{landscape}
of concurrent beliefs from different schools of thought, their relative
prominence in the society, and the interaction among the adherents
of different theories.
\begin{defn}
\label{def:Zeitdef}Fix theories $\Theta_{A}$ and $\Theta_{B}$.
A \emph{zeitgeist $\mathfrak{Z}=(\mu_{A}(G),\mu_{B}(G),p,\lambda,a(G))_{G\in\mathcal{G}}$
}consists of: (1) for each situation $G,$ a belief over models for
each theory, $\mu_{A}(G)\in\Delta(\Theta_{A})$ and $\mu_{B}(G)\in\Delta(\Theta_{B})$;
(2) relative sizes of the two groups in the society, $p=(p_{A},p_{B})$
with $p_{A},p_{B}\ge0,$ $p_{A}+p_{B}=1$; (3) a matching assortativity
parameter $\lambda\in[0,1]$; (4) for each situation $G,$ each group's
strategy when matched against each other group, $a=(a_{AA}(G),a_{AB}(G),a_{BA}(G),a_{BB}(G))$
where $a_{g,g^{'}}(G)\in\mathbb{A}$ is the strategy that an adherent
of $\Theta_{g}$ plays against an adherent of $\Theta_{g^{'}}$ in
situation $G$.
\end{defn}
A zeitgeist outlines the beliefs and interactions among agents with
heterogeneous theories living in the same society. Part (1) of this
definition captures the beliefs of each group. Parts (2) and (3) determine
social composition and social interaction---the relative prominence
of each theory and the probability of interacting with one's own group
versus with the population as a whole. In each period, every agent
is matched with an opponent from her own group with probability $\lambda,$
and matched uniformly by population proportion with probability $1-\lambda.$
Therefore, an agent from group $g$ has an overall probability of
$\lambda+(1-\lambda)p_{g}$ of being matched with an opponent from
her own group, and a complementary chance of being matched with an
opponent from the other group. Part (4) describes behavior in the
society. Note that a zeitgeist describes each group's situation-contingent
belief and behavior, since agents may learn different models and thus
adopt different subjective best replies in different situations.

\subsection{\label{subsec:EZDef}Equilibrium Zeitgeists}

To determine each theory's evolutionary fitness, we introduce our
equilibrium concept.

An equilibrium zeitgeist (EZ) imposes equilibrium conditions on the
beliefs and behavior in a zeitgeist. Specifically, it is a zeitgeist
that satisfies the optimality of inference and behavior in each situation
$G$, holding fixed the population shares $p$ and the matching assortativity
$\lambda.$ Optimality of behavior requires each player to best respond
given her beliefs, and optimality of inference requires that the support
of each player's belief only contains the ``best-fitting'' models
from her theory in the sense of minimizing Kullback-Leibler (KL) divergence.

In an EZ, we abstract away from the issues surrounding learning others'
strategies and focus on learning parameters of the environment. In
the example from the introduction where firms match into pairs to
play an incomplete-information Cournot duopoly, we can imagine that
each firm's strategy (i.e., mapping from signal realization to production
level) corresponds to a manufacturing process that automatically adjusts
production based on daily private information, and each firm learns
something about the matched rival firm's production process. We consider
an extension in Section \ref{sec:ABEE} that allows agents to be misspecified
about others' strategies and hold wrong beliefs about these strategies
in equilibrium: this corresponds to a learning environment where agents
only learn from the consequences $y_{i}$ and do not observe additional
feedback about others' strategies. (Online Appendix \ref{sec:Learning-Foundation}
contains a learning foundation of EZs and equilibrium zeitgeists with
strategic uncertainty, formalizing how the precision of the ex-post
feedback about others' strategies relates to which solution concept
emerges in the limit of the learning process.)

To formally give the definition of EZ, we require some new notation.

For two distributions over consequences, $\Phi,\Psi\in\Delta(\mathbb{Y})$
with density functions / probability mass functions $\psi,\phi$,
define the KL divergence from $\Psi$ to $\Phi$ as $D_{KL}(\Phi\parallel\Psi):=\int\phi(y)\ln\left(\frac{\phi(y)}{\psi(y)}\right)dy$.
Recall that every model $F$, like the true fundamental $F^{\bullet}$
in any situation $G$, outputs a distribution over consequences for
every profile of own play and opponent's play, $(a_{i},a_{-i})\in\mathbb{A}^{2}$.
For model $F,$ let $K(F;a_{i},a_{-i},G):=D_{KL}(F^{\bullet}(a_{i},a_{-i},G)\parallel F(a_{i},a_{-i}))$
be the KL divergence from the expected distribution of consequences
$F(a_{i},a_{-i})$ to the objective distribution of consequences $F^{\bullet}(a_{i},a_{-i},G)$
under the play $(a_{i},a_{-i})$ and situation $G$.

For a distribution $\mu$ over models, let $U_{i}(a_{i},a_{-i};\mu)$
represent $i$'s subjective expected utility under the belief that
the true model is drawn according to $\mu.$ That is, $U_{i}(a_{i},a_{-i};\mu):=\mathbb{E}_{F\sim\mu}(\mathbb{E}_{y\sim F(a_{i},a_{-i})}[\pi(y)])$.
\begin{defn}
\label{def:EZ}A zeitgeist $\mathfrak{Z}=(\mu_{A}(G),\mu_{B}(G),p,\lambda,a(G))_{G\in\mathcal{G}}$
is an\emph{ equilibrium zeitgeist (EZ)} if, for every $G\in\mathcal{G}$
and $g,g^{'}\in\{A,B\},$ $a_{g,g^{'}}(G)\in\underset{a_{i}\in\mathbb{A}}{\arg\max}\ U_{i}(a_{i},a_{g^{'},g}(G);\mu_{g}(G))$
and, for every $g\in\{A,B\},$ belief $\mu_{g}(G)$ is supported on
\begin{align*}
\underset{F\in\Theta_{g}}{\arg\min}\left\{ (\lambda+(1-\lambda)p_{g})\cdot K(F;a_{g,g}(G),a_{g,g}(G),G)+(1-\lambda)(1-p_{g})\cdot K(F;a_{g,-g}(G),a_{-g,g}(G),G)\right\}
\end{align*}
where $-g$ means the group other than $g$.
\end{defn}
In an EZ, each agent from group $g$ chooses a subjective best response
against each group $g^{'}$ of opponents, given her belief $\mu_{g}$
about the fundamental uncertainty. While the agent is allowed to choose
different actions against her own group and against the other group,
these choices are always made to selfishly maximize her individual
subjective utility function. Fixing the belief $\mu_{g}$, the agent
does not have different utility functions when facing different types
of opponents, so the framework does not advantage mutants with the
possibility of in-group collusive preference.

Each agent's belief $\mu_{g}$ is supported on the models in her theory
that minimize a weighted KL-divergence objective in situation $G$,
with the data from each type of match weighted by the probability
of confronting this type of opponent. The agent applies a misspecified
Bayesian inference procedure to her personal experience, so she assigns
more importance to matching the data generated by the equilibrium
strategy profile of the more frequent match type. We interpret the
ability to minimize the KL divergence separately across situations
as reflecting the true situation being sufficiently persistent, so
that agents have enough data between changes in the true situation
to establish new beliefs and behavior situation-by-situation.

An important assumption behind this framework is that agents (correctly)
believe the economic fundamentals (represented by $G$) do not vary
depending on which group they are matched against. That is, the mapping
$(a_{i},a_{-i})\mapsto\Delta(\mathbb{Y})$ describes the stage game
that they are playing, and agents know that they always play the same
stage game even though opponents from different groups may use different
strategies in the game. As a result, the agent's experience in games
against both groups of opponents jointly resolve the same fundamental
uncertainty about the environment. Generally, play between two groups
$g$ and $g^{'}$ is not a Berk-Nash equilibrium \citep{esponda2016berk},\footnote{A Berk-Nash equilibrium between groups $g$ and $g'$ would have each
player only draw inferences from the data generated in the match between
the two groups $g$ and $g'$.} as the individuals in group $g$ draw inferences about the game's
parameters not only from the matches against group $g^{'},$ but also
from the matches against the other group $-g^{'}$, who may use a
different strategy.

Even as agents adjust their beliefs and behavior to converge to an
EZ, the population proportions of different theories $p_{A},p_{B}$
remain fixed. We imagine a world where the relative prominence of
theories change much more slowly than the rate of convergence to an
EZ. Thus, an equilibrium zeitgeist provides snapshot of the society
in a given era, and the social transitions between different EZs as
$p$ evolves takes place on a longer timescale. This assumption about
the relative rate of change in the population sizes follows the previous
work on evolutionary game theory (See \citet*{Sandholm2001Preference}
or \citet*{Dekeletal2007}).

\subsection{Evolutionary Stability of Theories}

Given a distribution $q\in\Delta(\mathcal{G})$ over the true situation,
define the \emph{fitness} of each theory in an EZ as the objective
expected payoff of its adherents in the EZ when $G$ is drawn according
to $q$. Consider an evolutionary story where the relative prominence
of the two theories in the society rise and fall according to their
relative fitness. This could happen, for example, if the theories
are the basic heritable units of information passed down to future
agents via cultural transmission, and the school of thought whose
adherents have higher average payoff tends to acquire more resources
and attract a larger share of future adherents. We are interested
in a notion of stability based on this ``evolutionary'' process
where two co-existing rival theories compete to create intellectual
descendants in a payoff-monotonic way. Can the adherents of a \emph{resident
theory} $\Theta_{A}$, starting at a position of social prominence,
always repel an invasion from a small $\epsilon$ mass of agents who
adhere to a \emph{mutant theory} $\Theta_{B}$? The definition of
evolutionary stability formalizes this idea.

Evolutionary stability is based on the fitness of theories $\Theta_{A},\Theta_{B}$
in EZs with $p=(1,0).$ But it is motivated by environments where
a small but strictly positive population of theory $\Theta_{B}$ adherents
invade an otherwise homogeneous society all believing in theory $\Theta_{A}$.
In our applications, we directly analyze EZs with $p=(1,0)$, but
all these EZs can be written as the limit of EZs where the population
share of $\Theta_{B}$ is strictly positive but approaches 0. Online
Appendix \ref{sec:Existence-and-Continuity} provide conditions for
the existence of an EZ with $p=(1,0)$ and to ensure that any limit
of EZs with positive but diminishing fraction of $\Theta_{B}$ remains
an EZ with $p=(1,0)$.

Evolutionary stability is when $\Theta_{A}$ has higher fitness than
$\Theta_{B}$ in all EZs, and evolutionary fragility is when $\Theta_{A}$
has lower fitness in all EZs.\footnote{Both of these definitions presume the set of EZs is non-empty. If
this set is empty, then $\Theta_{A}$ is neither evolutionarily stable
nor evolutionarily fragile against $\Theta_{B}.$} These two cases give sharp predictions about whether a small share
of mutant-theory invaders might grow in size, across all equilibrium
selections. A third possible case, where $\Theta_{A}$ has lower fitness
than $\Theta_{B}$ in some but not all  EZs, correspond to a situation
where the mutant theory may or may not grow in the society, depending
on the equilibrium selection.
\begin{defn}
\label{def:stability} Say $\Theta_{A}$ is \emph{evolutionarily stable
{[}fragile{]} }against $\Theta_{B}$ under $\lambda$-matching if
there exists at least one EZ with theories $\Theta_{A},\Theta_{B}$,
$p=(1,0),$ matching assortativity $\lambda$ and, in all such EZs,
$\Theta_{A}$ has a weakly higher {[}strictly lower{]} fitness than
$\Theta_{B}$.
\end{defn}

\section{Learning Channel and New Stability Phenomena\label{sec:new_stability_phenomena}}

This section focuses on how the framework's learning channel leads
to new stability phenomena. A key feature of our theory-evolution
framework is that each agent interprets her observations through the
lens of her theory, thus drawing inferences about her environment
(i.e., game parameters). These inferences, in turn, shape her preference
over strategy profiles in the stage game. We call this phenomenon
the \emph{learning channel}: the subjective preferences that the adherents
of a given theory hold over stage-game strategy profiles is endogenously
determined in equilibrium.

We now explore implications of the learning channel for evolutionary
stability, exhibiting some new stability phenomena which only arise
when the indirect evolutionary approach is applied to theories, but
not when it is applied to preferences. Section \ref{subsec:When-Is-the}
shows that a rational society may be evolutionarily fragile against
some misspecified mutants who can draw (mis)inferences, but not fragile
against any mutants who do not learn. Sections \ref{subsec:Stability-Reversals}
and \ref{subsec:Non-Monotonic-Stability-in} show that the learning
channel can lead to a strong kind of multiplicity in the relative
stability of two theories, and a non-monotonicity of stability with
respect to the society's matching assortativity. All of these results
depend on the learning channel: the high-level reason is that we need
the mutant's subjective best response function to change depending
on the equilibrium data-generating process.

\subsection{\label{subsec:When-Is-the}When Is the Learning Channel Necessary
for Invading a Rational Society?}

\label{sec:TheoryOverPref}

We begin with an example that shows the ability to draw inferences
may be indispensable for a misspecified theory to invade a rational
society. Then, we provide conditions to characterize when this happens
in general.
\begin{example}
\label{exa:only_theory_invades}Consider a symmetric two-player game
where each player's strategy set is $\mathbb{A}=\{a_{1},a_{2},a_{3}\}$
and the possible consequences are $\text{\ensuremath{\mathbb{Y}}}=\{g,b\}.$
The utility function is $u(g)=1$ and $u(b)=0.$ There are two situations,
$G_{A},$ and $G_{B}$, equally likely. The two tables below show
the probabilities of receiving the $g$ consequence as P1 and P2 under
different strategy profiles in the two situations, which also correspond
to the objective expected utilities of the two players.
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
{\small{}$G_{A}$} & {\small{}$a_{1}$} & {\small{}$a_{2}$} & {\small{}$a_{3}$}\tabularnewline
\hline
\hline
{\small{}$a_{1}$} & {\small{}0.1, 0.1} & {\small{}0.1, 0.1} & {\small{}0.1, 0.11}\tabularnewline
\hline
{\small{}$a_{2}$} & {\small{}0.1, 0.1} & 0.3, 0.3 & {\small{}0.1, 0.1}\tabularnewline
\hline
{\small{}$a_{3}$} & {\small{}0.11, 0.1} & {\small{}0.1, 0.1} & {\small{}0.2, 0.2}\tabularnewline
\hline
\end{tabular}{\small{} \qquad{}}%
\begin{tabular}{|c|c|c|c|}
\hline
{\small{}$G_{B}$} & {\small{}$a_{1}$} & {\small{}$a_{2}$} & {\small{}$a_{3}$}\tabularnewline
\hline
\hline
{\small{}$a_{1}$} & {\small{}0.11, 0.11} & {\small{}0.5, 0.5} & {\small{}0.12, 0.4}\tabularnewline
\hline
{\small{}$a_{2}$} & {\small{}0.5, 0.5} & {\small{}0.12, 0.12} & {\small{}0.14, 0.55}\tabularnewline
\hline
{\small{}$a_{3}$} & {\small{}0.4, 0.12} & {\small{}0.55, 0.14} & {\small{}0.4, 0.4}\tabularnewline
\hline
\end{tabular}
\par\end{center}
Supposed $\lambda=0.$ The correctly specified theory is not evolutionarily
fragile against any singleton mutant theory $\Theta=\{F\}$. To see
this, note that since $(a_{2},a_{2})$, $(a_{3},a_{3})$ are objectively
Nash equilibria in the two situations, there is an EZ where the resident
correctly specified agents correctly learn the situations and get
payoff 0.3 in $G_{A}$ and 0.4 in $G_{B}$. Now we consider three
cases based on which strategy is a best response to $a_{3}$ under
the model $F$. If $a_{3}$ is a best response to $a_{3}$ under $F$,
then there is an EZ where the profile $(a_{3},a_{3})$ is played in
matches between the mutant and the resident in all situations, so
the mutant's fitness is 0.3. If $a_{2}$ is best response to $a_{3}$
under $F$, then there is an EZ where the profile $(a_{2},a_{3})$
is played by the mutant and the resident in $G_{B}$, so the fitness
of the mutant is no larger than 0.22. Finally, if $a_{1}$ is a best
response to $a_{3}$ under $F$, then there is an EZ where the profile
$(a_{1},a_{3})$ is played by the mutant and the resident in $G_{A}$,
so the fitness of the mutant is no larger than 0.325. We can construct
an EZ where the residents have fitness 0.35 but the mutants have fitness
no larger than 0.3, 0.22, or 0.325 in the three cases, thus the correctly
specified theory is not evolutionarily fragile.

Now consider the misspecified theory that contains two models, $F_{A}$
and $F_{B}$. For each of these models, the distribution of consequences
only depends on one's own strategy and not on the opponent's strategy.
Under model $F_{A},$ playing $a_{1},a_{2},a_{3}$ generate the $g$
consequence with probabilities 0.1, 0.3, and 0.2 respectively. Under
model $F_{B},$ playing $a_{1},a_{2},a_{3}$ generate the $g$ consequence
with probabilities 0.5, 0.14, and 0.4 respectively.

The resident correctly specified theory is evolutionarily fragile
against this misspecified theory. We first show that in every EZ with
$p=(1,0)$, the mutants must play $a_{2}$ in $G_{A}$ and $a_{1}$
in $G_{B}$. The mutants never choose $a_{3}$ in any situation in
EZ, since it is strictly dominated in every model in the theory. If
the mutants play $a_{1}$ in situation $G_{A}$, the residents must
best respond with $a_{3}$ in $G_{A}$ (as the residents correctly
learn the situation in every EZ) and the mutants learn $F_{A}$ in
$G_{A}$. But under $F_{A}$, the mutants believe they should deviate
to $a_{2}.$ If the mutants play $a_{2}$ in situation $G_{B},$ once
again the residents must best respond with $a_{3}$ in $G_{B}$ and
the mutants learn $F_{B}$ in $G_{B}.$ But under $F_{B},$ the mutants
believe they should deviate to $a_{1}.$ These arguments rule out
all other EZ behavior, so the mutants must play $a_{2}$ in $G_{A}$
and $a_{1}$ in $G_{B}$. The residents must best respond with $a_{2}$
in both situations. The mutants learn $F_{A}$ in $G_{A}$ and $F_{B}$
in $G_{B},$ so their actions are optimal in each situation given
their believed model. In this EZ,  mutants have strictly higher fitness
than the residents.
\end{example}
The next proposition generalizes this example and identifies a class
of environments where the mutants' ability to draw misinferences strictly
expands the scope for invasion against rationality. The idea is to
equip the mutants with the most beneficial misspecified theory against
the rational incumbents, featuring the same kind of ``illusion of
control'' bias as in the mutant theory from Example \ref{exa:only_theory_invades}.
There is one model $F_{G}$ for each situation $G\in\mathcal{G}$,
and all models in the misspecified theory stipulate that the mutant's
strategy alone fully determines her consequence, while the opponent's
action has no impact on her welfare. The dominant strategy under model
$F_{G}$ is the objective Stackelberg strategy in situation $G$.
We show that this misspecified theory allows the mutants to adopt
the optimal commitment situation-by-situation --- thus, the learning
channel is a mechanism that allows the mutants to tailor their commitment
to the true situation. By contrast, a mutant with only one model (i.e.,
an exogenous subjective preference) lacks the flexibility to play
differently in different situations. We provide conditions for when
such singleton theories cannot do better than the rationals due to
their inflexibility.

We introduce some notation to state the result. Given any situation
$G$, let $v_{G}^{\text{NE}}\in\mathbb{R}$ be the highest symmetric
Nash equilibrium payoff in the game with $G$, when agents choose
strategies from $\mathbb{A}$. For each $G$ and $a_{i}\in\mathbb{A}$,
we let $\underline{\text{BR}}(a_{i},G)$ be a rational best response
against the strategy $a_{i}$ in situation $G,$ breaking ties \emph{against}
the user of $a_{i}$. Let $\bar{v}_{G}\in\mathbb{R}$ be the Stackelberg
equilibrium payoff in situation $G$, with ties broken against the
Stackelberg leader, i.e.,

\begin{equation}
\bar{v}_{G}:=\max_{a_{i}}U_{i}(a_{i},\underline{\text{BR}}(a_{i},G),F^{\bullet}(G)).\label{eq:stackleberg}
\end{equation}

\noindent Call the strategy $\bar{a}_{G}$ that maximizes Equation
(\ref{eq:stackleberg}) the Stackelberg strategy in situation $G$.
We assume the Stackelberg strategy is unique in each situation, and
furthermore there is a unique rational best response to each Stackelberg
strategy $\bar{a}_{G}$ in each situation $G',$ where possibly $G\ne G'$.

For each correspondence $b:\mathbb{A}\rightrightarrows\mathbb{A}$,
let $v_{G}^{b}\in\mathbb{R}$ be defined as $i$'s lowest payoff across
all strategy profiles $(a_{i},a_{-i})$ such that $a_{i}\in b(a_{-i})$
and $a_{-i}$ is a rational response to $a_{i}$ in situation $G.$
(If no such profile exists, let $v_{G}^{b}=-\infty.)$ Interpret $v_{G}^{b}$
as the worst equilibrium payoff of an agent with the subjective best-response
correspondence $b$ when she plays against a rational opponent in
situation $G.$

We impose two identifiability conditions. The first condition says
under any strategy profile, a correctly specified agent can identify
the true situation. The second condition says playing the Stackelberg
strategy $\bar{a}_{G}$ in situation $G$ leads to different consequences
than playing the same strategy in a different situation $G'\ne G$,
provided the opponent chooses the rational best response to the strategy
(given the true situation).
\begin{defn}
\emph{Situation identifiability} is satisfied if for every $a_{i},a_{-i}\in\mathbb{A}$
and $G\ne G',$ we have $F^{\bullet}(a_{i},a_{-i},G)\ne F^{\bullet}(a_{i},a_{-i},G').$
\emph{Stackelberg identifiability} is satisfied if whenever $G\ne G'$
and $a_{-i}$, $a_{-i}'$ are rational best responses to $\bar{a}_{G}$
in situations $G$ and $G'$, we have $F^{\bullet}(\bar{a}_{G},a_{-i},G)\ne F^{\bullet}(\bar{a}_{G},a_{-i}',G')$.
\end{defn}
We can now state our result.
\begin{thm}
\label{thm:theoryneeded} Suppose $\lambda=0$, there are finitely
many situations, and there is a symmetric Nash equilibrium in $\mathbb{A}\times\mathbb{A}$
for every situation $G$.
\begin{enumerate}
\item If there is no point $(u_{G})_{G\in\mathcal{G}}$ in the convex hull
of $\{(v_{G}^{b})_{G\in\mathcal{G}}\mid b:\mathbb{A}\rightrightarrows\mathbb{A}\}$
with the property that $u_{G}\ge v_{G}^{\text{NE}}$ for every $G\in\mathcal{G},$
then there exists a full-support distribution $q\in\Delta(G)$ so
that the correctly specified theory is not evolutionarily fragile
against any singleton theory.
\item If $v_{G}^{\text{NE}}<\bar{v}_{G}$ for some $G$, situation identifiability
and Stackelberg identifiability hold, and there are finitely many
strategies, then there exists a theory $\hat{\Theta}$ such that the
correctly specified theory is evolutionarily fragile against $\hat{\Theta}$
under any full-support distribution $q\in\Delta(G)$.
\end{enumerate}
\end{thm}
In particular, when both conditions of Theorem \ref{thm:theoryneeded}
are satisfied, the correctly specified theory is evolutionarily fragile
against some mutant theory, but not evolutionarily fragile against
any \emph{singleton} mutant theory. In these environments, the ability
to draw inference is a necessary condition for an invading mutant
to displace the rational incumbent.  Hence, this result shows that
mutants with misspecified theories cannot in general be represented
simply as mutants with fixed subjective best-response functions. Another
implication of this result is that whenever the best Nash equilibrium
payoff is worse than the Stackelberg payoff in some situation, then
some misspecified theory can defeat rationality.

We can verify that Example \ref{exa:only_theory_invades} satisfies
both sets of conditions of Theorem \ref{thm:theoryneeded}. For any
$b:\mathbb{A}\rightrightarrows\mathbb{A},$ we have $v^{b}\le(0.3,0.55)$
coordinate-wise. If $b(a_{3})\ni a_{3},$ then $v^{b}\le(0.3,0.3).$
If $b(a_{3})\ni a_{2},$ then $v_{G_{B}}^{b}\le0.14.$ If $b(a_{3})\ni a_{1},$
then $v_{G_{A}}^{b}\le0.1$ This shows for every point $(u_{G})_{G\in\mathcal{G}}$
in the convex hull of $\{(v_{G}^{b})_{G\in\mathcal{G}}\mid b:\mathbb{A}\to\mathbb{A}\}$,
we have $u_{G_{A}}+u_{G_{B}}\le0.65.$ But $v_{G_{A}}^{\text{NE}}+v_{G_{B}}^{\text{NE}}=0.7,$
so $v^{\text{NE}}$ is not weakly coordinate-wise dominated by any
point in the convex hull of $\{(v_{G}^{b})_{G\in\mathcal{G}}\mid b:\mathbb{A}\rightrightarrows\mathbb{A}\}$.
Also, the best Nash payoff $v_{G_{B}}^{\text{NE}}$ of 0.4 is strictly
lower than the Stackelberg payoff $\bar{v}_{G_{B}}$ of 0.5 in situation
$G_{B}$, and it is straightforward to verify that situation identifiability
and Stackelberg identifiability hold for Example \ref{exa:only_theory_invades}.
Therefore, Theorem \ref{thm:theoryneeded} implies that there is some
$q\in\Delta(G)$ where the correctly specified theory is evolutionarily
fragile against some misspecified theory, but it is not evolutionarily
fragile against any singleton theory. In fact, the analysis of Example
\ref{exa:only_theory_invades} shows $q=(0.5,0.5)$ is one such distribution.

\subsection{\label{subsec:Stability-Reversals}Stability Reversals}

Stability reversal refers to a strong kind of multiplicity in the
relative stability of two theories $\Theta_{A}$ and $\Theta_{B}$
under uniform matching. For this section, suppose for simplicity that
the environment only has one situation, which we can think of as a
fixed stage game. In an EZ, the fitness of a theory is the objective
expected payoffs of its adherents, where this expectation averages
across expected payoffs in matches against each of the two groups.
Let a theory's \emph{conditional fitness against group $g$} refer
to the expected payoff of the theory's adherents in matches against
group $g.$
\begin{defn}
Suppose $|\mathcal{G}|=1.$ Two theories $\Theta_{A},\Theta_{B}$
exhibit \emph{stability reversal} if (i) in every EZ with $\lambda=0$
and $(p_{A},p_{B})=(1,0),$ $\Theta_{A}$ has strictly higher conditional
fitness than $\Theta_{B}$ against group A opponents and against group
B opponents, but also (ii) in every EZ with $\lambda=0$ and $(p_{A},p_{B})=(0,1),$
$\Theta_{B}$ has strictly higher fitness than $\Theta_{A}$.
\end{defn}
The first part in the definition of stability reversal is stronger
than requiring $\Theta_{A}$ to be evolutionarily stable against $\Theta_{B}$.
It imposes the more stringent condition that $\Theta_{A}$ outperforms
$\Theta_{B}$ not only on average, but also conditional on the opponent's
group. The next result states that stability reversal cannot take
place if both theories are singletons (i.e., if we are in the world
of preference evolution).
\begin{prop}
\label{prop:no_reversal}Two singleton theories (i.e., two subjective
preferences in the stage game) cannot exhibit stability reversal in
any stage game.
\end{prop}
Stability reversal is unique to the world of theory evolution. For
an example, consider a two-player investment game where player $i$
chooses an investment level $a_{i}\in\{1,2\}.$ A random productivity
level $P$ is realized according to $b^{\bullet}(a_{i}+a_{-i})+\epsilon$
where $\epsilon$ is a zero-mean noise term, $b^{\bullet}>0$. Player
$i$ gets $a_{i}\cdot P-1_{\{a_{i}=2\}}\cdot c$. So $P$ determines
the marginal return on investment, and $c>0$ is the cost for choosing
the higher investment level, with the cost of the lower investment
level normalized to 0. At the end of the game, players observe $y=(a_{i},a_{-i},P).$
The payoff matrix below displays the objective expected payoffs for
different investment profiles.
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
 & 1 & 2\tabularnewline
\hline
\hline
1 & $2b^{\bullet},2b^{\bullet}$ & $3b^{\bullet},6b^{\bullet}-c$\tabularnewline
\hline
2 & $6b^{\bullet}-c,3b^{\bullet}$ & $8b^{\bullet}-c,8b^{\bullet}-c$\tabularnewline
\hline
\end{tabular}
\par\end{center}
\begin{condition}
\label{cond:medium_cost}$5b^{\bullet}<c<6b^{\bullet}$.
\end{condition}
Condition \ref{cond:medium_cost} ensures that $a_{i}=1$ is a strictly
dominant strategy in the stage game, and the investment profile (2,2)
Pareto dominates the investment profile (1,1). Higher investment has
a positive externality as it also increases opponent's productivity.
Consider two theories in the society. Theory $\Theta_{A}$ is a correctly
specified singleton -- its adherents understand how investment profiles
translate into distributions over productivity. Theory $\Theta_{B}$
wrongly stipulates $P=b(x_{i}+x_{-i})-m+\epsilon$, where $m>0$ is
a fixed parameter of the theory and $b\in\mathbb{R}$ is a parameter
that the adherents infer. We require the following condition, which
is satisfied whenever $m>0$ is large enough --- that is, $\Theta_{B}$
is sufficiently misspecified.
\begin{condition}
\label{cond:large_misspec} $c<4b^{\bullet}+\frac{1}{3}m$ and $c<5b^{\bullet}+\frac{1}{4}m.$
\end{condition}
We show that in contrast to the impossibility result when all theories
are singletons, in this example theories $\Theta_{A}$ and $\Theta_{B}$
exhibit stability reversal.
\begin{example}
\label{exa:stability_reversal_example}In the investment game, under
Condition \ref{cond:medium_cost} and Condition \ref{cond:large_misspec},
$\Theta_{A}$ and $\Theta_{B}$ exhibit stability reversal.
\end{example}
The idea is that the adherents of $\Theta_{B}$ overestimate the complementarity
of investments, and this overestimation is more severe when they face
data generated from lower investment profiles. As a result, the match
between $\Theta_{A}$ and $\Theta_{B}$ plays out in a different way
depending on which theory is resident: it results in the investment
profile $(1,2)$ when $\Theta_{A}$ is resident, but results in $(1,1)$
when $\Theta_{B}$ is resident.

Let $b^{*}(a_{i},a_{-i})$ solve $\min_{b\in\mathbb{R}}D_{KL}(F^{\bullet}(a_{i},a_{-i})\parallel\hat{F}(a_{i},a_{-i};b,m))),$
where $F^{\bullet}(a_{i},a_{-i})$ is the objective distribution over
observations under the investment profile $(a_{i},a_{-i}),$ and $\hat{F}(a_{i},a_{-i};b,m)$
is the distribution under the same investment profile in the model
where productivity is given by $P=b(x_{i}+x_{-i})-m+\epsilon$. We
find that $b^{*}(a_{i},a_{-i})=b^{\bullet}+\frac{m}{a_{i}+a_{-i}}$.
That is, adherents of $\Theta_{B}$ end up with different beliefs
about the game parameter $b$ depending on the behavior of their typical
opponents, which in turn affects how they respond to different rival
investment levels. Stability reversal happens because when $\Theta_{A}$
is resident and the adherents of $\Theta_{B}$ always meet opponents
who play $a_{i}=1,$ they end up with a more distorted belief about
the fundamental than when $\Theta_{B}$ is resident.

In this example, stability reversal happens because the misspecified
agents hold different beliefs about a stage-game parameter depending
on which theory is resident. Also, note the stage game involves non-trivial
strategic interaction between the players --- the complementarity
in investment levels implies an agent's best response may vary with
the rival's strategy. Both of these turn out to be necessary conditions
for stability reversal in general stage games.
\begin{defn}
A theory $\Theta$ is \emph{strategically independent} if for all
$\mu\in\Delta(\Theta)$, $\underset{a_{i}\in\mathbb{A}}{\arg\max}\ U_{i}(a_{i},a_{-i};\mu)$
is the same for every $a_{-i}\in\mathbb{A}.$
\end{defn}
The adherents of a strategically independent theory believe that while
opponent's action may affect their utility, it does not affect their
best response.
\begin{prop}
\label{prop:reversal_inference_channel}Suppose $|\mathcal{G}|=1$,
suppose $\Theta_{A},\Theta_{B}$ exhibit stability reversal and $\Theta_{A}$
is the correctly specified singleton theory. Then, the beliefs that
the adherents of $\Theta_{B}$ hold in all EZs with $p=(1,0)$ and
the beliefs they hold in all EZs with $p=(0,1)$ form disjoint sets.
Also, $\Theta_{B}$ is not strategically independent.
\end{prop}
The first claim of Proposition \ref{prop:reversal_inference_channel}
shows that stability reversal must operate through the learning channel.
So in particular, it cannot happen if the group B agents simply have
a different subjective preference in the stage game. The second claim
shows that stability reversal can only happen if the misspecified
agents respond differently to different rival play. In particular,
it cannot happen in decision problems.

\subsection{\label{subsec:Non-Monotonic-Stability-in}Non-Monotonic Stability
in Matching Assortativity}

We now turn to the role of matching assortativity on the stability
of theories. For this section, again assume there is only one situation
for simplicity. In the world of preference evolution, fitness of a
group is linear in matching assortativity. This  implies that if a
theory $\Theta_{A}$ is evolutionarily stable against a theory $\Theta_{B}$
both under uniform matching $(\lambda=0)$ and under perfectly assortative
matching $(\lambda=1$), then the same must also hold under any intermediate
level of assortativity $\lambda\in(0,1).$
\begin{prop}
\label{prop:no_non_mono_lambda}Suppose $\Theta_{A},\Theta_{B}$ are
singleton theories (i.e., subjective preferences in the stage game)
and $\Theta_{A}$ is evolutionarily stable against $\Theta_{B}$ with
$\lambda$-matching for both $\lambda=0$ and $\lambda=1.$ Then,
$\Theta_{A}$ is also evolutionarily stable against $\Theta_{B}$
with $\lambda$-matching for any $\lambda\in[0,1]$.
\end{prop}
This result does not always hold with non-singleton general theories.
We use an example to show that stability need not be monotonic in
matching assortativity. In this example, a correctly specified singleton
theory is evolutionarily stable against another misspecified theory
both when $\lambda=0$ and when $\lambda=1$, but evolutionarily fragile
for some intermediate values of $\lambda.$

Consider a stage game where each player chooses an action from $\{a_{1},a_{2},a_{3}\}.$
Every player then receives a random prize, $y\in\{g,b\}$, which are
worth utilities $\pi(g)=1,$ $\pi(b)=0.$ The payoff matrix below
displays the objective expected utilities associated with different
action profiles, which also correspond to the probabilities that the
row and column players receive the good prize $g$.
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
 & $a_{1}$ & $a_{2}$ & $a_{3}$\tabularnewline
\hline
\hline
$a_{1}$ & 0.25, 0.25 & 0.50, 0.20 & 0.70, 0.15\tabularnewline
\hline
$a_{2}$ & 0.20, 0.50 & 0.40, 0.40 & 0.40, 0.20\tabularnewline
\hline
$a_{3}$ & 0.15, 0.70 & 0.20, 0.40 & 0.20, 0.20\tabularnewline
\hline
\end{tabular}
\par\end{center}

Let $\Theta_{A}$ be the correctly specified singleton theory. The
action $a_{1}$ is strictly dominant under the objective payoffs,
so an adherent of $\Theta_{A}$ always plays $a_{1}$ in all matches.
Let $\Theta_{B}$ be a misspecified theory $\Theta_{B}=\{F_{H},F_{L}\}$.
Each model $F_{H},F_{L}$ stipulates that the prize $g$ is generated
the probabilities in the following table, where $b$ and $c$ are
parameters that depend on the model. The model $F_{H}$ has $(b,c)=(0.8,0.2)$
and $F_{L}$ has $(b,c)=(0.1,0.4).$
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
 & $a_{1}$ & $a_{2}$ & $a_{3}$\tabularnewline
\hline
\hline
$a_{1}$ & 0.10, 0.10 & 0.10, $c$ & 0.10, 0.15\tabularnewline
\hline
$a_{2}$ & $c,$ 0.10 & $b,b$ & $b,$ 0.20\tabularnewline
\hline
$a_{3}$ & 0.15, 0.10 & 0.20, $b$ & 0.20, 0.20\tabularnewline
\hline
\end{tabular}
\par\end{center}

The learning channel for the biased mutants leads the correctly specified
theory to have non-monotonic evolutionarily stability in terms of
matching assortativity.
\begin{example}
\label{exa:non_mono_example}In this stage game, $\Theta_{A}$ is
evolutionarily stable against $\Theta_{B}$ under $\lambda$-matching
when $\lambda=0$ and $\lambda=1,$ but it is also evolutionarily
fragile under $\lambda$-matching when $\lambda\in(\lambda_{l},\lambda_{h})$,
where $0<\lambda_{l}<\lambda_{h}<1$ are $\lambda_{l}=0.25$, $\lambda_{h}\approx0.56$.
\end{example}
To understand the intuition, examine the match between two adherents
of $\Theta_{B}.$ If they believe in $F_{H}$, they will play the
action profile $(a_{2},a_{2})$ and generate the objective payoff
profile $(0.4,0.4)$, a Pareto improvement compared to the correctly
specified outcome $(a_{1},a_{1})$. The problem is that the data generated
from the $(a_{2},a_{2})$ profile provides a better fit for $F_{L}$
than $F_{H}$, since the objective 40\% probability of getting prize
$g$ is closer to $F_{L}$'s conjecture of 10\% than $F_{H}$'s conjecture
of 80\%. A belief in $F_{H}$ --- and hence the profile $(a_{2},a_{2})$
--- cannot be sustained if the mutants only play each other. On the
other hand, when an adherent of $\Theta_{B}$ plays a correctly specified
$\Theta_{A}$ adherent, both models $F_{H}$ and $F_{L}$ prescribe
a best response of $a_{2}$ against the $\Theta_{A}$ adherent's play
$a_{1}.$ The data generated from the $(a_{2},a_{1})$ profile lead
biased agents to the model $F_{H}$ that enables cooperative behavior
within the mutant community. But, these matches against correctly
specified opponents harm the mutant's welfare, as they only get an
objective payoff of 0.2.

Therefore, the most advantageous interaction structure for the mutants
is one where they can calibrate the model $F_{H}$ using the data
from matches against correctly specified opponents, then extrapolate
this optimistic belief about $b$ to coordinate on $(a_{2},a_{2})$
in matches against fellow mutants. This requires the mutants to match
with intermediate assortativity. Figure \ref{fig:3x3} depicts the
equilibrium fitness of the mutant theory $\Theta_{B}$ as a function
of assortativity. While payoffs of $\Theta_{B}$ adherents increase
in $\lambda$ at first, eventually they drop when mutant-vs-mutant
matches become sufficiently frequent that a belief in $F_{H}$ can
no longer be sustained. The preference evolution framework does not
allow this non-linear and even non-monotonic change in fitness with
respect to $\lambda,$ which the theory evolution framework accommodates.

\begin{figure}
\begin{centering}
\includegraphics[scale=0.5]{nonlinear_lambda}
\par\end{centering}
\vspace{-13bp}

\caption{\label{fig:3x3}The EZ fitness of $\Theta_{B}$ for different values
of matching assortativity $\lambda$ when $p_{B}=0$. (The EZ fitness
of the resident theory $\Theta_{A}$ is always 0.25.) In the blue
region, there is a unique EZ where the adherents of $\Theta_{B}$
infer $F_{H}$ and receive linearly increasing average payoffs across
all matches as $\lambda$ increases. In the red region, there is an
EZ where the adherents of $\Theta_{B}$ infer $F_{L}$ and receive
payoff 0.2 in all matches, regardless of $\lambda$.}
\end{figure}

While this analysis fixes the population sizes at $p_{A}=1,$ $p_{B}=0$
and varies the matching assortativity, we find a similar conclusion
if we fix the matching assortativity and vary the population sizes.
For instance, fix assortativity at $\lambda=0.5$ and consider the
equilibrium fitness of the misspecified theory $\Theta_{B}$ as a
function of its population share $x\in[0,1].$ The probability that
a misspecified agent meets another misspecified agent is $0.5+0.5x$,
which is the same as when the $\Theta_{A}$ theory is dominant and
$\lambda=0.5+0.5x$. By this equivalence, it is easy to show that
the fitness of $\Theta_{B}$ is non-monotonic in its population share.
Specifically, $\Theta_{B}$ has strictly higher EZ fitness than $\Theta_{A}$
whenever its share is lower than 0.128, but strictly lower EZ fitness
when its share is high enough.

\section{\label{sec:LQN}Higher-Order Misspecifications in LQN Games}

Our framework facilitates evolutionary analysis of new applications.
We illustrate this with a case study: the selection of perceptions
about the information structure in linear quadratic normal (LQN) games.
LQN games have been used to study comparative statics of equilibrium
outcomes with respect to changes in information (e.g., \citet{bergemann2013robust}).
Here, we exploit their tractability to study misspecifications about
the correlation in information between different players. While we
recover some familiar intuition for when ``mistakes'' might be selected,
our new insights relate to how learning interacts with these forces.
To maintain focus and streamline intuition, in the main text we focus
on a Cournot duopoly application. In Appendix \ref{subsec:LQNGeneral},
we show that the key ideas extend to incomplete-information games
more generally.

\subsection{Stage Game and Misperceptions of Information Structure}

At the start of the stage game, Nature's type (i.e., a demand state)
$\omega$ is drawn from $\mathcal{N}(0,\sigma_{\omega}^{2})$, where
$\mathcal{N}(\mu,\sigma^{2})$ is the normal distribution with mean
$\mu$ and variance $\sigma^{2}.$ Each of the two players $i$ (i.e.,
firms) receives a private signal $s_{i}=\omega+\epsilon_{i},$ then
chooses an action $q_{i}\in\mathbb{R}$ (i.e., a quantity). Market
price is then realized according to $P=\omega-r^{\bullet}\cdot\frac{1}{2}(q_{1}+q_{2})+\zeta$,
where $\zeta\sim\mathcal{N}(0,(\sigma_{\zeta}^{\bullet})^{2})$ is
an idiosyncratic price shock that is independent of all the other
random variables. Firm $i$'s profit in the game is $q_{i}P-\frac{1}{2}q_{i}^{2}.$

The stage game is parametrized by the strictly positive terms $\sigma_{\omega}^{2},r^{\bullet},$
and $(\sigma_{\zeta}^{\bullet})^{2},$ which represent variance in
market demand, the elasticity of market price with respect to average
quantity supplied, and the variance of price shocks. These parameters
remain constant through all matches (so, there is only one ``situation,''
in the language of Section \ref{sec:Environment-and-Stability}).
But in every match, demand state $\omega,$ signals $(s_{i})$, and
price shock $\zeta$ are redrawn, independently across matches. The
environment can be interpreted as a market with daily fluctuations
in demand, but the fluctuations are generated according to a fixed
set of fundamental parameters.

Note that market prices and quantity choices may be positive or negative.
To interpret, when $P>0,$ the market pays for each unit of good supplied,
and market price decreases in total supply. When $P<0,$ the market
pays for disposal of the good. Firms make money by submitting negative
quantities, which represent offers to remove the good from the market.
The per-unit disposal fee decreases as the firms offer to dispose
more. The cost $\frac{1}{2}q_{i}^{2}$ represents either a convex
production cost or a convex disposal cost, depending on the sign of
$q_{i}.$

We now turn to the information structure of the stage game --- that
is, the joint distribution of $(\omega,s_{i},s_{-i}).$ The firms'
signals $s_{i}=\omega+\epsilon_{i}$ are conditionally correlated
given $\omega.$ The error terms $\epsilon_{i}$ are generated by
$\epsilon_{i}=\frac{\kappa}{\sqrt{\kappa^{2}+(1-\kappa)^{2}}}z+\frac{1-\kappa}{\sqrt{\kappa^{2}+(1-\kappa)^{2}}}\eta_{i},$
where $\eta_{i}\sim\mathcal{N}(0,\sigma_{\epsilon}^{2})$ is the idiosyncratic
component of the error generated i.i.d. across $i,$ and $z\sim\mathcal{N}(0,\sigma_{\epsilon}^{2})$
is the common component for both $i.$ Here, $\kappa\in[0,1]$ parametrizes
the conditional correlation of the two firms' signals. Higher $\kappa$
leads to an information structure with higher conditional correlation.
When $\kappa=0,$ $s_{i}$ and $s_{-i}$ are conditionally uncorrelated
given the state. When $\kappa=1,$ we always have $s_{i}=s_{-i}$.
The functional form of $\epsilon_{i}$ ensures the variance of the
signals $\text{Var}(s_{i})$ remains constant across all values of
$\kappa.$

We consider a family of misspecifications about the information structure
parametrized by misperceptions of $\kappa$. The objective information
structure is given by $\kappa=\kappa^{\bullet}$. Note that a misspecified
information structure associated with a wrong $\kappa$ leads to a
higher-order misspecification about the state $\omega$ in the stage
game. Suppose agents are correct about the distributions of $\omega,$
$\eta_{i},$ and $z$. Write $\mathbb{E}_{\kappa}$ for expectation
under the information structure with correlational parameter $\kappa.$
Then $\mathbb{E}_{\kappa}[\omega\mid s_{i}]$ is the same for all
$\kappa$ --- in particular, even an agent who believes in some $\kappa\ne\kappa^{\bullet}$
makes a correct first-order inference about the expectation of the
market demand, given her own information. But, one can show (Lemma
\ref{lem:kappa} in Appendix \ref{subsect:LQNEZs}) there exists a
strictly increasing and strictly positive function $\psi(\kappa)$
so that $\mathbb{E}_{\kappa}[s_{-i}\mid s_{i}]=\psi(\kappa)\cdot s_{i}$
for all $s_{i}\in\mathbb{R},\kappa\in[0,1].$ The misspecified agent
holds a wrong belief about the rival's signal, and thus a wrong belief
about the rival's belief about $\omega.$

In experiments, subjects often do not form accurate beliefs about
the beliefs of others. We draw a connection between the misperception
we study and such statistical biases:
\begin{defn}
Let $\tilde{\kappa}$ be a player's perceived $\kappa$. A player
suffers from \emph{correlation neglect} if $\tilde{\kappa}<\kappa^{\bullet}$.
A player suffers from \emph{projection bias} if $\tilde{\kappa}>\kappa^{\bullet}$.
\end{defn}
\noindent Under correlation neglect, agents believe signals are more
independent from one another than they really are. Under projection
bias, agents ``project'' their own information onto others and exaggerate
the similarity between others' signals and their own signals. We are
agnostic about the origin of these misspecifications about correlation,
whether they emerge due to cognitive biases or more complex mechanisms.\footnote{For example, \citet*{hansen2021algorithmic} show that multiple agents
simultaneously conducting algorithmic price experiments in the same
market may generate correlated information which get misinterpreted
as independent information, a form of correlation neglect for firms.
\citet{goldfarb2019} structurally estimate a model of thinking cost
and find that bar owners over-extrapolate the effect of today's weather
shock on future profitability.} We instead ask whether such misspecifications could persist in the
society on an evolutionary basis once they appear.

\subsection{Formalizing Strategies and Theories}

This environment fits into the formalism from Section \ref{sec:Environment-and-Stability}
as follows. A strategy is a function $Q_{i}:\mathbb{R}\to\mathbb{R}$
that assigns a quantity $Q_{i}(s_{i})$ to every signal $s_{i}$.
We say a strategy is \emph{linear} if there exists an $\alpha_{i}\ge0$
so that $Q_{i}(s_{i})=\alpha_{i}s_{i}$ for every $s_{i}\in\mathbb{R}.$
Since the best response to any linear strategy is linear, regardless
of the agent's belief about the correlation parameter and market price
elasticity (Lemma \ref{lem:BR} in Appendix \ref{subsect:LQNEZs}),
we restrict attention to linear strategies and let $\mathbb{A}=[0,\bar{M}_{\alpha}]$
for $\bar{M}_{\alpha}<\infty,$ with $\alpha_{i}\in\mathbb{A}$ referring
to strategy $Q_{i}(s_{i})=\alpha_{i}s_{i}$.

We suppose all parameters of the stage game are common knowledge except
for $r^{\bullet},\kappa^{\bullet},$ and $\sigma_{\zeta}^{\bullet}$.
Our interest is in theories that are dogmatic and possibly wrong about
$\kappa,$ but allow agents to make inferences about $r$ and $\sigma_{\zeta}$.
We set the consequence space to be $\mathbb{Y}=\mathbb{R}^{3},$ where
$y=(s_{i},q_{i},P)\in\mathbb{Y}$ shows the agent's signal, quantity
choice, and the market price. Consequence $y$ delivers utility $\pi(y):=q_{i}P-\frac{1}{2}q_{i}^{2}.$
Theories are parametrized by $\kappa,$ writing $\Theta(\kappa):=\{F_{r,\kappa,\sigma_{\zeta}}:r\in[0,\bar{M}_{r}],\sigma_{\zeta}\in[0,\bar{M}_{\sigma_{\zeta}}]\}$
for some $\bar{M}_{r},\bar{M}_{\sigma_{\zeta}}<\infty.$ So, each
$\Theta(\kappa)$ is a set of models indexed by the parameters $(r,\kappa,\sigma_{\zeta}$),
all reflecting a dogmatic belief in the correlation parameter $\kappa$.
Each $F_{r,\kappa,\sigma_{\zeta}}:\mathbb{A}\times\mathbb{A}\to\Delta(\mathbb{Y})$
is such that $F_{r,\kappa,\sigma_{\zeta}}(\alpha_{i},\alpha_{-i})$
gives the distribution over $i$'s consequences in a stage game with
parameters $(r,\kappa,\sigma_{\zeta})$, when $i$ uses the linear
strategy $\alpha_{i}$ against an opponent using linear strategy $\alpha_{-i}.$

While agents learn about both $r$ and $\sigma_{\zeta}$, it is their
(mis)inferences about the elasticity $r$ that drives the main results.
Since each firm's profit is linear in the market price, an agent's
belief about the variance of the idiosyncratic price shock does not
change her expected payoffs or behavior. The parameter $\sigma_{\zeta}$
absorbs changes in the variance of market price, creating significant
tractability. To make inferences about $r$, it is only necessary
to consider the mean of the market price in the data, not its variance.

To guarantee EZ existence as to make statements about evolutionary
stability, we have assumed that the space of feasible linear strategies
$\alpha_{i}\in[0,\bar{M}_{\alpha}]$ and the domain of inference over
game parameters $r\in[0,\bar{M}_{r}],\sigma_{\zeta}\in[0,\bar{M}_{\sigma_{\zeta}}]$
are compact. To ensure behavior and beliefs are interior, we will
take the bounds $\bar{M}_{\alpha},\bar{M}_{r},\bar{M}_{\sigma_{\zeta}}$
are finite but sufficiently large, introducing the following shorthand:
\begin{notation}
A result is said to hold ``\emph{with high enough price volatility
and large enough strategy space and inference space}'' if, whenever
the strategy space $[0,\bar{M}_{\alpha}]$ has $\bar{M}_{\alpha}\ge\frac{1/\sigma_{\epsilon}^{2}}{1/\sigma_{\epsilon}^{2}+1/\sigma_{\omega}^{2}}$,
there exist $0<L_{1},L_{2},L_{3}<\infty$ so that for any objective
game $F^{\bullet}$ with $(\sigma_{\zeta}^{\bullet})^{2}\ge L_{1}$
and with theories where the parameter spaces $r\in[0,\bar{M}_{r}],$
$\sigma_{\zeta}\in[0,\bar{M}_{\sigma_{\zeta}}]$ are such that $\bar{M}_{\sigma_{\zeta}}^{2}\ge(\sigma_{\zeta}^{\bullet})^{2}+L_{2}$
and $\bar{M}_{r}\ge L_{3}$, the result is true.
\end{notation}
Our analysis relies on a number of technical lemmas, which we defer
to Appendix \ref{sec:LQNMore}. We show, for example, that the set
of EZs is non-empty and it is upper hemicontinuous in population sizes.
We also derive there closed-form expressions for the best-fitting
inference and optimal behavior of misspecified agents.

\subsection{The Impact of Misspecification: Some Intuition \label{subsec:CournotIntuition}}

Before presenting our results on the fragility of correct specifications,
we briefly describe what happens when players entertain a dogmatically
misspecified view of $\kappa$.

Most importantly, an agent's inference about $r$ is strictly decreasing
in her belief about the correlation parameter $\kappa.$ To understand
why, assume player $i$ uses the linear strategy $\alpha_{i}$ and
player $-i$ uses the linear strategy $\alpha_{-i}$. After receiving
a private signal $s_{i}$, player $i$ expects to face a price distribution
with a mean that is linearly increasing in $\mathbb{E}[s_{-i}\mid s_{i}]$,
which in turn is linearly increasing in $s_{i}$ (see Appendix \ref{subsect:LQNEZs}
for more details). Now, under projection bias $\kappa>\kappa^{\bullet},$
$\mathbb{E}_{\kappa}[s_{-i}\mid s_{i}]$ is excessively steep in $s_{i}$,
since the correlation is higher. For example, following a large and
positive $s_{i},$ the agent overestimates the similarity of $-i$'s
signal and wrongly predicts that $-i$ must also choose a very high
quantity, and thus becomes surprised when market price remains high.
As a result, the agent then wrongly infers that the market price elasticity
must be low. Therefore, in order to rationalize the average market
price conditional on own signal, an agent with projection bias must
infer $r<r^{\bullet}$. For similar reasons, an agent with correlation
neglect infers $r>r^{\bullet}.$

The fact that projection biased players believe the elasticity of
demand is lower than the truth suggests that they will behave more
aggressively than correctly specified players---with the converse
holding for correlation neglect players. Intuitively, if price reacts
less to quantity, then players should price more aggressively. While
this does turn out to be true---and drives many of the results below---things
are more subtle because increasing $\kappa$ has an \emph{a priori}
ambiguous impact on the agent's equilibrium aggressiveness. In fact,
in our characterization, we show that increasing $\kappa$ but holding
fixed the player's belief about price elasticity has the direct effect
of \emph{lowering} aggression. The results below show that the indirect
effect through the learning channel dominates, and the evolutionary
stability of correlational errors are driven by this channel. We show
in Section \ref{subsec:learningkey} that the conclusions are reversed
when we shut down the learning channel.

\subsection{Uniform Matching $(\lambda=0$) and Projection Bias}

We now discuss the evolutionary instability of correctly specified
beliefs about the information structure. We first study a society
where agents are uniformly matched, showing that a correctly specified
$\kappa$ will be evolutionarily fragile against some amount of projection
bias. A small amount of projection bias leads the mutants to have
higher payoffs in the resident-vs-mutant matches than the residents'
payoffs in the resident-vs-resident matches. As referenced above,
our proof uses our characterization of the \emph{asymmetric} equilibrium
strategy profile in matches between the correctly specified residents
and the projection-biased mutants.
\begin{prop}
\label{prop:LQN_lambda_0}Let $r^{\bullet}>0,$ $\kappa^{\bullet}\in[0,1]$
be given. With high enough price volatility and large enough strategy
space and inference space, there exist $\underline{\kappa}<\kappa^{\bullet}<\bar{\kappa}$
so that in societies with two theories $(\Theta_{A},\Theta_{B})=(\Theta(\kappa^{\bullet}),\Theta(\kappa))$
where $\kappa\in[\underline{\kappa},\bar{\kappa}]$, there is a unique
 EZ with uniform matching ($\lambda=0$) and $(p_{A},p_{B})=(1,0)$.
The equilibrium fitness of $\Theta(\kappa)$ is strictly higher than
that of $\Theta(\kappa^{\bullet})$ if $\kappa>\kappa^{\bullet}$,
and strictly lower if $\kappa<\kappa^{\bullet}$.
\end{prop}
 In other words, the correct specification is evolutionarily fragile
against a small amount of projection bias. We illustrate this in Figure
\ref{fig:uniformFitness}. Around the true $\kappa^{\bullet}=0.3$,
the payoff of the mutant increases as their $\kappa$ increases. However,
in this example, misperception only helps the mutant to a point, and
we see that the correct specification will in fact be evolutionarily
stable against sufficiently large $\kappa$.

\begin{figure}
\begin{centering}
\includegraphics[scale=0.4]{fitness_uniform}
\par\end{centering}
\vspace{-15bp}

\caption{\label{fig:uniformFitness}Fitness of the mutant theory as a function
of $\kappa$, with a correctly specified resident theory and uniform
matching. In this numerical example, $\kappa^{\bullet}=0.3$, $r^{\bullet}=1$,
$\sigma_{\omega}^{2}=\sigma_{\epsilon}^{2}=1$.}
\end{figure}

The intuition for this result follows from the intuition outlined
in Section \ref{subsec:CournotIntuition}--- projection bias generates
a commitment to aggression as it leads the biased agents to under-infer
market price elasticity. It is well known that in Cournot oligopoly
games, such commitment can be beneficial. For instance, if quantities
are chosen sequentially, the first mover obtains a higher payoff compared
to the case where quantities are chosen simultaneously. A similar
force is at work here, but the source of the commitment is different.
Misspecification about signal correlation leads to misinference about
$r^{\bullet}$, which causes the mutants to credibly respond to their
opponents' play in an overly aggressive manner. The rational residents,
who can identify the mutants in the population, back down and yield
a larger share of the surplus. While projection bias is beneficial
in small measure, it is also intuitive that excessive aggression would
be detrimental as well, as overproduction can be individually suboptimal.

\subsection{Fully Assortative Matching ($\lambda=1$) and Correlation Neglect}

We obtain the opposite result with perfectly assortative matching:
evolutionary stability now selects for theories with correlation neglect.
The fragility of the correct specification is even starker here, as
any level of correlation neglect leads to higher equilibrium fitness.
\begin{prop}
\label{prop:LQN_lambda_1}Let $r^{\bullet}>0,$ $\kappa^{\bullet}\in[0,1]$
be given. With high enough price volatility and large enough strategy
space and inference space, in societies with two theories $(\Theta_{A},\Theta_{B})=(\Theta(\kappa_{A}),\Theta(\kappa_{B}))$
where $\kappa_{A}\le\kappa_{B}$, the fitness of $\Theta_{A}$ is
weakly higher than that of $\Theta_{B}$ in every EZ with any population
proportion $p$ and perfectly assortative matching ($\lambda=1$).
\end{prop}
In other words, for any $\kappa_{A}<\kappa_{B},$ theory $\Theta(\kappa_{A})$
is evolutionarily stable against theory $\Theta(\kappa_{B}).$ Specializing
to $\kappa_{B}=\kappa^{\bullet}$, we conclude that the correct specification
is evolutionarily fragile against any level of correlation neglect.

Essentially, correlation neglect leads to the opposite effect of projection
bias: agents over-infer market price elasticity, enabling commitment
to more cooperative behavior (i.e., linear strategies with a smaller
coefficient $\alpha_{i}$). Rational opponents would take advantage
of such agents, but biased agents never match up against rational
opponents in a society with perfectly assortative matching. In the
uniform matching case, projection bias leads to higher payoff for
the mutant at the expense of the rational opponent's payoff. But with
perfectly assortative matching, correlation neglect Pareto improves
both biased agents' payoffs. This contrast is illustrated in Figure
\ref{fig:assortativeFitness}, which uses the same parameter choices
as Figure \ref{fig:assortativeFitness}. Here, we see that the misspecified
agents' payoffs are \emph{decreasing} in $\kappa$ around the $\kappa^{\bullet}$.
Also, mutant fitness is monotonically larger when their $\kappa$
is lower, as implied by Proposition \ref{prop:LQN_lambda_1}.

\begin{figure}
\begin{centering}
\includegraphics[scale=0.4]{fitness_assortative}
\par\end{centering}
\vspace{-15bp}

\caption{\label{fig:assortativeFitness}Fitness of the mutant theory as a function
of $\kappa$, with a correctly specified resident theory and perfectly
assortative matching. In this numerical example, $\kappa^{\bullet}=0.3$,
$r^{\bullet}=1$, $\sigma_{\omega}^{2}=\sigma_{\epsilon}^{2}=1$.}
\end{figure}

To understand why equilibrium fitness is a monotonically decreasing
function of $\kappa$ with perfectly assortative matching, let $\alpha^{TEAM}$
denote the symmetric linear strategy profile that maximizes the sum
of the two firms' expected objective payoffs. We can show that among
symmetric strategy profiles, players' payoffs strictly decrease in
their aggressiveness in the region $\alpha>\alpha^{TEAM}$. We can
also show that with $\lambda=1$ and any $\kappa\in[0,1],$ the equilibrium
play among two adherents of $\Theta(\kappa)$ strictly increases in
aggression as $\kappa$ grows, and it is always strictly more aggressive
than $\alpha^{TEAM}$. Lowering perception of $\kappa$ confers an
evolutionary advantage by bringing play monotonically closer to the
team solution $\alpha^{TEAM}$ in equilibrium.

\subsection{\label{subsec:learningkey}The Necessity of the (Mis)Learning Channel}

We have seen that misperceptions about $\kappa$ can confer an evolutionary
advantage through the learning channel: they cause the misspecified
agents to misinfer some other parameter of the stage game. This mislearning
commits the agents to certain behavior that increases their equilibrium
payoffs against their typical opponents, given the matching assortativity.

We establish two results to emphasize that the statistical biases
may not be beneficial on their own, but only become beneficial because
they cause misinference of another game parameter. First, in the environment
with a single true $r^{\bullet}$ that we have been working with so
far, we show that if players are dogmatically correct about $r=r^{\bullet}$
and did not need to infer it, then the predictions about the selection
of $\kappa$ could be reversed:
\begin{prop}
\label{prop:no_learning_channel}Let $r^{\bullet}>0,$ $\kappa^{\bullet}\in[0,1]$
be given. With high enough price volatility and large enough strategy
space and inference space, there exists $\epsilon>0$ so that for
any $\kappa_{l},\kappa_{h}\in[0,1]$, $\kappa_{l}<\kappa^{\bullet}<\kappa_{h}\le\kappa^{\bullet}+\epsilon$,
the correctly specified theory $\Theta(\kappa^{\bullet})$ is evolutionarily
stable against the singleton theory $\{F_{r^{\bullet},\kappa_{h},\sigma_{\zeta}^{\bullet}}\}$
under uniform matching ($\lambda=0$), and evolutionarily stable against
the singleton theory $\{F_{r^{\bullet},\kappa_{l},\sigma_{\zeta}^{\bullet}}\}$
under perfectly assortative matching ($\lambda=1$).
\end{prop}
This result shows that once the mislearning channel is shut down,
misperceptions about $\kappa$ that used to confer an evolutionary
advantage under a certain matching assortativity can no longer invade
a society of correctly specified residents. Intuitively, this is because
an error about $\kappa$ has the direct effect of lowering welfare,
but also causes mislearning about $r$ and hence a stronger, indirect
effect of increasing welfare. In the case of uniform matching, for
instance, the direct effect of an increase in the perceived correlation
$\kappa$ is for players to use less aggressive strategies, anticipating
that any favorable signal about market demand is also shared by the
opponent.

Now suppose, contrary to the analysis so far, that the environment
features multiple situations given by multiple feasible values of
$r^{\bullet}.$ Theorem \ref{thm:theoryneeded} does not apply directly
since we have infinitely many strategies, but the basic intuition
remains the same. Mistaken agents who do not learn  have a fixed belief
about $r$ that cannot be  beneficial in all situations, and so they
do not end up with  higher fitness than rational agents. But, misspecified
agents who can make different inferences about price elasticity in
different situations can invade a rational society. Instead of constructing
a theory that gets the Stackelberg payoff in every situation, we show
that simple and natural theories like the ones we have been working
with in this section can already beat the correctly specified theory.
\begin{prop}
\label{prop:LQNLearningNeeded} For every $\overline{r}\ge3$, there
exists a $q\in\Delta([0,\bar{r}])$ such that the correctly specified
theory is evolutionarily stable against any singleton theory with
a fixed $(r,\kappa)$ when $r^{\bullet}\sim q$. On the other hand,
for every $\bar{r}>0$, there exists a projection bias theory with
$\kappa>\kappa^{\bullet}$ so that the corrected specified theory
is evolutionarily fragile against it for any $\rho\in\Delta([0,\bar{r}])$.
\end{prop}

\section{\label{sec:ABEE}Evolutionary Stability of Analogy Classes}

The next application illustrates how our framework accommodates strategic
uncertainty and provides a justification for coarse thinking in games.
Coarse thinking can be evolutionarily beneficial as it allows players
to sustain more cooperative behavior. We show how game parameters
affect the prevalence of coarse thinking.

\citet{jehiel2005analogy} introduces the solution concept of analogy-based
expectation equilibrium (ABEE) in extensive-form games, where agents
group opponents' nodes in an extensive-form game into \emph{analogy
classes} and only keep track of aggregate statistics of opponents'
average behavior within each analogy class. An ABEE is a strategy
profile where agents best respond to the belief that at all nodes
in every analogy class, opponents behave according to the average
behavior in the analogy class. In the ensuing literature that applies
ABEE to different settings, analogy classes are usually exogenously
given and interpreted as arising from coarse feedback or agents' cognitive
limitations.\footnote{Section 6.2 of \citet{jehiel2005analogy} mentions that if players
could choose their own analogy classes, then the finest analogy classes
need not arise, but also says ``it is beyond the scope of this paper
to analyze the implications of this approach.'' In a different class
of games, \citet{jehiel1995limited} similarly observes that another
form of bounded rationality (having a limited forecast horizon about
opponent's play) can improve welfare.}

We use this setting  to highlight two  extensions of the general framework
outlined in Section \ref{sec:Environment-and-Stability}. First, accommodating
this bias requires us to relax the assumption that strategies are
observed, for this mistake is about wrong perception of opponent's
strategy. Second, we can determine the interior population shares
where analogy reasoners coexist with rational agents with both groups
having the same fitness, and describe how changes in the game parameter
affect the relative prominence of the two theories.

\subsection{Relaxing the Observability of Strategies}

Analogy-based reasoning involves players misperceiving their opponents'
strategies, and therefore requires  others' strategy to be embedded
within a theory so that agents can hold possibly wrong beliefs about
them. We introduce the concepts of extended models and extended theories.

\begin{defn}
An \emph{extended model} is a triplet $(a_{A},a_{B},F)$ with $a_{A},a_{B}\in\mathbb{A}$
and $F:\mathbb{A}^{2}\to\Delta(\mathbb{Y}).$ An\emph{ extended theory}
$\overline{\Theta}$ is a collection of extended models: that is,
a subset of $\mathbb{A}^{2}\times(\Delta(\mathbb{Y}))^{\mathbb{A}^{2}}$.
\end{defn}
\noindent Each extended model contains a conjecture $a_{A}$ about
how group A opponents act when matched with the agent, a conjecture
$a_{B}$ about how group B opponents act, and a conjecture $F$ about
how strategy profiles translate into consequences for the agent. We
assume the marginal of the extended theory on $(\Delta(\mathbb{Y}))^{\mathbb{A}^{2}}$
is metrizable. As before, we also assume each $F$ is given by a density
or probability mass function $f(a_{i},a_{-i}):\mathbb{Y}\to\mathbb{R}_{+}$
for every $(a_{i},a_{-i})\in\mathbb{A}^{2}$.

Given that models now include conjectures about the opponent's strategy,
we say that an extended theory $\overline{\Theta}$ is \emph{correctly
specified }if $\overline{\Theta}=\mathbb{A}^{2}\times\{F^{\bullet}(\cdot,\cdot,G)\}$,
so the agent can make unrestricted inferences about others' play and
does not rule out the correct data-generating process $F^{\bullet}(\cdot,\cdot,G)$
for any situation $G$.

Defining zeitgeists for extended models is immediate, as we can simply
replace ``theory'' with ``extended theory'' in Definition \ref{def:Zeitdef}.
The equilibrium notion, however, is subtly different:
\begin{defn}
A zeitgeist with strategic uncertainty $\overline{\mathfrak{Z}}=(\overline{\Theta}_{A},\overline{\Theta}_{B},\mu_{A}(G),\mu_{B}(G),p,\lambda,a(G))_{G\in\mathcal{G}}$
is an\emph{ equilibrium zeitgeist with strategic uncertainty (EZ-SU)}
if for every $G\in\mathcal{G}$ and $g,g^{'}\in\{A,B\},$ \\
 $a_{g,g^{'}}(G)\in\underset{\hat{a}\in\mathbb{A}}{\arg\max}\ \mathbb{E}_{(a_{A},a_{B},F)\sim\mu_{g}(G)}\left[\mathbb{E}_{y\sim F(\hat{a},a_{g^{'}})}(\pi(y))\right]$
and, for every $g\in\{A,B\},$ the belief $\mu_{g}(G)$ is supported
on
\begin{align*}
\underset{(\hat{a}_{A},\hat{a}_{B},\hat{F})\in\overline{\Theta}_{g}}{\arg\min}\left\{ \begin{array}{c}
(\lambda+(1-\lambda)p_{g})\cdot D_{KL}(F^{\bullet}(a_{g,g}(G),a_{g,g}(G),G)\parallel\hat{F}(a_{g,g}(G),\hat{a}_{g})))\\
+(1-\lambda)(1-p_{g})\cdot D_{KL}(F^{\bullet}(a_{g,-g}(G),a_{-g,g}(G),G)\parallel\hat{F}(a_{g,-g}(G),\hat{a}_{-g})
\end{array}\right\}
\end{align*}
where $-g$ means the group other than $g$.
\end{defn}
The  difference with Definition \ref{def:EZ} is that the KL divergence
is taken with respect to the \emph{conjectured} opponent's strategy,
which is part of the extended model. This is because we assume the
actual strategies are unobserved. In this environment with both fundamental
uncertainty and strategic uncertainty, agents still entertain some
belief over their conjectures (both about others' play and about parameters
of the stage game). An agent's subjective belief about the distribution
of consequences in a match depends on both of these dimensions. In
equilibrium, the ``best-fitting'' beliefs are the conjectures that
lead to subjective consequence distributions closest to the true ones
in the sense of KL divergence.

\subsection{Defining Stable Population Shares\label{sect:StableShares}}

Next, in what follows, we will be interested in solving for stable
population shares in a society that contains both rational and misspecified
players. To do this, we introduce the following solution concept.

\begin{defn}
Fix an interior population share, $p\in(0,1)$ and an EZ (or EZ-SU).
Then $p$ is said to be a\emph{ stable population share }given the
EZ (or EZ-SU) if both theories have the same fitness.
\end{defn}
Note that we have already defined EZ(-SU)s with interior population
shares, so we can calculate the fitness of a theory in terms of its
adherents' objective expected payoff. Also, recall that evolutionary
stability as given in Definition \ref{def:stability} reflects stability
at the population share of $(p_{A},p_{B})=(1,0)$. One theory outperforms
the other when the latter makes up a very small fraction of the society,
thus the mutants cannot invade the residents. By contrast, interior
population shares correspond to both theories being co-existing and
being equally successful.

\subsection{Centipede Games and Analogy-Based Reasoning}

We use these formalism to study analogy-based reasoning. Consider
a centipede game, shown in Figure \ref{fig:The-centipede-game} (there
is only one situation, given by the payoffs in this game). P1 and
P2 take turns choosing Across (A) or Drop (D). The non-terminal nodes
are labeled $n^{k}$, $1\le k\le K$ where $K$ is an even number.
P1 acts at nodes $n^{1},n^{3},...,n^{K-1}$, P2 acts at nodes $n^{2},n^{4},...,n^{K}$,
and choosing Drop at $n^{k}$ leads to the terminal node $z^{k}$.
If Across is always chosen, then the terminal node $z^{end}$ is reached.
If P1 chooses Drop at the first node, the game ends with the payoff
profile (0, 0). Every time a player $i$ chooses Across, the sum of
payoffs grows by $g>0,$ but if the next player chooses Drop then
$i$'s payoff is $\ell>0$ smaller than what $i$ would have gotten
by choosing Drop. If $z^{end}$ is reached, both get $Kg/2.$ That
is, if $u_{j}^{k}$ is the utility of $j$ at the terminal node $z^{k},$
and $i$ moves at $n^{k},$ then $u_{-i}^{k}=u_{-i}^{k-1}-\ell$ while
$u_{i}^{k}=(u_{i}^{k-1}+u_{-i}^{k-1}+g)-u_{-i}^{k}$. This works out
to $u_{j}^{k}=\frac{g(k-1)}{2}$ for both players when $k$ is odd,
and $u_{1}^{k}=\frac{k-2}{2}g-\ell$, $u_{2}^{k}=\frac{k}{2}g+\ell$
when $k$ is even.

\begin{figure}[h]
\begin{centering}
\includegraphics[scale=0.35]{centipede_K_moves}
\par\end{centering}
\vspace{-15bp}

\caption{\label{fig:The-centipede-game}The centipede game. There are $2K$
non-terminal nodes and players 1 (blue) and 2 (red) alternate in choosing
Across (A) or Drop (D). Payoff profiles are shown at the terminal
nodes.}
\end{figure}

While this is an asymmetric stage game, we study the symmetrized version
mentioned in Section \ref{subsec:Objective-Primitives}, where two
matched agents are randomly assigned into the roles of P1 and P2.
Let $\mathbb{A}=\{(d^{k})_{k=1}^{K}\in[0,1]^{K}\}$, so each strategy
is characterized by the probabilities of playing Drop at various nodes
in the game tree. When assigned into the role of P1, the strategy
$(d^{k})$ plays Drop with probabilities $d^{1},d^{3},...,d^{K-1}$
at nodes $n^{1},n^{3},...n^{K-1}$. When assigned into the role of
P2, it plays Drop with probabilities $d^{2},d^{4},...,d^{K}$ at nodes
$n^{2},n^{4},...n^{K}$. The set of consequences is $\mathbb{Y}=\{1,2\}\times(\{z_{k}:1\le k\le K\}\cup\{z_{end}\})$,
where the first dimension of the consequence returns the player role
that the agent was assigned into, and the second dimension returns
the terminal node reached. The objective distribution over consequences
as a function of play is $F^{\bullet}:\mathbb{A}^{2}\to\Delta(\mathbb{Y})$.

Consider a learning environment where agents know the game tree (i.e.,
they know $F^{\bullet}$), but some agents mistakenly think that when
their opponents are assigned into a role, these opponents play Drop
with the same probabilities at all of their nodes. Formally, define
the restricted space of strategies $\mathbb{A}^{An}:=\{(d^{k})\in[0,1]^{K}:d^{k}=d^{k'}\text{ if }k\equiv k^{'}\text{(mod 2)}\}\subseteq\mathbb{A}$.
The correctly specified extended theory is $\overline{\Theta}^{\bullet}:=\mathbb{A}\times\mathbb{A}\times\{F^{\bullet}\}.$
The misspecified theory with a restriction on beliefs about opponents'
play is $\overline{\Theta}^{An}:=\mathbb{A}^{An}\times\mathbb{A}^{An}\times\{F^{\bullet}\}$,
reflecting a dogmatic belief that opponents play the same mixed action
at all nodes in the analogy class. It is important to remember that
these restriction on strategies only exists in the subjective beliefs
of the theory $\overline{\Theta}^{An}$ adherents. All agents, regardless
of their theory, actually have the strategy space $\mathbb{A}$. Also,
consistent with our framework of theory evolution, agents are born
with a theory and cannot choose their own analogy classes.

\subsection{Results}

The next proposition provides a justification for why we might expect
agents with coarse analogy classes given by $\mathbb{A}^{An}$ to
persist in the society.
\begin{prop}
\label{prop:abee}Suppose $K\ge4$ and $g>\frac{2}{K-2}\ell$. For
any matching assortativity $\lambda\in[0,1],$ the correctly specified
extended theory $\overline{\Theta}^{\bullet}$ is evolutionarily stable
with strategic uncertainty against itself, but it is not evolutionarily
stable with strategic uncertainty against the misspecified extended
theory $\overline{\Theta}^{An}.$ Also, $\overline{\Theta}^{An}$
is not evolutionarily stable against $\overline{\Theta}^{\bullet}$,
unless $\lambda=1$.
\end{prop}
In contrast to the results about LQN games from the previous section,
which predict different biases may arise under different matching
assortativities, we find in this environment that the correctly specified
extended theory is not evolutionarily stable against the extended
theory $\overline{\Theta}^{An}$ with coarse analogy classes under
any level of assortativity. In the previous application to LQN games,
agents with projection bias commit to acting more aggressively, which
increases their equilibrium welfare in matches against rational agents
but decreases their equilibrium welfare in matches against other agents
with the same bias --- and vice versa for agents with correlation
neglect. But in the current application, the conditional fitness of
$\overline{\Theta}^{An}$ against both $\overline{\Theta}^{\bullet}$
and $\overline{\Theta}^{An}$ can strictly improve on the correctly
specified residents' equilibrium fitness. This is because the matches
between two adherents of $\overline{\Theta}^{\bullet}$ must result
in Dropping at the first move in equilibrium, while matches where
at least one player is an adherent of $\overline{\Theta}^{An}$ either
lead to the same outcome or lead to a Pareto dominating payoff profile
as the misspecified agent misperceives the opponent's continuation
probability and thus chooses Across at almost all of the decision
nodes.

But, $\overline{\Theta}^{An}$ is not evolutionarily stable against
$\overline{\Theta}^{\bullet}$ either. The correctly specified agents
can exploit the analogy reasoners' mistake and receive higher payoffs
in matches against them, compared with the payoffs that two misspecified
agents receive when matched with each other.

This result suggests that neither a homogeneous population of rationals
nor a homogeneous population of analogy reasoners can be stable in
the long-run, as the resident theory can have lower fitness than the
mutant theory in equilibrium. The remaining possibility is that the
two competing theories coexist, each with a sizable share of adherents,
as articulated in Section \ref{sect:StableShares}. Note that in the
Centipede games we study, there is always the trivial EZ-SU where
all agents in both groups always play Drop, and thus learn nothing
about others' play. We focus on the EZ-SU where Across is played as
often as possible.

Let there be uniform matching in a society with $\Theta_{A}=\Theta^{\bullet}$,
$\Theta_{B}=\Theta^{An}$, population sizes $(p_{A},p_{B})$, and
suppose $K\ge4$ and $g>\frac{2}{K-2}\ell$. Consider the\emph{ maximal
continuation EZ-SU} in the centipede game: this is the strategy profile
where $d_{AA}^{k}=1$, $d_{BB}^{k}=0$ for every $k<K$, $d_{BB}^{K}=1$,
$d_{AB}^{k}=0$ for $k\le K-2$, $d_{AB}^{K-1}=d_{AB}^{K}=1$, $d_{BA}^{k}=0$
for $k<K,$ and $d_{BA}^{K}=1.$ That is, agents always play Across,
except the misspecified agents always Drop at node $K$, the correctly
specified agent Drop at nodes $K-1$ and $K$ when matched with a
misspecified agent, and Drop at every node when matched with another
correctly specified agent. (The next result verifies that the maximal
continuation EZ-SU is indeed an EZ-SU.)

We find the\emph{ }stable population share under maximal continuation.
This is the unique population size $(p_{A}^{*},p_{B}^{*})$ such that
the two theories have the same fitness under the maximal continuation
EZ-SU. Intuitively, $p_{B}^{*}$ shows what fraction of the society
is expected to be analogy reasoners in the long run as the adherents
of $\Theta_{A}$ and $\Theta_{B}$ rise and fall according to their
fitness, assuming the focal equilibrium is the one with the highest
amount of continuation.
\begin{prop}
\textup{\label{prop:stable_pop_share} }\emph{Suppose $K\ge4$ and
$g>\frac{2}{K-2}\ell$.}\textup{\emph{ }}\textup{The two theories
have the same fitness in the maximal continuation EZ-SU of the centipede
game if and only if the population size of the misspecified theory
$\Theta_{B}$ is $p_{B}^{*}=1-\frac{\ell}{g(K-2)}$. This $p_{B}^{*}$
is strictly increasing in $g$ and $K$, strictly decreasing in $\ell.$}
\end{prop}
Under the maintained assumption $g>\frac{2}{K-2}\ell,$ the stable
population share of misspecified agents is strictly more than 50\%,
and the share grows when there are more periods and when the increase
in payoffs is larger across periods. The main intuition is that the
misspecified theory has a higher conditional fitness than the rational
theory against rational opponents. The former leads to many periods
of continuation and a high payoff for the biased agent when the rational
agent eventually drops, but the latter leads to 0 payoff from immediate
dropping. The size of the fitness advantage grows with $g$ and $K$.
On the other hand, the misspecified theory has a lower conditional
fitness than the rational theory against misspecified opponents. For
the two groups to have the same expected fitness, there must be fewer
rational opponents (i.e., a smaller stable population share $p_{A}^{*}$)
when $g$ and $K$ are higher.

In the centipede game, two successive periods of continuation lead
to a strict Pareto improvement in payoffs. Consider instead the dollar
game \citep{reny1993common} in Figure \ref{fig:The_dollar_game},
a variant of the centipede game with a more ``competitive'' payoff
structure, where an agent always gets zero payoff when the opponent
plays Drop, no matter how far along in the game tree. Total payoff
increases by 1 in each round. If the first player stops immediately,
the payoffs are (1, 0), and if the second player continues at the
final node $n^{K}$, payoffs are $(K+2,0).$

\begin{figure}[h]
\begin{centering}
\includegraphics[scale=0.35]{dollar_game_K_moves}
\par\end{centering}
\vspace{-10bp}

\caption{\label{fig:The_dollar_game}The dollar game. There are $K$ non-terminal
nodes and players 1 (blue) and 2 (red) alternate in choosing Across
(A) or Drop (D). Payoff profiles are shown at the terminal nodes.}
\end{figure}

\begin{prop}
\textup{\label{prop:stable_pop_share_dollar} For every population
size $(p,1-p)$ with $p\in[0,1],$ }\emph{the maximal continuation
EZ-SU}\textup{ is an EZ-SU where the fitness of $\overline{\Theta}^{\bullet}$
is strictly higher than that of $\overline{\Theta}^{An}$.}
\end{prop}
While the maximal continuation EZ-SU remains an EZ-SU in the dollar
game, the rational theory always has strictly higher fitness than
the misspecified theory with analogy classes with any population sizes.
So, provided the maximal continuation EZ-SU remains focal, then we
should expect no analogy reasoners in the long run with this different
stage game. This is because the misspecified theory now has a lower
conditional fitness than the rational theory against rational opponents.
A misspecified P1 gets 0 in the maximal continuation EZ-SU against
a rational P2 who drops at their penultimate decision node, whereas
a rational P1 gets 1 from immediate dropping.

In a recent survey article on ABEEs, \citet{jehiel2020survey} points
out that the misspecified Bayesian learning approach to analogy classes
should aim for ``a better understanding of how the subjective theories
considered by the players may be shaped by the objective characteristics
of the environment.''\footnote{\citet{jehiel2020survey} interprets ABEEs as players adopting the
``simplest'' explanations of observed aggregate statistics of play
in environments with physically or psychologically coarse feedback.
An objectively coarse feedback structure can lead agents to adopt
the subjective theory that others behave in the same way in all contingencies
in the same coarse analogy class. Our approach suggests that the very
prevalence of analogy classes may also be shaped by other objective
environmental primitives, including the payoff structure of the stage
game.} We find that a framework of theory evolution can deliver such results.
Taken together, the analyses of maximal continuation EZ-SUs in the
centipede game and the dollar game suggest that analogy-based reasoning
should be more prevalent in the former than the latter because the
payoff structure is ``less competitive'' in the former. Also, in
the centipede game, the prevalence of this bias should increase with
a longer horizon and with faster payoff growth.

\section{\label{sec:Concluding-Discussion}Concluding Discussion}

This paper presents an evolutionary selection criterion to endogenize
(mis)specifications when agents learn about a strategic environment.
We introduce the concept of a zeitgeist to capture the ambient social
structure where learning takes place: the prominence of different
theories in the society and the interaction patterns among their adherents.
These details matter because different types of opponents behave differently,
inducing different beliefs about the economic fundamentals for a misspecified
agent. Evolutionary stability of a theory is defined based on the
expected objective payoffs (fitness) of its adherents in equilibrium.

We have highlighted settings where the correct specification is not
evolutionarily stable against some misspecifications. We view our
main contributions as two fold. First, we point out that the learning channel strictly
expands the possibility for mistakes to invade a rational society. Second,
we show how the combination of learning, interaction structure, and payoff structure
affect which errors get selected, and provide intuition for when one should expect mistakes to emerge.
We also showed
that the learning channel we have studied, unique to a world where evolutionary forces
act on specifications (sets of feasible beliefs) instead of single
beliefs, generates novel stability phenomena.

Our framework evaluates whether a misspecification is likely to persist
once it emerges in a society, but does not account for which errors
appear in the first place. It is plausible that some first-stage filter
prevents certain obvious misspecifications from ever reaching the
stage that we study in the evolutionary framework. In the applications,
we have focused on misspecifications that seem psychologically plausible
or harder to detect, such as misspecified higher-order beliefs.

We have used the simplest evolutionary framework where fitness is
identified with the expectation of objective payoffs, as opposed to
some more exotic function of the payoffs. The goal of this paper is
not to identify the suitable definition of fitness to justify a particular
error (which is the focus for many of the papers that \citet{robson2011evolutionary}
survey). Rather, we hope that our stability notions are reasonably
simple and universal that they may become a part of the applied theory
toolkit in the future. Studies on the implications of misspecifications
in various strategic environments may further enrich our understanding
of these errors by paying more attention to their evolutionary stability.

\vspace{-30bp}

\begin{singlespace}
{\small{}{}{} \bibliographystyle{ecta}

redpajama_set_name: RedPajamaArXiv