# Heterogeneous Noise and Stable Miscoordination

Srinivas Arigapudi Faculty of Data and Decision Sciences, Technion.
<EMAIL_ADDRESS>Yuval Heller Department of Economics, Bar-
Ilan University and U. of California San Diego<EMAIL_ADDRESS>Amnon
Schreiber Department of Economics, Bar-Ilan University.
<EMAIL_ADDRESS>paper replaces an earlier working paper titled
“Sampling Dynamics and Stable Mixing in Hawk-Dove Games.”We are grateful to
Vince Crawford, Josef Hofbauer, Bill Zame, and to seminar and conference
participants at Bar-Ilan University, Haifa University, Learning, Evolution and
Games, SWET 2023 (UC Irvine), UC Los Angeles, UC Riverside, UC San Diego, and
University of South California, for many helpful comments. YH gratefully
acknowledges the financial support of the European Research Council (#677057),
the Israeli Science Foundation (#448/22), and the US-Israel Binational Science
Foundation (#2020022). SA is supported in part at the Technion by a Fine
Fellowship.

###### Abstract

Coordination games admit two types of equilibria: pure equilibria, where all
players successfully coordinate their actions, and mixed equilibria, where
players frequently experience miscoordination. The existing literature shows
that under many evolutionary dynamics, populations converge to a pure
equilibrium from almost any initial distribution of actions. By contrast, we
show that under plausible learning dynamics, where agents observe the actions
of a random sample of their opponents and adjust their strategies accordingly,
stable miscoordination can arise when there is heterogeneity in the sample
sizes. This occurs when some agents make decisions based on small samples
(anecdotal evidence) while others rely on large samples. Finally, we
demonstrate the empirical relevance of our results in a bargaining
application.

Keywords: sampling best response dynamics, action-sampling dynamics,
coordination games, hawk-dove games, evolutionary stability, logit dynamics.
JEL Classification: C72, C73.

##  1 Introduction

Many real-life situations can be modeled as coordination games, in which the
best response against an opponent’s action is to play the same action
(possibly after relabeling the actions of one of the players). Two-player,
two-action coordination games admit three equilibria of two distinct types:
two strict pure equilibria, where the players successfully coordinate their
actions, and a mixed equilibrium, where players frequently experience
miscoordination. A fundamental result in evolutionary game theory is that
under a broad set of learning dynamics, the mixed equilibrium is unstable and
populations in which agents are randomly matched to play coordination games
must converge to everyone playing one of the pure equilibria (as surveyed in
Section 2). By contrast, in this paper we show that the mixed equilibrium with
miscoordination can be stable if the populations are heterogeneous in the
sense that some (but not all) of the agents rely on anecdotal evidence induced
by small samples.

##### Highlights of the Model

Consider a setup in which pairs of agents from two infinite populations are
repeatedly randomly matched to play a (one-shot) coordination game.111
Appendix A.2 extends our results to symmetric coordination games played within
a single population. Agents occasionally die and are replaced by new agents
(or, alternatively, agents occasionally receive opportunities to revise their
actions). The new agents do not have precise information about the aggregate
behavior in the opponent’s population, and estimate this from sampling the
opponent’s population. Specifically, each population $i$ is characterized by a
distribution of sample sizes $\theta_{i}$, such that
$\theta_{i}\left(k\right)$ is the frequency of agents with sample size $k$.
Each such agent observes the behavior of $k$ random opponents, and then adopts
the action that is a best response to her sample (with an arbitrary tie-
breaking rule).222This behavior can be interpreted by each new agent utilizing
her own sample to calculate a maximum likelihood estimation for the overall
behavior of the opposing population.

These learning dynamics, which seem plausible in various setups, are called
_sampling best-response dynamics_(Sandholm, 2001; Osborne and Rubinstein,
2003; Oyama, Sandholm, and Tercieux, 2015, henceforth abbreviated as _sampling
dynamics_).

As explained in Appendix A.1, any two-action coordination game (such as the
battle of the sexes or the stag hunt) can be represented w.l.o.g. by the
payoff matrix presented in Table 1 that has two positive parameters
$\left(u_{1},u_{2}\right)$ that represent the players’ payoffs when both play
the first action: the payoffs when coordinating on the second action are
normalized to 1, and the payoffs for miscoordinating are normalized to 0.

Table 1: Standard Representation of Two-Action Coordination Game | _$a_{2}$_ | _$b_{2}$_
---|---|---
_$~{}~{}a_{1}~{}~{}$_ | _$\,\,{}_{\underset{\,}{{\color[rgb]{0,0,1}u_{1}}}}{\,{}^{\overset{\,}{u_{2}}}}$_ | _${{}_{\underset{\,}{0}}}\,\,^{{\color[rgb]{180,00,00}\overset{\,}{0}}}$_
_$~{}~{}b_{1}~{}~{}$_ | _${}_{\underset{\,}{{\color[rgb]{0,0,1}0}}}\,\,^{\overset{\,}{{\color[rgb]{180,00,00}0}}}$_ | _${}_{{\color[rgb]{0,0,1}1}}\,\,\,\,^{{\color[rgb]{180,00,00}1}}$_

The following definition will be helpful for presenting our results. An action
is $q$-dominant (Morris, Rob, and Shin, 1995) to player $i$ if it is the
player’s best response against any opponent’s mixed action that assigns a mass
of at least $q$ to the opponent playing the same action. In particular, action
$a_{i}$ (resp., $b_{i}$) is $\frac{1}{2}$-dominant for player $i$ if it is the
best response against an opponent’s uniform play, which is the case iff
$u_{i}>1$ (resp., $u_{i}<1$).

##### Global Convergence to Miscoordination

Theorem 1 presents a full characterization for environments in which the
populations converge to states with miscoordination from almost all initial
states. This happens if (and essentially only if): (1) each population has a
different $\frac{1}{2}$-dominant action (i.e., $u_{2}<1<u_{1}$), (2) the
product of the mass of agents with sample size 1 times the expected sample
size of the other population is larger than one (i.e., $\forall
i,\,\,\theta_{i}\left(1\right)\cdot\mathbb{E}\left(\theta_{j}\right)>1$ ), and
(3) each population has a $q$-dominant action for a sufficiently low $q$
(which is satisfied iff each $u_{i}$ is sufficiently far from 1).

The proof idea is as follows. Global convergence to miscoordination occurs iff
both pure equilibria are unstable. Assume that $u_{2}<1<u_{1}$. Consider a
slightly perturbed state near the equilibrium $\left(a_{1},a_{2}\right)$, in
which $\epsilon_{i}<<1$ of the agents in population $i$ plays $b_{i}$. Events
in which a new agent observes multiple occurrences of the opponent’s rare
action $b_{j}$ in her sample are negligible
($O\left(\epsilon_{j}^{2}\right)$). Neglecting these very rare events implies
that new agents of population $1$ will adopt the risk-dominated action $b_{1}$
only when they have sample size 1, and they have observed the rare action
$b_{2}$ (the probability of this is
$\theta_{1}\left(1\right)\cdot\epsilon_{2}$). By contrast, if $u_{2}<1$ is
sufficiently small, then a single occurrence of the rare action $b_{1}$ in a
sample of size $k$ (which occurs with probability of $k\cdot\epsilon_{1}$) is
sufficient to induce a new agent of population 2 to play her
$\frac{1}{2}$-dominant action $b_{2}$. This implies that the total share of
new agents of population 2 who play action $b_{2}$ is
$\mathbb{E}\left(\theta_{2}\right)\cdot\epsilon_{1}$. This, in turn, implies
that the product of the number of agents playing the rare action in each
population increases iff
$\theta_{1}\left(1\right)\cdot\mathbb{E}\left(\theta_{2}\right)>1$.

##### Heterogeneity and Stable Miscoordination

Our second main result (Theorem 2) shows that heterogeneity in the sample
sizes is necessary for stable miscoordination. Specifically, we show that if
all agents in each population $i$ have the same sample size $k_{i}\geq 2$,
then all states with miscoordination are unstable (the case in which everyone
has sample size 1 is discussed in Remark 1). Our final result (Theorem 3)
shows that many heterogeneous distributions of sample sizes in which some
agents have relatively small samples and others have sufficiently large
samples induce locally stable states with miscoordination if $u_{1}$ and
$u_{2}$ are not too close to 1.

The intuition for why heterogeneity in the sample sizes is important for the
stability of interior states with miscoordination is as follows. Consider
homogeneous populations with a fixed sample size $k$ in a stationary interior
state. In such a state the random sample of size $k$ frequently yields both
outcomes for which action $a_{i}$ is a best response and outcomes for which
$b_{i}$ is a best response. We show that in such situations the probability of
each action being a best response is sensitive to small perturbations in the
opponent’s distribution of actions. That is, if $\epsilon$ more of the
opponent’s population play $a_{j}$, it increases the probability that $a_{i}$
is the best response to the random sample by more than $\epsilon$.

Next consider a heterogeneous population in which some agents have relatively
small samples, while other agent have large samples. The stationary interior
state with miscoordination typically does not coincide with the mixed Nash
equilibrium, which implies that almost all agents with large samples play the
same action (the unique best response to the true distribution of the
opponents’ actions), and that their play is insensitive to small perturbations
of the opponents’ behavior. This allows the overall sensitivity of the entire
population to small perturbations to be sufficiently small to allow stable
miscoordination.

Theorem 2’s proof relies on deriving a property of binomial distributions
(Proposition 3), which may be of independent interest. This property states
that a composition of two binomial cumulative distributions has a unique
interior fixed point.

##### Numerical Analysis and Insights

In Section 7 we demonstrate our main insight that heterogeneity in the sample
sizes induces stable miscoordination does not depend on the specific details
of the sampling dynamics. Specifically, we numerically study the commonly used
logit dynamics, in which agents play a noisy best response to their opponents’
aggregate behavior, with $\eta_{i}$ describing the noise level in population
$i$. We first demonstrate that if the noise level in each population is
homogeneous, then one can induce stable miscoordination only with implausibly
high levels of noise. By contrast, when we introduce an extension of logit
dynamics that allows heterogeneity in the level of noise in each population,
we show that stable miscoordination can be supported by moderate heterogeneous
levels of noise.

Taken together, our results show that the conventional wisdom that
miscoordination is unstable is not accurate. Miscoordination can be stable in
heterogeneous populations in which some agents rely on anecdotal evidence or
noisy data, while other agents have access to more accurate data. The
experimentally testable implications of our results are discussed in Section
2.

##### Bargaining Application and Empirical Relevance

Demonstrating the empirical relevance of our theoretical findings requires
real-life scenarios in which it is plausible to have: (1) heterogeneous sample
sizes, (2) persistent miscoordination within an interior state, and (3) a
substantial proportion of agents not maximizing their payoffs with respect to
the aggregate behavior of the other population. We argue that these conditions
might exist in bargaining situations in both housing and used-car markets.
Coordination games can model simple bargaining in these markets, where each
agent has two strategies: a high-price strategy or low-price strategy. For
example, a seller can either demand a high price (with the risk of bargaining
failure) or agree to a low price, and a buyer can either agree to a high price
or insist on a low price (as modeled by hawk-dove coordination games; see
Example 3 in Appendix A.1). Heterogeneity in the sample sizes is likely to
exist in these markets, where both professional players (real-estate investors
and car dealers) and inexperienced players (those who have only bought or sold
houses or cars a couple of times) participate. Professional players are likely
to have access to large samples, while inexperienced players may rely on
anecdotal evidence.

Koster and Rouwendal (2021) conducted an empirical study on the Dutch housing
market. They found that a significant number of sellers set low list prices,
which can be consistent with profit maximization only if the sellers have high
annual discount factors of up to 50%. This behavior is consistent with an
interior stable state in our model in which some sellers rely on small, noisy
samples to choose non-payoff-maximizing low prices. Koster and Rouwendal’s
data does not allow separating between different types of sellers. By
contrast, Genesove and Mayer (2001) does allow separating the sellers to
owner-occupants and investors, and show that the two groups have systematic
differences in their list prices.

Larsen (2021) empirically studied the efficiency of bargaining following
wholesale used-car auctions in which the highest bid is lower than the reserve
price, and found that bargaining fails in about 35% of these cases (which is
consistent with an interior stable state with miscoordination). The data
suggests that this inefficiency results in substantial losses, representing
12%-23% of ex-post gains from trade. Furthermore, Larsen’s analysis suggests
that only a small part of this loss is due to incomplete information
constraints.

##### Structure

Section 2 presents the related literature. Our model is described in Section
3. Section 4 presents a “complete” characterization of global convergence to
states with miscoordination. In Section 5, we show that homogeneous
populations always converge to one of the pure equilibria. Section 6 shows
that many heterogeneous populations can induce locally stable states with
miscoordination. In Section 7, we numerically analyze logit dynamics, and
demonstrate that our main insights hold in this setup as well. We conclude in
Section 8. Formal proofs and additional results are presented in the appendix.
Appendix A.2 extends our results to one-population dynamics. Appendix A.3
states and proves an interesting general result on binomial distributions.

##  2 Related Literature

##### Instability of Miscoordination

It is well-known that strict equilibria satisfy strong stability refinements,
while mixed equilibria with miscoordination do not satisfy even weak stability
refinements. In particular, strict equilibria are evolutionarily stable
(Maynard Smith and Price, 1973), while the mixed equilibrium does not satisfy
neutral stability (Maynard Smith, 1982) or even the mild refinement of weak
stability (Heller, 2017). Moreover, it is well known that interior stationary
states in all multiple-population games cannot be asymptotically stable under
the widely studied replicator dynamics (see, e.g., Sandholm, 2010, Theorem
9.1.6). Instability of interior states is further shown for various classes of
learning dynamics in Crawford (1989).

Moreover, various papers in the literature have proven that populations
playing coordination games converge to one of the pure equilibria from almost
all initial states under various dynamics. Kaniovski and Young (1995) show
that there is global convergence to one of the pure equilibria for sampling
dynamics with sufficiently large samples. Oprea, Henwood, and Friedman (2011)
show that there is global convergence to one of the pure equilibria under
monotone dynamics (i.e., under any dynamics in which an action becomes more
frequent iff it yields a higher payoff than the alternative action).333Oprea,
Henwood, and Friedman (2011) proved this for hawk-dove games, but the proof
can be extended to all coordination games. Recently, Oyama, Sandholm, and
Tercieux (2015) proved the global stability of the $\frac{1}{2}$-dominant pure
equilibrium in symmetric coordination games for distributions of sample sizes
in which sufficiently many agents have sufficiently small samples. The
stochastic evolutionary dynamics literature (pioneered by Kandori, Mailath,
and Rob, 1993; Young, 1993; see also the recent application to hawk-dove games
in Bilancini, Boncinelli, Ille, and Vicario, 2022) shows that only pure
equilibria can be stochastically stable in large finite populations in which
agents most of the time best respond to a large sample, but occasionally
mistakenly play the other action.

Thus, taken together, the various existing studies suggest that states with
miscoordination are unstable. Our contribution is in showing that this is not
the case in plausible learning dynamics in which agents base their behavior on
sampling the actions of the other population, and there is substantial
heterogeneity in the sample sizes in each population.

##### Sampling Dynamics

Sampling best-response dynamics (henceforth, sampling dynamics) were pioneered
by Sandholm (2001) and Osborne and Rubinstein (2003). As argued by Oyama,
Sandholm, and Tercieux (2015), the deterministic nature of sampling dynamics
implies that when there is convergence to a stable state (which is always the
case in our setup; see Proposition 2) the convergence is fast.444Conditions in
which stochastic dynamics induce fast convergence are studied in Kreindler and
Young (2013) and Arieli, Babichenko, Peretz, and Young (2020). Recently,
Heller and Mohlin (2018) studied the conditions on the expected sample size
that implies global convergence for all payoff functions and all sampling
dynamics.555Hauert and Miekisz (2018) used the term “sampling dynamics” to
refer to a variant of the replicator dynamics, in which when an agent samples
another agent and mimics the other agent’s behavior, it is more likely that
these two agents will be matched with each other. This is less related to our
use of the notion of “sampling dynamics,” which is in line with the literature
cited above.

Salant and Cherry (2020) (see also Sawa and Wu, 2023) generalized sampling
dynamics by allowing new agents to use various procedures to infer from their
samples the aggregate behavior of the opponents (in addition to allowing for
payoff heterogeneity in the population). Salant and Cherry paid special
attention to _unbiased_ inference procedures in which the agent’s expected
belief about the share of opponents who play an action coincides with the
sample mean. Examples of unbiased procedures are maximum likelihood
estimation, beta estimation with a prior representing complete ignorance, and
a truncated normal posterior around the sample mean. In our setup, the payoffs
are linear in the share of agents who play action $a_{j}$, which implies that
the agent’s perceived best response depends only on the expectation of her
posterior belief. This implies that the behavior of agents in our model can be
interpreted as each new agent utilizes her own sample to calculate an unbiased
estimation procedure (such as, maximum likelihood estimation) for the overall
behavior of the opposing population. remains the same with any unbiased
inference procedure.

##### Experimental Literature and Testable Predictions

Our model yields a novel testable prediction: miscoordination can persist when
there is heterogeneity in the amount of information that the agents have about
the opponents’ behavior, and for each population one of the actions is
$q$-dominant for a sufficiently low $q$. Brunner, Camerer, and Goeree (2011)
compared the predictive power of various learning models when applied to the
experimental data of McKelvey, Palfrey, and Weber (2000), Goeree, Holt, and
Palfrey (2003), and Selten and Chmura (2008), in which (1) agents are randomly
and repeatedly matched to play a one-shot two-action two-player game against
an anonymous opponent, and (2) agents get feedback only about their most
recent opponent’s play (and can rely on their memory of her previous feedback
about past opponents). Brunner, Camerer, and Goeree’s findings show that
sampling dynamics explain well the aggregate experimental behavior, and that
sampling dynamics’ predictions are: (1) as good as the quantal response
equilibrium in all games and significantly better in some games, (2)
significantly better than the Nash equilibrium in all games, and (3)
significantly improve when focusing on the less noisy second halves of the
treatments (see, Brunner, Camerer, and Goeree, 2011, Figure 4).

Brunner, Camerer, and Goeree’s results suggest that sample sizes of 3–12 fit
the data best (which is roughly in line with the typical estimates of people’s
short-run memory capacity). The prediction of global convergence to
miscoordination (Theorem 1) requires that a substantial part of the population
to have a small sample size of 1. This may be induced by having some subjects
playing in different rounds different underlying games. For example, in each
round, one of several underlying games will be played. Each subject will be
informed about the payoff matrix of the current game, and will be reminded
about her most recent past opponent’s behavior in the previous period in which
the same game was played. It seems plausible that many subjects will rely only
on this feedback when deciding how to play (i.e., essentially have a sample of
size 1), while only a few subjects will exert effort to remember relevant past
feedback from less recent periods in which the same game was played.

Recently, Lyu, Li, and Xu (2022) applied a more elaborate experimental setup
that aims to implement sampling dynamics. In their implementation the subjects
are endowed with a specific “default” action to play in the first round. This
novel design component allows us to test dynamic predictions with respect to
specific initial states. At the end of each round each agent observes the $k$
most recent actions ($k=2$ and $k=7$ were applied in their treatments) of
random opponents among the 14 players in their matching group. Their
underlying games were symmetric coordination games with Pareto-ranked pure
equilibria. Lyu, Li, and Xu showed that action-sampling dynamics (with respect
to the most recent $k$ observed actions) fits about 80% of the subjects’
behavior.

A central reason why some subjects deviated from the predictions of sampling
dynamics (especially in early rounds) was that they played the action that is
part of the Pareto-dominant equilibrium in order to “teach” the other subjects
in the matching group to move from the Pareto-dominated equilibrium to the
Pareto-dominant equilibrium. These “teaching” incentives were justified, as
they often helped groups starting in the Pareto-dominated equilibrium to shift
to the Pareto-dominant equilibrium. We think that “teaching”incentives would
be substantially reduced, and the fit of sampling dynamics would be
substantially improved, if either (1) the strict equilibria of the underlying
coordination games were not Pareto-ranked (e.g., battle of the sexes), or (2)
the matching groups were substantially larger.

The predictions of Theorem 3 (local stability of miscoordination) require both
an appropriate initial state (which can be implemented $\grave{\textrm{a}}$ la
Lyu, Li, and Xu, 2022), and that some agents will have accurate information
about the opponents’ aggregate behavior. This can be implemented by providing
some agents with full feedback about the behavior of all opponents in the
previous round (while providing the remaining agents with feedback about the
behavior of only their own matched opponents).

##  3 Model

###  3.1 Coordination Games

Let $G=\\{A_{1}\times A_{2},u\\}$ denote a normal-form two-action two-player
coordination game. Let $i\in\left\\{1,2\right\\}$ be an index denoting one of
the players (“she”), and let $j$ denote her opponent (“he”). For each
$i\in\left\\{1,2\right\\}$, let $A_{i}=\left\\{a_{i},b_{i}\right\\}$ denote
the actions of player $i$.

The standard two-parameter payoff matrix of a coordination game is given in
Table 2: the players get a low payoff (normalized to 0) if they miscoordinate
(i.e., one player plays $a_{i}$ and the opponent plays $b_{j}$), they get a
high payoff (normalized to 1) if they coordinate on both playing
$\left(b_{1},b_{2}\right)$, and they get a payoff of $u_{i}>0$ if they
coordinate on both playing $\left(a_{1},a_{2}\right)$. By relabeling the
actions we can assume w.l.o.g. that action profile a is the weakly preferred
outcome of Player 1; i.e., we assume that $u_{1}\geq 1.$ We say that the
coordination game is (1) _symmetric_ if $u_{1}=u_{2}$, and (2) _antisymmetric_
if $u_{1}=\frac{1}{u_{2}}.$ Antisymmetric coordination games can be
interpreted as _battle of the sexes_ games, in which Player 1’s preference for
the first coordinated outcome a over the second coordinated outcome b is as
strong as Player 2’s preference for b over a.

In Appendix A.1 we formally show that this standard two-parameter
representation captures w.l.o.g. two-action coordination games (including
battle of the sexes, stag hunt, and hawk-dove games). This is so because our
dynamics (as defined in Eq. (3.1) and Eq. (A.1))) depend only on the
differences between the payoffs of a player in action profiles in which the
opponent plays the same action. This implies that the dynamics are invariant
to (1) adding a constant to the two payoffs of Player 1 (resp., Player 2) in
the same row (resp., column), and (2) dividing all the payoffs of a player by
a positive constant.

We extend the game to mixed actions in the standard linear way. We identify
each mixed action with the probability it assigns to the first action
($a_{i}$), and we denote it by $p_{i}\in\left[0,1\right]$. We identify the
degenerate mixed action $1$ (resp., $0$) with the pure action $a_{i}$ (resp.,
$b_{i}$). Observe that the coordination game admits three Nash equilibria: two
pure equilibria, $\left(a_{1},a_{2}\right)$ and $\left(b_{1},b_{2}\right),$ in
which the players coordinate their actions, and a mixed equilibrium
$\boldsymbol{p}^{NE}=\left(\frac{1}{1+u_{2}},\frac{1}{1+u_{1}}\right)$ in
which the players frequently experience miscoordination.

Table 2: Standard Representation of Two-Action Coordination Game | _$a_{2}$_ | _$b_{2}$_
---|---|---
_$~{}~{}a_{1}~{}~{}$_ | _$\,\,{}_{\underset{\,}{{\color[rgb]{0,0,1}u_{1}}}}{\,{}^{\overset{\,}{u_{2}}}}$_ | _${{}_{\underset{\,}{0}}}\,\,^{{\color[rgb]{180,00,00}\overset{\,}{0}}}$_
_$~{}~{}b_{1}~{}~{}$_ | _${}_{\underset{\,}{{\color[rgb]{0,0,1}0}}}\,\,^{\overset{\,}{{\color[rgb]{180,00,00}0}}}$_ | _${}_{{\color[rgb]{0,0,1}1}}\,\,\,\,^{{\color[rgb]{180,00,00}1}}$_
$\underset{\,}{{\color[rgb]{0,0,1}u_{1}\geq 1}}$,_ $u_{2}$>0_

#####  $\frac{1}{2}$-Dominance

Fix $q\in\left[0,1\right]$. We say that action $a_{i}$ (resp., $b_{i}$) is
_$q$ -dominant_ (Morris, Rob, and Shin, 1995; Oyama, Sandholm, and Tercieux,
2015) for player $i$ if it is a strict best response to any opponent’s mixed
action that assigns a mass of at least $q$ to the counterpart action $a_{j}$
(resp., $b_{j}$).

Notice that both actions are 1-dominant (which is equivalent to being part of
a strict equilibrium). Additionally, it can be noted that as $q$ decreases,
the $q$-dominance condition becomes more stringent. In other words, if an
action is $q$-dominant, it also satisfies $r$-dominance for any $q$ between
$r$ and 1. Lastly, it can be observed that action $a_{i}$ is
$\frac{1}{1+u_{i}}$-dominant, while action $b_{i}$ is
$\frac{u_{i}}{1+u_{i}}$-dominant.

We say that the pure equilibrium a (resp., b) is $\frac{1}{2}$-_dominant_ if
each action $a_{i}$ (resp., $b_{i}$) is $\frac{1}{2}$-dominant; in this case
we say that the remaining equilibrium is risk dominated. Observe that the game
admits a $\frac{1}{2}$-dominant equilibrium iff $u_{1},u_{2}>1$. By contrast,
if $u_{1}>1>u_{2}$, then $a_{1}$ is $\frac{1}{2}$-dominant for player 1 and
$b_{2}$ is $\frac{1}{2}$-dominant for player 2.

Observe that $q$-dominance depends only on the differences between the payoffs
a player can get by playing the different actions. This implies that
$q$-dominance is invariant to both of the payoff transformations mentioned
above (and described in detail in Appendix A.1). This implies that an action
is $q$-dominant in the standard representation of Table 2 iff it is
$q$-dominant in the original representation as a general coordination game (as
described in Appendix A.1). By contrast, payoff dominance is not invariant to
the first of these two transformations. Specifically, adding a constant to the
two payoffs of Player 1 (Player 2) in the same row (column) might change a
Pareto-dominated equilibrium into a Pareto-dominant equilibrium.

###  3.2 Evolutionary Dynamics

We assume that there are two unit-mass continuums of agents and that agents in
population 1 are randomly matched with agents in population 2\. Aggregate
behavior at time $t\in\mathbb{R}^{+}$ is described by a _state_
$\mathbf{p}\left(t\right)=\left(p_{1}\left(t\right),p_{2}\left(t\right)\right)\in\left[0,1\right]^{2}$,
which is equivalent to a mixed action profile (i.e., $p_{i}\left(t\right)$
represents the share of agents playing action $a_{i}$ at time $t$ in
population $i$). A state $\mathbf{p}=\left(p_{1},p_{2}\right)$ is _interior_
(i.e., mixed) if $p_{1},p_{2}\in(0,1).$

Agents die at a constant rate of 1, and are replaced by new agents (or,
equivalently, agents get opportunities to revise their actions). The
evolutionary process is represented by a function
$\textbf{w}:\left[0,1\right]^{2}\rightarrow\left[0,1\right]^{2}$, which
describes the frequency of new agents in each population who play action
$a_{i}$ as a function of the current state. Thus, the instantaneous change in
the share of agents in population $i$ who play $a_{i}$ is given by the
following dynamics (where $w_{i}$ is as defined in Eq. (A.1) below):

$\dot{p}_{i}=w_{i}\left(p_{j}\right)-p_{i}.$ (3.1)

##### Sample Sizes

We allow heterogeneity in the sample sizes used by new agents. Let
$\theta_{i}\in\Delta\left(\mathbb{Z}_{+}\right)$ denote the distribution of
sample sizes of new agents in population $i$. We assume that $\theta_{i}$ has
a finite support. A share of $\theta_{i}\left(k\right)$ of the new agents have
a sample of size $k$. Let $\text{supp}\left(\theta_{i}\right)$ denote the
support of $\theta_{i}$. If there exists some $k,$ for which
$\theta_{i}(k)=1,$ then we use $k$ to denote the degenerate (homogeneous)
distribution $\theta_{i}\equiv k$.

######  Remark 1.

We assume that some agents have sample sizes larger than 1 (i.e., $\forall i$
$\max(\textrm{supp}(\theta_{i}))>1$). This rules out the trivial case in which
all agents have sample size 1. In this case, the entire diagonal
$\left\\{\left(p,p\right)|p\in\left[0,1\right]\right\\}$ is Lyapunov stable,
and no state is asymptotically stable (as shown in Sethi, 2000, Example 7 in a
related setup).666See Appendix A.4 for the formal definitions of Lyapunov and
asymptotic stability. Lyapunov stability requires that populations starting
near the state remain close to the state, and asymptotic stability further
requires that the populations eventually converge to the state. The intuition
for the stability of the entire main diagonal is that new agents with sample
size 1 always adopt the action that they observe. If all new agents have
sample size 1, this implies that the share of new agents who play $a_{i}$
coincides with the share of incumbents of the other population who play
$a_{j}$. This, in turn, implies that the populations move from any initial
state toward the closest point in the main diagonal, until reaching a
stationary state in which both populations have the same aggregate behavior.

######  Definition 1.

An _environment_ is a tuple
$\left(\boldsymbol{u},\boldsymbol{\theta}\right)=\left(\left(u_{1},u_{2}\right),\left(\theta_{1},\theta_{2}\right)\right)$,
where $\left(u_{1},u_{2}\right)$ describe the payoffs of the underlying
coordination game, and $\left(\theta_{1},\theta_{2}\right)$ describes the
distributions of sample sizes in each population.

##### Sampling Best-Response Dynamics

Sampling best-response dynamics (henceforth _sampling dynamics_ ; Sandholm,
2001; Oyama, Sandholm, and Tercieux, 2015 ) fit situations in which agents do
not know the exact distribution of actions in the other population. New agents
estimate this unknown distribution by sampling the opponent’s actions.
Specifically, each new agent with sample size $k$ (henceforth, $k$-agent)
samples $k$ randomly drawn agents from the other population and then plays the
action that is the best response against the sample. One possible
interpretation of this behavior is that each new agent utilizes her own sample
to calculate a maximum likelihood estimation for the overall behavior of the
opposing population (or by any other unbiased estimation procedure,
$\grave{\textrm{a}}$ la Salant and Cherry, 2020).

To simplify notation, we assume that in case of the tie, the new agent plays
$a_{i}$. Our results are essentially the same for any tie-breaking rule.

Let $X(k,p_{j})\sim Bin\left(k,p_{j}\right)$ denote a random variable with
binomial distribution with parameters $k$ (number of trials) and $p_{j}$
(probability of success in each trial), which is interpreted as the number of
$a_{j}$-s in the sample. Observe that the sum of payoffs of playing action
$a_{i}$ against the sample is $u_{i}\cdot X(k,p_{j})$ and the sum of payoffs
of playing action $b_{i}$ against the sample is $k-X(k,p_{j})$.

This implies that action $a_{i}$ is a best response to a sample of size $k$
iff $u_{i}\cdot X(k,p_{j})\geq k-X(k,p_{j})\Leftrightarrow
X(k,p_{j})\geq\frac{k}{u_{i}+1}$. This, in turn, implies that the sampling
dynamics for environment $\left(\boldsymbol{u},\boldsymbol{\theta}\right)$ is
given by

$w_{i}\left(p_{j}\right)=\sum_{k\in\text{supp}\left(\theta_{i}\right)}\theta_{i}(k)\cdot\Pr\left(X\left(k,p_{j}\right)\geq\frac{k}{u_{i}+1}\right).$
(3.2)

Observe that
$\Pr\left(X\left(k,p_{j}\right)\geq\frac{k}{u_{i}+1}\right)=\sum_{l=\left\lceil\frac{k}{u_{i}+1}\right\rceil}^{k_{i}}\left(\begin{array}[]{c}k\\\
l\end{array}\right)p_{j}^{l}\left(1-p_{j}\right)^{k-l}$ , which is a
polynomial of $p_{j}$ of degree $k$. This implies that
$w_{i}\left(p_{j}\right)$ is a polynomial with a finite degree of
$\max\textrm{supp}\left(\theta_{i}\right)>1$.

######  Remark 2.

Our model deals with coordination games played between two different
populations (or, equivalently, with games played within a single population
where an agent can condition her play on the role she was assigned to in the
game: Player 1 or Player 2). In Appendix A.2 we show that our results remain
the same under the alternative one-population dynamics that can be applied to
symmetric coordination games ($u_{1}=u_{2}$) in which a player cannot
condition her play on her assigned role of Player 1 or Player 2.

##  4 Global Convergence to Miscoordination

As discussed in Section 2, various existing papers have shown that populations
playing coordination games will always converge to pure equilibria under
various dynamics. In this section, we fully characterize the conditions for
which the opposite result holds under sampling dynamics; i.e., the populations
converge from almost any initial state to interior states with
miscoordination.

###  4.1 Analysis of $w_{i}(\boldsymbol{\boldsymbol{p}})$

The characteristics of sampling dynamics are closely related to the properties
of the polynomials $w_{1}(p_{2})$ and $w_{2}(p_{1})$ and their intersection
points, which are analyzed in this subsection.

Figure 4.1 illustrates the phase plots of sampling dynamics and the properties
of the curves $w_{2}(p_{1})$ and $w_{1}(p_{2})$. We refer to the latter curve
also as $w_{1}^{-1}\left(p_{1}\right)$. The left panel illustrates a symmetric
coordination game with $u_{1}=u_{2}=3$, and the right panel illustrates an
antisymmetric coordination game with $u_{1}=3$ and $u_{2}=\frac{1}{3}$. In
both panels all agents have sample size of 3 ($\theta_{i}\equiv 3$). The
dashed orange curve is the polynomial $p_{2}=w_{2}(p_{1})$, which describes
the states in which $\dot{p}_{2}=0$. In all states above (resp., below) this
curve $\dot{p}_{2}<0$ (resp., $\dot{p}_{2}>0$). The solid purple curve is the
polynomial $p_{1}=w_{1}(p_{2}),$ which describes the states in which
$\dot{p}_{1}=0$. In all states to the right (resp., left) of this curve
$\dot{p}_{1}<0$ (resp., $\dot{p}_{1}>0$). Observe that in both panels, all
nonstationary initial states converge to a pure state. In the left panel there
is global convergence to $\left(1,1\right)$, while in the right panel some
states converge to $\left(1,1\right)$ and others to $\left(0,0\right)$.

Figure 4.1: Illustrative Phase Plots ($\theta_{i}\equiv 3=u_{1}$; $u_{2}$ is
either 3 or $\frac{1}{3}$ )

The figure illustrates the phase plots of sampling dynamics for two
environments: (1) $\forall i\,\,\theta_{i}\equiv 3=u_{i}$ (left panel, a
symmetric coordination game), and (2) $\forall i\,\,\theta_{i}\equiv
3=u_{1}=\frac{1}{u_{2}}$ (right panel, an antisymmetric coordination game).
The solid purple (resp., dashed orange ) curve of $w_{1}\left(p_{2}\right)$
(resp., $w_{2}\left(p_{1}\right)$) shows the states for which $\dot{p}_{1}=0$
(resp., $\dot{p}_{2}=0$ ). The intersection points of these curves are the
stationary states. A solid (resp., hollow) dot represents an asymptotically
stable (resp., unstable) stationary state.

The following fact is immediate given the basic properties of binomial random
variables.

######  Fact 1.

$w_{i}(p_{j})$ is a strictly increasing polynomial function that satisfies
$w_{i}(0)=0$ and $w_{i}(1)=1$. This implies that the inverse function
$w_{i}^{-1}:[0,1]\rightarrow[0,1]$ exists, is continuously differentiable, and
that $w_{i}^{-1}(0)=0$ and $w_{i}^{-1}(1)=1.$

Fact 1 implies that the two curves intersect at $(0,0)$ and $(1,1)$.

Appendix A.4 presents the standard definitions of stationary states,
asymptotically stable states, and unstable states. Observe that a state is
stationary (i.e., it is a fixed point of the dynamics) iff it is an
intersection point of the two curves $w_{1}$ and $w_{2}$.

######  Fact 2.

State $\boldsymbol{p}$ is stationary $\iff$ $p_{2}=w_{2}\left(p_{1}\right)$
and $p_{1}=w_{1}\left(p_{2}\right)$ $\implies$ $p_{1}=w_{1}(w_{2}(p_{1}))$ and
$p_{2}=w_{2}(w_{1}(p_{2})).$

###  4.2 Asymptotic Stability of Pure Equilibria

To state our next results, it will be helpful to consider the condition in
which a single appearance of a rare action can change the behavior of a new
agent. Specifically, consider a new agent in population $i$ with a sample size
of $k$. Observe that:

  1. 1.

Action $a_{i}$ induces a weakly higher payoff against a sample with an
opponent’s action $a_{j}$ (and $k-1$ opponents’ actions $b_{j})$ iff
$u_{i}\geq k-1\Leftrightarrow k\leq u_{i}+1$.

  2. 2.

Action $b_{i}$ induces a strictly777We require strictly higher payoffs for
action $b_{i}$ and weakly higher payoffs for action $a_{i}$ due to our tie-
breaking rule in favor of action $a_{i}$. higher payoff against a sample with
an opponent’s action $b_{j}$ (and $k-1$ opponents’ actions $a_{j})$ iff
$1>\left(k-1\right)u_{i}\Leftrightarrow k<\frac{1}{u_{i}}+1$.

Next we define $m$_-bounded expectation_ as the expected value of a
probability distribution when taking into account only values smaller than
$m$. Formally:

######  Definition 2.

The $m$_-bounded expectation_ $\mathbb{E}_{\leq m}$ (resp., $\mathbb{E}_{<m}$)
of distribution $\theta_{i}$ with support on integers is888Observe that in our
notation the parameter $k$ takes only (positive) integer values (although we
allow the upper bound $m$ to be a noninteger). $\mathbb{E}_{\leq
m}\left(\theta_{i}\right)=\sum_{1\leq k\leq m}\theta_{i}\left(k\right)\cdot k$
(resp., $\mathbb{E}_{<m}\left(\theta_{i}\right)=\sum_{1\leq
k<m}\theta_{i}\left(k\right)\cdot k$).

Proposition 1 characterizes the asymptotic stability of the pure equilibria.
It shows that asymptotic stability depends only on whether the product of the
bounded expectations of the distributions of sample sizes is larger or smaller
than one, where the bound of each distribution is the maximal sample size for
which a single appearance of a rare action can change the behavior of a new
agent. Formally:999Replacing the $a_{i}$-favorable tie-breaking rule with a
$b_{i}$-favorable one would replace the “<”-s and the “$\leq$”-s in the
bounded expectations in the statement of Proposition 1.

######  Proposition 1.

  1. 1.

$\mathbb{E}_{<\frac{1}{u_{1}}+1}\left(\theta_{1}\right)\cdot\mathbb{E}_{<\frac{1}{u_{2}}+1}\left(\theta_{2}\right)=\theta_{1}\left(1\right)\cdot\mathbb{E}_{<\frac{1}{u_{2}}+1}\left(\theta_{2}\right)>1\,\Rightarrow$
a=$\left(a_{1},a_{2}\right)$ is unstable;

  2. 2.

$\mathbb{E}_{<\frac{1}{u_{1}}+1}\left(\theta_{1}\right)\cdot\mathbb{E}_{<\frac{1}{u_{2}}+1}\left(\theta_{2}\right)=\theta_{1}\left(1\right)\cdot\mathbb{E}_{<\frac{1}{u_{2}}+1}\left(\theta_{2}\right)<1\,\Rightarrow$
$\boldsymbol{a}$ is asymptotically stable;

  3. 3.

$\mathbb{E}_{\leq u_{1}+1}\left(\theta_{1}\right)\cdot\mathbb{E}_{\leq
u_{2}+1}\left(\theta_{2}\right)>1\,\Rightarrow$ b=$\left(b_{1},b_{2}\right)$
is unstable; and

  4. 4.

$\mathbb{E}_{\leq u_{1}+1}\left(\theta_{1}\right)\cdot\mathbb{E}_{\leq
u_{2}+1}\left(\theta_{2}\right)<1\,\Rightarrow$ b=$\left(b_{1},b_{2}\right)$
is asymptotically stable.

###### Sketch of Proof.

Consider a slightly perturbed state $(1-\epsilon_{1},1-\epsilon_{2})$ near
$\boldsymbol{a}=\left(1,1\right)$ (the argument for $\boldsymbol{b}$ is
analogous) in which almost all agents play action $a_{i}$. The event of two
rare actions ($b_{i}$-s) appearing in a sample of a new agent has a negligible
probability of $O(\epsilon_{i}^{2})$. If a new agent has a sample size of $k$,
then the probability of a rare action appearing in the sample is approximately
$k\cdot\epsilon_{j}.$ This rare appearance changes the perceived best response
of a new agent in population $i$ iff $k$ is smaller than $\frac{1}{u_{i}}+1$.
Thus, the probability that a new agent in population $i$ adopts the rare
action $b_{i}$ is equal to
$\mathbb{E}_{<\frac{1}{u_{i}}+1}\left(\theta_{i}\right)$. This implies that
the product of the share of new agents adopting a rare action in each
population is
$\epsilon_{1}\cdot\mathbb{E}_{<\frac{1}{u_{1}}+1}\left(\theta_{1}\right)\cdot\epsilon_{2}\cdot\mathbb{E}_{<\frac{1}{u_{1}}+1}\left(\theta_{2}\right)$.
This shows that the share of agents playing rare actions gradually increases
(resp., decreases) if
$\mathbb{E}_{<\frac{1}{u_{1}}+1}\left(\theta_{1}\right)\cdot\mathbb{E}_{<\frac{1}{u_{1}}+1}\left(\theta_{2}\right)>1$
(resp.,
$\mathbb{E}_{<\frac{1}{u_{1}}+1}\left(\theta_{1}\right)\cdot\mathbb{E}_{<\frac{1}{u_{1}}+1}\left(\theta_{2}\right)<1),$
which implies instability (resp., asymptotic stability). Finally, observe that
our assumption that $u_{1}\geq 1$ implies that $\frac{1}{u_{1}}+1\leq 2$,
which, in turn, implies that
$\theta_{1}\left(1\right)=\mathbb{E}_{<\frac{1}{u_{1}}+1}\left(\theta_{1}\right)$.
See Appendix A.5 for a formal proof. ∎

An interesting implication of Proposition 1 is the substantial difference in
the stability of a $\frac{1}{2}$-dominant equilibrium and a risk-dominated
equilibrium (related results for symmetric coordination games are derived in
Oyama, Sandholm, and Tercieux, 2015). A risk-dominated equilibrium (say,
$\boldsymbol{b}$) is unstable as long as sufficiently many agents have samples
that are not too large (i.e., samples below $u_{i}+1$). The process inducing
instability is as follows. A small perturbation of a few agents who play
$\boldsymbol{a}_{i}$ induces a slightly larger number of new agents to observe
$\boldsymbol{a}_{i}$ at least once in their samples, which induces them to
play $a_{i}$ as well, which allows the small perturbations to gradually
increase, until, at the end of the process, everyone plays the
$\frac{1}{2}$-dominant action $\boldsymbol{a}_{i}$.

By contrast, any $\frac{1}{2}$-dominant equilibrium is asymptotically stable.
To see this, observe that if $\boldsymbol{a}$ is a weakly
$\frac{1}{2}$-dominant equilibrium (i.e., if $u_{2}\geq 1$, where the weak
inequality is due to the tie-breaking rule in favor of $a_{i}$) then
$\frac{1}{u_{2}}+1\leq 2$, which implies that
$\mathbb{E}_{<\frac{1}{u_{2}}+1}\left(\theta_{2}\right)=\theta_{2}\left(1\right)<1$.
This implies that
$\mathbb{E}_{<\frac{1}{u_{1}}+1}\left(\theta_{1}\right)\cdot\mathbb{E}_{<\frac{1}{u_{2}}+1}\left(\theta_{2}\right)=\theta_{1}\left(1\right)\cdot\theta_{2}\left(1\right)<1$,
and thus $\boldsymbol{a}$ is asymptotically stable. This implies that:

######  Corollary 1.

Any $\frac{1}{2}$-dominant equilibrium is asymptotically stable.

Thus, only in games in which neither equilibrium is $\frac{1}{2}$-dominant
(i.e., those in which $u_{2}<1\leq u_{1}$), might it be possible for both pure
equilibria to be unstable.

###  4.3 Global Convergence Results

The following definition of neighboring stationary states will be helpful for
stating our results.

######  Definition 3.

Two stationary states $\underline{\boldsymbol{p}},\boldsymbol{\overline{p}}$
are said to be neighbors if there does not exist any stationary state
$\hat{\boldsymbol{p}}$ such that
$\underline{p}_{i}<\hat{p}_{i}<\overline{p}_{i}$.

We first show that the populations always converge to a stationary
state.101010One could present an alternative proof to Proposition 2 that
relies, in part, on the Bendixson–Dulac Theorem (see Theorem 9.A.6 of
Sandholm, 2010) to show that there are no closed orbits. However, we prefer
presenting a direct proof due to its simplicity, and because some of its
arguments are used later in the results of Section 5.

######  Proposition 2.

$\lim_{t\rightarrow\infty}\mathbf{p}\left(t\right)$ converges to a stationary
state for any $\textbf{p}(0).$

###### Proof.

Since the function $w_{i}(\cdot)$ is a polynomial of degree
$\max(\textrm{supp}\left(\theta_{i}\right))>1,$ it follows that
$w_{i}\left(w_{j}\left(\cdot\right)\right)$ is a polynomial of a finite degree
strictly larger than 1. As the stationary states are characterized by the
intersection of the curves $p_{1}=w_{1}(p_{2})$ and $p_{2}=w_{2}(p_{1}),$ we
can conclude that there are a finite number of stationary states. The result
now follows from the following claims whose proofs are presented in Appendix
A.6.

######  Claim 1 (Any trajectory reaches the area between the curves).

Either $\lim_{t\rightarrow\infty}\mathbf{p}\left(t\right)$ converges to a
stationary state, or there exists $t<\infty$ such that either
$w_{1}^{-1}\left(p_{1}\left(t\right)\right)\leq p_{2}\left(t\right)\leq
w_{2}\left(p_{1}\left(t\right)\right)$ or
$w_{2}\left(p_{1}\left(t\right)\right)\leq p_{2}\left(t\right)\leq
w_{1}^{-1}\left(p_{1}\left(t\right)\right)$.

The intuition for Claim 1 (see Figure 4.1) is that populations starting at an
initial state below (resp., above) the two curves move upward and to the left
(resp., downward and to the right) until reaching the area between the two
curves.

######  Claim 2 (Convergence from the area between the curves to a neighboring
stationary state).

Let $\boldsymbol{p}\left(t\right)$ be an interior state for some $t\geq 0$ and
let $\underline{\boldsymbol{p}},\boldsymbol{\overline{p}}$ be neighboring
stationary states with $\underline{p}_{i}\leq p_{i}(t)\leq\overline{p}_{i}$.
We have,

  1. 1.

If
$p_{2}\left(t\right)\in\left[w_{1}^{-1}\left(p_{1}\left(t\right)\right),w_{2}\left(p_{1}\left(t\right)\right)\right]$,
then
$\lim_{t\rightarrow\infty}\mathbf{p}\left(t\right)=\overline{\boldsymbol{p}}$,
and

  2. 2.

If
$p_{2}\left(t\right)\in\left[w_{2}\left(p_{1}\left(t\right)\right),w_{1}^{-1}\left(p_{1}\left(t\right)\right)\right]$,
then
$\lim_{t\rightarrow\infty}\mathbf{p}\left(t\right)=\underline{\boldsymbol{p}}$.∎

The intuition for Claim 2 (see Figure 4.1) is that in the area between the two
curves in the interval between $\underline{\boldsymbol{p}}$ and
$\boldsymbol{\overline{p}}$, the populations either always move upward and to
the right (in which case they must converge to $\boldsymbol{\overline{p}}$),
or they always move downward and to the left (in which case they must converge
to $\boldsymbol{\underline{p}}$).

Next we show that if any initial interior state converges to one of the pure
equilibria, then this equilibrium is asymptotically stable (as defined in
Appendix A.4).

######  Lemma 1.

Assume that $\textbf{p}(0)\neq(0,0)$ and
$\lim_{t\rightarrow\infty}\mathbf{p}\left(t\right)=(0,0)$; then $(0,0)$ is
asymptotically stable. The same result holds when $(1,1)$ replaces $(0,0).$

###### Proof.

See Appendix A.7. ∎

We now show that the populations converge from almost any initial state to an
interior stationary state if (and essentially only if) the product of the
bounded expectation of the distribution of sample sizes in each population and
the share of agents with sample size 1 in the other population is larger than
1.

######  Theorem 1.

Assume that $u_{2}<1\leq u_{1}$ (no $\frac{1}{2}$-dominant equilibrium). Then

  1. 1.

Global convergence to miscoordination: Assume that

$\theta_{1}\left(1\right)\cdot\mathbb{E}_{<\frac{1}{u_{2}}+1}\left(\theta_{2}\right)>1\,\,\textrm{\,\,and}\,\,\,\,\theta_{2}\left(1\right)\cdot\mathbb{E}_{\leq
u_{1}+1}\left(\theta_{2}\right)>1.$

If $\boldsymbol{p}(0)\notin\left\\{(0,0),(1,1)\right\\}$, then
$\lim_{t\rightarrow\infty}\mathbf{p}\left(t\right)\notin\left\\{(0,0),(1,1)\right\\}$.

  2. 2.

Local convergence to coordination: Assume that

$\theta_{1}\left(1\right)\cdot\mathbb{E}_{<\frac{1}{u_{2}}+1}\left(\theta_{2}\right)<1\,\,\,\,\textrm{or}\,\,\,\,\theta_{2}\left(1\right)\cdot\mathbb{E}_{\leq
u_{1}+1}\left(\theta_{2}\right)<1.$

Then at least one of the pure equilibria is asymptotically stable.

###### Proof.

  1. 1.

Proposition 1 implies that both pure stationary states are unstable.
Proposition 2 and Lemma 1 imply that from almost any initial state, the
population converges to an interior stationary state.

  2. 2.

Proposition 1 implies that at least one of the pure equilibria is
asymptotically stable, which implies that some interior initial states
converge to a pure equilibrium. ∎

Theorem 1 shows that global convergence to miscoordination requires
heterogeneity in the sample sizes in each population that includes both agents
with a small sample size of one, and agents with larger samples (but not too
large, as they must be below the bound for which a single observation of a
rare action can influence behavior). Specifically, in each population it is
required that the product of (1) the share of agents with a sample size of 1
and (2) the bounded expected sample size should be sufficiently large.

Observe that the farther the $u_{i}$-s are from 1, the higher (i.e., less
restrictive) the bounded expected value is. That is, games in which for each
population one of the actions is much riskier than the other action (i.e., for
population 2, action $a_{2}$ is much riskier than action $a_{1},$ while for
population 1, action $a_{1}$ is much riskier than action $a_{2}$) are more
likely have stable miscoordination.

The following example demonstrates global convergence to miscoordination, and
the fact that the stability of pure states is nonmonotone in the sample sizes.

######  Example 1 (Nonmonotone Impact of Sample Size).

Consider a battle of the sexes game in which $u_{1}=5$ and $u_{2}=0.2$ and
both populations have the same distribution of sample sizes. Consider 3
distributions of sample sizes, in each of which half of the players in the
population have sample size 1\. In the first distribution (left panel of
Figure 4.2) the remaining half of the players have sample size 3, in the
second distribution (middle panel) they have sample size 5, and in the third
distribution (right panel of Figure 4.2) they have sample size 7. Observe that
the second distribution satisfies the condition for global convergence to
miscoordination, i.e.,

$\theta_{1}\left(1\right)\cdot\mathbb{E}_{<\frac{1}{u_{2}}+1}\left(\theta_{2}\right)=\theta_{1}\left(1\right)\cdot\mathbb{E}_{<6}\left(\theta_{2}\right)=\theta_{1}\left(1\right)\cdot\mathbb{E}\left(\theta_{2}\right)=0.5\cdot
3>1$

(and the same holds for population 2). Indeed, the middle phase plot shows
that the populations converge from any interior state to the state
$\left(0.37,0.63\right)$ with substantial miscoordination (specifically, the
agents miscoordinate and get a payoff of zero in 46%$\approx 2\cdot 0.37\cdot
0.63$ of the interactions). By contrast, either decreasing the larger sample
size from 5 to 2, or increasing it to 7, yields a product
$\theta_{1}\left(1\right)\cdot\mathbb{E}_{<\frac{1}{u_{2}}+1}\left(\theta_{2}\right)$
that is strictly smaller than 1
($\theta_{1}\left(1\right)\cdot\mathbb{E}\left(\theta_{2}\right)=0.5\cdot
1.5<1$ in the first case, and
$\theta_{1}\left(1\right)\cdot\mathbb{E}_{<6}\left(\theta_{2}\right)=\theta_{1}\left(1\right)\cdot\theta_{1}\left(1\right)=0.5\cdot
0.5<1$ in the second case). The left and right panels of Figure 4.2 illustrate
that in both cases almost all initial states converge to a pure state. Thus,
changing sample sizes of agents has a nonmonotone impact on the stability of
miscoordination.

Figure 4.2: Illustration of Theorem 1 and Example 1

The figure illustrates the phase plots for three environments. In each
environment, the underlying coordination game is antisymmetric with $u_{1}=5$
and $u_{2}=0.2$, and 50% of the agents in each population have sample size 1.
In the environment illustrated in the left (resp., middle, right) panel the
remaining half of the players in the population have sample size 2 (resp., 5,
7). The middle panel shows global convergence to the interior state with
miscoordination $\left(0.37,.0.63\right)$, while the other two panels show
global convergence to one of the pure equilibria.

##  5 Homogeneity and Unstable Miscoordination

In this section, we show that heterogeneity is necessary for stable
miscoordination. Specifically, we show that any environment in which all
agents in each population have the same sample size admits at most one
interior stationary state, and that this state is unstable.

##### Auxiliary Results

We begin by showing that a stationary state $\hat{\boldsymbol{p}}$ is
asymptotically stable iff the curve $w_{2}$ is above the curve $w_{1}^{-1}$ in
a left neighborhood of $\hat{\boldsymbol{p}}$, and it is below the curve
$w_{1}^{-1}$ in a right neighborhood of $\hat{\boldsymbol{p}}$.

######  Lemma 2.

Let $\hat{\boldsymbol{p}}$ be a stationary state. $\hat{\boldsymbol{p}}$ is
asymptotically stable if both of the following conditions hold:

  1. 1.

Left neighborhood: If $\hat{p}_{1}>0$ then there exists
$\underline{p}_{1}\in\left(0,\hat{p}_{1}\right)$ such that
$w_{2}\left(p_{1}\right)>w_{1}^{-1}\left(p_{1}\right)$ for any
$p_{1}\in\left(\underline{p}_{1},\hat{p}_{1}\right)$, and

  2. 2.

Right neighborhood: If $\hat{p}_{1}<1,$ then there exists
$\overline{p}_{1}\in\left(\hat{p}_{1},1\right)$ such that
$w_{2}\left(p_{1}\right)<w_{1}^{-1}\left(p_{1}\right)$ for any
$p_{1}\in\left(\hat{p}_{1},\overline{p}_{1}\right)$.

Moreover, if either of the above two conditions is not satisfied, then
$\hat{\boldsymbol{p}}$ is unstable.

###### Proof.

See Appendix A.8. ∎

Lemma 2 implies that the neighbor of an asymptotically stable state must be
unstable.

######  Corollary 2.

Let $\underline{\boldsymbol{p}}\neq\overline{\boldsymbol{p}}$ be two
neighboring stationary states. If $\underline{\boldsymbol{p}}$ is
asymptotically stable, then $\overline{\boldsymbol{p}}$ is unstable.

###### Proof.

Without loss of generality assume that $\underline{p}_{1}<\overline{p}_{1}$.
Due to Claim 2, the fact that $\underline{\boldsymbol{p}}$ is asymptotically
stable implies that $w_{2}\left(p_{1}\right)>w_{1}^{-1}\left(p_{1}\right)$ for
any $p_{1}\in\left(\underline{p}_{1},\overline{p}_{1}\right)$ which, in turn
implies that $\overline{\boldsymbol{p}}$ is unstable.∎

##### A Property of Binomial Distributions

The main result is implied by a property of binomial distributions (which may
be of independent interest). Recall our notation of $X(k,p)\sim
Bin\left(k,p\right)$ denoting a random variable with a binomial distribution.
Define $F_{m}^{k}\left(p\right)\equiv\Pr\left(X\left(k,p\right)\geq m\right)$
as the probability of having at least $m$ successes in $k$ trials when the
probability of success in each trial is $p$. Observe that
$F_{m}^{k}\left(0\right)=0$, $F_{m}^{k}\left(1\right)=1,$ and
$\left(F_{m}^{k}\right)^{\prime}>0$. It is known that $F_{m}^{k}$ has at most
one interior fixed point, i.e.,

######  Fact 3 (Green, 1983, Theorem 1).

Fix arbitrary integers $0<m\leq k$. Then there is at most one
$p\in\left(0,1\right)$ such that $F_{m}^{k}\left(p\right)=p.$

We extend it by showing that the same is true also for a composition of any
two cumulative binomial distributions, i.e., that $F_{m_{1}}^{k_{1}}\circ
F_{m_{2}}^{k_{2}}$ has at most one interior fixed point.

######  Proposition 3.

Fix arbitrary integers satisfying $0<m_{1}\leq k_{1}$ and $0<m_{2}\leq k_{2}$.
There exists at most one $p\in\left(0,1\right)$ such that
$\left(F_{m_{1}}^{k_{1}}\circ F_{m_{2}}^{k_{2}}\right)\left(p\right)=p$.

The proof, which is detailed in Appendix A.3, shows that
$F_{m_{1}}^{k_{1}}\circ F_{m_{2}}^{k_{2}}$ has at most one inflection point,
which implies that there exists at most one interior fixed point.

##### Main Result

We now present the main result of this section: any environment with
homogeneous sample sizes admits at most one interior stationary state that is
unstable. This implies that almost all initial states converge to one of the
pure equilibria.

######  Theorem 2.

Assume that $\theta_{i}\equiv k_{i}>1$ for each $i\in\left\\{1,2\right\\}$.
There exists at most one interior stationary state, and this state (if it
exists) is unstable.

###### Proof.

The fact that $\theta_{i}\equiv k_{i}>1$ implies that
$w_{i}\left(p\right)=F_{m_{i}}^{k_{i}}\left(p\right)$ for some $1\leq
m_{i}\leq k_{i}$. This implies that any stationary state
$\boldsymbol{p}^{\ast}$ must satisfy $\left(F_{m_{1}}^{k_{1}}\circ
F_{m_{2}}^{k_{2}}\right)\left(p_{1}^{\ast}\right)=p_{1}^{\ast}$. Proposition 3
implies that this holds for at most one interior state $\hat{\boldsymbol{p}}$.
Therefore, the stationary state $\hat{\boldsymbol{p}}$ (if it exists) is a
neighbor of both pure stationary states $\left(0,0\right)$ and
$\left(1,1\right)$. Proposition 1 and the fact that no agents have sample size
of 1 implies that at least one of these pure states is asymptotically stable.
Finally, Corollary 2 implies that $\hat{\boldsymbol{p}}$ is unstable. ∎

##  6 Heterogeneity and Stable Miscoordination

The conditions presented for global convergence to miscoordination in Section
4 are somewhat narrow in the sense of requiring sufficiently many agents with
sample size one. In this section we show that a much broader set of
heterogeneous distributions of sample sizes can induce asymptotically stable
states with miscoordination.

Specifically, we show that essentially any distribution of sample sizes that
combines agents with small samples and agents with large samples induces
locally stable miscoordination if the $u_{i}$-s are not too close to one. As
demonstrated below, this type of heterogeneity in the sample sizes is
plausible in various setups.

######  Example 2.

In housing markets (in which the bargaining situations can be modeled as hawk-
dove coordination games), it is often the case that each population includes
two types of agents: (1) professional real-estate investors, and (2) people
who buy/sell houses only a couple of times during their life. It seems
plausible that the real-estate investors have reliable information on the
aggregate behavior (captured in our model by having large samples), while the
remaining agents are likely to have limited, anecdotal evidence about the
aggregate behavior (modeled by having small samples).

##### Result

Recall (Theorem 2) that all interior equilibria are unstable if all agents in
each population have the same size. Our final result shows that, perhaps
surprisingly, one can always add agents with large samples, and obtain an
asymptotically stable interior equilibrium with miscoordination, provided that
the $u_{i}$-s are not too close to one.

We formally define adding agents with large samples as follows. Given two
sample sizes $k,\bar{k}$ and $\alpha\in(0,1)$, let $k\alpha\bar{k}$ be the
distribution of sample sizes that assigns mass $\alpha$ to sample size $k$ and
mass $1-\alpha$ to sample size $\bar{k}$.

######  Theorem 3.

Fix any pair of sample sizes $k_{1},k_{2}>1$. Then there exist
$r,\alpha_{1},\alpha_{2}\in(0,1)$, such that for any
$u_{1},u_{2}\notin(r,\frac{1}{r})$ the environment
$\left(\boldsymbol{u},\left(k_{1}\alpha_{1}\bar{k},k_{2}\alpha_{2}\bar{k}\right)\right)$
admits an asymptotically stable interior state for any sufficiently large
$\bar{k}$.

###### Proof idea (see Appendix A.9 for a formal proof).

We present one argument for games with a $\frac{1}{2}$-dominant equilibrium,
and another for those without. The arguments are illustrated in the phase
plots of Figure 6.1.

  1. 1.

_Games with a $\frac{1}{2}$-dominant equilibrium_: Assume that $u_{i}>k_{i}$
for each $i$. Proposition 1 implies that the risk-dominated equilibrium
$\boldsymbol{b}$ is unstable when all agents have sample size $k_{i}$. Observe
that adding agents with large samples decreases the average probability of new
agents playing $a_{i}$ in state $\left(\epsilon_{1},\epsilon_{2}\right)$ near
$\boldsymbol{b}$. We choose $\alpha_{i}$, such that the share of new agents
playing action $a_{i}$ (1) is above $\epsilon_{j}$ in state
$\left(\epsilon_{1},\epsilon_{2}\right)$, and (2) is below $p_{j}$ in an
interior state $\boldsymbol{\hat{p}}$ (this is possible because
$\frac{w_{i}\left(p_{j}\right)}{p_{j}}$ is decreasing in small $p_{j}$-s).
This, in turn, implies that there is a stable interior state between
$\boldsymbol{b}$ and $\boldsymbol{\hat{p}}$.

  2. 2.

_Games without a $\frac{1}{2}$-dominant equilibrium_: Observe that
$w_{i}^{k_{i}}\left(p_{j}\right)$ remains the same for all values of $u_{i}$
that are sufficiently far from one ($u_{1}>k_{1}$, $u_{2}<\frac{1}{k_{2}}$),
while the interior Nash equilibrium $p^{NE}_{1}$ (resp., $p^{NE}_{2}$)
converges to 0 (resp., 1) as $u_{1}$ converges to infinity (resp., as $u_{2}$
converges to zero). Thus, for $u_{i}$-s sufficiently far from 1,
$p_{2}^{NE}<\frac{w_{2}^{\theta_{2}}\left(\frac{1}{2}\right)}{2}$. We show
that this implies that there must be an unstable stationary state with
$p_{1}<\frac{1}{2}$ and $p_{2}\simeq p_{2}^{NE}<\frac{1}{2}$. By an analogous
argument, there is an unstable stationary state with $p_{1}\simeq
p_{1}^{NE}>\frac{1}{2}$ and $p_{2}>\frac{1}{2}$. This implies that there must
be an asymptotically stable interior state between these two unstable states.
∎

Figure 6.1: Illustrative Phase Plots for Theorem 3 (Locally Stable
Miscoordination)

The left panel illustrates the symmetric game $u_{1}=u_{2}=2.5$, where 40%
(resp., 60%) of the agents in each population have sample size 3 (resp.,
1,000). The pure risk-dominated equilibrium $\left(0,0\right)$ is unstable,
and its neighbor the interior stationary state with miscoordination
$\left(0.18,0.18\right)$ is asymptotically stable. The right panel illustrates
the antisymmetric game $u_{1}=\frac{1}{u_{2}}=5$, where 50% (resp., 50%) of
the agents have sample size 2 (resp., 1,000). The mixed Nash equilibrium of
this game is $\left(\frac{5}{6},\frac{1}{6}\right)$. The environment admits
three interior stationary states: two unstable interior stationary states
where one of the coordinates is close to the Nash equilibrium:
$\left(0.6,0.18\right)$ and $\left(0.82,0.3\right)$, and a stable interior
stationary state at $\left(0.73,0.27\right)$.

######  Remark 3.

For simplicity, Theorem 3 deals with a heterogeneous population that combines
two sample sizes, $k_{i}$ and $\bar{k}.$ The proof’s argument can be adapted
in a straightforward way to populations combining some agents who have
arbitrary sample sizes (i.e., an arbitrary distribution $\theta_{i}$ replaces
the arbitrary $k_{i}$), and some agents who have sufficiently large sample
sizes (i.e., not necessarily all of the agents with large samples have the
same sample size of $\bar{k}).$

###### Non-Payoff-Maximizing Behavior of Agents with Small Samples

Observe the interior stable state is (generically) not a mixed Nash
equilibrium. For example, in the environment illustrated in the left panel of
Figure 3 in which $u_{i}=2.5$, $\theta_{i}\left(3\right)=40\%$ and
$\theta_{i}\left(1,000\right)=60\%$ for each population $i$, the interior
stable state is $\left(0.17,0.17\right)$, while the mixed Nash equilibrium is
$\left(0.28,0.28\right)$. This implies that one of the actions is the unique
true best response to the opponent’s behavior (action $b_{i}$ in the example).
This unique best response is played essentially by all agents with a large
sample size of 1,000. By contrast, about $45\%$ of the agents with a small
sample size of $3$ observe action $a_{-i}$ at least once in their sample, and
play action $a_{i}$ due to an erroneous belief (based on their small sample)
that the frequency of $a_{-i}$ is at least $\frac{1}{3}$. As a result, the
average payoff to agents with a small sample is only 0.75, which is 8% smaller
then the payoff of 0.82 to agents with a large sample.

This raises the question of why the agents with small samples do not learn to
increase their samples, or why they are not driven out of the population. We
suggest two explanations for this. The first is setups like our motivating
example of a housing market that combine both inexperienced players (who have
little experience in buying/selling houses) and experienced players (real-
estate investors). In such setups, it seems likely that the inexperienced
agents might have very limited (or costly) access to large samples (and a
continuous inflow of inexperienced traders prevents them from being driven out
of the market). Another reason why some players do not exert effort to
increase their sample sizes is the “the law of small numbers,” which is the
commonly believed bias that small samples well represent large populations
(Tversky and Kahneman, 1971).

##  7 Comparison with Logit Dynamics

The main candidate in the existing literature to induce stable miscoordination
in coordination games is logit dynamics. In this section we numerically
demonstrate that (1) standard logit dynamics with a homogeneous level of noise
in each population can induce stable miscoordination only with high levels of
noise that seem implausible, and (2) a variant of logit dynamics with
heterogeneity in the noise level can induce stable miscoordination with
substantially lower levels of noise. This suggests that our key insight that
stable miscoordination is induced by heterogeneous noise might hold in various
classes of dynamics, and not only in sampling dynamics.111111 Preliminary
calculations suggest that this insight holds also under best experienced
payoff dynamics (see, e.g., Sethi, 2000) and under dynamics induced by
representative sampling $\grave{\textrm{a}}$ la Danenberg and Spiegler (2022).

##### Standard (Homogeneous) Logit Dynamics

Logit dynamics (introduced in Fudenberg and Levine, 1995; see Sandholm, 2010,
Section 6.2.3 for a textbook exposition) are characterized by a single
parameter $\eta_{i}$ that describes the _noise level_ for each population $i$.
If player $i$ plays action $a_{i}$, she will get a payoff of $p_{j}\cdot
u_{j}$. If she plays action $b_{i}$ she will get a payoff of
$\left(1-p_{j}\right)\cdot 1$. Logit dynamics assume that the probability of
revising agents playing action $a_{i}$ is proportional to
$e^{\frac{\textrm{Payoff of $a_{i}$}}{\eta}}$. Specifically, logit dynamics
are given by

$w_{i}\left(p_{j}\right)\equiv
w_{i}\left(\textbf{p}\right)=\frac{e^{\frac{p_{j}\cdot
u_{j}}{\eta_{i}}}}{e^{\frac{1-p_{j}}{\eta_{i}}+}e^{\frac{p_{j}\cdot
u_{j}}{\eta_{i}}}}.$ (7.1)

Trivially, logit dynamics can induce substantial miscoordination by having
high values of noise. The interesting question is whether stable
miscoordination can be supported by a low level of noise. Our numerical
analysis suggests that the answer is negative. In what follows we demonstrate
that this is indeed the case. For example, when we revisit the the two
examples of Figure 6.1 ($u_{1}=u_{2}=2.5$ and $u_{1}=\frac{1}{u_{2}}=5$), then
the minimal level of noise that is required to sustain an asymptotically
stable interior state in which each action is played with a probability of at
least 10% is $\eta=1$ (see the left panel of Figure 7.1 for an illustration of
the case of $u_{1}=u_{2}=2.5$). Such a high level of noise implies that 27% of
the revising agents make the obvious mistake of playing $a_{i}$ when facing a
population in which almost everyone plays $b_{j}$; by contrast, this obvious
mistake is never made under action-sampling dynamics. Moreover, the average
expected payoff obtained by revising agents who follow logit dynamics against
an opponent population in which the share of agents playing action $a_{i}$ is
distributed uniformly is 85% (resp., 71%) of the maximal payoff that can be
obtained by payoff-maximizing agents in the first (resp., second) environment
with $u_{1}=u_{2}=2.5$ (resp., $u_{1}=\frac{1}{u_{2}}=5$). By contrast, this
average expected payoff is 98% (resp., 95%) of the maximal payoff under the
sampling dynamics. Thus, stable cooperation can be supported by standard
(homogeneous) logit dynamics only when the agents have high levels of noise.

##### Heterogeneous Logit Dynamics

Figure 7.1: Supporting Stable Coordination with Heterogeneous Logit Dynamics

The figure revisits the symmetric game presented in the left panel of Figure
6.1 in which $u_{1}=u_{2}=2.5$. The left panel shows the phase plot of the
minimal homogeneous level of noise, $\eta_{i}=1$, that sustains an
asymptotically stable state in which each action is played with a probability
of at least 10%. The right panel shows the phase plot of a heterogeneous
variant of logit dynamics in which 55% of the the agents in each population
have a moderate level of noise $\eta_{i}=0.55$ and 45% have a small level of
noise $\eta_{i}=0.01$.

Next, consider a variant of logit dynamics in which there is heterogeneity in
the level of noise for agents in each population. Specifically, in a
population in which there are $n$ groups, the size of the $l$-th group is
$\mu_{i}^{l}$ and its members have a noise level of $\eta_{i}^{l},$ the
heterogeneous logit dynamics are given by

$w_{i}\left(p_{j}\right)\equiv
w_{i}\left(\textbf{p}\right)=\sum_{l}\mu_{i}^{l}\cdot\frac{e^{\frac{p_{j}\cdot
u_{j}}{\eta_{i}^{l}}}}{e^{\frac{1-p_{j}}{\eta_{i}^{l}}+}e^{\frac{p_{j}\cdot
u_{j}}{\eta_{i}^{l}}}}.$ (7.2)

The numerical calculations demonstrate that heterogeneous noise levels can
induce asymptotically stable miscoordination with relatively low levels of
noise. Specifically, in both of the above examples ($u_{1}=u_{2}=2.5$, which
is illustrated in the right panel of Figure 7.1, and
$u_{1}=\frac{1}{u_{2}}=5$), populations in which 55% of the agents have a
moderate level of noise (i.e., $\eta=0.55$) and 45% have a small level of
noise (i.e., $\eta=0.01$) induce asymptotically stable states with
miscoordination ($\left(0.21,0.21\right)$ in the right panel of Figure 7.1).
Given these heterogeneous levels of noise, only 8% of the agents make the
mistake of playing action $a_{i}$ when facing a population in which everyone
plays $a_{j}$, and the average expected payoff obtained by playing against
opponent populations in which the share of agents playing action $a_{i}$ is
distributed uniformly is 96% (resp., 89%) of the maximal payoff that can be
obtained by payoff-maximizing revising agents in the environment with
$u_{1}=u_{2}=2.5$ (resp., $u_{1}=\frac{1}{u_{2}}=5$).

##  8 Conclusion

The conventional wisdom, which is supported by key results in evolutionary
game theory, is that only pure (coordinated) outcomes are reasonable long-run
predictions of behavior in coordination games. By contrast, we show that
plausible learning dynamics, in which new agents rely on samples to estimate
and best respond to the behavior of the opponents’ population, can induce
stable miscoordination. This happens if there is heterogeneity in the sample
sizes: some agents have accurate information based on large samples of the
opponents’ aggregate behavior, while other agents rely on anecdotal evidence
induced by small samples. We further show that stable miscoordination holds
under a broad set of heterogeneous distributions of sample sizes, if for each
player one of the actions in the underlying game is $q$-dominant for a
sufficiently small $q$.

Although our analytical results focus on the specific family of sampling
dynamics, the numerical results for logit dynamics suggest that qualitatively
similar results are likely to hold under other learning dynamics, i.e.,
heterogeneous noise levels can induce stable miscoordination. Such
heterogeneity is plausible in many applications, such as bargaining in housing
markets, where some participants are professional investors, while other
participants are inexperienced. The predictions of our model can be
experimentally tested, as outlined Section 2.

##  Appendix A Appendix

###  A.1 Standard Representation of General Coordination Games

In this subsection we explain why the two-parameter standard representation
(Table 2) captures w.l.o.g. any two-action coordination game (in line with
Harsanyi and Selten’s () Axiom 2 of best-reply invariance).

The most general definition of a two-action coordination game is a game that
admits two strict Nash equilibria. By relabeling the actions of player 1, we
can assume w.l.o.g. that these two pure equilibria are
$\left(a_{1},a_{2}\right)$ and $\left(b_{1},b_{2}\right)$ (i.e., if the two
pure equilibria are $\left(a_{1},b_{2}\right)$ and $\left(b_{1},a_{2}\right)$,
then we switch the labels of player 1’s actions:
$a_{1}$$\leftrightarrow$$b_{1}).$ This implies that the left panel of Table 3
shows a parametric representation of all coordination games.

Table 3: Normalization of General Two-Action Coordination Games Original Representation |  | Standard Representation
---|---|---
| _$a_{2}$_ | _$b_{2}$_ | $\begin{array}[]{c}\\\ \\\ \Rightarrow\\\ \\\ \\\ \end{array}$ |  | _$a_{2}$_ | _$b_{2}$_
_$~{}~{}a_{1}~{}~{}$_ | _$\,\,{}_{\underset{\,}{{\color[rgb]{0,0,1}u_{11}}}}{\,{}^{\overset{\,}{v_{11}}}}$_ | _$\,\,{}_{\underset{\,}{{\color[rgb]{0,0,1}u_{12}}}}{\,{}^{\overset{\,}{v_{12}}}}$_ | _$~{}~{}a_{1}~{}~{}$_ | _${}_{\frac{{\color[rgb]{0,0,1}u_{11}}-{\color[rgb]{0,0,1}u_{21}}}{{\color[rgb]{0,0,1}u_{22}-}{\color[rgb]{0,0,1}u_{1\underset{\,}{2}}}}}{\,{}^{\overset{}{\frac{v_{11}-v_{12}}{v_{22}-v_{21}}}}}$_ | _$\,\,\,\,{}_{\underset{\,}{{\color[rgb]{0,0,1}0}}}{\,\,\,\,{}^{\overset{\,}{0}}}$_
_$~{}~{}b_{1}~{}~{}~{}$_ | _$\,\,{}_{\underset{\,}{{\color[rgb]{0,0,1}u_{21}}}}{\,{}^{\overset{\,}{v_{21}}}}$_ | _$\,\,{}_{\underset{\,}{{\color[rgb]{0,0,1}u_{22}}}}{\,{}^{\overset{\,}{v_{22}}}}$_ | _$~{}~{}b_{1}~{}~{}~{}$_ | _$\,\,{}_{\underset{\,}{{\color[rgb]{0,0,1}0}}}{\,\,\,\,\,\,\,\,\,{}^{\overset{\,}{0}}}$_ | _$\,\,{}_{\underset{\,}{{\color[rgb]{0,0,1}1}}}\,\,\,{\,{}^{\overset{\,}{1}}}$_
$\underset{}{{\color[rgb]{0,0,1}u_{11}>u_{21}},u_{22}>u_{12}}$,
$\underset{}{v_{11}>v_{12}},\underset{}{v_{22}>v_{21}}$

Sampling dynamics (as defined in 3.1 and A.1) depend only on the differences
between the payoffs a player can get by playing different actions (the same
property holds for best-response dynamics and logit dynamics, which implies
that the sets of Nash equilibria, quantal response equilibria, and
evolutionary stable strategies depend only on these differences).121212A
notable exception is best experienced payoff dynamics (see, e.g., Osborne and
Rubinstein, 1998; Cárdenas, Mantilla, and Sethi, 2015; Mantilla, Sethi, and
Cárdenas, 2018; Sandholm, Izquierdo, and Izquierdo, 2019, 2020; Sethi, 2021;
Arigapudi, Heller, and Milchtaich, 2021), where the dynamics depend directly
on the payoffs, and not only on payoff differences. These differences are
invariant to subtracting a constant from all the payoffs of a player while
fixing the opponent’s action (e.g., subtracting $u_{21}$ from all of Player
1’s first-column payoffs). Moreover, sampling dynamics (as well as all the
other dynamics and solution concepts mentioned above) are invariant to
dividing all of a player’s payoff by a positive constant (which preserves the
vN–M utility). The left matrix in Table 3 is reduced to the right matrix by
the following steps (none of which affect the sampling dynamics):

  1. 1.

Three changes to Player 1’s payoffs: (I) subtracting $u_{21}$ from Player 1’s
payoffs in her first column, (II) subtracting $u_{12}$ from Player 1’s payoffs
in her second column, and (III) dividing Player 1’s payoff by
${\color[rgb]{0,0,1}u_{22}-}u_{12}$; and

  2. 2.

Three changes to Player 2’s payoffs: (I) subtracting $v_{12}$ from Player 1’s
payoffs in her first row, (II) subtracting ${\color[rgb]{180,00,00}v_{21}}$
from Player 1’s payoffs in her first row, and (III) dividing Player 1’s payoff
by ${\color[rgb]{180,00,00}v_{22}-v_{21}}$.

Observe that the assumption that
${\color[rgb]{0,0,1}u_{1}}=\frac{{\color[rgb]{0,0,1}u_{11}}-{\color[rgb]{0,0,1}u_{21}}}{{\color[rgb]{0,0,1}u_{22}-}{\color[rgb]{0,0,1}u_{12}}}\geq
1$ in the standard representation of Table 2 is w.l.o.g.. If
$\frac{{\color[rgb]{0,0,1}u_{11}}-{\color[rgb]{0,0,1}u_{21}}}{{\color[rgb]{0,0,1}u_{22}-}{\color[rgb]{0,0,1}{\color[rgb]{0,0,1}u_{12}}}}<1$,
then we can multiply all of Player 1’s payoffs by
$\frac{{\color[rgb]{0,0,1}u_{22}-}{\color[rgb]{0,0,1}{\color[rgb]{0,0,1}u_{12}}}}{{\color[rgb]{0,0,1}u_{11}}-{\color[rgb]{0,0,1}u_{21}}}$
and all of Player 2’s payoffs by
${\color[rgb]{180,00,00}\frac{v_{22}-v_{21}}{v_{11}-v_{12}}}$, relabel the
actions $a_{i}\leftrightarrow b_{i}$ for both players, and obtain a standard
representation as in Table 2 in which $u_{1}\geq 1$.

######  Example 3 (Hawk-Dove Games).

Consider a hawk-dove (aka, chicken) game, which can be interpreted as a game
of bargaining over the price of an asset (e.g., a house) between a buyer and a
seller. Each player can either insist on a more favorable price (“hawk”) or
agree to a less favorable price in order to close the deal (“dove”). The left
panel of Table 4 shows the payoffs of a hawk-dove game. Two doves agree on an
equally favorable price (which gives both players a relatively high payoff
normalized to one). A hawk obtains a favorable price when matched with a dove
(which increases the payoff to the hawk by $g\in(0,1)$, while reducing the
dove’s payoff by $l\in(0,1)$), but faces a high probability of bargaining
failure when matched with another hawk (which would yield a low payoff of zero
to both hawks).

Observe that a hawk-dove game can be transformed to our standard
representation of a coordination game (the right panel of Table 4) as follows:
(1) relabel the actions of player 1 such that $a_{1}=d_{1}$ and $b_{1}=h_{1}$
(while keeping the actions of Player 2 as $a_{2}=h_{2}$ and $b_{2}=d_{2}$),
(2) subtract a payoff of 1 from Player 1’s payoffs in her second column and
from Player 2’s payoffs in her first column, and (3) divide all the payoffs of
Player 1 by $g$, and all payoffs of Player 2 by $1-l$. Observe that the
induced standard representation is antisymmetric, i.e.,
$u_{1}=\frac{{\color[rgb]{0,0,1}1-l}}{g}=\frac{1}{u_{2}}$.

Table 4: Normalization of Hawk-Dove Games ($g,l\in\left(0,1\right)$) Original Representation |  | Standard Representation
---|---|---
| _$h_{2}$_ | _$d_{2}$_ | $\begin{array}[]{c}\\\ \\\ \Rightarrow\\\ \\\ \\\ \end{array}$ |  | _$a_{2}=h_{2}$_ | _$b_{2}=d_{2}$_
_$~{}h_{1}$_ | _$\,\,{}_{\underset{\,}{{\color[rgb]{0,0,1}0}}}{\,{}^{\overset{\,}{0}}}$_ | ${}_{\underset{\,}{{\color[rgb]{0,0,1}1+g}}}\,\,\,{\,{}^{\overset{\,}{1-l}}}$ | _$~{}~{}~{}a_{1}=d_{1}~{}$_ | $\,\,{}_{\underset{\,}{{\color[rgb]{0,0,1}\frac{{\color[rgb]{0,0,1}1-l}}{g}}}}\,\,\,{\,{}^{\overset{\,}{\frac{g}{1-l}}}}$ | _$\,{}_{\underset{\,}{{\color[rgb]{0,0,1}0}}}{\,{}^{\overset{\,}{0}}}$_
_$~{}d_{1}$_ | $\,\,{}_{\underset{\,}{{\color[rgb]{0,0,1}1-l}}}\,\,\,{\,{}^{\overset{\,}{1+g}}}$ | _${}_{\underset{\,}{{\color[rgb]{0,0,1}1}}}\,\,\,{\,{}^{\overset{\,}{1}}}$_ | _$~{}~{}~{}b_{1}=h_{1}~{}~{}$_ | _$\,\,{}_{\underset{\,}{{\color[rgb]{0,0,1}0}}}{\,\,\,\,\,\,\,{}^{\overset{\,}{0}}}$_ | _$\,\,{}_{\underset{\,}{{\color[rgb]{0,0,1}1}}}{\,{}^{\overset{\,}{1}}}$_

###  A.2 One-Population Dynamics

In the main text, we derived results for general coordination games under two-
population dynamics. In this appendix, we show that analogous results hold for
symmetric coordination games under one-population dynamics.

##### Adapted Definitions

An environment $((u_{1},u_{2}),(\theta_{1},\theta_{2}))$ is symmetric if
$u_{1}=u_{2}$ and $\theta_{1}=\theta_{2}$. As we focus on symmetric
environments here, we omit the subscript $i$ referring to the population from
the parameters $u_{i},\theta_{i}$, the actions $a_{i},b_{i}$, and the sampling
dynamics $w_{i}.$ Thus, a symmetric environment is denoted by $(u,\theta)$,
where $u$ is the payoff to each player when coordinating on $a$, and $\theta$
is the distribution of sample size of each player.

Under one-population dynamics, the state is denoted by a single number
$p\in(0,1)$ that describes the share of players who play action $a$ in the
population. At each time $t,$ a constant flow (normalized to one) of agents
die, and are replaced by new agents. The sample sizes of the new agents are
distributed according to $\theta$. Each $k$-agent (a new agent with sample
size $k$) randomly and independently samples $k$ players and observes the
actions that they played. Thus, the one-dimensional sampling dynamics is given
by $\dot{p}=w(p)-p$, where

$w\left(p\right)=\sum_{k\in\text{supp}\left(\theta\right)}\theta(k)\cdot\Pr\left(X\left(k,p\right)\geq\frac{k}{u+1}\right).$
(A.1)

In the above equation, $X(k,p)$ denotes a binomial random variable with
parameters $k$ and $p.$ Analogous arguments to those presented in Appendix A.1
imply that any symmetric coordination game can be normalized w.l.o.g. to the
standard form with a single parameter $u$.131313Under one-population dynamics
it is important that the strict equilibria are obtained by both players
playing the same action (i.e., being on the main diagonal of the payoff
matrix), as one cannot relabel the actions of only one of the players in the
game without breaking the symmetry. Thus, under one-population dynamics, hawk-
dove games (in which the pure equilibria are off the main diagonal) are no
longer coordination games (by contrast, they are asymmetric coordination games
under two-population dynamics). As demonstrated in Herold and Kuzmics (2020),
introducing pre-play cheap-talk can induce some hawk-dove games to essentially
be coordination games also in one-population dynamics.

##### Adaptation of Results

We next show that in symmetric coordination games, the set of stationary
states and the set of asymptotically stable stationary states coincide under
the one-population dynamics and the two-population dynamics. This implies that
all our main-text results hold also for one-population dynamics in symmetric
coordination games.

We first show that the sets of stationary states coincide in both sets of
dynamics.

######  Proposition 4.

Let $(u,\theta)$ be a symmetric environment. State $(p_{1},p_{2})$ is
stationary under two-population dynamics iff the state is symmetric (i.e.,
$p\equiv p_{1}=p_{2}$), and $p$ is stationary under one-population dynamics.

###### Proof.

Assume that $(p_{1},p_{2})$ is a stationary state under two-population
dynamics. Assume to the contrary that $p_{1}\neq p_{2}$. w.l.o.g., assume that
$p_{1}<p_{2}$. The fact that $w$ is strictly increasing implies that
$p_{2}=w(p_{1})<w(p_{2})=p_{1}$, and we get a contradiction. Thus, we know
that all stationary states under two-population dynamics are symmetric (i.e.,
of the form $(p,p)$ in symmetric environments). Observe that a symmetric state
$(p,p)$ is stationary under two-population dynamics iff $p$ is stationary
under one-population dynamics, because stationarity under both dynamics is
characterized by the equation $p=w(p)$. ∎

The fact that one-population dynamics are one-dimensional implies the
following simple and well-known fact:

######  Fact 4.

Stationary state $p$ is asymptotically stable under one-population dynamics
iff both of the following conditions hold:

  1. 1.

If $p>0$, then $w(p)>p$ in a left neighborhood of $p.$

  2. 2.

If $p<1$, then $w(p)<p$ in a right neighborhood of $p.$

We conclude this section by showing that the sets of asymptotically stable
states coincide under both one-population dynamics and two-population
dynamics. Fact 4 and Propositions 4–5 are illustrated in Figure A.1.

Figure A.1: Illustration of Equivalence Results for One- and Two-Population
Dynamics

The figure illustrates the phase plots of three symmetric environments for
two-population dynamics (upper panels) and the corresponding one-population
dynamics (lower panels). Notice that the phase plots of one-population
dynamics are similar to the part of phase plots of the corresponding two-
population dynamics along the main diagonal (marked with red arrows). In all
environments $u=2.4.$ In the left panels all agents have sample size $k=3$, in
the middle panels all agents have sample size 7, and in the right panels
$40\%$ of the agents have sample size 3, and $60\%$ have sample size 1000.
Observe that in all cases, the dynamic predictions under both one-population
dynamics and two-population dynamics are the same.

######  Proposition 5.

Let $(u,\theta)$ be a symmetric environment. State $(p_{1},p_{2})$ is
asymptotically stable under two-population dynamics iff the state is symmetric
(i.e., $p\equiv p_{1}=p_{2}$), and $p$ is asymptotically stable under one-
population dynamics.

###### Proof.

From Proposition 4, it follows that all asymptotically stable states under
two-population dynamics are symmetric. The strict monotonicity of $w$ implies
that state $p$ satisfies conditions (1) and (2) in Fact 4 iff the symmetric
state $(p,p)$ satisfies conditions (1) and (2) in Lemma 2. This, in turn,
implies that state $p$ is asymptotically stable under one-population dynamics
iff the symmetric state $(p,p)$ is asymptotically stable under two-population
dynamics. ∎

###  A.3 General Result for Binomial Distributions

Recall our notation of $X(k,p)\sim Bin\left(k,p\right)$ denoting a random
variable with binomial distribution, and of the function
$F_{m}^{k}\left(p\right)\equiv\Pr\left(X\left(k,p\right)\geq m\right)$.

######  Proposition.

3 Fix arbitrary integers satisfying $0<m_{1}\leq k_{1}$ and $0<m_{2}\leq
k_{2}$. There exists at most one $p\in\left(0,1\right)$ such that
$\left(F_{m_{1}}^{k_{1}}\circ F_{m_{2}}^{k_{2}}\right)\left(p\right)=p$.

###### Proof.

Let $w_{i}\left(p\right)\equiv F_{m_{i}}^{k_{i}}\left(p\right)$ for each
$i\in\left\\{1,2\right\\}$, $F\left(p\right)\equiv\left(F_{m_{1}}^{k_{1}}\circ
F_{m_{2}}^{k_{2}}\right)\left(p\right)\equiv\left(w_{1}\circ
w_{2}\right)\left(p\right),$ and $G\left(p\right)=F\left(p\right)-p$. In what
follows we show that the function $G(\cdot)$ has at most one
$p\in\left(0,1\right)$ such that $G\left(p\right)=0,$ which proves the result.

We have $G\left(0\right)=G\left(1\right)=0$. Assume to the contrary that there
exist two different interior points $0<\underline{p}<\overline{p}<1$ such that
$G\left(\underline{p}\right)=G\left(\overline{p}\right)$. Then $G$ equals zero
at four points in the interval $\left[0,1\right]$. By Rolle’s theorem, this
implies that $G^{\prime}$ is equal to zero in at least three points in the
interval $\left(0,1\right),$ which further implies that $G^{\prime\prime}$ is
equal to zero in at least two interior points in the interval
$\left(0,1\right)$. Observe that $G^{\prime\prime}\equiv F^{\prime\prime}$.
Thus, in order to obtain a contradiction we have to show that
$F^{\prime\prime}\left(p\right)=0$ in at most one interior point. Recall that
(see, e.g., Green, 1983, Eq. (5)):

$w^{\prime}_{i}\left(p\right)=m_{i}\left(\begin{array}[]{c}k_{i}\\\
m_{i}\end{array}\right)p^{m_{i}-1}\left(1-p\right)^{k_{i}-m_{i}}\,\Rightarrow\,\frac{w^{\prime\prime}_{i}\left(p\right)}{w^{\prime}_{i}\left(p\right)}=\frac{m_{i}-1}{p}-\frac{k_{i}-m_{i}}{1-p}.$
(A.2)

For $p\in(0,1),$ using Eq. (A.2), we compute as follows:

$\displaystyle F^{\prime}\left(p\right)$
$\displaystyle=w_{1}^{\prime}\left(w_{2}\left(p\right)\right)w_{2}^{\prime}\left(p\right)$
$\displaystyle F^{\prime\prime}\left(p\right)$
$\displaystyle=w_{1}^{\prime\prime}\left(w_{2}\left(p\right)\right)\left(w_{2}^{\prime}\left(p\right)\right)^{2}+w_{1}^{\prime}\left(w_{2}\left(p\right)\right)w_{2}^{\prime\prime}\left(p\right)$
$\displaystyle=w_{1}^{\prime}\left(w_{2}\left(p\right)\right)w_{2}^{\prime}\left(p\right)\left[\left(\frac{m_{1}-1}{w_{2}\left(p\right)}-\frac{k_{1}-m_{1}}{1-w_{2}\left(p\right)}\right)w_{2}^{\prime}\left(p\right)+\frac{m_{2}-1}{p}-\frac{k_{2}-m_{2}}{1-p}\right].$

The fact that each $w_{i}\left(p\right)$ is strictly increasing implies that
$F^{\prime\prime}\left(p\right)=0$ iff

$\left(\frac{m_{1}-1}{w_{2}\left(p\right)}-\frac{k_{1}-m_{1}}{1-w_{2}\left(p\right)}\right)w_{2}^{\prime}\left(p\right)=\frac{k_{2}-m_{2}}{1-p}-\frac{m_{2}-1}{p}\Leftrightarrow$
$m_{2}\left(\begin{array}[]{c}k_{2}\\\
m_{2}\end{array}\right)\left(\frac{m_{1}-1}{w_{2}\left(p\right)}-\frac{k_{1}-m_{1}}{1-w_{2}\left(p\right)}\right)p^{m_{2}-1}\left(1-p\right)^{k_{2}-m_{2}}=\frac{k_{2}-m_{2}}{1-p}-\frac{m_{2}-1}{p}\Leftrightarrow$
$\\!\\!\\!\\!\\!\\!\,\,\\!\\!m_{2}\left(\begin{array}[]{c}k_{2}\\\
m_{2}\end{array}\right)\left(\frac{\left(m_{1}-1\right)p^{m_{2}}\left(1-p\right)^{k_{2}-m_{2}}}{\sum_{l=m_{2}}^{k_{2}}\left(\begin{array}[]{c}k_{2}\\\
l\end{array}\right)p^{l}\left(1-p\right)^{k_{2}-l}}-\frac{\left(k_{1}-m_{1}\right)p^{m_{2}}\left(1-p\right)^{k_{2}-m_{2}}}{\sum_{l=0}^{m_{2}-1}\left(\begin{array}[]{c}k_{2}\\\
l\end{array}\right)p^{l}\left(1-p\right)^{k_{2}-l}}\right)\frac{1}{p}=\frac{k_{2}-m_{2}}{1-p}-\frac{m_{2}-1}{p}\Leftrightarrow$
$m_{2}\left(\begin{array}[]{c}k_{2}\\\
m_{2}\end{array}\right)\left(\frac{\left(m_{1}-1\right)}{\sum_{l=m_{2}}^{k_{2}}\left(\begin{array}[]{c}k_{2}\\\
l\end{array}\right)\left(\frac{p}{1-p}\right)^{l-m_{2}}}-\frac{k_{1}-m_{1}}{\sum_{l=0}^{m_{2}-1}\left(\begin{array}[]{c}k_{2}\\\
l\end{array}\right)\left(\frac{1-p}{p}\right)^{m_{2}-l}}\right)\frac{1}{p}=\frac{k_{2}-m_{2}}{1-p}-\frac{m_{2}-1}{p}$

One could verify that the left-hand side of the above equation is strictly
decreasing and the right-hand side is strictly increasing in $p.$ Therefore,
there can be at most one point $p^{\ast}\in(0,1)$ where
$F^{\prime\prime}(p^{\ast})=0.$ This completes the proof.∎

###  A.4 Standard Definitions of Dynamic Stability

For completeness, we present in this appendix the standard definitions of
dynamic stability that are used in the paper (see, e.g., Weibull, 1997,
Chapter 5).

A state is said to be stationary if it is a rest point of the dynamics.

######  Definition 4.

State $\mathbf{p}^{*}\in\left[0,1\right]^{2}$ is a _stationary state_ if
$w_{i}\left(\mathbf{p}^{*}\right)=p_{i}^{*}$ for each
$i\in\left\\{1,2\right\\}$.

Let $\mathcal{E}\left(w\right)$ denote the set of stationary states of $w,$
i.e.,
$\mathcal{E}\left(w\right)=\\{\mathbf{p}^{*}|w_{i}\left(\mathbf{p}^{*}\right)=p_{i}^{*}\\}.$
Under monotone dynamics, an interior (mixed) state
$\mathbf{p}^{*}\in(0,1)^{2}$ is a stationary state iff it is a Nash
equilibrium (Weibull, 1997, Prop. 4.7). By contrast, under nonmonotone
dynamics (such as the sampling dynamics analyzed below) the two notions
differ.

A state $\mathbf{p}^{*}$ is Lyapunov stable if a population beginning near
$\mathbf{p}^{*}$ stays close to it. $\mathbf{p}^{*}$ is asymptotically stable
if it is Lyapunov stable and, in addition, nearby populations eventually
converge to it. A state is unstable if it is not Lyapunov stable. It is well
known (see, e.g., Weibull, 1997, Section 6.4) that every Lyapunov stable state
must be a stationary state. Formally:

######  Definition 5.

A stationary state $\mathbf{p}^{*}\in\left[0,1\right]^{2}$ is _Lyapunov
stable_ if for every neighborhood $U$ of $\mathbf{p}^{*}$ there is a
neighborhood $V\subseteq U$ of $\mathbf{p}^{*}$ such that if the initial state
$p\left(0\right)\in V$, then $\mathbf{p}\left(t\right)\in U$ for all $t>0$. A
state is _unstable_ if it is not Lyapunov stable.

######  Definition 6.

A stationary state $\mathbf{p}^{*}\in\left[0,1\right]^{2}$ is _asymptotically
stable_ (or _locally stable_) if it is Lyapunov stable and there is some
neighborhood $U$ of $\mathbf{p}^{*}$ such that all trajectories initially in
$U$ converge to $\mathbf{p}^{*},$ i.e., $\mathbf{p}\left(0\right)\in U$
implies $\lim_{t\rightarrow\infty}\mathbf{p}\left(t\right)=\mathbf{p}^{*}$.

###  A.5 Proof of Proposition 1 (Stability of Pure States)

We are interested in deriving conditions for the stability of pure stationary
states. In what follows, we compute the Jacobian of sampling dynamics in the
pure state $\boldsymbol{a}=\left(0,0\right)$ (resp.,
$\boldsymbol{b}=\left(1,1\right)$). For this, we consider a slightly perturbed
state with a “very small” $\epsilon_{i}$ share of agents playing $b_{i}$
(resp., $a_{i})$ in population $i$. By “very small,” we mean that higher-order
terms of $\epsilon_{i}$ and $\epsilon_{j}$ are neglected.

Consider a new agent in population $i$ who has a sample size of $k_{i}.$
Action $b_{i}$ (resp., $a_{i}$) has a weakly (resp., strictly) higher mean
payoff against a sample size of $k_{i}$ iff (neglecting rare events of having
multiple $b_{j}$-s (resp., $a_{j}$-s) in the sample): (1) the sample includes
the opponent’s action $b_{j}$ (resp., $a_{j}$), and (2)
$k_{i}<\frac{1}{u_{i}}+1$ (resp., $k_{i}\leq u_{i}+1$). The probability of (1)
is $k_{i}\cdot\epsilon_{j}+o(\epsilon_{j})$, where $o(\epsilon_{j})$ denotes
terms that are sublinear in $\epsilon_{j}$, and, thus, it will not affect the
Jacobian as $\epsilon_{j}\rightarrow 0$. This implies that the probability
that a new agent in population $i$ (with a random sample size distributed
according to $\theta_{i}$) who has a higher mean payoff for action $b_{i}$
(resp., $a_{i}$) against her sample is
$w_{i}(1-\epsilon_{j})=\epsilon_{j}\cdot\mathbb{E}_{<\frac{1}{u_{i}}+1}\left(\theta_{i}\right)+o(\epsilon_{j})$
(resp., $w_{i}(\epsilon_{j})=\epsilon_{j}\cdot\mathbb{E}_{\leq
u_{i}+1}\left(\theta_{i}\right)+o(\epsilon_{j})$ ). Therefore, the sampling
dynamics at $(\epsilon_{1},\epsilon_{2})$ (resp.,
$(1-\epsilon_{1},1-\epsilon_{2})$) can be written as follows (ignoring the
higher-order terms of $\epsilon_{1}$ and $\epsilon_{2}$):

$\dot{\epsilon}_{i}=\epsilon_{j}\cdot\mathbb{E}_{<\frac{1}{u_{i}}+1}\left(\theta_{i}\right)-\epsilon_{i}\,\,\,(\textrm{resp.,}\,\dot{\epsilon}_{i}=\epsilon_{i}-\epsilon_{j}\cdot\mathbb{E}_{\leq
u_{i}+1}\left(\theta_{i}\right)).$ (A.3)

Define:
$a_{\theta_{i}}=\mathbb{E}_{<\frac{1}{u_{i}}+1}\left(\theta_{i}\right)$
(resp., $b_{\theta_{i}}=\mathbb{E}_{\leq u_{i}+1}\left(\theta_{i}\right)$).
The Jacobian of the above system of Equations (A.3) is then given by
$J_{a}=\left(\begin{array}[]{cc}-1&a_{\theta_{1}}\\\
a_{\theta_{2}}&-1\end{array}\right)$ (resp.,
$J_{a}=\left(\begin{array}[]{cc}1&-b_{\theta_{1}}\\\
-b_{\theta_{2}}&1\end{array}\right)$). The eigenvalues of $J_{a}$ (resp.,
$J_{b}$) are $-1-\sqrt{a_{\theta_{1}}a_{\theta_{2}}}$ and
$-1+\sqrt{a_{\theta_{1}}a_{\theta_{2}}}$ (resp.,
$-1-\sqrt{b_{\theta_{1}}b_{\theta_{2}}}$ and
$-1+\sqrt{b_{\theta_{1}}b_{\theta_{2}}}$). Observe that: (1) if
$a_{\theta_{1}}a_{\theta_{2}}<1$ (resp., $b_{\theta_{1}}b_{\theta_{2}}$>1)
then both eigenvalues are negative, which implies that the pure state
$\boldsymbol{a}$ (resp., $\boldsymbol{b}$) is asymptotically stable, and (2)
if $a_{\theta_{1}}a_{\theta_{2}}>1$ (resp., $b_{\theta_{1}}b_{\theta_{2}}$>1)
then one of the eigenvalues is positive, which implies that this state is
unstable (see, e.g., Perko, 2013, Theorems 1 and 2 in Section 2.9).

###  A.6 Proof of Proposition 2

####  A.6.1 Proof of Claim 1 (Reaching the Area between the Curves)

We say that state p is _above_ (resp., _below_) curve $w_{2}(p_{1})$ if
$p_{2}>w_{2}(p_{1})$ (resp., ($p_{2}<w_{2}(p_{1})$). Similarly, we say that
state p is to the _right_ (resp., _left_) of the curve $w_{1}(p_{2})$ if
$p_{1}>w_{1}(p_{2})$ (resp., $p_{1}<w_{1}(p_{2})$). We say that the state p is
on the curve $w_{i}(p_{j})$ if $p_{i}=w_{i}(p_{j}).$ Due to the fact that the
two curves are strictly increasing, we identify the notion of being above a
curve and being to the left of the curve, and similarly we identify the notion
of being below a curve and being to the right of the curve. The states on the
curve $w_{2}(p_{1})$ (resp., $w_{1}(p_{2})$) are characterized by having
$\dot{p_{2}}=0$ (resp., $\dot{p_{1}}=0$). Observe that $\dot{p_{2}}>0$ (resp.,
$\dot{p_{2}}<0$) in any state p above and to the left (resp., below and to the
right) of the curve $w_{2}(p_{1})$. Similarly, $\dot{p_{1}}>0$ (resp.,
$\dot{p_{1}}<0$) in any state p above and to the left (resp., below and to the
right) of the curve $w_{1}(p_{2})$.

Any state $\textbf{p}\in[0,1]$ can be classified in one of $9=3\cdot 3$
classes, depending on its relative location with respect to the two curves,
i.e., whether p is below, above, or on each of the two curves $w_{i}(p_{j})$.
If state p is on (resp., above, below) the curve $w_{2}(p_{1})$, then
$\dot{p_{2}}$ is zero (resp., negative, positive). Similarly, if state p is on
(resp., above, below) the curve $w_{1}(p_{2})$, then $\dot{p_{1}}$ is zero
(resp., positive, negative). In particular, any state p that is above (resp.,
below) both curves must satisfy $\dot{p_{1}}>0>\dot{p_{2}}$. This implies that
any trajectory that begins above (resp., below) both curves must always move
downward and to the right. This (together with the fact that
$\dot{\boldsymbol{p}}\left(t\right)=(0,0)$ only in the intersections of the
two curves) implies that the trajectory must either converge to a stationary
state (i.e., an intersection of the two curves), or cross one of the curves,
and reach a state $\boldsymbol{p}\left(t\right)$ that satisfies either
$w_{1}^{-1}\left(p_{1}\left(t\right)\right)\leq p_{2}\left(t\right)\leq
w_{2}\left(p_{1}\left(t\right)\right)$ or
$w_{2}\left(p_{1}\left(t\right)\right)\leq p_{1}\left(t\right)\leq
w_{1}^{-1}\left(p_{1}\left(t\right)\right)$.

####  A.6.2 Proof of Claim 2 (Convergence to Stationary States)

We first show why we can assume w.l.o.g. that $\boldsymbol{p}\left(t\right)$
is strictly between the two curves. If $\boldsymbol{p}\left(t\right)$ crosses
one of the curves and is strictly above (resp., below) the remaining curve,
then the dynamics must take the populations to a state that is strictly below
one of the curves and strictly above the remaining curve. This is so because
on the crossing point one of the $\dot{p_{i}}$-s is zero and the remaining
derivative $\dot{p}_{j}$ is negative (resp., positive), which implies that the
dynamics take the trajectory below and to the right (resp., above and to the
left) of the curve that was crossed.

Next assume that
$p_{i}\left(t\right)\in\left(w_{1}^{-1}\left(p_{1}\left(t\right)\right),w_{2}\left(p_{1}\left(t\right)\right)\right)$
(resp.,
$p_{i}\left(t\right)\in\left(w_{2}\left(p_{1}\left(t\right)\right),w_{1}^{-1}\left(p_{1}\left(t\right)\right)\right).$
By the classification presented in the proof of Claim 1, the trajectory must
move upward and to the right, i.e., $\dot{p_{1}},\dot{p_{2}}>0$ (resp.,
downward and to the left, i.e., $\dot{p_{1}},\dot{p_{2}}<0$). This implies
that the trajectory must cross one of the curves. The crossing point cannot be
only on the curve of $w_{2}(p_{1})$ (resp., $w_{1}(p_{2})$), because at such a
point the trajectory moves horizontally to the left (vertically upward), which
implies that it must cross the lower (resp., higher) curve $w_{2}(p_{1})$
(resp.,$w_{1}(p_{2})$) from the left side (resp., from below), and we get a
contradiction. This implies that the crossing point must be the closest
intersection points of the two curves to the right (resp., left) of
$p_{i}\left(t\right)$, namely, $\overline{\boldsymbol{p}}$ (resp.,
$\boldsymbol{\underline{p}}$).

###  A.7 Proof of Lemma 1 (Convergence to Pure States)

If p is below (resp., above) both curves, then by the classification presented
in the proof of Claim 1 it must be that $\dot{p_{2}}>0$ (resp.,
$\dot{p_{1}}>0$), which implies that convergence to (0,0) is possible only if
the trajectory passes through a state that is strictly between the two curves,
and that the closest intersection point of the two curves to the left of this
state is (0,0). By the classification presented in the proof of Claim 2 it
must be that the curve of $w_{2}(p_{1})$ is strictly below the curve of
$w_{1}(p_{2})$ in a right neighborhood of (0,0), which implies that (0,0) is
asymptotically stable because any sufficiently close initial state would
converge to (0,0).

###  A.8 Proof of Lemma 2

Assume that Conditions (1) and (2) hold. Let
$\boldsymbol{p}\neq\hat{\boldsymbol{p}}$ be any sufficiently close state. By
Claim 1 any trajectory beginning at $\boldsymbol{p}$ will enter one of the two
areas between the two curves on either side of $\hat{\boldsymbol{p}}$. By
Claim 2, Condition (1) (resp., (2)) implies convergence to
$\hat{\boldsymbol{p}}$ if the trajectory has entered the area between the
curves to the left (resp., right) of $\hat{\boldsymbol{p}}.$ This implies that
any trajectory that starts sufficiently close to $\hat{\boldsymbol{p}}$ must
converge to $\hat{\boldsymbol{p}}$, and, thus, $\hat{\boldsymbol{p}}$ is
asymptotically stable.

Next assume that $\hat{p}_{1}>0$ and Condition (1) (resp., $\hat{p}_{1}<1$ and
Condition (2)) is not satisfied. This implies that
$w_{2}\left(p_{1}\right)<w_{1}^{-1}\left(p_{1}\right)$ (resp.,
$w_{2}\left(p_{1}\right)>w_{1}^{-1}\left(p_{1}\right)$) for any $p_{1}$ that
is sufficiently close to $\hat{p}_{1}$ from the left (resp., right). By Claim
2, this implies that a trajectory starting at $\left(p_{1},p_{2}\right)$ with
$p_{1}$ sufficiently close to $\hat{p}_{1}$ from the left (resp., right) and
with
$p_{2}\in\left(w_{2}\left(p_{1}\right),w_{1}^{-1}\left(p_{1}\right)\right)$
(resp.,
$p_{2}\in\left(w_{1}^{-1}\left(p_{1}\right),w_{2}\left(p_{1}\right)\right)$)
converges to the neighboring stationary point on the left (resp., right) side
of $\hat{\boldsymbol{p}}$. Thus, $\hat{\boldsymbol{p}}$ is unstable.

###  A.9 Proof of Theorem 3 (Stability of Miscoordination)

Let $w_{i}^{k_{i}}\left(p_{j}\right)$ (resp.,
$w_{i}^{\theta_{i}}\left(p_{j}\right)$) denote the sampling dynamics induced
by an agent with sample size $k_{i}$ (resp., agents with the distribution
$\theta_{i}$).

  1. 1.

Fix any $r<\min\left(\frac{1}{k_{1}},\frac{1}{k_{2}}\right)$. Assume that
$u_{i}>\frac{1}{r}$ for each $i$ (which implies that $u_{i}>k_{i}$). Observe
that

$w_{i}^{k_{i}}\left(p_{j}\right)=\Pr\left(X\left(k_{i},p_{j}\right)\geq
1\right)=1-\left(1-p_{j}\right)^{k_{i}}=k_{i}p_{j}-\left(\begin{array}[]{c}k_{i}\\\
2\end{array}\right)\left(p_{j}\right)^{2}+O\left(\left(p_{j}\right)^{3}\right).$

Fix a sufficiently small $\epsilon>0$. Let
$\hat{p}_{1}\in\left(0,p_{1}^{NE}-\epsilon\right)$ be sufficiently small such
that the term $O\left(\left(p_{j}\right)^{3}\right)<\epsilon$ is negligible
for any $p_{j}<\hat{p}_{1}$. For each $i\in\left\\{1,2\right\\}$, let
$\alpha_{i}\in\left(0,1\right)$ be such that: (1) $\alpha_{i}k_{i}>1$ and (2)
$\alpha_{i}\left(k_{i}-\left(\begin{array}[]{c}k_{i}\\\
2\end{array}\right)p_{j}\right)<1-2\epsilon$. This implies that
$w_{i}^{k_{i}}\left(p_{j}\right)>\frac{p_{j}}{\alpha_{i}}$ in a right
neighborhood of zero, and
$w_{i}^{k_{i}}\left(p_{j}\right)<\frac{p_{j}}{\alpha_{i}}$ in a left
neighborhood of $\hat{p}_{1}$. Observe that
$\lim_{k\rightarrow\infty}w_{2}^{k}\left(\hat{p}_{1}\right)=0$ and
$\lim_{k\rightarrow\infty}\left(w_{1}^{k}\right)^{-1}\left(\hat{p}_{1}\right)=p_{2}^{NE}$.
This implies that there exists $\overline{k}$ sufficiently large such that
$w_{2}^{\overline{k}}\left(\hat{p}_{1}\right)<\epsilon$ and
$\left(w_{1}^{\overline{k}}\right)^{-1}\left(\hat{p}_{1}\right)>p_{2}^{NE}-\epsilon>\hat{p}_{1}$.
Observe that $w_{i}^{k_{i}\alpha_{i}\overline{k}}\left(p_{j}\right)>p_{j}$ in
a right neighborhood of zero, and
$w_{i}^{k_{i}\alpha_{i}\overline{k}}\left(p_{j}\right)<p_{j}$ in a left
neighborhood of $\hat{p}_{1}$. This, in turn, implies that
$w_{2}^{k_{2}\alpha_{2}\overline{k}}\left(p_{1}\right)>p_{1}>\left(w_{1}^{k_{2}\alpha_{2}\overline{k}}\right)^{-1}\left(p_{1}\right)$
in a right neighborhood of zero, and
$w_{2}^{k_{2}\alpha_{2}\overline{k}}\left(p_{1}\right)<p_{1}<\left(w_{1}^{k_{1}\alpha_{1}\overline{k}}\right)^{-1}\left(p_{1}\right)$
in a left neighborhood of $\hat{p}_{1}$. Thus, there exists a stationary state
$\tilde{\boldsymbol{p}}$ that satisfies $0<\tilde{p}_{1}<\hat{p}_{1}<1$, and
$w_{2}^{k_{i}\alpha_{i}\overline{k}}\left(p_{1}\right)>\left(w_{1}^{k_{i}\alpha_{i}\overline{k}}\right)^{-1}\left(p_{1}\right)$
(resp.,
$w_{2}^{k_{i}\alpha_{i}\overline{k}}\left(p_{1}\right)<\left(w_{1}^{k_{i}\alpha_{i}\overline{k}}\right)^{-1}\left(p_{1}\right)$)
in a left (resp., right) neighborhood of $\tilde{p}_{1}$. This implies, by
Claim 2, that $\tilde{\boldsymbol{p}}$ is asymptotically stable. The argument
in this case is illustrated in the left panel of Figure 6.1.

  2. 2.

Observe that

$w_{1}^{k_{1}}\left(p_{2}\right)=\Pr\left(\textrm{a new agent in population 1
has at least one }a_{2}\textrm{ in her sample}\right)$

is the same for all values of $u_{1}>k_{1}$, and, similarly,

$w_{2}^{k_{2}}\left(p_{1}\right)=\Pr\left(\textrm{a new agent in population 2
has no }b_{1}\textrm{-s}\right.\left.\textrm{in the sample}\right)$

is the same for all values of $u_{2}<\frac{1}{k_{2}}$. Fix a sufficiently
small $\epsilon>0$. Let $\bar{u}_{1}>k_{1},\bar{u}_{2}<\frac{1}{k_{2}}$ be
payoffs satisfying
$p_{2}^{NE}=\frac{1}{1+\bar{u}_{1}}<\frac{w_{2}^{k_{2}}\left(\frac{1}{2}\right)}{2}-\epsilon$
and
$1-p_{1}^{NE}=\frac{\bar{u}_{2}}{1+\bar{u}_{2}}<\frac{1-w_{1}^{k_{1}}\left(\frac{1}{2}\right)}{2}-\epsilon$.
Let
$r=\min\left(\frac{1}{1+\bar{u}_{1}},\frac{\bar{u}_{2}}{1+\bar{u}_{2}}\right)$.
Fix any payoff profile $u_{1}>\frac{1}{r}$ and $u_{2}<r$. Let $\overline{k}$
be sufficiently large such that
$w_{i}^{\overline{k}}\left(p_{j}^{NE}-\epsilon\right)<\epsilon$,
$w_{i}^{\overline{k}}\left(p_{j}^{NE}+\epsilon\right)>1-\epsilon$. In what
follows, we show that the environment
$\left((u_{1},u_{2}),(k_{1}\frac{1}{2}\bar{k},k_{2}\frac{1}{2}\bar{k})\right)$
admits an asymptotically stable interior state (i.e., we let
$\alpha_{1}=\alpha_{2}=\frac{1}{2}$).
By Proposition 1, the two pure equilibria are asymptotically stable and
$w_{2}^{k_{2}\frac{1}{2}\bar{k}}\left(p_{1}\right)<\left(w_{1}^{k_{1}\frac{1}{2}\bar{k}}\right)^{-1}\left(p_{1}\right)$
(resp.,
$w_{2}^{k_{2}\frac{1}{2}\bar{k}}\left(p_{1}\right)>\left(w_{1}^{k_{1}\frac{1}{2}\bar{k}}\right)^{-1}\left(p_{1}\right)$)
in a right (resp., left) neighborhood of zero (resp., one). Next observe that
$w_{2}^{k_{2}\frac{1}{2}\bar{k}}\left(\frac{1}{2}\right)>\frac{1}{2}w_{2}^{k_{2}}\left(\frac{1}{2}\right)>p_{NE}^{2}+\epsilon>\left(w_{1}^{k_{1}\frac{1}{2}\bar{k}}\right)^{-1}\left(\frac{1}{2}\right)$,
which implies that there is an (unstable) interior stationary state
$\hat{\boldsymbol{p}}$ that satisfies: (1) $\hat{p}_{1}<0.5$, (2)
$p_{NE}^{2}-\epsilon<\hat{p}_{2}<p_{NE}^{2}+\epsilon<0.5$, and (3)
$w_{2}^{k_{2}\frac{1}{2}\bar{k}}\left(p_{1}\right)>\left(w_{1}^{k_{1}\frac{1}{2}\bar{k}}\right)^{-1}\left(p_{1}\right)$
in a right neighborhood of $\hat{p}_{1}$. By an analogous argument, there is
also an (unstable) interior stationary state $\tilde{\boldsymbol{p}}$ that
satisfies: (1) $\tilde{p}_{2}>0.5$, (2)
$p_{1}^{NE}-\epsilon<\tilde{p}_{1}<p_{1}^{NE}+\epsilon<0.5$, and (3)
$w_{2}^{k_{2}\frac{1}{2}\bar{k}}\left(p_{1}\right)<\left(w_{1}^{k_{1}\frac{1}{2}\bar{k}}\right)^{-1}\left(p_{1}\right)$
in a left neighborhood of $\tilde{p}_{1}$. This implies that must be an
interior stationary state $\grave{\boldsymbol{p}}$ between
$\hat{\boldsymbol{p}}$ and $\tilde{\textbf{p}}$ such that
$w_{2}^{k_{2}\frac{1}{2}\bar{k}}\left(p_{1}\right)>\left(w_{1}^{k_{1}\frac{1}{2}\bar{k}}\right)^{-1}\left(p_{1}\right)$
(resp.,
$w_{2}^{k_{2}\frac{1}{2}\bar{k}}\left(p_{1}\right)>\left(w_{1}^{k_{1}\frac{1}{2}\bar{k}}\right)^{-1}\left(p_{1}\right)$)
in a left (resp., right) neighborhood of $\grave{\boldsymbol{p}}$. Finally,
Claim 2 implies that $\grave{\boldsymbol{p}}$ is asymptotically stable.

## References

  * Arieli et al. (2020) Arieli, I., Y. Babichenko, R. Peretz, and H. P. Young (2020).  The speed of innovation diffusion in social networks.  Econometrica 88(2), 569–594.
  * Arigapudi et al. (2021) Arigapudi, S., Y. Heller, and I. Milchtaich (2021).  Instability of defection in the prisoner’s dilemma under best experienced payoff dynamics.  Journal of Economic Theory 197, 105174.
  * Bilancini et al. (2022) Bilancini, E., L. Boncinelli, S. Ille, and E. Vicario (2022).  Memory retrieval and harshness of conflict in the hawk–dove game.  Economic Theory Bulletin 10(2), 333–351.
  * Brunner et al. (2011) Brunner, C., C. F. Camerer, and J. K. Goeree (2011).  Stationary concepts for experimental 2$\times$ 2 games: Comment.  American Economic Review 101(2), 1029–1040.
  * Cárdenas et al. (2015) Cárdenas, J., C. Mantilla, and R. Sethi (2015).  Stable sampling equilibrium in common pool resource games.  Games 6(3), 299–317.
  * Crawford (1989) Crawford, V. P. (1989).  Learning and mixed-strategy equilibria in evolutionary games.  Journal of Theoretical Biology 140(4), 537–550.
  * Danenberg and Spiegler (2022) Danenberg, T. and R. Spiegler (2022).  Representative sampling equilibrium.  Working Paper.
  * Fudenberg and Levine (1995) Fudenberg, D. and D. K. Levine (1995).  Consistency and cautious fictitious play.  Journal of Economic Dynamics and Control 19(5–7), 1065–1089.
  * Genesove and Mayer (2001) Genesove, D. and C. Mayer (2001).  Loss aversion and seller behavior: Evidence from the housing market.  Quarterly Journal of Economics 116(4), 1233–1260.
  * Goeree et al. (2003) Goeree, J. K., C. A. Holt, and T. R. Palfrey (2003).  Risk averse behavior in generalized matching pennies games.  Games and Economic Behavior 45(1), 97–113.
  * Green (1983) Green, D. N. (1983).  Fixed point properties of the binomial function.  Journal of the Franklin Institute 316(1), 51–62.
  * (12) Harsanyi, J. C. and R. Selten.  A General Theory of Equilibrium Selection in Games.  MIT Press.
  * Hauert and Miekisz (2018) Hauert, C. and J. Miekisz (2018).  Effects of sampling interaction partners and competitors in evolutionary games.  Physical Review E 98(5), 052301.
  * Heller (2017) Heller, Y. (2017).  Instability of belief-free equilibria.  Journal of Economic Theory 168, 261–286.
  * Heller and Mohlin (2018) Heller, Y. and E. Mohlin (2018).  Social learning and the shadow of the past.  Journal of Economic Theory 177, 426–460.
  * Herold and Kuzmics (2020) Herold, F. and C. Kuzmics (2020).  The evolution of taking roles.  Journal of Economic Behavior & Organization 174, 38–63.
  * Kandori et al. (1993) Kandori, M., G. J. Mailath, and R. Rob (1993).  Learning, mutation, and long run equilibria in games.  Econometrica 61(1), 29–56.
  * Kaniovski and Young (1995) Kaniovski, Y. M. and H. P. Young (1995).  Learning dynamics in games with stochastic perturbations.  Games and Economic Behavior 11(2), 330–363.
  * Koster and Rouwendal (2021) Koster, H. and J. Rouwendal (2021).  Housing market discount rates: Evidence from bargaining and bidding wars.  Working Paper.
  * Kreindler and Young (2013) Kreindler, G. E. and H. P. Young (2013).  Fast convergence in evolutionary equilibrium selection.  Games and Economic Behavior 80, 39–67.
  * Larsen (2021) Larsen, B. J. (2021).  The efficiency of real-world bargaining: Evidence from wholesale used-auto auctions.  Review of Economic Studies 88(2), 851–882.
  * Lyu et al. (2022) Lyu, J., Z. Li, and Y. Xu (2022).  Inexact information, strategic sophistication and equilibrium transition: A quasi-continuous-time experiment.  Working Paper.
  * Mantilla et al. (2018) Mantilla, C., R. Sethi, and J. C. Cárdenas (2018).  Efficiency and stability of sampling equilibrium in public goods games.  Journal of Public Economic Theory 22(2), 355–370.
  * Maynard Smith (1982) Maynard Smith, J. (1982).  Evolution and the Theory of Games.  Cambridge University Press.
  * Maynard Smith and Price (1973) Maynard Smith, J. and G. R. Price (1973).  The logic of animal conflict.  Nature 246(5427), 15–18.
  * McKelvey et al. (2000) McKelvey, R. D., T. R. Palfrey, and R. A. Weber (2000).  The effects of payoff magnitude and heterogeneity on behavior in 2$\times$ 2 games with unique mixed strategy equilibria.  Journal of Economic Behavior & Organization 42(4), 523–548.
  * Morris et al. (1995) Morris, S., R. Rob, and H. S. Shin (1995).  p-dominance and belief potential.  Econometrica 63(1), 145–157.
  * Oprea et al. (2011) Oprea, R., K. Henwood, and D. Friedman (2011).  Separating the hawks from the doves: Evidence from continuous time laboratory games.  Journal of Economic Theory 146(6), 2206–2225.
  * Osborne and Rubinstein (1998) Osborne, M. J. and A. Rubinstein (1998).  Games with procedurally rational players.  American Economic Review 88(4), 834–847.
  * Osborne and Rubinstein (2003) Osborne, M. J. and A. Rubinstein (2003).  Sampling equilibrium, with an application to strategic voting.  Games and Economic Behavior 45(2), 434–441.
  * Oyama et al. (2015) Oyama, D., W. H. Sandholm, and O. Tercieux (2015).  Sampling best response dynamics and deterministic equilibrium selection.  Theoretical Economics 10(1), 243–281.
  * Perko (2013) Perko, L. (2013).  Differential Equations and Dynamical Systems, Volume 7.  Springer Science & Business Media.
  * Salant and Cherry (2020) Salant, Y. and J. Cherry (2020).  Statistical inference in games.  Econometrica 88(4), 1725–1752.
  * Sandholm (2001) Sandholm, W. H. (2001).  Almost global convergence to $p$-dominant equilibrium.  International Journal of Game Theory 30(1), 107–116.
  * Sandholm (2010) Sandholm, W. H. (2010).  Population Games and Evolutionary Dynamics.  MIT Press.
  * Sandholm et al. (2019) Sandholm, W. H., S. S. Izquierdo, and L. R. Izquierdo (2019).  Best experienced payoff dynamics and cooperation in the centipede game.  Theoretical Economics 14(4), 1347–1385.
  * Sandholm et al. (2020) Sandholm, W. H., S. S. Izquierdo, and L. R. Izquierdo (2020).  Stability for best experienced payoff dynamics.  Journal of Economic Theory 185, 104957.
  * Sawa and Wu (2023) Sawa, R. and J. Wu (2023).  Statistical inference in evolutionary dynamics.  Games and Economic Behavior 137, 294–316.
  * Selten and Chmura (2008) Selten, R. and T. Chmura (2008).  Stationary concepts for experimental 2x2-games.  American Economic Review 98(3), 938–66.
  * Sethi (2000) Sethi, R. (2000).  Stability of equilibria in games with procedurally rational players.  Games and Economic Behavior 32(1), 85–104.
  * Sethi (2021) Sethi, R. (2021).  Stable sampling in repeated games.  Journal of Economic Theory 197, 105343.
  * Tversky and Kahneman (1971) Tversky, A. and D. Kahneman (1971).  Belief in the law of small numbers.  Psychological Bulletin 76(2), 105.
  * Weibull (1997) Weibull, J. W. (1997).  Evolutionary Game Theory.  MIT Press.
  * Young (1993) Young, H. P. (1993).  The evolution of conventions.  Econometrica 61(1), 57–84.