# Sampling Dynamics and Stable Mixing in Hawk–Dove Games

Srinivas Arigapudi  Faculty of Industrial Engineering and Management,
Technion<EMAIL_ADDRESS>Yuval Heller Department of
Economics, Bar-Ilan University<EMAIL_ADDRESS> Amnon Schreiber
Department of Economics, Bar-Ilan University<EMAIL_ADDRESS>We
thank Daniel Friedman, Itay Kavaler, Ron Peretz, Ernst Schulte-Geers, Daniel
Stephenson, Jiabin Wu, and seminar audiences at Bar-Ilan University and the
University of Haifa for various helpful comments. We thank Luis R. Izquierdo
and Segismundo S. Izquierdo for their help in implementing the simulations
using ABED software. YH and SA gratefully acknowledge the financial support of
the European Research Council (#677057) and the Israeli Science Foundation
(#2443/19). SA is supported in part at the Technion by a Fine Fellowship.

###### Abstract

The hawk–dove game admits two types of equilibria: an asymmetric pure
equilibrium in which players in one population play “hawk” and players in the
other population play “dove,” and an inefficient symmetric mixed equilibrium,
in which hawks are frequently matched against each other. The existing
literature shows that populations will converge to playing one of the pure
equilibria from almost any initial state. By contrast, we show that plausible
sampling dynamics, in which agents occasionally revise their actions by
observing either opponents’ behavior or payoffs in a few past interactions,
can induce the opposite result: convergence to one of the inefficient mixed
stationary states.

Keywords: Chicken game, learning, evolutionary stability, payoff sampling
dynamics, action sampling dynamics, binomial distributions. JEL codes: C72,
C73.

##  1 Introduction

The hawk–dove game is widely used to study situations of conflict between
strategic participants.111 A few examples of the applications of the hawk–dove
game (also known as the chicken game; see, e.g., Rapoport and Chammah, 1966;
Aumann, 1987) are: provision of public goods (Lipnowski and Maital, 1983),
nuclear deterrence between superpowers (Brams and Kilgour, 1987; Dixit et al.,
2019), industrial disputes (Bornstein et al., 1997), bargaining problems
(Brams and Kilgour, 2001), conflicts between countries over contested
territories (Baliga and Sjöström, 2012, 2020), and task allocation among
members of a team (Herold and Kuzmics, 2020). As a simple motivating example,
consider a situation in which a buyer (Player 1) and a seller (Player 2)
bargain over the price of an asset (e.g., a house). Each player has two
possible bargaining strategies (actions): insisting on a more favorable price
(referred to as being a “hawk”), or agreeing to a less favorable price in
order to close the deal (being a “dove”). The payoffs of the game are
presented in Table 1. Two doves agree on a price that is equally favorable to
both sides, and obtain a relatively high payoff, which is normalized to 1. A
hawk obtains a favorable price when being matched with a dove, which yields
her an additional gain of $g\in(0,1)$, at the expense of her dovish
opponent.222 Our formal model studies a broader class of generalized hawk–dove
games, in which the gain of a hawkish player might differ from the loss of the
dovish opponent (as shown in Table 2 in Section 3). Finally, two hawks obtain
the lowest payoff of 0, due to a substantial probability of bargaining
failure.333 Our one-parameter payoff matrix is equivalent to the commonly used
two-parameter payoff matrix (Maynard-Smith, 1982), in which a dove obtains
$\frac{V}{2}$ against another dove and 0 against a hawk, and a hawk obtains
$\frac{V-C}{2}$ against another hawk and $V$ against a dove. Specifically, our
one-parameter matrix is obtained from the two-parameter matrix by the affine
transformation of adding the constant $\frac{C-V}{2}$ and dividing all payoffs
by $\frac{C}{2},$ followed by substituting $g\equiv\frac{V}{C}.$ Observe that
large values of $g$ (close to 1) correspond to environments that are
advantageous to hawks (i.e., being a hawk yields a higher expected payoff
against an opponent who might play either action with equal probability),
small values of $g$ correspond to environments that are advantageous to doves,
and values of $g$ that are close to 0.5 correspond to approximately balanced
environments.

Table 1: Payoff Matrix of the Standard Hawk–Dove Game ($g\in\left(0,1\right)$)  |  | Player 2
---|---|---
|  | ${\color[rgb]{180,00,00}h}$ | ${\color[rgb]{180,00,00}d}$
Player 1 | ${\color[rgb]{0,0,1}h}$ | ${\color[rgb]{0,0,1}0},{\color[rgb]{180,00,00}0}$ | ${\color[rgb]{0,0,1}1+g},{\color[rgb]{180,00,00}1-g}$
${\color[rgb]{0,0,1}d}$ | ${\color[rgb]{0,0,1}1-g},{\color[rgb]{180,00,00}1+g}$ | ${\color[rgb]{0,0,1}1},{\color[rgb]{180,00,00}1}$

The hawk–dove game admits three Nash equilibria: two asymmetric pure
equilibria, and an inefficient symmetric mixed equilibrium. In the pure
equilibria (in which one of the players plays hawk while the opponent plays
dove), all conflicts are avoided at the cost of inequality, as the payoff of
the hawkish player is substantially higher than that of the dovish opponent.
By contrast, in the symmetric mixed equilibrium both players obtain the same
expected payoff, yet this payoff is relatively low due to the positive
probability of a conflict arising between two hawks.

A natural question is to ask which equilibrium is more likely to obtain.
Standard game theory is not helpful in answering this question, as all these
Nash equilibria satisfy all the standard refinements (e.g., perfection). By
contrast, the dynamic (evolutionary) approach can yield sharp predictions (for
textbook expositions, see Weibull, 1997; Sandholm, 2010).

#### Revision dynamics

Consider a setup in which pairs of agents from two infinite populations are
repeatedly matched at random times (each such match of an agent from
population 1 is against a new opponent from population 2).444Our paper focuses
on two-population dynamics in which players condition their play on their role
in the game. By contrast, in a one-population model, players cannot condition
their play on their role in the game. The predictions of the one-population
model are briefly discussed in Remark 1. Agents occasionally die (or,
alternatively, agents occasionally receive opportunities to revise their
actions). New agents observe some information about the aggregate behavior and
the payoffs, and use this information to choose the action they will play in
all future encounters. We are interested in characterizing the stable rest
points of such revision dynamics, which can be used as an equilibrium
refinement.

Most existing models assume that the revision dynamics are _monotone_ (also
known as sign-preserving) with respect to the payoffs: the frequency of the
strategy that yields the higher payoff (among the two feasible strategies)
increases. A key result in evolutionary game theory is that in a hawk–dove
game, all monotone (two-population) revision dynamics converge to the
asymmetric pure equilibria from almost any initial state (henceforth, _global
convergence_ ; see Maynard-Smith and Parker, 1976, for the classic analysis,
Maynard-Smith, 1982, for the textbook presentation, Sugden, 1989, for the
economic implications, and Oprea et al., 2011, for the general dynamic
result.) Thus, the existing literature predicts that an efficient convention
will emerge in which trade always occurs and most of the surplus goes to one
side of the market. Casual observation suggests that this prediction might not
fit well the behavior in situations such as the motivating example, in which
the surplus of trade is typically divided relatively equally between the two
sides of the market, and in which bargaining frequently fails.

In many applications, precise information about the aggregate behavior in the
population may be difficult or costly to obtain. In such situations, new
agents have to infer the aggregate behavior in the population from a small
sample of other players. In what follows, we study two plausible inference
procedures, both of which violate monotonicity. The first procedure is the
_action-sampling dynamics_ (also known as sampling best-response dynamics;
Sandholm, 2001; Osborne and Rubinstein, 2003). In these dynamics, each new
agent observes the behavior of $k$ random opponents, and then adopts the
action that is a best reply to her sample (with an arbitrary tie-breaking
rule).

In some applications, new agents may not be able to observe opponents’
actions, or they may lack information about the payoff matrix. Plausible
revision dynamics in such situations are the _payoff-sampling dynamics_ (also
known as best experienced payoff dynamics; Osborne and Rubinstein, 1998;
Sethi, 2000). In these dynamics, each new agent observes the payoffs obtained
by incumbents of her own population in $k$ interactions in which these
incumbents played hawk, and in $k$ interactions in which these incumbents
played dove. Following these observations, the new agent adopts the action
that yielded the higher mean payoff.

We analyze both sampling dynamics in the hawk–dove game. In our analysis, we
allow agents to have heterogeneous sample sizes (i.e., each new agent is
endowed with a sample size of $k$ that is randomly chosen from an exogenous
distribution). It is relatively straightforward to show that these dynamics
admit at least three stationary states: two asymmetric pure states, and an
inefficient symmetric state.555 Although the symmetric stationary state does
not coincide with the symmetric Nash equilibrium, they share similar
qualitative properties: namely, symmetry between the two populations, and
inefficiency induced by frequent matching of two hawks., 666In some cases, the
dynamics admit also asymmetric mixed states, as is demonstrated in Section 7.

#### Global convergence to mixed states

We show that sampling dynamics can yield qualitatively different results
compared to monotone dynamics in the hawk–dove game. Specifically, our first
main result (Theorem 1 and Corollary 3) presents a simple condition for
sampling dynamics to induce the opposite result (relative to monotone
dynamics), namely, convergence to one of the mixed stationary states from any
interior state. Roughly speaking, this happens iff the following three
conditions hold: (1) sufficiently many agents have small sample sizes, (2) the
population includes some agents with relatively large sample sizes (though
still below a threshold that is increasing in $|g-0.5|$), and (3) $g$ is not
too close to 0.5.

The key to this result is to characterize when populations that start near one
of the pure equilibria get away from this equilibrium. Assume that initially
almost all ($1-\epsilon$) buyers are hawks, and almost all ($1-\epsilon$)
sellers are doves. Consider a new agent who bases her decision on a sample of
$k$ actions. The sample will have a single occurrence of the rare action with
a probability of about $k\cdot\epsilon$ (while the probability of having two
or more occurrences of the rare action is negligible). The presence of agents
with small samples is necessary for moving away from the pure equilibrium,
because for dove-favorable games with $g<0.5$ a single occurrence of a rare
action can induce new sellers to be hawks only if their samples are small
(similarly, only new buyers with small samples change their behavior after a
single occurrence of a rare action in hawk-favorable games with $g>0.5$).

In dove-favorable games, a single occurrence of a rare action can induce a new
buyer to be a dove even for relatively large sample sizes up to a threshold,
where this threshold is larger the farther $g$ is from 0.5 (and the same holds
for inducing new sellers to be hawks in hawk-favorable games). Having a larger
sample size $k$ (up to the above-mentioned threshold) helps to increase the
frequency in which rare actions are observed, which explains why having a
sufficiently large expected sample size (Condition (2)) is necessary for
moving away from the pure equilibrium. Finally, when the game is balanced
between the two actions (i.e., $g$ is close to 0.5), a single occurrence of a
rare action can change the behavior of a new agent (in both populations) only
if her sample is relatively small, but in this case, the total frequency in
which rare actions occur in the samples might be too small. This is why having
a less balanced game (i.e., condition (3) of $g$ not being too close to 0.5)
induces the population to move away from a pure equilibrium.

#### Heterogeneous populations and stable symmetric mixed states

Our next result focuses on the action-sampling dynamics. We show that the
behavior of the population is qualitatively different for homogeneous
populations (in which all agents have the same sample size) than for
heterogeneous populations. Specifically, we show (Theorem 2) that all
homogeneous populations (with a sample size of at least 2) converge from
almost all initial states to one of the pure states under the action-sampling
dynamics. By contrast, Theorem 3 shows that the symmetric stationary mixed
state (in which both populations obtain the same expected payoff, and
bargaining frequently fails) is asymptotically stable in a large class of
heterogeneous populations in which some agents have relatively small samples,
while the remaining agents have large samples. Interestingly, _the symmetric
mixed state is asymptotically stable even in populations in which most agents
perfectly observe the opposing population’s true distribution of play, while a
small minority relies on small samples_ (provided that $g$ is sufficiently far
from 0.5).

The argument that the symmetric stationary state is unstable in homogeneous
populations and stable in heterogeneous populations is sketched as follows. A
symmetric stationary state under the action-sampling dynamics is characterized
by solving the equation $p=\Pr(X(k,p)\leq{m})$, where $X(k,p)$ is a binomial
random variable and $m\in\\{0,1,\dots,k-1\\}$. This is so because the rate of
hawks ($p$) among the new agents is equal to the probability that the number
of the opponents’ hawkish actions observed by a new agent (which has a
binomial distribution) is below some threshold ($m$). One can show that this
solution is close to the peak of the unimodal density of $X(k,p),$ which
implies that a small perturbation in $p$ has a strong effect on
$\Pr(X(k,p)\leq{m})$, and thus it substantially changes the new agents’
behavior, which takes the population away from the symmetric state. By
contrast, in heterogeneous populations the stationary state falls in between
the peaks of the two distributions, which implies that small perturbations
have a weak effect on the agents’ behavior.

Our final result (Theorem 4) shows that the conditions for the stability of
the symmetric mixed state are broader under the payoff-sampling dynamics, and
that populations with relatively small sample sizes (either homogeneous or
heterogeneous) can have an asymptotically stable symmetric mixed state.

Taken together, our results show that when some agents have limited
information about the aggregate behavior of the population (which seems
plausible in various real-life applications, such as the motivating example of
buyers and sellers of houses), then an egalitarian, yet inefficient,
convention may arise in which bargaining frequently fails.

Our proofs rely on deriving new properties of binomial distributions, which
may be of independent interest. One such new property (Proposition 4) is that
there do not exist any two different probabilities $p\neq q\in(0,1)$ such that
the probability of obtaining at most $m$ successes out of $k$ trials is $q$
when the success probability in each trial is $p$, and at the same time, the
probability of obtaining at most $m$ successes out of $k$ trials is $q$ when
the success probability in each trial is $p$.

#### Structure

Section 2 presents the related literature. Our model is described in Section
3. Section 4 presents a “complete” characterization for global convergence to
one of the mixed stationary states. In Section 5 we show that homogeneous
populations converge to pure stationary states under the action-sampling
dynamics. Section 6 presents various sufficient conditions for the stability
of the symmetric stationary state. The analytic results of the paper are
supplemented by a numeric analysis in Section 7. We conclude in Section 8.
Formal proofs are presented in the appendix. Appendix A.1 describes our
results for binomial distributions.

##  2 Related Literature

#### Related theoretical literature

The action-sampling dynamics were pioneered by Sandholm (2001) and Osborne and
Rubinstein (2003). Oyama et al. (2015) applied these dynamics to prove global
convergence results in supermodular games. 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.777 Hauert and
Miekisz (2018) use 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 in the main
text.

Salant and Cherry (2020) (see also Sawa and Wu, 2021) generalized the action-
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 pay
special attention to _unbiased_ inference procedures in which the agent’s
expected belief about the share of opponents who play hawk 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 hawk, which implies that the
agent’s perceived best reply depends only on the expectation of her posterior
belief. This implies that our results hold for any unbiased inference
procedure.

The present paper, similar to the papers cited above, studies deterministic
dynamics in infinite populations. When there is convergence to a stable
stationary state in such dynamics, the convergence is fast (Oyama et al.,
2015). By contrast, stochastic evolutionary models (see, e.g., the seminal
contribution of Young, 1993, and the recent hawk–dove application in Bilancini
et al., 2021), which are also based on revising agents observing a finite
sample of opponents’ actions, focus on the very long-run behavior of
stochastic processes when players’ choice rules include the possibility of
rare “mistakes” (sufficient conditions for stochastic evolutionary models to
yield fast convergence are studied in Kreindler and Young, 2013; Arieli et
al., 2020).

The payoff-sampling dynamics were pioneered by Osborne and Rubinstein (1998)
and Sethi (2000) and later generalized in various respects by Sandholm et al.
(2020). It has been used in a variety of applications, including price
competition with boundedly rational consumers (Spiegler, 2006), common-pool
resources (Cárdenas et al., 2015), contributions to public goods (Mantilla et
al., 2018), centipede games (Sandholm et al., 2019), finitely repeated games
(Sethi, 2021) and the prisoner’s dilemma (Arigapudi et al., 2021). The
existing literature assumes that all agents have the same sample size.

A methodological contribution of the present paper is in extending the setup
of payoff-sampling dynamics to analyze heterogeneous populations in which new
agents differ in their sample sizes, and this heterogeneity leads to
qualitatively new results.

It is well known that mixed stationary states in multiple-population games
cannot be asymptotically stable under the commonly used replicator dynamics
(see, e.g., Sandholm, 2010, Theorem 9.1.6). By contrast, we show that both
classes of sampling dynamics, which are plausible in various real-life
applications, can induce asymptotically stable mixed stationary states in a
two-population hawk–dove game.

#### Related experimental literature

Selten and Chmura (2008) experimentally tested the predictive power of various
solution concepts in two-action, two-player games with a unique completely
mixed Nash equilibrium. They show that both the payoff-sampling equilibrium
and the action-sampling equilibrium outperform the predictions of both the
Nash equilibrium and the quantal-response equilibrium.

Recently, Stephenson (2019) tested the predictive validity of various
evolutionary models in coordinated attacker–defender games.888Experiments that
directly test the dynamic predictions of evolutionary game theory are quite
scarce. Two notable exceptions are the experiments showing the good fit of the
dynamic predictions in the rock–paper–scissors game (Cason et al., 2014;
Hoffman et al., 2015). Stephenson’s experimental design is very favorable for
monotone dynamics because each participant is shown the exact (population-
dependent) payoff that would be obtained by each action at each point in time.
Nevertheless, subjects frequently violate monotonicity: 10%–20% of the
subjects switch from higher-performing strategies to lower-performing
strategies.

The key prediction of monotone dynamics for hawk–dove games (in which agents
from one population are randomly matched with agents from another population)
is experimentally tested in Oprea et al. (2011) and Benndorf et al. (2016).
Both experiments apply an interface that is favorable to monotonicity (i.e.,
each participant is shown the exact population-dependent payoff of each
action). Both experiments show that the prediction of monotone dynamics holds
in this setup, and that the populations converge to an asymmetric pure
equilibrium in which one population (say, the buyers) plays hawk and the other
population (say, the sellers) plays dove.

Consider a revised experimental design, where an agent observes only the
behavior of her own opponent, rather than the aggregate behavior of the
opposing population. An interesting testable prediction of our model is that
in this experimental design, the populations are likely to converge to the
symmetric stationary state in the relevant parameter domain (in particular,
when $g$ is not too far from 1; see Figure 7.1).999Benndorf et al. (2016,
2021) studied a more general setup in which each participant in each round is
randomly matched with an opponent from the other population with probability
$\kappa\in(0,1)$, and is randomly matched with an opponent from her own
population with the remaining probability $1-\kappa$. Our theoretical
predictions fit the setup of $\kappa$ close to one.

##  3 Model

###  3.1 The Hawk–Dove Game

Let $G=\\{A,u\\}$ denote a symmetric two-player hawk–dove game, where:

  1. 1.

$A=\\{h,d\\}$ is the set of actions of each player, and

  2. 2.

$u:A^{2}\rightarrow\mathbb{R}$ is the payoff function of each player.

Let $i\in\left\\{1,2\right\\}$ be an index referring to one of the players,
and let $j=\left\\{1,2\right\\}\backslash\left\\{i\right\\}$ be an index
referring to the opponent. We interpret action $h$ as the hawkish (more
aggressive) action and $d$ as the dovish action. The payoff matrix
$u(\cdot,\cdot)$ of a generalized hawk–dove game is given in Table 2. When
both agents are dovish, they obtain a relatively high payoff, which is
normalized to 1. When both agents are hawkish, they obtain their lowest
feasible payoff, which is normalized to 0. Finally, when one of the players is
hawkish and her opponent is dovish, the hawkish player _gains_ $g\in(0,1)$
(relative to the payoff 1 obtained by two dovish players), while her dovish
opponent _loses_ 101010Herold and Kuzmics (2020) allow a broader domain in
which the assumption of $g,l\in(0,1)$ is replaced with the weaker assumption
of $g>0$, $l<1,$ and $l+g>0$. All of our results hold in this extended setup.
$l\in(0,1)$. The game admits three Nash equilibria: two asymmetric pure Nash
equilibria: $(h,d)$ and $(d,h),$ and a symmetric mixed Nash equilibrium in
which each player plays $h$ with probability $\frac{g}{1+g-l}$, and obtains a
relatively low expected payoff of $\frac{(1+g)(1-l)}{1+g-l}<1$.

Table 2: Payoff Matrix of a Generalized Hawk–Dove Game $g,l\in\left(0,1\right)$ |  | Player 2
---|---|---
|  | ${\color[rgb]{180,00,00}h}$ | ${\color[rgb]{180,00,00}d}$
Player 1 | ${\color[rgb]{0,0,1}h}$ | ${\color[rgb]{0,0,1}0},{\color[rgb]{180,00,00}0}$ | ${\color[rgb]{0,0,1}1+g},{\color[rgb]{180,00,00}1-l}$
${\color[rgb]{0,0,1}d}$ | ${\color[rgb]{0,0,1}1-l},{\color[rgb]{180,00,00}1+g}$ | ${\color[rgb]{0,0,1}1},{\color[rgb]{180,00,00}1}$

An important special subclass is the _standard_ hawk–dove games, in which
$g=l$ (see Table 1); i.e., the gain of the hawkish player is equal to the loss
of her dovish opponent.

###  3.2 Evolutionary Process

We assume that there are two unit-mass continuums of agents (e.g., buyers and
sellers) and that agents in population 1 are randomly matched with agents in
population 2\. Aggregate behavior in the populations at time
$t\in\mathbb{R}^{+}$ is described by a _state_
$\mathbf{p}\left(t\right)=\left(p_{1}\left(t\right),p_{2}\left(t\right)\right)\in\left[0,1\right]^{2}$,
where $p_{i}\left(t\right)$ represents the share of agents playing the hawkish
action $h$ at time $t$ in population $i$. We extend the payoff function $u$ to
states (which have the same representation as mixed strategy profiles) in the
standard linear way. Specifically, $u(p_{i},p_{j})$ denotes the average payoff
of population $i$ (in which a share $p_{i}$ of the population plays $h$) when
randomly matched against population $j$ (in which a share $p_{j}$ plays $h$).
With a slight abuse of notation, we use $d$ (resp., $h$) to denote a
degenerate population in which all of its agents play action $d$ (resp., $h$).
A state $\mathbf{p}=\left(p_{1},p_{2}\right)$ is _symmetric_ if $p_{1}=p_{2}$.
A state $\mathbf{p}=\left(p_{1},p_{2}\right)$ is _mixed_ (or _interior_) if
$p_{1},p_{2}\in(0,1).$

Agents occasionally die and are replaced by new agents (equivalently, agents
occasionally receive opportunities to revise their actions). Let $\delta>0$
denote the death rate of agents in each population, which we assume to be
independent of the currently used actions. It turns out that $\delta$ does not
have any effect on the dynamics, except to multiply the speed of convergence
by a constant. The evolutionary process is represented by a continuous
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 adopt
action $h$ as a function of the current state. That is,
$w_{i}\left(\mathbf{p}\right)$ describes the share of new agents of population
$i$ who adopt action $h$, given state $\mathbf{p}$. Thus, the instantaneous
change in the share of agents of population $i$ that play hawk is given by
$\dot{p}_{i}=\delta\cdot\left(w_{i}\left(\mathbf{p}\right)-p_{i}\right).$

######  Remark 1.

Our _two-population_ dynamics fit situations in which each player can
condition her play on her role in the game (being Player 1 or Player 2).
Common examples of such situations are (1) when sellers are matched with
buyers, as in the motivating example, and (2) when each player observes if she
has arrived slightly earlier or slightly later at a contested resource
(Maynard-Smith, 1982). The two-population dynamics essentially induce the same
results as one-population dynamics over a larger game with $2\times 2$ “role-
conditioned” actions (see, e.g., Weibull, 1997, end of Section 5.1.1): being a
hawk in both roles, being a dove in both roles, being a hawk as Player 1 and a
dove as Player 2 (the “bourgeois” strategy of Maynard-Smith, 1982), and being
a hawk as Player 2 and a dove as Player 1.

By contrast, in _one-population_ dynamics of the original two-action game, an
agent cannot condition her play on her role. It is well known that all
monotone one-population dynamics converge to the unique mixed Nash equilibrium
in hawk–dove games (see, e.g., Weibull, 1997, Section 4.3.2). It is relatively
straightforward to establish that one-population sampling dynamics lead to
qualitatively similar results (convergence is to a somewhat different interior
state than in the mixed Nash equilibrium, but the comparative statics with
respect to the payoff parameters remain similar).

#### Monotone Dynamics

The most widely studied dynamics are those that are monotone with respect to
the payoffs. A dynamic is monotone if the share of agents playing an action
increases iff the action yields a higher payoff than the alternative
action.111111In games with more than two actions, there are various
definitions that capture different aspects of monotonicity. All these
definitions coincide for two-action games. In particular, Definition 1
coincides in two-action games with Weibull’s (1997, Section 5.5) textbook
definitions of payoff monotonicity, payoff positivity, sign preserving, and
weak payoff positivity.,121212The best-known example of payoff monotone
dynamics is the standard replicator dynamic (Taylor, 1979), which is given by
$\dot{p}_{i}=w_{i}\left(\textbf{p}\right)-p_{i}=p_{i}\left(u\left(h,p_{j}\right)-u\left(p_{i},p_{j}\right)\right).$

######  Definition 1.

The dynamic $\textbf{w}:\left[0,1\right]^{2}\rightarrow\left[0,1\right]^{2}$
is _monotone_ if for any player $i$, any interior $p_{i}\in\left(0,1\right)$,
and any $p_{j}\in\left[0,1\right]$:
$\dot{p}_{i}>0\,\,\Leftrightarrow\,\,u\left(h,p_{j}\right)>u\left(d,p_{j}\right).$

Oprea et al. (2011) showed that under monotone dynamics, from almost any
initial state, the populations converge to one of the two asymmetric pure
equilibria in which one population always plays $h$ and the other population
always plays $d$ (generalizing the seminal analysis of Maynard-Smith and
Parker, 1976).

###  3.3 Sampling Dynamics

In what follows, we study two plausible nonmonotone dynamics, in which new
agents base their choice on inference from small samples.

#### Distribution of sample sizes

We allow heterogeneity in the sample sizes used by new agents. Let
$\theta\in\Delta\left(\mathbb{Z}_{+}\right)$ denote the distribution of sample
sizes of new agents. A share of $\theta\left(k\right)$ of the new agents have
a sample of size $k$. Let $\text{supp}\left(\theta\right)$ denote the support
of $\theta$, let $\max($supp$(\theta))$ denote the maximal sample size in the
support of $\theta$, and let $\max($supp($(\theta))=\infty$ if $\theta$’s
support is unbounded. If there exists some $k,$ for which $\theta(k)=1,$ then
we use $k$ to denote the degenerate (homogeneous) distribution $\theta\equiv
k$.

######  Definition 2.

An _environment_ is a tuple $E=(g,l,\theta)$ where $g,l\in(0,1)$ describe the
underlying hawk–dove game, and $\theta\in\Delta\left(\mathbb{Z}_{+}\right)$
describes the distribution of sample sizes.

#### Action-sampling dynamics

The action-sampling dynamics (Sandholm, 2001) fit situations in which agents
do not know the exact distribution of actions being played in the opponent’s
population, but know the payoffs of the underlying game. Agents estimate the
unknown distribution of actions by sampling a few opponents’ actions.
Specifically, each new agent with sample size $k$ (henceforth, a $k$-agent)
samples $k$ randomly drawn agents from the opponent’s population and then
adopts the action that is the best reply against the sample. _To simplify
notation, we assume that in case of a tie, the new agent plays action $d$_.
Our results are qualitatively 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
$h$-s in the sample. Observe that the sum of payoffs of playing action $h$
against the sample is $(1+g)\cdot(k-X(k,p_{j}))$ and the sum of payoffs of
playing action $d$ against the sample is
$(k-X(k,p_{j}))+X(k,p_{j})\cdot(1-l)=k-lX(k,p_{j})$.

This implies that action $h$ is the unique best reply to a sample of size $k$
iff
$(1+g)(k-X(k,p_{j})>k-lX(k,p_{j})\Leftrightarrow\frac{X(k,p_{j})}{k}<\frac{g}{1+g-l}$.
This, in turn, implies that the action-sampling dynamic in environment
$(g,l,\theta)$ is given by

$w_{\theta}^{A}\left(p_{j}\right)\equiv
w_{i}^{A}\left(\textbf{p}\right)=\sum_{k\in\text{supp}\left(\theta\right)}\theta(k)\cdot\Pr\left(\frac{X\left(k,p_{j}\right)}{k}<\frac{g}{1+g-l}\right).$
(3.1)

#### Payoff-sampling dynamics

The payoff-sampling dynamics (Osborne and Rubinstein, 1998) fit situations in
which agents either do not know the payoff matrix or do not have feedback
about the actions being played in the opponent’s population. Specifically, a
new $k$-agent observes for each of her feasible actions the mean payoff
obtained by playing this action in $k$ interactions (with each play of each
action being against a newly drawn opponent), and then chooses the action
whose mean payoff was highest. One possible interpretation for these
observations is that each new $k$-agent tests each of the available actions
$k$ times, and then adopts for the rest of her life the action with the
highest mean payoff during the testing phase. As above, we assume that a tie
induces a new agent to play action $d$ (and the results are qualitatively the
same for any tie-breaking rule).

We refer to the sample against which action $h$ (resp., $d$) is tested as the
$h$-sample (resp., $d$-sample). Let $X(k,p_{j}),Y(k,p_{j})\sim
Bin\left(k,p_{j}\right)$ denote two iid random variables with a binomial
distribution with parameters $k$ and $p_{j}$. Random variable $X(k,p_{j})$
(resp., $Y(k,p_{j})$) is interpreted as the number of times in which the
opponents have played action $h$ in the $h$-sample (resp., $d$-sample).
Observe that the sum of payoffs of playing action $h$ (resp., $d$) against its
$h$-sample (resp., $d$-sample) is $(1+g)(k-X(k,p_{j}))$ (resp.,
$k-lY(k,p_{j}$)). This implies that action $h$ has the highest mean payoff iff
$(1+g)(k-X(k,p_{j}))>k-lY(k,p_{j})\Leftrightarrow(1+g)X(k,p_{j})<gk+lY(k,p_{j})$.
Thus the payoff-sampling dynamic in environment $(g,l,\theta)$ is given by

$w_{\theta}^{P}\left(p_{j}\right)\equiv
w_{i}^{P}\left(\textbf{p}\right)=\sum_{k\in\text{supp}\left(\theta\right)}\theta(k)\cdot\Pr((1+g)X(k,p_{j})<gk+lY(k,p_{j})).$
(3.2)

Henceforth, we omit the superscript $A$ / $P$ (i.e., we write
$w_{\theta}(p_{j})$) when referring to properties that hold for both action-
sampling and payoff-sampling dynamics, or when it is clear from the context
which class of sampling dynamics we are dealing with.

##  4 Global Convergence to Mixed States

Recall that under monotone dynamics the population converges from almost any
initial state to one of the pure equilibria. 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 one
of the mixed (interior) stationary states.

###  4.1 Analysis of $w(p_{1})$ and Preliminary Results

The dynamic characteristics of both classes of sampling dynamics are closely
related to the properties of the polynomial $w_{\theta}(p_{1})$, its inverse
$w^{-1}_{\theta}(p_{1}),$ and their intersection points, which are analyzed in
this subsection.

Figure 4.1 illustrates the phase plots of the sampling dynamics and the
properties of the polynomial $w_{\theta}(p_{1})$ and its inverse
$w_{\theta}^{-1}(p_{1})$ for the environment in which $g=l=0.25$ and
$\theta\equiv 3$. The green solid curve is the polynomial
$p_{2}=w_{\theta}(p_{1})$, which describes the states in which
$\dot{p}_{1}=0$. The orange dashed curve is the polynomial
$p_{2}=w_{\theta}^{-1}(p_{1}).$

Figure 4.1: Illustrative Phase Plots ($g=l=0.25$ and $\theta\equiv 3$)

The figure illustrates the phase plots of the action-sampling dynamics (left
panel) and the payoff-sampling dynamics (right panel) for the environment in
which $g=l=0.25$ and $\theta\equiv 3$. The green solid (resp., orange dashede)
curve shows the states for which $\dot{p}_{1}=0$. 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 (for both classes of sampling dynamics) from
the basic properties of binomial random variables.

######  Fact 1.

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

Fact 1 implies that the two curves intersect at $(0,1)$, $(1,0)$ and at a
unique symmetric state $(p,p)$. The curves might admit additional intersection
points.

Appendix A.2 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 $w_{\theta}$ and $w_{\theta}^{-1}$.

######  Fact 2.

State $(p_{1},p_{2})$ is stationary iff
$w_{\theta}(p_{1})=w_{\theta}^{-1}(p_{1})=p_{2}$.

Our first result shows that the population converges to one of the stationary
states from any initial state.131313 One could also prove Proposition 1 by
relying on the Bendixson–Dulac theorem (see Theorem 9.A.6 of Sandholm, 2010).
We present a direct proof as the arguments are helpful in the proof of
Proposition 2.

######  Proposition 1.

$\lim_{t\rightarrow\infty}\mathbf{p}\left(t\right)$ exists for any
$\textbf{p}(0),$ and it is a stationary state.

###### Sketch of Proof.

Observe that in any trajectory that begins above (resp., below) both curves,
the share of hawks in both populations decreases (increases), which implies
that the trajectory moves downward and to the left (resp., upward and to the
right) until it intersects with one of the curves. Thus, we can focus on
trajectories that begin between the two curves. Assume w.l.o.g. that the
initial state is below $w(p_{1})$ and above $w^{-1}(p_{1})$ (the argument in
the opposite case is analogous). In any such state the share of hawks in
population 1 (resp., 2) increases (resp., decreases). This implies that the
trajectory moves upward and to the left until meeting one of the curves. This
meeting point must be an intersection point of both curves, because if the
meeting point were only with $w(p_{1})$ (resp., $w^{-1}(p_{1})$), the
trajectory there would have been horizontal to the left (vertical upward),
which implies that the trajectory would come from the right side of the curve
(from below the curve), leading to a contradiction. See Appendix A.3 for a
formal proof. ∎

Our second result shows that if any initial state converges to one of the pure
equilibria, then this equilibrium must be asymptotically stable (as defined in
Appendix A.2).

######  Proposition 2.

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

###### Sketch of Proof.

Due to analogous arguments to the above sketch of proof we can assume w.l.o.g.
that $\textbf{p}(0)$ is between the two curves, and that the trajectory moves
upward and to the left (which is required for convergence to (1,0)) iff the
curve $w(p_{1})$ is above $w^{-1}(p_{1})$, and the closest intersection point
of the two curves to the left of $\textbf{p}(0)$ is (1,0). These two
conditions imply that any initial state sufficiently close to (1,0) converges
to (1,0), which implies that (1,0) is asymptotically stable. See Appendix A.4
for the formal proof. ∎

###  4.2 Single Appearance of a Rare Action

The following lemma characterizes when a single appearance of a rare action in
a new agent’s sample can change the agent’s behavior (see the proof in
Appendix A.5; we require strictly higher payoffs for action $h$ and weakly
higher payoffs for action $d$ due to our tie-breaking rule in favor of action
$d$).

######  Lemma 1.

Consider a new agent in population $i$ with a sample size of $k$.

  1. 1.

Action-sampling dynamics: (I) Action $h$ induces a strictly higher payoff
against a sample with a single opponent’s action $d$ iff $k<\frac{1+g-l}{1-l}$
and (II) Action $d$ induces a weakly higher payoff against a sample with a
single $h$ iff $k\leq\frac{1+g-l}{g}$.

  2. 2.

Payoff-sampling dynamics: (I) a $h$-sample with a single $d$ induces a
strictly higher mean payoff than a $d$-sample with no $d$-s iff
$k<\frac{1+g}{1-l}$ and (II) a $d$-sample with no $h$-s induces a weakly
higher mean payoff than an $h$-sample with a single $h$ iff
$k\leq\frac{1+g}{g}$.

Lemma 1 allows us to define the upper bounds on the sample size in which a
single appearance of a rare action can change the behavior of a new agent.

######  Definition 3.

Let
$\,m_{h}^{A}=1+\frac{g}{1-l},\,m_{d}^{A}=1+\frac{1-l}{g},m_{h}^{P}=\frac{1+g}{1-l},\,m_{d}^{P}=\frac{1+g}{g}.$

We omit the superscript ${A,P}$ when stating a result that is true for both
dynamics; for example, we write $m_{h}$, which denotes $m_{h}^{P}$ when the
underlying dynamics is payoff sampling, and which denotes $m_{h}^{A}$ when the
underlying dynamics is action sampling.

The parameter $m_{h}$ is the upper bound on the sample size for which a single
appearance of $d$ in the sample, when all other sampled actions are $h$,
induces a new agent to adopt action $h$. Similarly, $m_{d}$ is the upper bound
on the sample size for which a single appearance of $h$ in the sample, when
all other sampled actions are $d$, can induce a new agent to adopt action $d$.

We conclude this subsection by presenting a definition of $m$_-bounded
expectation_ of a probability distribution with support on the set of positive
integers. It is the expected value of the probability distribution by
restricting its support to $m.$ Formally,

######  Definition 4.

The $m$_-bounded expectation_ $\mathbb{E}_{\leq m}$ (resp., $\mathbb{E}_{<m}$)
of distribution $\theta$ with support on integers is141414 Observe that in our
notation the parameter $k$ takes only (positive) integer values (although we
allow the upper bound $m$ to be a non-integer). $\mathbb{E}_{\leq
m}\left(\theta\right)=\sum_{1\leq k\leq m}\theta\left(k\right)\cdot k$ (resp.,
$\mathbb{E}_{<m}\left(\theta\right)=\sum_{1\leq k<m}\theta\left(k\right)\cdot
k$).

###  4.3 Asymptotic Stability of Pure Equilbria

Our next result characterizes the asymptotic stability of the pure states. It
shows that the asymptotic stability depends only on whether the product of the
bounded expectations of the distribution 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 (where replacing the tie-breaking rule with an $h$-favorable
one would replace the “<”-s and the “$\leq$”-s in the statement):

######  Proposition 3.

  1. 1.

$\mathbb{E}_{<m_{h}}\left(\theta\right)\cdot\mathbb{E}_{\leq
m_{d}}\left(\theta\right)>1\,\Rightarrow\,$ both pure stationary states are
unstable.

  2. 2.

$\mathbb{E}_{<m_{h}}\left(\theta\right)\cdot\mathbb{E}_{\leq
m_{d}}\left(\theta\right)<1\,\Rightarrow\,$ both pure stationary states are
asymptotically stable.

###### Sketch of Proof.

Consider a slightly perturbed state $(\epsilon,1-\epsilon)$ near the pure
equilibrium $(0,1).$ Observe that almost all agents in population 1 (resp., 2)
play $d$ (resp., $h$). The event of two rare actions appearing in a sample of
a new agent has a negligible probability of $O(\epsilon^{2})$. If a new agent
has a sample size of $k$, then the probability of a rare action appearing in
the sample is approximately $k\cdot\epsilon.$ This rare appearance changes the
perceived best reply of a new agent of population 1 iff $k$ is smaller than
$m_{h}$. Thus, the total probability that a new agent of population 1 (resp.,
2) adopts a rare action is equal to $\mathbb{E}_{<m_{h}}\left(\theta\right)$
(resp., $\mathbb{E}_{\leq m_{d}}\left(\theta\right)$). This implies that the
product of the share of new agents adopting a rare action in each population
is
$\epsilon\cdot\mathbb{E}_{<m_{h}}\left(\theta\right)\cdot\epsilon\cdot\mathbb{E}_{\leq
m_{d}}\left(\theta\right)$. This shows that the share of agents playing rare
actions gradually increases (resp., decreases) if
$\mathbb{E}_{<m_{h}}\left(\theta\right)\cdot\mathbb{E}_{\leq
m_{d}}\left(\theta\right)>1$ (resp.,
$\mathbb{E}_{<m_{h}}\left(\theta\right)\cdot\mathbb{E}_{\leq
m_{d}}\left(\theta\right)<1),$ which implies instability (resp., asymptotic
stability). See Appendix A.6 for a formal proof. ∎

Observe that the fact that $l>0$ immediately implies that
$m_{h}^{A}<m_{h}^{P}$ and $m_{d}^{A}<m_{d}^{P}$, which, in turn, implies that
instability under the action-sampling dynamics holds in a strictly smaller set
of distributions than under the payoff-sampling dynamics.

######  Corollary 1.

If the pure stationary states are unstable under the action-sampling dynamics,
then they are also unstable under the payoff-sampling dynamics.

Next we observe that the pure stationary states are stable under the action-
sampling dynamics in populations in which all agents have sample sizes of at
least 2.

######  Corollary 2.

Assume that $\theta(1)=0$. The pure stationary states are stable under the
action-sampling dynamics.

###### Proof.

Observe that either $\min(m_{h}^{A},m_{d}^{A})<2$ or $m_{h}^{A}=m_{d}^{A}=2$.
Both conditions imply that one of the expressions in the product of
$\mathbb{E}_{<m_{h}}\left(\theta\right)\cdot\mathbb{E}_{\leq
m_{d}}\left(\theta\right)$ is equal to $\theta(1),$ which implies (by
Proposition 3) that the pure stationary states are stable if $\theta(1)=0$. ∎

###  4.4 Main Result

Combining Propositions 1–3 yields the main result of this section. It shows
that the population converges from almost any initial state to an interior
stationary state if (and essentially only if) the product of the bounded
expectations of the distribution of sample sizes is larger than 1.

######  Theorem 1.

  1. 1.

Assume that $\mathbb{E}_{<m_{h}}\left(\theta\right)\cdot\mathbb{E}_{\leq
m_{d}}\left(\theta\right)>1.$ If
$\boldsymbol{p}(0)\notin\left\\{(0,1),(1,0)\right\\}$, then
$\lim_{t\rightarrow\infty}\mathbf{p}\left(t\right)\in{(0,1)}^{2}$.

  2. 2.

Assume that $\mathbb{E}_{<m_{h}}\left(\theta\right)\cdot\mathbb{E}_{\leq
m_{d}}\left(\theta\right)<1.$ Then there exist
$\boldsymbol{p}(0),\boldsymbol{\hat{p}}(0)\notin\left\\{(0,1),(1,0)\right\\}$
such that $\lim_{t\rightarrow\infty}\mathbf{p}\left(t\right)=(0,1)$ and
$\lim_{t\rightarrow\infty}\mathbf{\hat{p}}\left(t\right)=(1,0)$.

###### Proof.

  1. 1.

Assume that $\mathbb{E}_{<m_{h}}\left(\theta\right)\cdot\mathbb{E}_{\leq
m_{d}}\left(\theta\right)>1.$ Proposition 3 implies that both pure stationary
states are unstable. Combining Propositions 1–2 implies that from almost any
initial state, the population converges to an interior stationary state.

  2. 2.

Assume that $\mathbb{E}_{<m_{h}}\left(\theta\right)\cdot\mathbb{E}_{\leq
m_{d}}\left(\theta\right)<1.$ Proposition 3 implies that both pure stationary
states are stable, which implies part (2). ∎

Next we apply Theorem 1 to standard hawk–dove games in which the parameter
$g=l$ describes both the gain of a hawkish player and the loss of her dovish
opponent. In order to simplify the statement we assume that the various $m$-s
are non-integers (to avoid the weak dependency on the tie-breaking rule).

######  Corollary 3.

Assume that $g=l$ and that $\frac{1}{g}$, $\frac{1}{1-g}$, $\frac{1+g}{g}$,
and $\frac{1+g}{1-g}$ are non-integers. The population converges to a mixed
stationary state from any interior state iff:

  1. 1.

Action-sampling dynamics:
$\theta(1)\cdot\mathbb{E}_{\leq\max\left(\frac{1}{g},\frac{1}{1-g}\right)}\left(\theta\right)>1.$

  2. 2.

Payoff-sampling dynamics: $\begin{aligned} \text{either }(a)\ g&<\frac{1}{3}\
\text{and}\
\theta(1)\cdot\mathbb{E}_{\leq\frac{1+g}{g}}\left(\theta\right)>1\\\ \text{or
}(b)\ g&\geq\frac{1}{3}\ \text{and}\
(\theta(1)+2\theta(2))\mathbb{E}_{\leq\max\left(3,\frac{1+g}{1-g}\right)}\left(\theta\right)>1.\end{aligned}$

The straightforward proof, which relies on substituting $g=l$ in Definition 3,
is presented in Appendix A.7. Corollary 3 implies that global convergence to
one of the mixed stationary states holds iff: (1) sufficiently many agents
have a sample size of 1, (2) the expected sample size (conditional on the
sample being below a threshold that is increasing in $|g-0.5|$) is
sufficiently large, and (3) $g$ is not too close to 0.5. Under the payoff-
sampling dynamics (and assuming $g>\frac{1}{3}$), global convergence holds
also if one relaxes Condition (1) by requiring that (1’) sufficiently many
agents have a sample size of at most 2. As demonstrated in Section 7, these
conditions hold for many environments with relatively small samples under the
payoff-sampling dynamics, and for a somewhat limited set of environments under
the action-sampling dynamics.

######  Remark 2.

Theorem 1 implies that if the sample sizes of all agents in the population are
sufficiently large (larger than all the $m$-s in Definition 3), then the pure
stationary states are stable. Perhaps surprisingly, a moderate increase in the
sample sizes of some agents in the population can yield the opposite effect,
that of destabilizing the pure stationary states. For example, when
$g=l=0.25$, the pure stationary states are stable under the action-sampling
dynamics given a population in which $75\%$ of the agents have sample size 1
and $25\%$ have sample size 2 (because
$\theta(1)\cdot\mathbb{E}_{\leq\max\left(\frac{1}{g},\frac{1}{1-g}\right)}\left(\theta\right)=\frac{3}{4}\cdot\frac{5}{4}=\frac{15}{16}<1$).
By contrast, if one increases the samples of the latter group of agents from
size 2 to size 3, then the pure stationary states become unstable (because
$\theta(1)\cdot\mathbb{E}_{\leq\max\left(\frac{1}{g},\frac{1}{1-g}\right)}\left(\theta\right)=\frac{3}{4}\cdot\frac{3}{2}=\frac{9}{8}>1$);
moreover, one can show that in this case the symmetric mixed state is globally
stable.

##  5 Global Convergence to Pure States

In this section, we show that _homogeneous_ populations (in which all agents
have the same sample size) behave under the _action-sampling dynamics_ in the
same way as under monotone dynamics: namely, they converge from almost any
initial state to one of the pure equilibria. By contrast, In Section 6 we will
show that convergence to the symmetric mixed stationary state is achieved in
many cases in which we have either: (1) heterogeneous populations or (2)
payoff-sampling dynamics.

Theorem 2 shows that if all agents have the same sample size, then the action-
sampling dynamics admit a unique interior stationary state, which is unstable.
This implies (together with Proposition 1) that the phase plots of all such
dynamics are qualitatively equivalent to the left panel of Figure 4.1, and
that _all initial populations eventually converge to one of the pure
equilibria_ (in the sense that even if a population initially converges to the
unique unstable interior equilibrium, an arbitrarily small perturbation will
still take the population from there to one of the pure equilibria).

######  Theorem 2.

For151515 It is straightforward to show that the case in which all agents have
sample size 1 induces an environment in which the stationary states are
$(p,1-p)$ for any $p\in[0,1]$, and that all these states are neither
asymptotically stable nor unstable (i.e., they are Lyapunov stable) for any
$g,l$ and under both sampling dynamics. $\theta\equiv k>1,$ the action-
sampling dynamics admit a unique interior stationary state, which is unstable.

It turns out that Theorem 2 is implied by a new general property of binomial
distributions, which may be of independent interest. Proposition 4 states that
there do not exist any two different probabilities $p\neq q\in(0,1)$ such that
if the success probability in each trial is $p$, the probability of obtaining
at most $m$ successes is $q$, and the same holds when swapping $p$ and $q$.
This is formalized as follows. (Recall that $X(k,p)$ denotes a _binomial
random variable_ with $k\geq 1$ trials and probability of success $p$ in each
trial.)

######  Proposition 4.

Fix any $k\geq 2$ and any $0\leq m<k$. Then:

$\Pr\left(X(k,p)\leq m\right)=q\in\left(0,1\right),\,\Pr\left(X(k,q)\leq
m\right)=p\,\,\Rightarrow\,\,p=q.$

The proof of Proposition 4 is presented in Appendix A.1.

###### Proof of Theorem 2.

Each pair of parameters $g,l$ induces a threshold $0\leq m<k$, such that
playing hawk is the best reply against a sample of size $k$ iff the sample
includes at most $m$ hawkish actions. That is, $w_{k}(p_{1})$ must coincide
with the curve of $\Pr\left(X(p_{1},k)\leq m\right)$ for some $0\leq m<k$.
This implies that an interior state p is stationary iff
$\Pr\left(X(p_{1},k)\leq m\right)=p_{2}$ and $\Pr\left(X(p_{2},k)\leq
m\right)=p_{1}.$ Thus, Proposition 4 implies that there are no asymmetric
interior stationary states. Corollary 2 implies that the pure stationary
states are asymptotically stable, and that the curve $w(p_{1})$ is above
(resp., below) the curve $w^{-1}(p_{1})$ near state $(1,0)$ (resp., $(0,1)$).
This implies that curve $w(p_{1})$ must be above (resp., below) the curve
$w^{-1}(p_{1})$ in a left (right) environment of the symmetric stationary
state, which implies that this stationary state must be unstable, as
illustrated in the left panel of Figure 4.1. Another, direct argument for that
the symmetric stationary state is unstable is implied by combining Proposition
6 ($\left|w^{\prime}_{k}(p)\right|=\left|\frac{d\Pr\left(X(k,p)\leq
m\right)}{dp}\right|>1$) and Proposition 5 (state $(p,p)$ is asymptotically
stable if $|w^{\prime}_{k}(p)|>1$). ∎

##  6 Stability of the Symmetric Stationary State

In this section, we present two sufficient conditions for the asymptotic
stability of _symmetric_ mixed stationary states. Symmetric stationary states
are qualitatively very different from pure stationary states. In particular,
in symmetric states both populations have the same expected payoff, and the
bargaining fails frequently (whenever two hawks are matched together). By
contrast, _asymmetric_ mixed stationary states might be similar to pure
stationary states in the sense of having $p_{1}$ close to 1, $p_{2}$ close to
0, and a large payoff difference between the two populations.

###  6.1 Auxiliary Result for the Stability of Symmetric States

We begin with a simple auxiliary result showing that the symmetric stationary
state is asymptotically stable iff the derivative of $w_{\theta}(p)$ is larger
than one. Formally

######  Proposition 5.

Symmetric stationary state $(p,p)$ is unstable if
$|w^{\prime}_{\theta}(p)|>1$.

###### Proof.

In order to assess the asymptotic stability, we compute the Jacobian $J$ of
$\dot{p_{i}}=\delta\cdot(w_{\theta}(p_{j})-p_{i})$ at the symmetric rest point
$(p,p)$ (ignoring the constant $\delta$, which plays no role in the dynamics,
other than multiplying the speed of convergence by a constant):

$J=\left(\begin{array}[]{cc}-1&w^{\prime}_{\theta}(p)\\\
w^{\prime}_{\theta}(p)&-1\end{array}\right).$

The eigenvalues of $J$ are $-1+w^{\prime}_{\theta}(p)$ and
$-1-w^{\prime}_{\theta}(p).$ A sufficient condition for instability at $(p,p)$
is that there exists a positive eigenvalue, which is implied by
$|w^{\prime}_{\theta}(p)|>1.$ ∎

###  6.2 Action-Sampling Dynamics

Our next result shows that the symmetric mixed stationary state is often
asymptotically stable under the action-sampling dynamics if the population is
heterogeneous, such that some agents have a relatively small samples, while
the other agents have large samples. Specifically, the symmetric stationary
state is asymptotically stable for any population in which a positive, yet
sufficiently small, share of the population have sample $k$, the remaining
agents have sufficiently large samples, and either $g$ is sufficiently small
or $l$ is sufficiently large.

######  Theorem 3.

Fix any $k>1$ and any $0<q<\frac{1}{k}$:

  1. 1.

For any $0<l<1$, there exists $0<\overline{g}<1$ and
$\overline{k}\in\mathbb{N}$, such that the symmetric stationary state is
asymptotically stable under the action-sampling dynamics for any
$0<g<\overline{g}$ and any distribution $\theta$ satisfying $\theta(k)=q$ and
$\sum_{n\geq\overline{k}}\theta(n)=1-q.$

  2. 2.

For any $0<g<1$, there exists $0<\overline{l}<1$ and
$\overline{k}\in\mathbb{N}$, such that the symmetric stationary state is
asymptotically stable under the action-sampling dynamics for any
$\overline{l}<l<1$ and any distribution $\theta$ satisfying $\theta(k)=q$ and
$\sum_{n\geq\overline{k}}\theta(n)=1-q.$

###### Sketch of Proof.

To simplify the sketch, we treat the agents with large samples
($n\geq\overline{k}$) as best responders (i.e., playing the best reply to the
population’s true distribution). For sufficiently small $g$ (resp.,
sufficiently large $l$), the symmetric stationary state $p$ has the property
that only agents with sample size $k$ play hawk (resp., dove), while all the
agents with large samples play dove (resp., hawk), and they continue to do so
even after a small perturbation in the share of hawks. Proposition 5 implies
that the symmetric stationary state $p$ is asymptotically stable if
$|w^{\prime}_{\theta}(p)|<1$. This latter inequality is implied by the fact
that the best responders continue playing dove (resp., hawk) after any small
perturbation and that $q$ is sufficiently small:
$|w^{\prime}_{\theta}(p)|=(1-q)\cdot 0+q\cdot|w^{\prime}_{k}(p)|<1.$ See
Appendix A.8 for a formal proof. ∎

Figure 6.1: Illustrative Phase Plot for Theorem 3: $g=l=0.95,$
$\theta_{2}=0.3,$ and $\theta_{50}=0.7$

Figure 6.1 illustrates the phase plot of a heterogeneous population that
satisfies the conditions of Theorem 3: the games is “hawk-favorable” with
parameter $g=l=0.95$, $70\%$ of the agents have large samples of size 50,
while the remaining $30\%$ of the agents have small samples of size 2. Numeric
analysis shows that the basin of attraction of the symmetric stationary state
in this example covers approximately $88\%$ of the unit square.

It is straightforward to adapt Theorem 3 to show that the stationary symmetric
state is asymptotically stable in various heterogeneous populations for the
payoff-sampling dynamics as well. We omit the formal statement and proof for
brevity.

Observe that Theorem 3 implies that _introducing a small share of agents with
a finite sample size can stabilize the symmetric equilibrium of a population
of (exact) best responders, as long as either $g$ is sufficiently small or $l$
is sufficiently large._

###  6.3 Payoff-Sampling Dynamics

In this subsection, we show that the symmetric stationary state is
asymptotically stable under the payoff-sampling dynamics for homogeneous
populations (and populations in which all agents have relatively small sample
sizes). For tractability in the analysis of payoff-sampling dynamics, we focus
on the cases where the gain of a hawkish player and the loss of her dovish
opponent are large, namely, $l,g>\frac{1}{\max(\textrm{supp}(\theta))}$. Our
result shows that in this domain, the symmetric stationary state is
asymptotically stable in various populations in which agents have relatively
small samples:

  1. 1.

for any homogeneous distribution of sample sizes $\theta\equiv k<20$; or

  2. 2.

for any heterogeneous distribution with a maximal sample size of at most 5.

The threshold of $k=20$ is binding. The symmetric stationary state becomes
unstable if the sample size is $k\geq 20.$ By contrast, the bound of a maximal
size of 5 for heterogeneous distributions of sample sizes is only a constraint
of our proof technique. Numeric analysis (see Section 7) suggests that the
stability of the symmetric stationary state:

  1. 1.

holds for many distributions of types with larger maximal sample sizes (in
particular, it holds for uniform distributions over
$\left\\{1,\dots,k\right\\}$ for any $k\leq 20$); and

  2. 2.

is often global (i.e., in many cases almost all initial states converge to
this state).

######  Theorem 4.

Assume that
$l,g\in\left(\frac{1}{\max\left(\emph{supp}(\theta\right))},1\right)$, and
either (1) $\theta\equiv k<20$, or (2)
$\max\left(\emph{supp}(\theta\right))\leq 5.$ Then, the game admits an
asymptotically stable symmetric stationary state under the payoff-sampling
dynamics.

###### Sketch of Proof.

When $l$ and $g$ are sufficiently large, the payoff of action $h$ is slightly
below twice the number of $d$-s in the $h$-sample, and the payoff of action
$d$ is slightly above the number of $d$-s in the $d$-sample. This implies that
action $h$ has a higher mean payoff than action $d$ iff the number of $d$-s in
the $h$-sample is strictly greater than half the number of $d$-s in the
$d$-sample.

Figure 6.2: The Function $w_{k}(p)$ for Various Values of $k$

Thus, we can write $w_{k}(p)$ as follows:

$w_{k}(p)=P\left(\underbrace{k-X(k,p)}_{\\#d_{j}\,\textrm{in}\,h\textrm{{-}sample}}>\frac{1}{2}\underbrace{(k-Y(k,p))}_{\\#d_{j}\,\textrm{in}\,d\textrm{-sample}}\right)=P(2X(k,p)-Y(k,p)<k),$
(6.1)

where $X(k,p)$ and $Y(k,p)$ are iid binomial random variables with parameters
$k$ and $p.$

In the formal proof (see Appendix A.9), we show that for any $k<20,$
$w_{k}(p)$ has a unique fixed point $p^{(k)}$ such that
$|w_{k}^{\prime}(p^{(k)})|<1$ (see Figure 6.2). This implies, by the same
argument as in the sketch of proof of Theorem 2, that the symmetric stationary
state is asymptotically stable. (By contrast, one can verify that
$|w_{k}^{\prime}(p^{k})|>1$ for $k\geq 20,$ which implies that the symmetric
stationary state is unstable for large $k\geq 20$.)

Next, we verify in the formal proof that for any
$k\in\left\\{1,2,3,4,5\right\\}$ it holds that (I) the fixed points are all in
the interval $(0.5,0.68)$, and (II) $|w_{k}^{\prime}(p)|<1$ for any
$k\in\left\\{1,...,5\right\\}$ and any $p\in(0.5,0.68).$ Let $\theta$ be any
distribution with $\max(\textrm{supp}(\theta))\leq 5$. The fact that
$w_{\theta}(p)$ is a weighted average of the various $w_{k}(p)$ implies that
(I) the fixed point $p^{(\theta)}$ of $w_{\theta}(p)$ is in $(0.5,0.68)$, and
(II) $|w_{\theta}^{\prime}(p^{(\theta)})|<1$ $\Rightarrow$
$(p^{(\theta)},p^{(\theta)})$ is asymptotically stable. ∎

##  7 Numeric Analysis

We present numeric results that complement the analytic results of the
previous sections.

#### Methodology and Parameter Values

The analysis focuses on the standard hawk–dove games, in which the gain of a
hawkish player is equal to the loss of her dovish opponent, i.e., $g=l$ for
each $i\in\left\\{1,2\right\\}$. We have tested the following $360=10\times
27$ combinations of parameter values for each of the two sampling dynamics:

  1. 1.

10 values for $g$: 0.05, 0.15, 0.25, 0.35, 0.45, 0.55, 0.65, 0.75, 0.85, 0.95.

  2. 2.

36 distributions of sample sizes:

    1. (a)

9 _homogeneous populations_ , in which all agents have sample size $k$, for
each $k\in\\{2,3,4,5,7,10,15,20,30\\}$ (the case of $k=1$ is discussed in
Footnote 15).

    2. (b)

9 _uniform distributions_ over $\left\\{1,..,\overline{k}\right\\}$, for each
$\overline{k}\in\\{2,3,4,5,7,10,15,20,30\\}$.

    3. (c)

9 distributions with support $\\{1,5\\}$, in which a share
$q\in\left\\{10\%,20\%,...,90\%\right\\}$ have sample size 1, and the
remaining agents have sample size 5.

    4. (d)

9 distributions with support $\\{2,30\\}$, in which a share
$q\in\left\\{10\%,20\%,...,90\%\right\\}$ have sample size 2, and the
remaining agents have sample size 30.

For each set of parameters, we have numerically calculated the phase portrait
and the curves for which $\dot{p}{}_{1}=0$ and $\dot{p}{}_{2}=0$, and used
this to determine the dynamic behavior. The code is provided in the online
supplementary material.161616 Our numeric analysis is based on deterministic
dynamics in a continuum population. We have randomly chosen 10 of these
combinations of parameter values, and tested each of them by running it 100
times in the stochastic dynamics induced by a finite population of 1,000
agents, using ABED software (Izquierdo et al., 2019). The results for finite
populations are qualitatively the same.

Figure 7.1: Summary of Results of the Numeric Analysis

#### Results

The numeric results are summarized in Figure 7.1. The _action-sampling
dynamics_ yield global convergence to the pure stationary states (orange
shaded region in Figure 7.1) in $90\%$ of the cases. In the remaining $10\%$
of the cases, the dynamics globally (or almost globally) converge to the
symmetric stationary state (green shaded region), where these cases involve
heterogeneous populations (some agents have small samples, and others have
relatively large samples) and values of $g$ that are far from 0.5.

The _payoff-sampling dynamics_ yield global convergence to the pure stationary
states in about $50\%$ of the cases. In about $40\%$ of the cases the payoff-
sampling dynamics globally (or almost globally) converge to the symmetric
stationary state, where this occurs (1) in all cases in which the action-
sampling dynamics converge to the symmetric stationary state, and (2) in many
cases with large values of $g$. Finally, in the remaining $10\%$ of the cases
the dynamics either converge to asymmetric mixed states (blue shaded region in
Figure 7.1), or that both the pure stationary states and the symmetric
stationary state have sizable basins of attractions (yellow shaded region).

##  8 Conclusion

A key result in evolutionary game theory is that two populations that are
matched to play a hawk–dove game converge to one of the pure equilibria from
almost any initial state. We demonstrate that this result crucially depends on
the revision dynamics being monotone. Specifically, we show that two plausible
classes of dynamics, in which some agents base their chosen actions on
sampling the actions of a few agents in the opponent population (action-
sampling dynamics) or on sampling the payoffs of a few agents in their own
population (payoff-sampling dynamics) can lead to the opposite prediction:
convergence to a symmetric mixed stationary state.

Our results provide a new explanation of why in bargaining situations, such as
the motivating example of buying and selling houses, players in both
populations tend to play hawkish strategies, and bargaining frequently fails.
Our model assumes that all players in each population have the same payoff
matrix. Heterogeneity in the payoffs, and private information regarding one’s
payoff, are important aspects of many real-life bargaining situations. An
interesting direction for future research is to apply the analysis of sampling
dynamics in richer models that incorporate heterogeneous payoffs.

##  Appendix A Appendix

###  A.1 Results for Binomial Random Variables

In this section, we state and prove results for binomial random variables
(result that to the best of our knowledge, do not appear in the existing
literature). These results are used in the proof of Theorem 2, and may also be
of independent interest. Our first result shows that if the success
probability $p$ in a single trial coincides with the probability of succeeding
at most $m$ times out of $k$ trials, then the derivative of the latter
probability with respect to $p$ must be larger than 1. This is formalized as
follows. (Recall that $X(k,p)$ denotes a binomial random variable with $k\geq
1$ trials and probability of success $p$ in each trial.)

######  Proposition 6.

For any $k\geq 2$ and $0\leq m<k$, there is a unique value $r_{m}\in(0,1)$ of
$p$ for which $\Pr\left(X(k,p)\leq m\right)=p.$ Moreover, at $p=r_{m}$,

$\left|\frac{d\Pr\left(X(k,p)\leq m\right)}{dp}\right|_{p=r_{m}}>1.$

###### Proof.

Fix some $k\geq 2$, and let

$f_{m}(p)\equiv P(X(k,p)\leq m)=\sum_{i=0}^{m}\binom{k}{i}p^{i}(1-p)^{k-i}.$
(A.1)

We have

$\displaystyle f_{m}(p)+f_{k-m-1}(1-p)$ $\displaystyle=P(X(k,p)\leq
m)+P(X(k,1-p)\leq k-m-1)$ $\displaystyle=P(X(k,p)\leq m)+P(X(k,p)\geq
k-(k-m-1))$ $\displaystyle=P(X(k,p)\leq m)+P(X(k,p)\geq m+1)=1.$

Let $r_{m}$ denote the fixed point of the function $f_{m}(\cdot),$ i.e.,
$f_{m}(r_{m})=r_{m}.$ The fact that $f_{k-m-1}(1-p)=1-f_{m}(p)$ implies that
$r_{k-m-1}=1-r_{m}$ and $f_{k-m-1}^{\prime}(r_{k-m-1})=f_{m}^{\prime}(r_{m}).$
Without loss of generality, we can therefore focus on analyzing the cases of
$m$ for which $m\leq\lfloor\frac{k-1}{2}\rfloor.$ To complete the proof, we
need to show that $|f_{m}^{\prime}(r_{m})|>1$ for
$m\in\\{0,1,\dots,\lfloor\frac{k-1}{2}\rfloor\\}.$ In what follows, we show
this.

We compute as follows:

$\displaystyle f_{m}^{\prime}(p)$
$\displaystyle=-\binom{k}{m}(k-m)p^{m}(1-p)^{k-m-1}=-k\binom{k-1}{m}p^{m}(1-p)^{k-m-1}$
(A.2) $\displaystyle f_{m}^{\prime\prime}(p)$
$\displaystyle=\binom{k}{m}(k-m)p^{m-1}(1-p)^{k-m-2}((k-m-1)p-m(1-p)).$

From the above computations, it follows that the function $f_{m}(\cdot)$ is
concave for values of $p\leq p^{\ast}$ and convex for values of $p\geq
p^{\ast}$ where $p^{\ast}=\frac{m}{k-1}.$ This is because
$f_{m}^{\prime\prime}(p^{\ast})=0.$ Either the concave part or the convex part
of the function $f_{m}(\cdot)$ intersects the $45^{\circ}$ line. Suppose that
the concave part of the function $f_{m}(\cdot)$ intersects the $45^{\circ}$
line from the origin, i.e., $r_{m}\leq p^{\ast}.$ Since $m\leq\frac{k-1}{2},$
it follows that $r_{m}\leq 0.5.$ By concavity, we have

$f_{m}^{\prime}(r_{m})<\frac{f_{m}(0)-f_{m}(r_{m})}{0-r_{m}}=-\left(\frac{1-r_{m}}{r_{m}}\right)\leq-1.$

Therefore, we are done in cases where $r_{m}\leq p^{\ast}.$

We now consider the cases where $r_{m}>p^{*}$. By definition,

$\displaystyle r_{m}$
$\displaystyle=(1-r_{m})^{k}+\binom{k}{1}r_{m}(1-r_{m})^{k-1}+\dots+\binom{k}{m}r_{m}^{m}(1-r_{m})^{k-m}.$
(A.3)

For $j=0,1,2,\dots,m,$ let $a_{j}$ denote the $j^{\text{th}}$ term of the sum
on the RHS of Eq. (A.3). For $j=1,2,\dots,m,$ we compute as follows:

$\displaystyle\frac{a_{j}}{a_{j-1}}=\frac{\binom{k}{j}r_{m}^{j}(1-r_{m})^{k-j}}{\binom{k}{j-1}r_{m}^{j-1}(1-r_{m})^{k-j+1}}=\left(\frac{k-j+1}{j}\right)\left(\frac{r_{m}}{1-r_{m}}\right)$
$\displaystyle\frac{a_{j}}{a_{j-1}}\geq 1\iff(k-j+1)r_{m}\geq j(1-r_{m})\iff
r_{m}\geq\frac{j}{k+1}.$

Since $\frac{m}{k-1}>\frac{j}{k+1}$ for $j=1,2,\dots,m,$ we have
$r\geq\frac{j}{k+1}$ and thus $a_{j}\geq a_{j-1}$. This implies that
$a_{j}\leq a_{m}$ for $j=1,2\dots,m-1.$ By Eq.(A.3), we can thus infer the
following:

$r_{m}\leq(m+1)\binom{k}{m}r_{m}^{m}(1-r_{m})^{k-m}.$ (A.4)

By Eqs. (A.2) and (A.4), we have

$|f_{m}^{\prime}(r_{m})|=(k-m)\binom{k}{m}r_{m}^{m}(1-r_{m})^{k-m-1}\geq\left(\frac{k-m}{m+1}\right)\frac{r_{m}}{1-r_{m}}.$

Using the above set of equations, we can write a sufficient condition for
$|f_{m}^{\prime}(r_{m})|>1$ as follows:

$\left(\frac{k-m}{m+1}\right)\frac{r_{m}}{1-r_{m}}>1\iff
r_{m}>\frac{m+1}{k+1}.$ (A.5)

We will now establish that $|f_{m}^{\prime}\left(\frac{m+1}{k+1}\right)|>1.$
By Eq. (A.2), we have

$\displaystyle|f_{m}^{\prime}(p)|$
$\displaystyle=k\binom{k-1}{m}p^{m}(1-p)^{k-m-1}=k\cdot\Pr(X(k-1,p)=m).$

where $X(k-1,p)$ is a binomial distribution with parameters $k-1$ and $p.$ It
is well known that the binomial distribution’s mode with parameters $k-1$ and
$p$ is attained at $\lfloor kp\rfloor.$ For $p=\frac{m+1}{k+1},$ we have

$m\leq k\cdot\left(\frac{m+1}{k+1}\right)<m+1.$

The above inequalities imply that $\left\lfloor
k\cdot\left(\frac{m+1}{k+1}\right)\right\rfloor=m.$ The binomial distribution
$X_{k-1}(\cdot)$ has $k$ possible values and thus the probability of the
occurrence of the mode has to be greater than $\frac{1}{k},$ i.e.,

$\Pr\left(X\left(k-1,\frac{m+1}{k+1}\right)=m\right)>\frac{1}{k}\implies\left|f_{m}^{\prime}\left(\frac{m+1}{k+1}\right)\right|>1.$

We need to consider the following two possible cases:

Case 1: $\frac{m}{k-1}<r_{m}\leq\frac{m+1}{k+1}$. For $1\leq
m\leq\frac{k-1}{2},$ we know that $|f_{m}^{\prime}(\cdot)|$ attains its
maximum value at $\frac{m}{k-1}$ and that it is strictly decreasing for
$p>\frac{m}{k-1}.$ For $m=0,$ $|f_{m}^{\prime}(\cdot)|$ is strictly decreasing
for $p>0.$ Thus, we have

$|f_{m}^{\prime}(r_{m})|>\left|f_{m}^{\prime}\left(\frac{m+1}{k+1}\right)\right|>1.$

Case 2: $r_{m}>\frac{m+1}{k+1}.$ Here, we are done by the sufficient condition
of Eq. (A.5).

∎

Recall that Proposition 4 states that there are no $p\neq q\in(0,1)$, such
that $\Pr\left(X(k,p)\leq m\right)=q\in\left(0,1\right)$ and
$\Pr\left(X(k,q)\leq m\right)=p.$ By contrast, Theorem 3 implies that this
result does not hold if one replaces the binomial distribution with a mixture
of binomial distributions. Formally:

######  Corollary 4.

For any $k\geq 2$ and any $\alpha<\frac{1}{k}$, there exist $0\leq m<k$,
$0\leq m^{\prime}<k^{\prime}$, and $p\neq q\in(0,1)$, such that
$\alpha\cdot\Pr\left(X(k,p)\leq
m\right)+(1-\alpha)\cdot\Pr\left(X(k^{\prime},p)\leq m^{\prime}\right)=q$
and $\alpha\cdot\Pr\left(X(k,q)\leq
m\right)+(1-\alpha)\cdot\Pr\left(X(k^{\prime},q)\leq m^{\prime}\right)=p.$

We conclude this appendix with a proof of Proposition 4:

###### Proof of Proposition 4.

Fix $k\geq 2$ and $m<k$, and for the same reason as in the proof of
Proposition (6), we can assume without loss of generality that
$m\leq\lfloor\frac{k-1}{2}\rfloor.$ Let $f(p)\equiv\Pr\left(X(k,p)\leq
m\right)$, and $r$ be the value of $p$ for which $f(r)=r$. Let $F(p)\equiv
f(f(p))$. It is easy to see that if $f(p_{1})=p_{2}$ and $f(p_{2})=p_{1}$,
then both $p_{1}$ and $p_{2}$ are fixed points of $F$. Moreover, since $f$ is
decreasing, if $p_{1}<r$, then $p_{2}>r$. Therefore, it is enough to show that
$F$ has no fixed points in $(0,r)$.

We claim first that if $p^{*}\equiv\frac{m}{k-1}\geq r$, then there is no
fixed point $p_{1}<r$ for which $p_{2}\equiv f(p_{1})\leq p^{*}$. In other
words, $F$ has no fixed point in the range $[f^{-1}(p^{*}),r)$. To see this,
assume by contradiction that there are such fixed points $p_{1}$ and $p_{2}$.
Recall that the function $f$ is concave on $(0,p^{*})$, and since $r<0.5$ and
$|f^{\prime}(r)|>1$ (Proposition (6)), the absolute value of the slope of the
line connecting the points $(0,1)$ and $(p_{2},p_{1})$ is larger than one.
Because of the concavity of $f$ on this range, the graph of $f$ is above this
line, and the absolute value of the slope of the line connecting
$(p_{1},p_{2})$ with $(p_{2},p_{1})$ (which is obviously equal to one) must be
even larger of that of the line connecting the points $(0,1)$ and
$(p_{2},p_{1})$, i.e., larger than
$1=\left|\frac{p_{1}-p_{2}}{p_{2}-p_{1}}\right|$, a contradiction.

Now, the first and second derivatives of $F$ are

$\displaystyle F^{\prime}(p)$ $\displaystyle=f^{\prime}(f(p))\cdot
f^{\prime}(p)$ $\displaystyle F^{\prime\prime}(p)$
$\displaystyle=f^{\prime\prime}(f(p))\cdot
f^{\prime}(p)^{2}+f^{\prime}(f(p))\cdot f^{\prime\prime}(p).$

To complete the case of $p^{*}\geq r$ it is left to show that there are no
fixed points of $F$ on (0,$f^{-1}(p^{*})$). Indeed, for all
$p\in(0,f^{-1}(p^{*}))$, $f^{\prime\prime}(f(p))>0$, which implies that
$F^{\prime\prime}(p)>0$ (as $f^{\prime\prime}(p)<0$). Thus, on
$(0,f^{-1}(p^{*}))$, $F(x)$ is convex and since $F(0)=0$, in this range there
could be at most one fixed point of $F(p)$. Now, as was shown before, there
are no fixed points of $F$ in $[f^{-1}(p^{*}),r)$, and by Proposition (6),
$F^{\prime}(r)>1$. This implies that the graph of $F(p)$ must be below the
diagonal for any $p\in[f^{-1}(p^{*}),r)$. This could not happen if $F(p)$ were
above the diagonal at some $p\in(0,f^{-1}(p^{*}))$. We conclude that there is
no such fixed point, and this completes the proof for the case of $p^{*}\geq
r$.

Now assume $p^{*}<r.$ In this case, we will show that $F$ is convex on
$(0,r)$. On the range $(0,p^{*})$, $f^{\prime\prime}(f(p))>0$ and
$f^{\prime\prime}(p)<0$, implying that $F^{\prime\prime}(p)>0$ and therefore
$F$ are convex. On the range $(p^{*},r)$, the condition for convexity is

$f^{\prime\prime}(f(p))\cdot f^{\prime}(p)^{2}+f^{\prime}(f(p))\cdot
f^{\prime\prime}(p)>0,\textrm{ or}$
$\frac{f^{\prime\prime}(f(p))}{-f^{\prime}(f(p))}>\frac{f^{\prime\prime}(p)}{f^{\prime}(p)^{2}}.$

Since $|f^{\prime}(r)|>1$ (Proposition 6), $|f^{\prime}(p)|>1$ for any
$p\in(p^{*},r)$, and therefore it is enough to show that for $p\in(p^{*},r)$,

$\frac{f^{\prime\prime}(f(p))}{-f^{\prime}(f(p))}>\frac{f^{\prime\prime}(p)}{-f^{\prime}(p)}.$

Let $G(p)\equiv\frac{f^{\prime\prime}(p)}{-f^{\prime}(p)}$, and we will show
that for any $p_{1}\in(p^{*},r)$, $G(p)$ increases on $(p_{1},f(p_{1}))$.
Indeed, setting the explicit expressions of Eq. (A.2) in $G$, we have

$G(p)=\frac{(k-1)p-m}{p(1-p)}=\frac{k-1-m}{1-p}-\frac{m}{p},$

and since $m<k-1$, $G$ is increasing.

To summarize: since $F(0)=0$ and $F(r)=r$, when $p^{*}<r$, the convexity of
$F$ on $(0,r)$ that we just showed implies that $F$ does not intersect the
diagonal on $(0,r)$, and this completes the proof.

∎

###  A.2 Standard Definitions of Dynamic Stability

For completeness, we present in this appendix the standard definitions of
dynamic stability that is 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 5.

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 is Lyapunov stable if a population beginning near it remains close,
and it is asymptotically stable if, in addition, it eventually converges 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 6.

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 7.

A stationary state $\mathbf{p}^{*}\in\left[0,1\right]^{2}$ is _asymptotically
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.3 Proof of Proposition 1 (Convergence to Stationary States)

We say that state p is _above_ (resp., _below_) curve $w(p_{1})$ if
$p_{2}>w(p_{1})$ (resp., ($p_{2}<w(p_{1})$). Similarly, we say that state p is
to the _right_ (resp., _left_) of the curve $w(p_{1})$ if
$w^{-1}(p_{2})<p_{1}$ (resp., $w^{-1}(p_{2})<p_{1}$). We say that the state p
is on the curve $w(p_{1})$ if $p_{2}=w(p_{1}).$

The fact that the two curves of $w(p_{1})$ and $w^{-1}(p_{1})$ are strictly
decreasing implies that a state is above one of these curves iff it is to the
right of that curve. The states on the curve $w(p_{1})$ (resp.,
$w^{-1}(p_{1})$) are characterized by having $\dot{p_{1}}=0$ (resp.,
$\dot{p_{2}}=0$). Observe that in any state p above and to the right (resp.,
below and to the left) of the curve $w(p_{1})$, the share of hawks in
population 2 is higher relative to the corresponding state on the curve with
the same share of hawks in population 1, which implies that $\dot{p_{1}}<0$
(resp., $\dot{p_{1}}>0$). Similarly, in any state p above (resp., below) the
curve $w^{-1}(p_{1})$, the share of hawks in population 1 is higher relative
to the corresponding state on the curve with same share of hawks in population
2, which implies that $\dot{p_{1}}<0$ (resp., $\dot{p_{1}}>0$).

Any state $\textbf{p}\in[0,1]$ can be classified in one of 3*3 classes,
depending on its relative location with respect to the two curves, i.e.,
whether p is below (i.e., to the left of), above (i.e., to the right of) or on
the curve $w(p_{1})$, and similarly whether p is below, above, or on the curve
$w^{-1}(p_{1})$. The above argument implies that this classification
determines the signs of $\dot{p_{1}}$ and $\dot{p_{2}}$. If state p is on
(resp., above, below) the curve $w(p_{1})$, then $\dot{p_{2}}$ is zero (resp.,
negative, positive). Similarly, if state p is on (resp., above, below) the
curve $w^{-1}(p_{1})$, then $\dot{p_{1}}$ is zero (resp., negative, positive).

In particular, any state p that is above (resp., below) both curves must
satisfy that $\dot{p_{1}},\dot{p_{2}}$ are both negative (resp., positive).
This implies that any trajectory that begins above (resp., below) both curves
(namely, $w(p_{1})$ and $w^{-1}(p_{1})$) must always move downward and to the
left. This (together with the fact that all stationary states are in the
intersection of the two curves) implies that the trajectory must cross one of
the curves. If this crossing point is an intersection of both curves, then it
is a stationary state.

Next observe that any trajectory that starts on one of the curves and strictly
above/below the remaining curve must move 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 $\dot{p_{i}}$ is zero and the remaining
derivative $\dot{p_{j}}$ is negative (resp., positive), which implies that the
trajectory must move below and to the left (resp., above and to the right) of
the curve that was crossed.

Thus, we have established that any trajectory (which has not converged to a
stationary state) must reach a state that is strictly below one of the curves
and strictly above the remaining curve. If such a state is strictly above the
curve $w(p_{1})$, then by the classification mentioned above the trajectory
must move both downward and to the right; i.e., $\dot{p_{1}},\dot{p_{2}}<0$
(resp., both upward 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 $w(p_{1})$ (resp., $w^{-1}(p_{1})$), because at
such a point the trajectory moves horizontally to the left (vertically
downward), which implies that it must cross the lower (resp., higher) curve
$w(p_{1})$ (resp.,$w^{-1}(p_{1})$) from the left side (resp., from above) and
we get a contradiction. The argument in case the state is strictly below the
curve $w(p_{1})$ is analogous.

###  A.4 Proof of Proposition 2 (Convergence to Pure Equilibria)

If p is below (resp., above) both curves, then by the classification presented
above it must that $\dot{p_{1}}>0$ (resp., $\dot{p_{2}}<2$), which implies (by
a similar argument to the proof of Proposition 1) that convergence to (1,0) 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 (1,0). By the classification presented
above it must be that the curve of $w^{-1}(p_{1})$ is strictly below the curve
of $w(p_{1})$ in a right neighborhood of (1,0), which implies that (1,0) is
asymptotically stable because any sufficiently close initial state would
converge to (1,0)

###  A.5 Proof of Lemma 1 (Maximal Sample Sizes)

  1. 1.

(I) The sum of payoffs of action $h$ (resp., $d$) against a sample with a
single $d$ is $1+g$ (resp., $1+\left(k-1\right)(1-l)$). The mean payoff of $h$
is strictly higher than the mean payoff of $d$ iff $k<\frac{1+g-l}{1-l}.$ (II)
The sum of payoffs of action $d$ (resp., $h$) against a sample with a single
$h$ is $k-1+1-l$ (resp., $\left(k-1\right)\left(1+g\right)$). The mean payoff
of $h$ is higher than the mean payoff of $d$ iff $k<\frac{1+g-l}{g}$.

  2. 2.

(I) The sum of payoffs of an $h$-sample with a single $d$ is equal to $1+g$.
The sum of payoffs of a $d$-sample with no $d$ is equal to $k\cdot l$. The
former sum is higher than the latter iff $k<\frac{1+g}{1-l}.$ (II) The sum of
payoffs of a $d$-sample with no $h_{j}$-s is equal to $k$. The sum of payoffs
of an $h$-sample with a single $h_{j}$ is equal to $(k-1)\left(1+g\right)$.
The former sum is higher than the latter iff $k<\frac{1+g}{g}$.

###  A.6 Proof of Proposition 3 (Stability of Pure Equilibria)

We are interested in deriving conditions for the stability of the pure
stationary states. In what follows, we compute the Jacobian of the sampling
dynamics for the state in which all agents of population $i$ play $d$ and all
agents of population $j$ play $h$. For this, we consider a slightly perturbed
state with a “very small” $\epsilon_{i}$ share of hawks in population $i$ and
a “very small” $\epsilon_{j}$ share of doves in population $j.$ By “very
small,” we mean that higher-order terms of $\epsilon_{i}$ and $\epsilon_{j}$
are neglected.

Consider a new agent of population $i$ with a sample size of $k_{i}.$ Action
$h$ has a higher mean payoff against a sample size of $k_{i}$ iff (neglecting
rare events of having multiple $d$-s in the sample): (1) the sample includes
the single action $d$ of an opponent, and (2) $k_{i}\leq m_{h}$ (due to Lemma
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 of population $i$ (with a random
sample size distributed according to $\theta$) has a higher mean payoff for
action $h$ against her sample is
$w_{\theta}(1-\epsilon_{j})=\epsilon_{j}\cdot\sum_{k_{i}=1}^{m_{h}}\theta(k_{i})\cdot
k_{i}+o(\epsilon_{j})$. An analogous argument implies that the probability
that a new agent of population $j$ has a higher mean payoff for action $d_{j}$
against her sample is
$w_{\theta_{j}}(\epsilon_{i})=\epsilon_{i}\cdot\sum_{k_{j}=1}^{m_{d}}\theta_{j}(k_{j})\cdot
k_{j}+o(\epsilon_{i})$. Therefore, the sampling dynamics at
$(\epsilon_{i},1-\epsilon_{j})$ can be written as follows (ignoring the
higher-order terms of $\epsilon_{i}$ and $\epsilon_{j}$):

$\dot{\epsilon}_{i}=\epsilon_{j}\cdot\sum_{k_{i}=1}^{m_{h}}\theta(k_{i})\cdot
k_{i}-\epsilon_{i},\,\,\,\,\,\,\,\,\,\,\,\,\dot{\epsilon}_{j}=\epsilon_{i}\cdot\sum_{k_{j}=1}^{m_{d}}\theta_{j}(k_{j})\cdot
k_{j}-\epsilon_{j}.$ (A.6)

Define: $a_{\theta}=\sum_{k_{i}=1}^{m_{h}}\theta(k_{i})\cdot k_{i}$ and
$b_{\theta}=\sum_{k_{j}=1}^{m_{d}}\theta_{j}(k_{j})\cdot k_{j}$. The Jacobian
of the above system of Equations (A.6) is then given by
$J=\left(\begin{array}[]{cc}-1&a_{\theta}\\\ b_{\theta}&-1\end{array}\right).$
The eigenvalues of $J$ are $-1-\sqrt{a_{\theta}b_{\theta}}$ and
$-1+\sqrt{a_{\theta}b_{\theta}}.$ Observe that: (1) if
$a_{\theta}b_{\theta}<1$ then both eigenvalues are negative, which implies
that the state in which all agents in population $i$ (resp., $j$) are doves
(resp., hawks) is asymptotically stable, and (2) if $a_{\theta}b_{\theta}>1$
then one of the eigenvalues is positive, which implies that this state is
unstable (see, e.g., Perko, 2013, Theorems 1–2 in Section 2.9).

###  A.7 Proof of Corollary 3 (Standard Case of $g=l$)

Substituting $g=l$ in the bounds presented in Definition 3 yields the
following bounds:
$\,m_{h}^{A}=\left\lfloor\frac{1}{1-g}\right\rfloor,\,m_{d}^{A}=\left\lfloor\frac{1}{g}\right\rfloor,m_{h}^{P}=\left\lfloor\frac{1+g}{1-g}\right\rfloor,\,m_{d}^{P}=\left\lfloor\frac{1+g}{g}\right\rfloor.$
Observe that $\min(m_{h}^{A},m_{d}^{A})=\min(\frac{1}{1-g},\frac{1}{g})=1$.
This implies that $\mathbb{E}_{\leq
m_{h}^{A}}\left(\theta\right)\cdot\mathbb{E}_{\leq
m_{d}^{A}}\left(\theta\right)=\theta(1)\cdot\mathbb{E}_{\leq\max\left(\frac{1}{g},\frac{1}{1-g}\right)}\left(\theta\right),$
which proves part 1. Next observe that if $g<\frac{1}{3}$, then
$\min(m_{h}^{P},m_{d}^{P})=\min(\frac{1+g}{1-g},\frac{1+g}{g})=1$ and
$\max(m_{h}^{P},m_{d}^{P})=\max(\frac{1+g}{1-g},\frac{1+g}{g})=\frac{1+g}{g}$,
which proves part (2-a). Finally, observe that if $g\geq\frac{1}{3}$, then
$\min(m_{h}^{P},m_{d}^{P})=\min(\frac{1+g}{1-g},\frac{1+g}{g})=2$ and
$\max(m_{h}^{P},m_{d}^{P})=\max(\frac{1+g}{1-g},\frac{1+g}{g})=\max(3,\frac{1+g}{1-g})$,
which proves part (2-b).

###  A.8 Proof of Theorem 3 (Stable Symmetric State, ASD)

The following notation will be helpful. For each $k>1$ and $q\in(0,1)$, let
$\underline{p}_{k}^{q}$ (resp., $\overline{p}_{k}^{q}$) be the unique solution
in $(0,1)$ to the equation $p=q\cdot(1-p)^{k}$ (resp., $p=1-q\cdot p^{k}$). We
interpret $\underline{p}_{k}^{q}$ (resp., $\overline{p}_{k}^{q}$) as the
symmetric stationary state when $g$ and $l$ are close to zero (resp., close to
one) in a population in which a share $q$ of the population have samples of
size $k$, while the remaining agents always play dove (resp., hawk).

Let $\overline{g}\in(0,1)$ (resp., $\overline{l}\in(0,1)$) be sufficiently
small (resp., large) such that

$(k-1)\overline{g}<1-l\textrm{ and
}\frac{\overline{g}}{1+\overline{g}-l}<\underline{p}_{k}^{q}$
$(\textrm{resp.,}(k-1)(1-\overline{l})<g\textrm{ and
}\frac{g}{1+g-\overline{l}}>\overline{p}_{k}^{q}).$

Fix any $0<g<\overline{g}$ (resp., any $\overline{l}<l<1$). The definition of
$\underline{p}_{k}^{q}$ (resp., $\overline{p}_{k}^{q}$) implies that the
symmetric stationary state $(p_{\theta},p_{\theta})$ in any population with
$\theta(k)=q$ must satisfy $p_{\theta}>\underline{p}_{k}^{q}$ (resp.,
$p_{\theta}<\overline{p}_{k}^{q}$). Observe that the best reply to the true
distribution of hawks in the population in state $(p_{\theta},p_{\theta})$ is
to play $d$ (resp., $h$) because $\frac{g}{1+g-l}<p_{\theta}$ (resp., because
$\frac{g}{1+g-l}>p_{\theta}$). The central limit theorem implies that new
agents with sufficiently large samples almost always play $d$ (resp., $h$) in
a neighborhood of $p_{\theta}.$ This implies that for each $\epsilon>0,$ there
exists $\overline{k}$ such that $|w^{\prime}_{n}(p_{\theta})|<\epsilon$ for
any $n\geq\overline{k}$. Let $\epsilon>0$ be sufficiently small such that
$q\cdot k<1-\epsilon$. This implies that:

$|w^{\prime}_{\theta}(p_{\theta})|<(1-q)\cdot\epsilon+q\cdot|w^{\prime}_{k}(p_{\theta})|<\epsilon+q\cdot|((1-p)^{k})^{\prime}|=\epsilon+q\cdot|k\cdot(1-p)^{k-1}|<\epsilon+q\cdot
k<1.$
$\left(|w^{\prime}_{\theta}(p)|=<(1-q)\cdot\epsilon+q\cdot|w^{\prime}_{k}(p_{\theta})|<\epsilon+q\cdot|(1-p^{k})^{\prime}|=\epsilon+q\cdot|k\cdot{p}^{k-1}|<\epsilon+q\cdot
k<1\right.)$

This implies that $(p_{\theta},p_{\theta})$ is asymptotically stable due to
Proposition 3.

###  A.9 Proof of Theorem 4 (Stable Symmetric State, PSD)

Recall that the payoff-sampling dynamics in state $(p_{1},p_{2})$ are given by
$\dot{p}_{1}=\delta(w_{\theta}(p_{2})-p_{1}$ and
$\dot{p}_{2}=\delta(w_{\theta}(p_{1})-p_{2}),$ and that a symmetric state
$(p^{(\theta)},p^{(\theta)})$ is stationary iff $p^{(\theta)}$
$w_{\theta}(p^{(\theta)})=p^{(\theta)}$.

We now establish some properties of the payoff-sampling dynamics and the
symmetric rest points for symmetric distributions of types $\theta\equiv k$.
If $l,1-g\in\left(0,\frac{1}{\max(\textrm{supp}(\theta))}\right),$ action
$h_{i}$ has a higher mean payoff iff the number of $d_{j}$-s in the
$h_{i}$-sample is strictly greater than half the number of $d_{j}$-s in the
$d_{i}$-sample. To express $w_{k}(p)$ concisely in this case, we define
$X(k,p)$ and $Y(k,p)$ to be independent and identically distributed binomial
random variables with parameters $k$ and $p.$ We can then write $w_{k}(p)$ as
follows:

$w_{k}(p)=P\left(k-X(k,p)>\frac{1}{2}(k-Y(k,p))\right)=P(2X(k,p)-Y(k,p)<k).$
(A.7)

Observe that $w_{k}(p)$ is a polynomial in $p$ of degree at most $2\cdot k.$
We have verified the following facts about these polynomials for $k<20$ (for
an illustration see Figure 6.2; the Mathematica code is given in the online
supplementary material, and the explicit values of the rest points and the
derivatives are presented in Table 3):

  * •

For $k\in\left\\{1,2,\dots,18,19\right\\},$ $w_{k}(p)$ is decreasing in $p.$

  * •

For $k\in\left\\{1,2,\dots,18,19\right\\},$ $w_{k}(p)$ has a unique fixed
point $p^{(k)}$.
Moreover, $0.5<p^{(k)}<0.68$ for any $k\in\left\\{1,2,3,4,5\right\\}.$

  * •

$|w_{1}^{\prime}(p)|\equiv 1,$ and $|w_{k}^{\prime}(p)|<1$ for any
$k\in\left\\{2,3,4,5\right\\}$ and $0.5<p<0.68.$

Recall that $w_{\theta}(p)$ is a convex combination of the $w_{k}(p)$ for the
$k$-s in its support (i.e., $w_{\theta}(p)=\sum_{k}\theta\left(k\right)\cdot
w_{k}\left(p\right)$). From the above facts, it follows that:

  1. 1.

For $\theta\equiv k<20,$ the function $w_{k}(p)$ has a unique fixed point
$p^{(k)}$ such that $|w^{\prime}_{k}(p^{(k)})|<1,$ which implies that
$(p^{(k)},p^{(k)})$ is asymptotically stable.

  2. 2.

For $\max(\textrm{supp}(\theta))\leq 5,$ the function $w_{\theta}(p)$ has a
unique fixed point $p^{(\theta)}$ such that $p^{(\theta)}\in(0.5,0.68)$ and
$|w^{\prime}_{\theta}(p^{(\theta)})|<1$ if $\theta(1)\neq 1,$ which implies
that $(p^{(\theta)},p^{(\theta)})$ is asymptotically stable.

Table 3: Fixed Points of the Function $w_{k}(p)$ in the Proof of Theorem 4

$k$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
---|---|---|---|---|---|---|---|---|---|---
$p^{\left(k\right)}$ | 0.500 | 0.579 | 0.620 | 0.649 | 0.672 | 0.690 | 0.706 | 0.720 | 0.731 | 0.741
$|w^{\prime}_{k}(p^{\left(k\right)})|$ | 1 | 0.690 | 0.618 | 0.645 | 0.690 | 0.730 | 0.763 | 0.793 | 0.818 | 0.840

$k$ | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20
---|---|---|---|---|---|---|---|---|---|---
$p^{\left(k\right)}$ | 0.750 | 0.758 | 0.765 | 0.773 | 0.778 | 0.784 | 0.789 | 0.794 | 0.799 | 0.803
$|w^{\prime}_{k}(p^{\left(k\right)})|$ | 0.861 | 0.88 | 0.899 | 0.916 | 0.932 | 0.948 | 0.963 | 0.978 | 0.991 | 1.001

Table 4: Values of $|w_{k}^{\prime}(p^{(j)})|$ for $k,j\in\\{1,2,3,4,5\\}.$ ${}_{k}\backslash^{p^{(j)}}$ |  $p^{(1)}$ |  $p^{(2)}$ |  $p^{(3)}$ |  $p^{(4)}$ |  $p^{(5)}$
---|---|---|---|---|---
1 |  1 |  1 |  1 |  1 |  1
2 |  0.5 |  0.690 |  0.812 |  0.905 |  0.981
3 |  0.562 |  0.560 |  0.618 |  0.687 |  0.759
4 |  0.625 |  0.616 |  0.623 |  0.645 |  0.679
5 |  0.605 |  0.642 |  0.659 |  0.673 |  0.690

## 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.
  * Aumann (1987) Aumann, R. J. (1987).  Correlated equilibrium as an expression of Bayesian rationality.  Econometrica 55(1), 1–18.
  * Baliga and Sjöström (2012) Baliga, S. and T. Sjöström (2012).  The strategy of manipulating conflict.  American Economic Review 102(6), 2897–2922.
  * Baliga and Sjöström (2020) Baliga, S. and T. Sjöström (2020).  The strategy and technology of conflict.  Journal of Political Economy 128(8), 3186–3219.
  * Benndorf et al. (2016) Benndorf, V., I. Martinez-Martinez, and H.-T. Normann (2016).  Equilibrium selection with coupled populations in hawk–dove games: Theory and experiment in continuous time.  Journal of Economic Theory 165, 472–486.
  * Benndorf et al. (2021) Benndorf, V., I. Martínez-Martínez, and H.-T. Normann (2021).  Games with coupled populations: An experiment in continuous time.  Journal of Economic Theory, 105281.
  * Bilancini et al. (2021) Bilancini, E., L. Boncinelli, S. Ille, and E. Vicario (2021).  Memory length and harshness of conflict in the hawk–dove game.  Mimeo.
  * Bornstein et al. (1997) Bornstein, G., D. Budescu, and S. Zamir (1997).  Cooperation in intergroup, $n$-person, and two-person games of chicken.  Journal of Conflict Resolution 41(3), 384–406.
  * Brams and Kilgour (1987) Brams, S. J. and D. M. Kilgour (1987).  Threat escalation and crisis stability: A game-theoretic analysis.  American Political Science Review 81(3), 833–850.
  * Brams and Kilgour (2001) Brams, S. J. and D. M. Kilgour (2001).  Fallback bargaining.  Group Decision and Negotiation 10(4), 287–316.
  * 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.
  * Cason et al. (2014) Cason, T. N., D. Friedman, and E. Hopkins (2014).  Cycles and instability in a rock–paper–scissors population game: A continuous time experiment.  Review of Economic Studies 81(1), 112–136.
  * Dixit et al. (2019) Dixit, A., D. McAdams, and S. Skeath (2019).  “We haven’t got but one more day”: The Cuban missile crisis as a dynamic chicken game.  Available at SSRN 3406265.
  * 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 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.
  * Hoffman et al. (2015) Hoffman, M., S. Suetens, U. Gneezy, and M. A. Nowak (2015).  An experimental investigation of evolutionary dynamics in the rock–paper–scissors game.  Scientific Reports 5(1), 1–7.
  * Izquierdo et al. (2019) Izquierdo, L. R., S. S. Izquierdo, and W. H. Sandholm (2019).  An introduction to ABED: Agent-based simulation of evolutionary game dynamics.  Games and Economic Behavior 118, 434–462.
  * 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.
  * Lipnowski and Maital (1983) Lipnowski, I. and S. Maital (1983).  Voluntary provision of a pure public good as the game of “chicken”.  Journal of Public Economics 20(3), 381–386.
  * 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 Parker (1976) Maynard-Smith, J. and G. A. Parker (1976).  The logic of asymmetric contests.  Animal Behaviour 24(1), 159–175.
  * 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.  Berlin: Springer Science & Business Media.
  * Rapoport and Chammah (1966) Rapoport, A. and A. M. Chammah (1966).  The game of chicken.  American Behavioral Scientist 10(3), 10–28.
  * Salant and Cherry (2020) Salant, Y. and J. Cherry (2020).  Statistical inference in games.  Econometrica 88(4), 1725–52.
  * 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.  Cambridge, MA: 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), 1347–85.
  * 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 (2021) Sawa, R. and J. Wu (2021).  Statistical inference in evolutionary dynamics.  SSRN 3767635.
  * 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.
  * Spiegler (2006) Spiegler, R. (2006).  Competition over agents with boundedly rational expectations.  Theoretical Economics 1(2), 207–231.
  * Stephenson (2019) Stephenson, D. (2019).  Coordination and evolutionary dynamics: When are evolutionary models reliable?  Games and Economic Behavior 113, 381–395.
  * Sugden (1989) Sugden, R. (1989).  Spontaneous order.  Journal of Economic Perspectives 3(4), 85–97.
  * Taylor (1979) Taylor, P. D. (1979).  Evolutionarily stable strategies with two types of player.  Journal of Applied Probability 16, 76–83.
  * Weibull (1997) Weibull, J. W. (1997).  Evolutionary Game Theory.  Cambridge, MA:MIT press.
  * Young (1993) Young, H. P. (1993).  The evolution of conventions.  Econometrica 61(1), 57–84.