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Entropic Risk-Sensitive Evolutionary Learning and Equilibrium Selection in Coordination Games
Solaleh Mohammadi, Xiang Gao, Kaiqing Zhang
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 95%
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Summary
This paper investigates risk-sensitive evolutionary learning dynamics in coordination games, utilizing the entropic risk measure (ERM) to model agent risk attitudes. It analyzes equilibrium selection in single-population symmetric and two-population asymmetric settings under best response with mutations (BRM) and logit choice protocols. Key findings indicate that risk-seeking agents favor payoff-dominant equilibria, while risk-averse agents favor maximin equilibria, effectively using risk attitude as a control knob for long-run outcomes. Super-dominant equilibria remain stochastically stable regardless of risk attitude. The analysis extends to k-action games, showing that sufficiently risk-seeking/averse agents uniquely select strongly payoff-dominant/maximin equilibria in large populations.
Entities (13)
Relation Signals (8)
Risk-Seeking → favors → Payoff-Dominant Equilibrium
confidence 96% · a greater risk-seeking attitude favors the payoff-dominant equilibrium
Risk-Averse → favors → Maximin Equilibrium
confidence 96% · a greater risk-averse attitude favors the maximin equilibrium.
Entropic Risk Measure → models → Risk Attitude
confidence 95% · Agents' risk attitudes enter through the classical entropic risk measure
Risk Sensitivity → steers → Equilibrium selection
confidence 95% · entropic risk sensitivity may serve as a systematic mechanism for steering equilibrium selection
Super-Dominant Equilibrium → isstablyselectedby → All Risk Attitudes
confidence 94% · any super-dominant equilibrium is stochastically stable for all risk attitudes
Risk-Neutral → favors → Risk-Dominant Equilibrium
confidence 93% · unlike the risk-neutral case where the dynamics are known to favor the risk-dominant equilibrium
Logit Choice → isusedin → Coordination Games
confidence 92% · noisy best responses under two standard revision protocols: best response with mutations and logit choice.
Best Response with Mutations → isusedin → Coordination Games
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Abstract
Abstract:We study risk-sensitive evolutionary learning dynamics and their long-run equilibrium selection behaviors in coordination games. Agents' risk attitudes enter through the classical entropic risk measure, which evaluates opponent-induced payoff uncertainty and feeds into noisy best responses under two standard revision protocols: best response with mutations and logit choice. We first analyze $2\times 2$ coordination games in both single-population symmetric and two-population asymmetric settings. In the single-population setting, unlike the risk-neutral case where the dynamics are known to favor the risk-dominant equilibrium, we show that risk sensitivity can change the stochastically stable outcome: a greater risk-seeking attitude favors the payoff-dominant equilibrium, while a greater risk-averse attitude favors the maximin equilibrium. Thus, the population's risk attitude may act as a control knob for long-run equilibrium selection. In both population settings, we also identify a robust regime: any super-dominant equilibrium is stochastically stable for all risk attitudes, under both protocols, and across populations. We further extend the single-population analysis to symmetric $k$-action games, which include symmetric $k$-action coordination games as a special case, under risk-sensitive best response with mutations. In this setting, we show that, for sufficiently large populations, sufficiently risk-seeking agents uniquely select the strongly payoff-dominant equilibrium when it exists, whereas sufficiently risk-averse agents uniquely select the strongly maximin equilibrium when it exists. These results show that entropic risk sensitivity may serve as a systematic mechanism for steering equilibrium selection in evolutionary games, beyond the classical risk-neutral benchmark.
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- Source: https://arxiv.org/abs/2609.08677v1
- Canonical: https://arxiv.org/abs/2609.08677v1
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Entropic Risk-Sensitive Evolutionary Learning and Equilibrium Selection in Coordination Games Solaleh Mohammadi Email: solalehm@umd.edu Affiliation: University of Maryland, College Park Xiang Gao Email: xgao242424@gmail.com Affiliation: University of Illinois Urbana-Champaign Kaiqing Zhang Email: kaiqing@umd.edu Affiliation: University of Maryland, College Park Abstract We study risk-sensitive evolutionary learning dynamics and their long-run equilibrium selection behaviors in coordination games. Agents’ risk attitudes enter through the classical entropic risk measure, which evaluates opponent-induced payoff uncertainty and feeds into noisy best responses under two standard revision protocols: best response with mutations and logit choice. We first analyze 2×22× 2 coordination games in both single-population symmetric and two-population asymmetric settings. In the single-population setting, unlike the risk-neutral case where the dynamics are known to favor the risk-dominant equilibrium, we show that risk sensitivity can change the stochastically stable outcome: a greater risk-seeking attitude favors the payoff-dominant equilibrium, while a greater risk-averse attitude favors the maximin equilibrium. Thus, the population’s risk attitude may act as a control knob for long-run equilibrium selection. In both population settings, we also identify a robust regime: any super-dominant equilibrium is stochastically stable for all risk attitudes, under both protocols, and across populations. We further extend the single-population analysis to symmetric k-action games, which include symmetric k-action coordination games as a special case, under risk-sensitive best response with mutations. In this setting, we show that, for sufficiently large populations, sufficiently risk-seeking agents uniquely select the strongly payoff-dominant equilibrium when it exists, whereas sufficiently risk-averse agents uniquely select the strongly maximin equilibrium when it exists. These results show that entropic risk sensitivity may serve as a systematic mechanism for steering equilibrium selection in evolutionary games, beyond the classical risk-neutral benchmark. keywordsRisk Sensitivity, Evolutionary Dynamics, Coordination Games, Equilibrium Selection 1 Introduction In game theory, strategic interactions and learning processes among agents often come with substantial uncertainty, driven by opponents’ behavior, environmental randomness, and/or noise in payoffs and decision-making. For this reason, earlier literature argued that it is unrealistic to assume agents are always risk-neutral against such uncertainties, and behave as expectation maximizers Goeree et al. (2003); Fiat and Papadimitriou (2010). Instead, agents may be risk-averse or risk-seeking, and such attitudes are commonly modeled using risk measures that go beyond expected payoffs. More recently, risk sensitivity has also been incorporated in game theory for more tractable equilibrium computation Mazumdar et al. (2025) and for promoting cooperation in multi-agent systems Noorani and Baras (2022); Zhang et al. (2025). In particular, these works studied games whose payoff or value functions are transformed by risk sensitivity, and then analyzed the existence Fiat and Papadimitriou (2010) and computation Mazumdar et al. (2025); Noorani and Baras (2022); Zhang et al. (2025) of equilibria in the resulting transformed games. What remains less understood is whether risk attitudes can be incorporated directly into agents’ learning dynamics, and how doing so changes equilibrium selection in the original game, which asks the question: which outcome will agents coordinate on through learning, and why Harsanyi and Selten (1988); Sandholm (2010b)? Towards addressing this question, we study the stochastic stability of risk-sensitive evolutionary dynamics, in caonical 2×22× 2 coordination games Harsanyi and Selten (1988); Kandori et al. (1993); Young (1993). We consider both the single-population symmetric setting and the two-population asymmetric setting, allowing different risk attitudes across populations. At each step, one random agent revises its action according to a risk-sensitive noisy best-response rule. Risk sensitivity enters through the classical entropic risk measure (ERM) Föllmer and Schied (2002), which replaces the risk-neutral expected payoffs used in learning. We focus on two standard revision protocols from evolutionary game theory: best response with mutations (BRM) Kandori et al. (1993) and logit choice Blume (1993). Prior work in evolutionary game theory showed that, with risk-neutral agents, the risk-dominant equilibrium is often uniquely selected in the long run Kandori et al. (1993); Young (1993); Staudigl (2012). Few results have studied the impact of risk sensitivity on evolutionary learning, except, e.g., Sawa and Wu (2018); Nax and Newton (2019). They study how risk dominance and stochastic stability are preserved under reference-dependent preferences Sawa and Wu (2018) and different risk attitudes Nax and Newton (2019), particularly when risk dominance coincides with maximin or payoff dominance. By comparison, our results systematically characterize how varying entropic risk sensitivity in the evolutionary dynamics changes long-run equilibrium selection. More precisely, in the single-population symmetric setting, and under both protocols, when the payoff-dominant and maximin equilibria differ, increasing the risk-sensitivity parameter shifts stochastic stability from the maximin equilibrium to the payoff-dominant equilibrium. This is also consistent with the recent empirical observations that more risk-seeking learning agents are more likely to coordinate at the higher-payoff equilibrium Noorani and Baras (2022); Zhang et al. (2025), and our results can thus be viewed as a formalization of these insights through the lens of coordination games and evolutionary dynamics. By contrast, if there exists a super-dominant equilibrium Sawa and Wu (2018), which is both weakly maximin and weakly payoff-dominant, then it is stochastically stable for all risk attitudes under both protocols and in both population settings. Compared with Sawa and Wu (2018); Nax and Newton (2019), our insight on ERM aligns with theirs in the single-population BRM setting, regarding the robustness of super-dominance, although we start from a different modeling framework of risk-sensitive learning. Beyond this setting, we also analyze the single-population BRM in cases where payoff dominance and maximin differ, and thus a super-dominant equilibrium does not exist; analyze the logit choice rule and its robustness, which was not covered in Nax and Newton (2019); and extend the analyses to asymmetric two-population games with heterogeneous risk parameters. Furthermore, we show that the same mechanism extends beyond the two-action case. In single-population symmetric k-action games under risk-sensitive BRM, if a strongly payoff-dominant equilibrium exists, then it is uniquely selected by sufficiently risk-seeking agents for all sufficiently large populations; similarly, if a strongly maximin equilibrium exists, then it is uniquely selected by sufficiently risk-averse agents for all sufficiently large populations. Thus, the risk attitudes may provide a systematic way to steer long-run equilibrium selection for evolutionary dynamics: optimism toward high payoff realizations (i.e., risk-seeking dynamics) may promote payoff-dominant outcome, while pessimism toward low payoff realizations (i.e., risk-averse dynamics) may promote maximin outcome and thus robust behaviors. 2 Preliminaries A 2×22× 2 bimatrix game is specified by the payoff matrices Mp=(apbpcpdp),p∈1,2,M^p= pmatrixa^p&b^p\\ c^p&d^p pmatrix, p∈\1,2\, where MpM^p denotes the payoff matrix of agent p, and both agents have action set 1,2\1,2\. In each MpM^p, rows index agent p’s own action and columns index the opponent’s action. The game is symmetric if agents are interchangeable, i.e., M1=M2=(abcd).M^1=M^2= pmatrixa&b\\ c&d pmatrix. Definition 1 (2×22× 2 coordination games). A 2×22× 2 bimatrix game is a coordination game if ap>cp,dp>bp,p∈1,2.a^p>c^p, d^p>b^p, p∈\1,2\. In the symmetric setting, these conditions reduce to a>ca>c and d>bd>b. Under these conditions, the pure action profiles (1,1)(1,1) and (2,2)(2,2) are strict Nash equilibria of the game. We refer to them as coordinated equilibria. Unless stated otherwise, we focus on such 2×22× 2 coordination games. We further introduce the following standard refinements of such Nash equilibria. Definition 2 (Risk dominance). The coordinated equilibrium (1,1)(1,1) is risk-dominant Harsanyi and Selten (1988) if (a1−c1)(a2−c2)>(d1−b1)(d2−b2).(a^1-c^1)(a^2-c^2)>(d^1-b^1)(d^2-b^2). In the symmetric setting, this condition reduces to a−c>d−ba-c>d-b. The coordinated equilibrium (2,2)(2,2) is risk-dominant if the reverse inequality holds. Definition 3 (Payoff dominance). The coordinated equilibrium (1,1)(1,1) is payoff-dominant Harsanyi and Selten (1988) if it Pareto-dominates the other coordinated equilibrium, i.e., ap>dp,p∈1,2.a^p>d^p, p∈\1,2\. It is weakly payoff-dominant if the above inequalities are non-strict, i.e., ap≥dpa^p≥ d^p for all p∈1,2p∈\1,2\. In the symmetric setting, payoff dominance reduces to a>da>d, and weak payoff dominance reduces to a≥da≥ d. The coordinated equilibrium (2,2)(2,2) is payoff-dominant, respectively weakly payoff-dominant, if the reverse strict, respectively non-strict, inequalities hold. Definition 4 (Maximin action and maximin equilibrium). For agent p, a maximin action is an action that maximizes the agent’s worst-case payoff over the opponent’s actions. Thus, action 11 is weakly maximin for agent p if minap,bp≥mincp,dp. \a^p,b^p\≥ \c^p,d^p\. It is strictly maximin if the above inequality is strict. Similarly, action 22 is weakly maximin, respectively strictly maximin, for agent p if the reverse non-strict, respectively strict, inequality holds. A coordinated equilibrium is weakly maximin if each agent’s equilibrium action is weakly maximin. Definition 5 (Super-dominant equilibrium). A coordinated equilibrium is super-dominant Sawa and Wu (2018) if it is weakly maximin for both agents and weakly payoff-dominant, with at least one of these conditions holding strictly. Equivalently, (1,1)(1,1) is super-dominant if for each p∈1,2p∈\1,2\, ap≥dp,bp≥cp,a^p≥ d^p, b^p≥ c^p, and at least one of these inequalities is strict for each p. The coordinated equilibrium (2,2)(2,2) is super-dominant if the reverse inequalities hold, with at least one strict inequality. We then consider the standard evolutionary game setting as in Sandholm (2010b), with either one or two populations, i.e., ∈1,1,2.P∈ \\1\,\1,2\ \. Each population p∈p contains Np∈ℕN^p agents, and each agent chooses an action from the set Sp=1,2S^p=\1,2\. Let xp∈Xp,Npx^p∈ X^p,N^p denote the fraction of population p playing action 11, where Xp,Np:=0,1Np,…,1.X^p,N^p:= \0, 1N^p,…,1 \. The aggregate state is x=(xp)p∈XN:=∏p∈Xp,Np.x=(x^p)_p ∈ X^N:= _p X^p,N^p. In the single-population setting, the state is one-dimensional. In this case, we write N:=N1N:=N^1, XN:=X1,NX^N:=X^1,N, and x:=x1x:=x^1, and suppress the population index throughout for notational simplicity. In the two-population setting, we write N:=(N1,N2),XN≡XN1,N2:=X1,N1×X2,N2.N:=(N^1,N^2), X^N≡ X^N^1,N^2:=X^1,N^1× X^2,N^2. Payoffs used in evolutionary learning are given by Fp:XN→ℝ2F^p:X^N ^2, where Fp(x)=(F1p(x),F2p(x)),F^p(x)=(F_1^p(x),F_2^p(x)), and Fip(x)F_i^p(x) is the payoff to action i∈Spi∈ S^p for population p at state x. Agents are repeatedly and randomly matched to play the coordination game. In the single-population setting, matching is with replacement, so the opponent is sampled from the same population according to the current state, and thus x−p=x^-p=x. In the two-population setting, agents in population p are matched against population −p-p, and payoffs depend on the opponent state x−px^-p, where −p=2-p=2 if p=1p=1 and −p=1-p=1 if p=2p=2. 3 Risk-Sensitive Evolutionary Dynamics & Stochastic Stability 3.1 Risk-Sensitive Evolutionary Dynamics To model uncertainty in agents’ learning process, we consider risk-sensitive agents who evaluate payoffs via the entropic risk measure Howard and Matheson (1972); Whittle (1981); Föllmer and Schied (2002), rather than risk-neutral expected payoffs. Definition 6 (Entropic Risk Measure (ERM)). For a real-valued random payoff R and parameter β∈ℝβ , the entropic risk measure is defined as ℛβ(R):=1βlog[eβR],R_β(R):= 1β \! [e^β R ], with the continuous extension ℛ0(R)=[R]R_0(R)=E[R]. For β near zero, the Taylor expansion of the ERM gives ℛβ(R)=[R]+β2Var(R)+β26κ3(R)+O(β3),R_β(R)=E[R]+ β2Var(R)+ β^26 _3(R)+O(β^3), where κ3(R)=[(R−[R])3] _3(R)=E[(R-E[R])^3]. Thus, the ERM departs from expected payoff by accounting first for variance and then for skewness. A positive β tilts evaluations toward favorable payoff realizations and corresponds to risk-seeking behavior, whereas a negative β emphasizes adverse realizations and corresponds to risk-averse behavior. In our model, agents within each population p share a common risk-sensitivity parameter βp∈ℝβ^p , although this parameter may differ across populations. Conditioning on the population state x, the uncertainty faced by a revising agent comes from random matching. Hence, for the 2×22× 2 bimatrix game, for each p∈p , the ERM-adjusted payoffs to actions 11 and 22 are Fβp,1p(x) F_β^p,1^p(x) =1βplog(x−peβpap+(1−x−p)eβpbp), = 1β^p \! (x^-pe^β^pa^p+(1-x^-p)e^β^pb^p ), (1) Fβp,2p(x) F_β^p,2^p(x) =1βplog(x−peβpcp+(1−x−p)eβpdp), = 1β^p \! (x^-pe^β^pc^p+(1-x^-p)e^β^pd^p ), with the continuous extension at βp=0β^p=0 given by the expected payoffs F0,1p(x)=apx−p+bp(1−x−p),F0,2p(x)=cpx−p+dp(1−x−p). F_0,1^p(x)=a^px^-p+b^p(1-x^-p), F_0,2^p(x)=c^px^-p+d^p(1-x^-p). For each fixed βpβ^p, these functions are continuous in x and smooth on the interior. We thus define the ERM payoff difference for population p as Δβp(x):=Fβp,1p(x)−Fβp,2p(x), _β^p^p(x):=F_β^p,1^p(x)-F_β^p,2^p(x), (2) which further leads to the following definition. Definition 7 (ERM-adjusted best response). For population p, an ERM-adjusted best response at state x is an action that maximizes the ERM-adjusted payoff against the population state. Equivalently, action 11 is an ERM-adjusted best response if Δβp(x)≥0, _β^p^p(x)≥ 0, and action 22 is an ERM-adjusted best response if Δβp(x)≤0. _β^p^p(x)≤ 0. When Δβp(x)=0 _β^p^p(x)=0, both actions are ERM-adjusted best responses. We now introduce our risk-sensitive evolutionary dynamics. Note that our focus is on the effect of risk-sensitivity on the learning process/dynamics, rather than that on the equilibrium concept of the underlying game (as in e.g., Fiat and Papadimitriou (2010); Mazumdar et al. (2025)). At each time step k∈1,2,…k∈\1,2,…\, a uniformly random agent receives a revision opportunity and updates its action according to the following risk-sensitive noisy best-response protocol: ρ1,2p(x)=ση(−Δβp(x−p)),ρ2,1p(x)=ση(Δβp(x−p)), _1,2^p(x)=σ^η\! (- _β^p^p(x^-p) ), _2,1^p(x)=σ^η\! ( _β^p^p(x^-p) ), (3) with the convention that, in the single-population setting, we drop the superscript p. Here, ση:ℝ→(0,1)σ^η:R→(0,1) is parameterized by a noise level η>0η>0, and ρi,jp(x) _i,j^p(x) denotes the probability that an agent in population p switches from action i to action j at state x. As the noise level vanishes, limη→0ση(a)=1,a>0,0,a<0. _η→ 0σ^η(a)= cases1,&a>0,\\ 0,&a<0. cases Hence, ση(⋅)σ^η(·) converges to the ERM-adjusted best response in the vanishing-noise limit. We call the dynamics generated by (3) the risk-sensitive dynamics for short. To characterize how the probability of choosing a suboptimal action decays as the noise vanishes, we impose the following assumption. Assumption 1. Let ση>0\σ^η\_η>0 be a family of noisy best-response choice functions, with ση:ℝ→(0,1)σ^η:R→(0,1). For each d∈ℝd , assume that c(d):=−limη→0ηlogση(−d)c(d):=- _η→ 0η σ^η(-d) exists and is finite. The induced function c:ℝ→[0,∞)c:R→[0,∞) satisfies c(d)=0for d≤0,c(d)>0for d>0,c(d)=0 d≤ 0, c(d)>0 d>0, and is nondecreasing on ℝR. Moreover, limη→0ση(0)=q∈(0,1). _η→ 0σ^η(0)=q∈(0,1). We consider two classical noisy best-response protocols from evolutionary game theory: best response with mutations Kandori et al. (1993); Young (1993) and logit choice Blume (1993). 3.1.1 Best Response with Mutations (BRM) Under BRM, a strict ERM-adjusted best response is chosen with probability 1−e−1/η1-e^-1/η, and the alternative action is chosen with probability e−1/ηe^-1/η; ties are broken uniformly: ση(a)=1−exp(−1/η),a>0,12,a=0,exp(−1/η),a<0.σ^η(a)= cases1- (-1/η),&a>0,\\ 12,&a=0,\\ (-1/η),&a<0. cases This choice rule satisfies Assumption 1 with c(d)=d>0c(d)=1_\d>0\. We refer to the resulting risk-sensitive dynamics as RS-BRM and 2Pop-RS-BRM in the single- and two-population settings, respectively. 3.1.2 Logit Choice Under the logit choice rule, ση(a)=ea/η1+ea/η,η>0,σ^η(a)= e^a/η1+e^a/η, η>0, such that the actions with higher ERM-adjusted payoffs are chosen with higher probability, and σησ^η converges to the ERM-adjusted best response as η→0η→ 0. This protocol satisfies Assumption 1 with c(d)=(d)+:=max0,dc(d)=(d)_+:= \0,d\. We refer to the resulting risk-sensitive dynamics as RS-logit and 2Pop-RS-logit in the single- and two-population settings, respectively. For both of these protocols, limη→0ση(0)=12 _η→ 0σ^η(0)= 12. We conclude this section with two lemmas that will be used repeatedly later, whose proofs are given in Appendices 8.1 and 8.2, respectively. Lemma 1. For each population p, the function z↦Δβp(z)z _β^p^p(z) in (2) is strictly increasing on (0,1)(0,1) and has a unique zero xβp−p,∗∈(0,1)x_β^p^-p,*∈(0,1). For βp≠0β^p≠ 0, this zero is given by xβp−p,∗=eβpdp−eβpbp(eβpap−eβpcp)+(eβpdp−eβpbp).x_β^p^-p,*= e^β^pd^p-e^β^pb^p (e^β^pa^p-e^β^pc^p )+ (e^β^pd^p-e^β^pb^p ). (4) At βp=0β^p=0, it has the continuous extension x0−p,∗=dp−bp(ap−cp)+(dp−bp). x_0^-p,*= d^p-b^p(a^p-c^p)+(d^p-b^p). Moreover, Δβp(z)<0 _β^p^p(z)<0 if and only if z<xβp−p,∗z<x_β^p^-p,*, and Δβp(z)>0 _β^p^p(z)>0 if and only if z>xβp−p,∗z>x_β^p^-p,*. Thus, the ERM-adjusted best response changes from action 22 to action 11 as z crosses xβp−p,∗x_β^p^-p,*, with both actions being best responses at z=xβp−p,∗z=x_β^p^-p,*. Lemma 2. Fix a population p, and let xβp−p,∗x_β^p^-p,* be the unique threshold obtained by solving (4). Then, the map βp↦xβp−p,∗β^p x_β^p^-p,* is continuous on ℝR. Moreover: (I) If bp>cpb^p>c^p and dp>apd^p>a^p, then βp↦xβp−p,∗β^p x_β^p^-p,* is strictly increasing, with limβp→−∞xβp−p,∗=0,limβp→∞xβp−p,∗=1. _β^p→-∞x_β^p^-p,*=0, _β^p→∞x_β^p^-p,*=1. (I) If bp<cpb^p<c^p and dp<apd^p<a^p, then βp↦xβp−p,∗β^p x_β^p^-p,* is strictly decreasing, with limβp→−∞xβp−p,∗=1,limβp→∞xβp−p,∗=0. _β^p→-∞x_β^p^-p,*=1, _β^p→∞x_β^p^-p,*=0. (I) If bp≥cpb^p≥ c^p and dp≤apd^p≤ a^p, with at least one strict inequality, then xβp−p,∗<12for all βp∈ℝ.x_β^p^-p,*< 12 all β^p . (IV) If bp≤cpb^p≤ c^p and dp≥apd^p≥ a^p, with at least one strict inequality, then xβp−p,∗>12for all βp∈ℝ.x_β^p^-p,*> 12 all β^p . In the degenerate case where ap=dpa^p=d^p and bp=cpb^p=c^p, we have xβp−p,∗≡12x_β^p^-p,*≡ 12. In the single-population setting, the superscripts p and −p-p are omitted for notational convenience. 3.2 A Unified Framework for Stochastic-Stability Analysis We now introduce a stochastic-stability analysis framework, adapted from Sandholm (2010a); Staudigl (2012), that will unify our subsequent analyses in the single-population and two-population settings. Throughout this subsection, N is fixed. For each η>0η>0, the risk-sensitive dynamics induce a discrete-time Markov chain XN,η:=XkN,ηk≥0X^N,η:=\X_k^N,η\_k≥ 0 on the finite state space XNX^N, where XkN,ηX_k^N,η denotes the population state after k revision opportunities. For the noisy best-response protocols studied here, each action is chosen with strictly positive probability for every η>0η>0. Hence, from any state, every adjacent state can be reached with positive probability whenever the corresponding transition is feasible, and the chain also has positive self-loop probability. Therefore, the chain is irreducible and aperiodic on XNX^N. Since XNX^N is finite, the standard finite-state Markov-chain theorem implies that the chain admits a unique stationary distribution, denoted by μN,ημ^N,η; see e.g., Stroock (2005) for the standard argument on this result. Definition 8 (Stochastic stability). A state x∈XNx∈ X^N is stochastically stable if limη→0μN,η(x)>0, _η→ 0μ^N,η(x)>0, and uniquely stochastically stable if limη→0μN,η(x)=1. _η→ 0μ^N,η(x)=1. Intuitively, a stochastically stable state retains strictly positive stationary probability as the noise level vanishes. Unique stochastic stability means that the stationary distribution concentrates entirely on that state. This is the standard stochastic-stability criterion for finite-state small-noise Markov chains; see e.g., Young (1993); Kandori et al. (1993). To analyze stochastic stability, we record the exponential decay rates of one-step transition probabilities. For states x,y∈XNx,y∈ X^N, let PN,η(x,y):=ℙ(Xk+1N,η=y|XkN,η=x)P^N,η(x,y):=P\! (X_k+1^N,η=y\, |\,X_k^N,η=x ) denote the one-step transition probability. Since the noisy best-response choice functions satisfy ση(a)∈(0,1)σ^η(a)∈(0,1) for every a∈ℝa and every η>0η>0, the support of PN,ηP^N,η is independent of η. Thus, if a one-step transition has positive probability for some η>0η>0, then it has positive probability for every η>0η>0. Definition 9 (Transition costs). For distinct states x≠yx≠ y, we write x→yx→ y if y is reachable from x in one feasible revision step, which is equivalent to PN,η(x,y)>0P^N,η(x,y)>0 for every η>0η>0. Under Assumption 1, the small-noise cost of a feasible one-step transition x→yx→ y is κN(x,y):=limη→0[−ηlogPN,η(x,y)].κ^N(x,y):= _η→ 0 [-η P^N,η(x,y) ]. (5) For infeasible transitions, we set κN(x,y)=+∞κ^N(x,y)=+∞. A feasible transition x→yx→ y is called a zero-cost transition if κN(x,y)=0κ^N(x,y)=0. The limit in (5) exists because the non-exponential factors in PN,η(x,y)P^N,η(x,y), such as the probability of selecting a particular population or action type, do not affect the value of −ηlogPN,η(x,y)-η P^N,η(x,y) as η→0η→ 0, while Assumption 1 controls the exponential decay rate of the revision probability. Under the revision protocols considered here, zero-cost transitions are exactly the one-step updates that either follow an ERM-adjusted best response or occur along a tie. Definition 10 (Paths and path costs). A path is a finite sequence of states ϕ=(ϕ0,ϕ1,…,ϕK)φ=( _0, _1,…, _K) in XNX^N such that ϕk→ϕk+1 _k→ _k+1 for all k=0,…,K−1k=0,…,K-1. The cost of a path ϕφ is CN(ϕ):=∑k=0K−1κN(ϕk,ϕk+1).C^N(φ):= _k=0^K-1κ^N( _k, _k+1). For states x,y∈XNx,y∈ X^N, the minimal path cost from x to y is CN(x,y):=minCN(ϕ):ϕ0=x,ϕK=y. C^N(x,y):= \C^N(φ): _0=x,\ _K=y \. More generally, for a state x∈XNx∈ X^N and a nonempty set E⊆XNE X^N, the minimal path cost from x to E is CN(x,E):=minCN(ϕ):ϕ0=x,ϕK∈E. C^N(x,E):= \C^N(φ): _0=x,\ _K∈ E \. The zero-cost transitions induce a directed graph on XNX^N. We refer to this graph as the zero-noise dynamics, because it describes the limiting motion of the noisy best response dynamics as η→0η→ 0. Definition 11 (Attracting state, stability set, and exit boundary). A state A∈XNA∈ X^N is an attracting state if A\A\ is a singleton sink recurrent class of the zero-noise dynamics; equivalently, there is no zero-cost transition from A to any state outside A\A\. The stability set of A, denoted by ℬ0N(A)B_0^N(A), is the largest subset of XNX^N containing A such that every state in ℬ0N(A)B_0^N(A) reaches A along a zero-cost path, and no state in ℬ0N(A)B_0^N(A) has a zero-cost path to XN∖ℬ0N(A)X^N _0^N(A). The corresponding exit boundary of A is ∂outN(A):=x∈XN∖ℬ0N(A):∃y∈ℬ0N(A)such that y→x. _out^N(A):= \x∈ X^N _0^N(A):∃\,y _0^N(A)\ such that y→ x \. In coordination games, the zero-noise dynamics have two attracting states corresponding to the two coordinated equilibria. The following assumption isolates this structure. Assumption 2. The zero-noise dynamics have exactly two attracting states, denoted by A and B. Moreover: (I) Every state x∈XN∖A,Bx∈ X^N \A,B\ has a zero-cost path to at least one of A or B. (I) If x∉ℬ0N(A)x _0^N(A), then CN(x,B)=0C^N(x,B)=0, and if x∉ℬ0N(B)x _0^N(B), then CN(x,A)=0C^N(x,A)=0. Assumption 2 will be verified explicitly in both the single-population and two-population settings. Under this assumption, stochastic-stability analysis reduces to comparing the minimal costs of escaping the two stability sets, as formalized in the following theorem. Theorem 3 (Stochastic stability via escape costs). Fix the population-size parameter N, and consider the Markov chain XN,ηX^N,η induced by the risk-sensitive dynamics. Suppose Assumptions 1 and 2 hold, with attracting states A and B. Define γAN:=CN(B,∂outN(B)),γBN:=CN(A,∂outN(A)). _A^N:=C^N\! (B, _out^N(B) ), _B^N:=C^N\! (A, _out^N(A) ). Then, we have limη→0ηlogμN,η(B)μN,η(A)=γAN−γBN. _η→ 0η μ^N,η(B)μ^N,η(A)= _A^N- _B^N. (6) Consequently, A is uniquely stochastically stable if γAN<γBN _A^N< _B^N, while B is uniquely stochastically stable if γBN<γAN _B^N< _A^N. Theorem 3 follows from the standard Freidlin–Wentzell tree representation used in stochastic-stability analysis; see e.g., Freidlin and Wentzell (1998); Staudigl (2012); Sandholm and Staudigl (2016). For completeness, we provide a short proof in Appendix 8.3. 4 Single-Population Symmetric Setting Since only one agent revises at each time step, the feasible one-step transitions are x→x+1Nx→ x+ 1N, generated by an agent switching from action 22 to action 11, and x→x−1Nx→ x- 1N, generated by an agent switching from action 11 to action 22. Recalling the one-step cost function c(⋅)c(·) from Assumption 1, (5) gives κN(x,x+1N)=c((−Δβ(x))+),κN(x,x−1N)=c((Δβ(x))+). κ^N\! (x,x+ 1N )=c\! ((- _β(x))_+ ), κ^N\! (x,x- 1N )=c\! (( _β(x))_+ ). (7) Let xβ∗∈(0,1)x_β^*∈(0,1) be the unique threshold from Lemma 1. Define the adjacent grid points x¯βN:=⌊Nxβ∗⌋N,x¯βN:=⌈Nxβ∗⌉N. x_β^N:= Nx_β^* N, x_β^N:= Nx_β^* N. The next proposition verifies Assumption 2. Proposition 4. Under the single-population risk-sensitive dynamics induced by (3), with transition costs given by (7), the zero-noise dynamics have exactly two attracting states, namely 00 and 11. If Nxβ∗∉ℤNx_β^* , then ℬ0N(0)=0,1N,…,x¯βN,ℬ0N(1)=x¯βN,x¯βN+1N,…,1,B_0^N(0)= \0, 1N,…, x_β^N \, _0^N(1)= \ x_β^N, x_β^N+ 1N,…,1 \, with ∂outN(0)=x¯βN,∂outN(1)=x¯βN. _out^N(0)=\ x_β^N\, _out^N(1)=\ x_β^N\. If Nxβ∗∈ℤNx_β^* , so that xβ∗∈XNx_β^*∈ X^N, then ℬ0N(0)=0,1N,…,xβ∗−1N,ℬ0N(1)=xβ∗+1N,xβ∗+2N,…,1,B_0^N(0)= \0, 1N,…,x_β^*- 1N \, _0^N(1)= \x_β^*+ 1N,x_β^*+ 2N,…,1 \, with ∂outN(0)=∂outN(1)=xβ∗. _out^N(0)= _out^N(1)=\x_β^*\. In either case, Assumption 2 holds with A=0A=0 and B=1B=1. Proof. By Lemma 1, Δβ(x)<0 _β(x)<0 for x<xβ∗x<x_β^*, and Δβ(x)>0 _β(x)>0 for x>xβ∗x>x_β^*. Hence, by (7), every state x<xβ∗x<x_β^* has a zero-cost downward move x→x−1Nx→ x- 1N, while every state x>xβ∗x>x_β^* has a zero-cost upward move x→x+1Nx→ x+ 1N. If xβ∗∈XNx_β^*∈ X^N, then Δβ(xβ∗)=0 _β(x_β^*)=0, so both moves xβ∗→xβ∗−1N,xβ∗→xβ∗+1Nx_β^*→ x_β^*- 1N, x_β^*→ x_β^*+ 1N have zero cost. Define E0:=x∈XN:x<xβ∗,E1:=x∈XN:x>xβ∗.E_0:=\x∈ X^N:x<x_β^*\, E_1:=\x∈ X^N:x>x_β^*\. If Nxβ∗∉ℤNx_β^* , then E0=0,1N,…,x¯βN,E1=x¯βN,x¯βN+1N,…,1.E_0= \0, 1N,…, x_β^N \, E_1= \ x_β^N, x_β^N+ 1N,…,1 \. If Nxβ∗∈ℤNx_β^* , then E0=0,1N,…,xβ∗−1N,E1=xβ∗+1N,xβ∗+2N,…,1.E_0= \0, 1N,…,x_β^*- 1N \, E_1= \x_β^*+ 1N,x_β^*+ 2N,…,1 \. Every state in E0E_0 reaches 00 along repeated downward zero-cost moves. Moreover, no state in E0E_0 has a zero-cost path to XN∖E0X^N E_0, since any such path must contain an upward move from some state x<xβ∗x<x_β^*, which has positive cost. Therefore, ℬ0N(0)=E0B_0^N(0)=E_0. Similarly, every state in E1E_1 reaches 11 along repeated upward zero-cost moves, and no state in E1E_1 has a zero-cost path to XN∖E1X^N E_1. Hence, ℬ0N(1)=E1B_0^N(1)=E_1. The formulas for the exit boundaries follow directly. If Nxβ∗∉ℤNx_β^* , the only one-step transition from ℬ0N(0)B_0^N(0) to its complement is x¯βN→x¯βN x_β^N→ x_β^N, so ∂outN(0)=x¯βN _out^N(0)=\ x_β^N\. Similarly, the only one-step transition from ℬ0N(1)B_0^N(1) to its complement is x¯βN→x¯βN x_β^N→ x_β^N, so ∂outN(1)=x¯βN _out^N(1)=\ x_β^N\. If Nxβ∗∈ℤNx_β^* , then the first state outside either stability set that can be reached in one step is xβ∗x_β^*, and thus ∂outN(0)=∂outN(1)=xβ∗. _out^N(0)= _out^N(1)=\x_β^*\. It remains to verify Assumption 2. The zero-noise dynamics move downward on E0E_0 and upward on E1E_1, while at xβ∗x_β^*, when xβ∗∈XNx_β^*∈ X^N, both adjacent moves have zero cost. Hence, the only attracting states are 00 and 11. Assumption 2 (I) holds because every state x∈XN∖0,1x∈ X^N \0,1\ either lies in E0E_0, in which case it has a zero-cost path to 00, or lies in E1E_1, in which case it has a zero-cost path to 11, or equals xβ∗x_β^*, in which case it has zero-cost paths to both 00 and 11. Assumption 2 (I) holds because if x∉ℬ0N(0)x _0^N(0), then x≥xβ∗x≥ x_β^*, so x has a zero-cost path to 11; similarly, if x∉ℬ0N(1)x _0^N(1), then x≤xβ∗x≤ x_β^*, so x has a zero-cost path to 00. ∎ With the above notation, the escape costs from the two stability sets are γ0N:=CN(1,∂outN(1))=CN(1,x¯βN),γ1N:=CN(0,∂outN(0))=CN(0,x¯βN). _0^N:=C^N\! (1, _out^N(1) )=C^N\! (1,\ x_β^N\ ),~~~ _1^N:=C^N\! (0, _out^N(0) )=C^N\! (0,\ x_β^N\ ). (8) We also define the selection coefficient SN:=γ1N−γ0N.S^N:= _1^N- _0^N. (9) By Theorem 3, state 11 is uniquely stochastically stable if SN<0S^N<0, whereas state 00 is uniquely stochastically stable if SN>0S^N>0. We next establish the stochastic stability results under both risk-sensitive dynamics of RS-BRM and RS-logit. Theorem 5 (Stochastic stability under RS-BRM). Fix β∈ℝβ , and let xβ∗∈(0,1)x_β^*∈(0,1) be the unique threshold from Lemma 1. Then, there exists N0(β)N_0(β) such that, for every N≥N0(β)N≥ N_0(β), the unique stochastically stable state under RS-BRM is 11 if xβ∗<12x_β^*< 12, and is 00 if xβ∗>12x_β^*> 12. Consequently, for every fixed β, the following conclusions hold for all N≥N0(β)N≥ N_0(β): (I) If a>da>d and b<cb<c, or a<da<d and b>cb>c, then there exists a unique β†∈ℝβ such that the payoff-dominant equilibrium is uniquely stochastically stable when β>β†β>β , whereas the maximin equilibrium is uniquely stochastically stable when β<β†β<β . (I) If a≥da≥ d and b≥cb≥ c, or a≤da≤ d and b≤cb≤ c, with at least one strict inequality, then the super-dominant equilibrium is uniquely stochastically stable. Proof. By Proposition 4, Assumption 2 holds with attracting states 00 and 11. Under RS-BRM, the cost function is c(d)=d>0c(d)=1_\d>0\. Hence, a transition that moves opposite to the unique ERM-adjusted best response has cost 11, whereas a transition that follows an ERM-adjusted best response, or occurs at a tie, has cost 00. Starting from 11, any path to x¯βN=⌊Nxβ∗⌋/N x_β^N= Nx_β^* /N must contain at least N−⌊Nxβ∗⌋N- Nx_β^* downward steps. The monotone path (1, 1−1N,…,x¯βN) (1,\ 1- 1N,\ …,\ x_β^N ) attains this lower bound. Each downward step on this path starts from a state strictly larger than xβ∗x_β^*. Indeed, the last such step starts from x¯βN+1N>xβ∗ x_β^N+ 1N>x_β^* if Nxβ∗∉ℤNx_β^* , and from xβ∗+1N>xβ∗x_β^*+ 1N>x_β^* if Nxβ∗∈ℤNx_β^* . At such states, action 11 is the unique ERM-adjusted best response, so a downward step, which decreases the share of action 11, moves opposite to the ERM-adjusted best-response direction and has cost 11. Therefore, we have γ0N=N−⌊Nxβ∗⌋. _0^N=N- Nx_β^* . Similarly, any path from 00 to x¯βN=⌈Nxβ∗⌉/N x_β^N= Nx_β^* /N must contain at least ⌈Nxβ∗⌉ Nx_β^* upward steps. The monotone path (0,1N,…,x¯βN) (0,\ 1N,\ …,\ x_β^N ) attains this lower bound. Each upward step on this path starts from a state strictly smaller than xβ∗x_β^*. At such states, action 22 is the unique ERM-adjusted best response, so an upward step, which increases the share of action 11, moves opposite to the ERM-adjusted best-response direction and has cost 11. Thus, we have γ1N=⌈Nxβ∗⌉. _1^N= Nx_β^* . By Theorem 3, state 11 is uniquely stochastically stable if γ1N<γ0N _1^N< _0^N, while state 00 is uniquely stochastically stable if γ0N<γ1N _0^N< _1^N. Since γ1N=⌈Nxβ∗⌉N→xβ∗,γ0N=1−⌊Nxβ∗⌋N→1−xβ∗, _1^NN= Nx_β^* N→ x_β^*, _0^NN=1- Nx_β^* N→ 1-x_β^*, the sign of γ1N−γ0N _1^N- _0^N agrees with the sign of xβ∗−12x_β^*- 12 for all sufficiently large N, provided xβ∗≠12x_β^*≠ 12. This proves the first statement. The remaining claims follow from Lemma 2. Suppose first that a<da<d and b>cb>c. Then (2,2)(2,2), corresponding to state 00, is payoff-dominant, while action 11 is maximin, so (1,1)(1,1), corresponding to state 11, is the maximin equilibrium. By Lemma 2, β↦xβ∗β x_β^* is continuous and strictly increasing, with limits 00 and 11 as β→−∞β→-∞ and β→+∞β→+∞, respectively. Hence, there exists a unique β†β such that xβ†∗=12x_β ^*= 12. For every fixed β<β†β<β , we have xβ∗<12x_β^*< 12, so state 11, the maximin equilibrium, is stochastically stable for all sufficiently large N. For every fixed β>β†β>β , we have xβ∗>12x_β^*> 12, so state 00, the payoff-dominant equilibrium, is stochastically stable for all sufficiently large N. Now suppose that a>da>d and b<cb<c. Then (1,1)(1,1), corresponding to state 11, is payoff-dominant, while action 22 is maximin, so (2,2)(2,2), corresponding to state 00, is the maximin equilibrium. In this case, Lemma 2 implies that β↦xβ∗β x_β^* is continuous and strictly decreasing, again crossing 12 12 at a unique β†β . Thus, for every fixed β<β†β<β , we have xβ∗>12x_β^*> 12, so state 00, the maximin equilibrium, is stochastically stable for all sufficiently large N. For every fixed β>β†β>β , we have xβ∗<12x_β^*< 12, so state 11, the payoff-dominant equilibrium, is stochastically stable for all sufficiently large N. This proves (I). Finally, suppose that a≥da≥ d and b≥cb≥ c, with at least one strict inequality. Then, by definition, (1,1)(1,1) is super-dominant, and Lemma 2 gives xβ∗<12x_β^*< 12 for all β∈ℝβ . Hence, state 11 is stochastically stable for every fixed β, once N is sufficiently large. The case a≤da≤ d and b≤cb≤ c, with at least one strict inequality, is symmetric: (2,2)(2,2) is super-dominant, xβ∗>12x_β^*> 12 for all β, and state 00 is stochastically stable for every fixed β, once N is sufficiently large. This proves (I) and completes the proof. ∎ Theorem 6 (Stochastic stability under RS-logit). Define S¯logit(β):=−∫01Δβ(x)dx. S_logit(β):=- _0^1 _β(x)\,dx. (10) Fix β∈ℝβ and assume S¯logit(β)≠0 S_logit(β)≠ 0. Then, there exists N0(β)N_0(β) such that, for every N≥N0(β)N≥ N_0(β), the unique stochastically stable state under RS-logit is 11 if S¯logit(β)<0 S_logit(β)<0, and 00 if S¯logit(β)>0 S_logit(β)>0. Consequently, for every fixed β, the following conclusions hold for all N≥N0(β)N≥ N_0(β): (I) If a>da>d and b<cb<c, or a<da<d and b>cb>c, then there exists a unique β†∈ℝβ such that the payoff-dominant equilibrium is uniquely stochastically stable for β>β†β>β , while the maximin equilibrium is uniquely stochastically stable for β<β†β<β . (I) If a≥da≥ d and b≥cb≥ c, or a≤da≤ d and b≤cb≤ c, with at least one strict inequality, then the super-dominant equilibrium is uniquely stochastically stable. Proof. By Proposition 4, Assumption 2 holds with attracting states 00 and 11. Under RS-logit, the cost function is c(d)=(d)+c(d)=(d)_+. Therefore, by (7), κN(x,x+1N)=(−Δβ(x))+,κN(x,x−1N)=(Δβ(x))+.κ^N\! (x,x+ 1N )=(- _β(x))_+, κ^N\! (x,x- 1N )=( _β(x))_+. Since the state space is one-dimensional and all step costs are nonnegative, the minimum-cost paths to the exit boundaries are the monotone paths. Hence, γ0N=∑m=⌊Nxβ∗⌋+1NΔβ(mN),γ1N=∑m=0⌈Nxβ∗⌉−1[−Δβ(mN)]. _0^N= _m= Nx_β^* +1^N _β\! ( mN ), _1^N= _m=0 Nx_β^* -1 [- _β\! ( mN ) ]. The summands are nonnegative: the first sum runs over states x>xβ∗x>x_β^*, where Δβ(x)>0 _β(x)>0, and the second runs over states x<xβ∗x<x_β^*, where Δβ(x)<0 _β(x)<0. Recalling the definition SN=γ1N−γ0NS^N= _1^N- _0^N from (9), define S¯logitN(β):=1NSN. S_logit^\,N(β):= 1NS^N. The above expressions imply S¯logitN(β)=−1N∑m=0NΔβ(mN), S_logit^\,N(β)=- 1N _m=0^N _β\! ( mN ), (11) where, if Nxβ∗∈ℤNx_β^* , the threshold term contributes zero. Since Δβ _β is continuous on [0,1][0,1], this is a Riemann sum, and therefore S¯logitN(β)⟶S¯logit(β)as N→∞. S_logit^\,N(β) S_logit(β) N→∞. Thus, whenever S¯logit(β)≠0 S_logit(β)≠ 0, the sign of SNS^N agrees with the sign of S¯logit(β) S_logit(β) for all sufficiently large N. The first claim then follows from Theorem 3. It remains to identify the selected equilibrium in the payoff-dominant/maximin and super-dominant cases. By Lemma 12, S¯logit S_logit is continuous and monotone on ℝR, with limβ→−∞S¯logit(β) _β→-∞ S_logit(β) =−mina,b+minc,d, =- \a,b\+ \c,d\, limβ→+∞S¯logit(β) _β→+∞ S_logit(β) =−maxa,b+maxc,d. =- \a,b\+ \c,d\. In fact, when payoff-dominant and maximin equilibria differ, the coordination-game inequalities imply |a−b|≠|c−d||a-b|≠|c-d|: specifically, a<da<d and b>cb>c imply |a−b|<|c−d||a-b|<|c-d|, while a>da>d and b<cb<c imply |a−b|>|c−d||a-b|>|c-d|. Hence, S¯logit S_logit is strictly monotone in these cases. Suppose first that a<da<d and b>cb>c. Then (2,2)(2,2), which corresponds to state 00, is payoff-dominant, while (1,1)(1,1), which corresponds to state 11, is the maximin equilibrium. In this case, the two endpoint limits of S¯logit S_logit have opposite signs. Hence, by Lemma 12, there exists a unique β†∈ℝβ such that S¯logit(β†)=0. S_logit(β )=0. For every fixed β<β†β<β , we have S¯logit(β)<0 S_logit(β)<0, so state 11, the maximin equilibrium, is stochastically stable for all sufficiently large N. For every fixed β>β†β>β , we have S¯logit(β)>0 S_logit(β)>0, so state 00, the payoff-dominant equilibrium, is stochastically stable for all sufficiently large N. Now suppose that a>da>d and b<cb<c. Then (1,1)(1,1), which corresponds to state 11, is payoff-dominant, while (2,2)(2,2), which corresponds to state 00, is the maximin equilibrium. Again, Lemma 12 gives a unique β†β at which S¯logit S_logit changes sign. For every fixed β<β†β<β , state 00, the maximin equilibrium, is stochastically stable for all sufficiently large N, while for every fixed β>β†β>β , state 11, the payoff-dominant equilibrium, is stochastically stable for all sufficiently large N. This proves (I). Finally, suppose that a≥da≥ d and b≥cb≥ c, with at least one strict inequality. Then, equilibrium (1,1)(1,1) is super-dominant. In this case, mina,b≥minc,d,maxa,b≥maxc,d, \a,b\≥ \c,d\, \a,b\≥ \c,d\, and S¯logit(0)<0 S_logit(0)<0. By Lemma 12, it follows that S¯logit(β)<0for all β∈ℝ. S_logit(β)<0 all β . Hence, state 11 is stochastically stable for every fixed β, once N is sufficiently large. The case a≤da≤ d and b≤cb≤ c, with at least one strict inequality, is symmetric: (2,2)(2,2) is super-dominant, S¯logit(β)>0 S_logit(β)>0 for all β∈ℝβ , and state 00 is stochastically stable for every fixed β, once N is sufficiently large. This proves (I). ∎ Remark 1. The conclusions of Theorems 5 and 6 are pointwise in β: N0(β)N_0(β) need not be uniformly bounded and may diverge as β approaches the corresponding critical value β†β . Theorems 5 and 6 show that, despite their differences, RS-BRM and RS-logit induce the same equilibrium-selection pattern in symmetric 2×22× 2 coordination games. When one equilibrium is super-dominant, long-run equilibrium selection is robust to both agents’ risk attitudes and the choice of noisy best-response protocols. Intuitively, a super-dominant equilibrium aligns payoff dominance with worst-case robustness: it yields higher coordinated payoffs while also protecting agents against unfavorable mismatches. Thus, when such an equilibrium exists, there is no tension between efficiency and safety, and the learning dynamics select it regardless of whether agents are risk-averse, risk-neutral, or risk-seeking. This robustness is desirable because the selected outcome does not depend sensitively on the precise risk preferences of the population or on the specific noise model used in the revision protocol. For example, consider M=(4102).M= pmatrix4&1\\ 0&2 pmatrix. Here, (1,1)(1,1) is payoff-dominant because 4>24>2, and action 11 is also maximin because min4,1=1>0=min0,2. \4,1\=1>0= \0,2\. Hence, (1,1)(1,1) is super-dominant. Theorems 5 and 6 imply that (1,1)(1,1) is stochastically stable for every risk-sensitivity parameter β, under both RS-BRM and RS-logit, once the population is sufficiently large. On the other hand, when the payoff-dominant and maximin equilibria differ, there is a fundamental tension between efficiency and robustness. In this case, the risk-sensitivity parameter can act as a knob for equilibrium selection: as agents become more risk-seeking, the payoff-dominant equilibrium, which Pareto-dominates the other coordinated equilibrium and yields higher coordinated payoffs, becomes stochastically stable; whereas as agents become more risk-averse, the maximin equilibrium, which provides better protection against worst-case opponent actions, becomes stochastically stable. For example, consider the symmetric coordination game M=(4023).M= pmatrix4&0\\ 2&3 pmatrix. Here, (1,1)(1,1) is payoff-dominant because 4>34>3, while (2,2)(2,2) is maximin because min4,0=0<2=min2,3. \4,0\=0<2= \2,3\. Thus, efficiency and robustness point to different equilibria. Theorems 5 and 6 imply that, under both RS-BRM and RS-logit, sufficiently risk-averse agents select the maximin equilibrium (2,2)(2,2), whereas sufficiently risk-seeking agents select the payoff-dominant equilibrium (1,1)(1,1), for all sufficiently large populations. The cutoff value of the risk-sensitivity parameter may differ across the two protocols, but the transition is the same: increasing β shifts long-run selection from the “safer” and more robust maximin equilibrium to the more “efficient” payoff-dominant equilibrium. These insights align with recent empirical observations in Noorani and Baras (2022); Zhang et al. (2025) that risk-seeking learning agents can improve coordination efficiency, and our results may be viewed as a formalization of these insights through the lens of coordination games and evolutionary dynamics. 5 Extension: Two-Population Asymmetric Setting We now consider a two-population setting with an asymmetric coordination game. This setting is natural because many coordination problems involve distinct agent roles with different payoffs, such as the Battle of the Sexes Osborne and Rubinstein (1994). Notably, this setting also allows for heterogeneous risk attitudes: agents are homogeneous within each population, sharing a common risk-sensitivity parameter βpβ^p, while the two populations may have different risk parameters, so β1β^1 and β2β^2 need not be equal. Moving from one to two populations requires more than a change in notation. In the single-population model, the aggregate state is one-dimensional, and a single risk-dependent threshold separates the two best-response regions, yielding closed-form escape costs. In the two-population model, the state is two-dimensional, and each population’s best response depends on the state of the other. Although BRM remains tractable, transition costs under logit dynamics are state- and path-dependent, requiring a genuinely two-dimensional analysis. We characterize stochastic selection under super-dominance, while the general risk-sensitive two-population logit case remains open. In the two-population setting, the aggregate population state is x=(x1,x2)∈XN,x=(x^1,x^2)∈ X^N, where xpx^p is the fraction of population p playing action 11. A one-step update changes the action of only one agent in one population. Let e1=(1,0)e_1=(1,0) and e2=(0,1)e_2=(0,1). Then, for p∈1,2p∈\1,2\, the nontrivial feasible one-step transitions are x→x+1Npep,x→x−1Npep,x→ x+ 1N^pe_p, x→ x- 1N^pe_p, whenever the resulting state remains in XNX^N. Here, x+1Npepx+ 1N^pe_p corresponds to an agent in population p switching from action 22 to action 11, while x−1Npepx- 1N^pe_p corresponds to an agent in population p switching from action 11 to action 22. Applying the small-noise cost definition in (5) to the two-population state space, we write the corresponding transition costs as κNκ^N, which may also be denoted as κN1,N2κ^N^1,N^2 when we want to emphasize the two population sizes. Recalling the risk-sensitive revision protocol (3) and the cost function c(⋅)c(·) from Assumption 1, we obtain κN(x,x+1Npep)=c((−Δβp(x−p))+),κN(x,x−1Npep)=c((Δβp(x−p))+). κ^N\! (x,x+ 1N^pe_p )=c\! ((- _β^p^p(x^-p))_+ ),~κ^N\! (x,x- 1N^pe_p )=c\! (( _β^p^p(x^-p))_+ ). (12) Next, let xβ12,∗∈(0,1)x_β^1^2,*∈(0,1) and xβ21,∗∈(0,1)x_β^2^1,*∈(0,1) be the unique thresholds from Lemma 1, defined by Δβ11(xβ12,∗)=0,Δβ22(xβ21,∗)=0. _β^1^1\! (x_β^1^2,* )=0, _β^2^2\! (x_β^2^1,* )=0. Define the associated grid points by x¯1:=⌊N1xβ21,∗⌋N1,x¯1:=⌈N1xβ21,∗⌉N1, x^1:= N^1x_β^2^1,* N^1, x^1:= N^1x_β^2^1,* N^1, and x¯2:=⌊N2xβ12,∗⌋N2,x¯2:=⌈N2xβ12,∗⌉N2. x^2:= N^2x_β^1^2,* N^2, x^2:= N^2x_β^1^2,* N^2. For p=1,2p=1,2, let x¯p,− x^p,- and x¯p,+ x^p,+ denote the grid points immediately below and above xβ−p,∗x_β^-p^p,*, respectively. Thus, if Npxβ−p,∗∉ℤN^px_β^-p^p,* , then x¯p,−=x¯p,x¯p,+=x¯p. x^p,-= x^p, x^p,+= x^p. If Npxβ−p,∗∈ℤN^px_β^-p^p,* , then x¯p,−=x¯p−1Np,x¯p,+=x¯p+1Np. x^p,-= x^p- 1N^p, x^p,+= x^p+ 1N^p. The next proposition verifies Assumption 2. Figure 1: Zero-noise geometry for the two-population dynamics when the thresholds do not lie on the population grids, i.e., Npxβ−p,∗∉ℤN^px_β^-p^p,* for p=1,2p=1,2. The lower-left and upper-right rectangles are the stability sets ℬ0N(A)B_0^N(A) and ℬ0N(B)B_0^N(B), respectively, and the highlighted boundary segments indicate the exit boundaries ∂outN(A) _out^N(A) and ∂outN(B) _out^N(B). Proposition 7. Under the two-population risk-sensitive dynamics induced by (3), with transition costs given by (12), the zero-noise dynamics have exactly two attracting states, namely A=(0,0)A=(0,0) and B=(1,1)B=(1,1). With N=(N1,N2)N=(N^1,N^2), their stability sets are ℬ0N(A)=([0,x¯1,−]×[0,x¯2,−])∩XN,ℬ0N(B)=([x¯1,+,1]×[x¯2,+,1])∩XN, _0^N(A)= ([0, x^1,-]×[0, x^2,-] )∩ X^N, _0^N(B)= ([ x^1,+,1]×[ x^2,+,1] )∩ X^N, and their exit boundaries are ∂outN(A) _out^N(A) =[(x¯1×[0,x¯2,−])∪([0,x¯1,−]×x¯2)]∩XN, = [ (\ x^1\×[0, x^2,-] )\;∪\; ([0, x^1,-]×\ x^2\ ) ]∩ X^N, ∂outN(B) _out^N(B) =[(x¯1×[x¯2,+,1])∪([x¯1,+,1]×x¯2)]∩XN. = [ (\ x^1\×[ x^2,+,1] )\;∪\; ([ x^1,+,1]×\ x^2\ ) ]∩ X^N. Consequently, Assumption 2 holds. The geometry is illustrated in Fig. 1. Proof. By Lemma 1, for each p∈1,2p∈\1,2\, Δβp(z)<0if z<xβp−p,∗,Δβp(z)>0if z>xβp−p,∗, _β^p^p(z)<0 z<x_β^p^-p,*, _β^p^p(z)>0 z>x_β^p^-p,*, and Δβp(z)=0 _β^p^p(z)=0 at z=xβp−p,∗z=x_β^p^-p,*. Hence, by the two transition-cost identities in (12), if x−p<xβp−p,∗x^-p<x_β^p^-p,*, then decreasing xpx^p by 1/Np1/N^p has zero cost; if x−p>xβp−p,∗x^-p>x_β^p^-p,*, then increasing xpx^p by 1/Np1/N^p has zero cost; and if x−p=xβp−p,∗x^-p=x_β^p^-p,*, then both moves have zero cost. Define E00:=([0,x¯1,−]×[0,x¯2,−])∩XN,E11:=([x¯1,+,1]×[x¯2,+,1])∩XN.E_00:= ([0, x^1,-]×[0, x^2,-] )∩ X^N, E_11:= ([ x^1,+,1]×[ x^2,+,1] )∩ X^N. Every state in E00E_00 reaches A=(0,0)A=(0,0) along repeated downward zero-cost moves. Moreover, no state in E00E_00 has a zero-cost path to XN∖E00X^N E_00: any first exit from E00E_00 must either increase x1x^1 at a state with x2≤x¯2,−<xβ12,∗x^2≤ x^2,-<x_β^1^2,*, or increase x2x^2 at a state with x1≤x¯1,−<xβ21,∗x^1≤ x^1,-<x_β^2^1,*, and each such move has strictly positive cost. Therefore, ℬ0N(A)=E00.B_0^N(A)=E_00. By the symmetric argument, every state in E11E_11 reaches B=(1,1)B=(1,1) along repeated upward zero-cost moves, and no state in E11E_11 has a zero-cost path to XN∖E11X^N E_11. Hence, ℬ0N(B)=E11.B_0^N(B)=E_11. We next show that all states outside these two stability sets are transient under the zero-noise dynamics. As depicted in Fig. 1, the sets E00E_00 and E11E_11 are the lower-left and upper-right rectangles, respectively. Consider any state x∉E00x∉ E_00. Then at least one coordinate is at or above its threshold. If x1≥x¯1x^1≥ x^1, then population 22 has a zero-cost upward move, so x2x^2 can be increased until it reaches the upper region. Once x2≥x¯2x^2≥ x^2, population 11 has zero-cost upward moves, and the state can reach B=(1,1)B=(1,1). The case x2≥x¯2x^2≥ x^2 is symmetric. Thus, every state outside E00E_00 has a zero-cost path to B. Similarly, every state outside E11E_11 has a zero-cost path to A. Indeed, if x1≤x¯1x^1≤ x^1, then population 22 has zero-cost downward moves, so x2x^2 can be decreased until it reaches the lower region; once x2≤x¯2x^2≤ x^2, population 11 has zero-cost downward moves, and the state can reach A=(0,0)A=(0,0). The case x2≤x¯2x^2≤ x^2 is symmetric. If a coordinate lies exactly on a threshold grid line, the corresponding population is indifferent, so both upward and downward moves in the other coordinate have zero cost; the same argument applies after one such zero-cost move. Therefore, every state in XN∖(E00∪E11)X^N (E_00∪ E_11) has zero-cost paths to both A and B. Since A and B themselves have no outgoing zero-cost transitions, they are the only recurrent states of the zero-noise dynamics, i.e., the only attracting states. The formulas for the exit boundaries follow directly from the descriptions of E00E_00 and E11E_11. From E00E_00, the first one-step exits occur by increasing either x1x^1 to x¯1 x^1 while x2≤x¯2,−x^2≤ x^2,-, or x2x^2 to x¯2 x^2 while x1≤x¯1,−x^1≤ x^1,-. This gives the stated expression for ∂outN(A) _out^N(A). The expression for ∂outN(B) _out^N(B) follows analogously by considering the first downward exit from E11E_11. Finally, Assumption 2 (I) holds because every state outside A,B\A,B\ has a zero-cost path to at least one of A or B. Assumption 2 (I) holds because every state outside ℬ0N(A)=E00B_0^N(A)=E_00 has a zero-cost path to B, and every state outside ℬ0N(B)=E11B_0^N(B)=E_11 has a zero-cost path to A. ∎ We define the escape costs from the two stability sets as γ00N:=CN((1,1),∂outN(B)),γ11N:=CN((0,0),∂outN(A)). _00^N:=C^N\! ((1,1), _out^N(B) ), _11^N:=C^N\! ((0,0), _out^N(A) ). (13) We also define the selection coefficient SN:=γ11N−γ00N.S^N:= _11^N- _00^N. By Theorem 3, the state (1,1)(1,1) is uniquely stochastically stable if SN<0S^N<0, whereas (0,0)(0,0) is uniquely stochastically stable if SN>0S^N>0. Theorem 8 (Stochastic stability under 2Pop-RS-BRM). Consider an asymmetric two-population coordination game under 2Pop-RS-BRM. For fixed N=(N1,N2)N=(N^1,N^2), the escape costs defined in (13) satisfy γ00N=minN1(1−x¯1),N2(1−x¯2),γ11N=minN1x¯1,N2x¯2. _00^N= \! \N^1(1- x^1),\,N^2(1- x^2) \, _11^N= \! \N^1 x^1,\,N^2 x^2 \. Moreover, if (1,1)(1,1) is super-dominant, then for every fixed pair (β1,β2)(β^1,β^2), there exists some N0(β1,β2)∈ℕN_0(β^1,β^2) such that (1,1)(1,1) is uniquely stochastically stable for all N1,N2≥N0(β1,β2)N^1,N^2≥ N_0(β^1,β^2). The analogous statement holds for (0,0)(0,0). Proof. By the definition of 2Pop-RS-BRM, the cost function in Assumption 1 is c(d)=d>0c(d)=1_\d>0\. Let B=(1,1)B=(1,1). By Proposition 7, the exit boundary of ℬ0N1,N2(B)B_0^N^1,N^2(B) is ∂outN1,N2(B)=[(x¯1×[x¯2,+,1])∪([x¯1,+,1]×x¯2)]∩XN1,N2. _out^N^1,N^2(B)= [ (\ x^1\×[ x^2,+,1] )\;∪\; ([ x^1,+,1]×\ x^2\ ) ]∩ X^N^1,N^2. Consider any path from B to ∂outN1,N2(B) _out^N^1,N^2(B), and truncate it at its first hitting time of ∂outN1,N2(B) _out^N^1,N^2(B). Since transition costs are nonnegative, this truncation cannot increase the path cost. Before the terminal step, the path lies in ℬ0N1,N2(B)B_0^N^1,N^2(B). If the truncated path ends in x¯1×[x¯2,+,1]\ x^1\×[ x^2,+,1], then its first coordinate must have decreased from 11 to x¯1 x^1. Hence, the path contains at least N1(1−x¯1)N^1(1- x^1) downward moves in population 11. Each such move starts from a state in ℬ0N1,N2(B)B_0^N^1,N^2(B), where x2≥x¯2,+>xβ12,∗x^2≥ x^2,+>x_β^1^2,*. Therefore, action 11 is the unique ERM-adjusted best response for population 11, and a downward move is a strict mistake with cost 11. Thus, the cost of such a path is at least N1(1−x¯1)N^1(1- x^1). Similarly, if the truncated path ends in [x¯1,+,1]×x¯2[ x^1,+,1]×\ x^2\, then it contains at least N2(1−x¯2)N^2(1- x^2) downward moves in population 22, each of cost 11. Therefore, every path from B to ∂outN1,N2(B) _out^N^1,N^2(B) has cost at least minN1(1−x¯1),N2(1−x¯2). \! \N^1(1- x^1),\,N^2(1- x^2) \. This lower bound is attained by the monotone path that decreases only the coordinate achieving the minimum, while keeping the other coordinate equal to 11. Hence γ00N1,N2=minN1(1−x¯1),N2(1−x¯2). _00^N^1,N^2= \! \N^1(1- x^1),\,N^2(1- x^2) \. The same argument applies to A=(0,0)A=(0,0). Any path from A to ∂outN1,N2(A) _out^N^1,N^2(A), truncated at its first hitting time of the exit boundary, must either increase x1x^1 from 00 to x¯1 x^1, or increase x2x^2 from 00 to x¯2 x^2. While the path remains in ℬ0N1,N2(A)B_0^N^1,N^2(A), each such upward move is a strict mistake under 2Pop-RS-BRM and has cost 11. The corresponding monotone coordinate paths attain these costs. Therefore γ11N1,N2=minN1x¯1,N2x¯2. _11^N^1,N^2= \! \N^1 x^1,\,N^2 x^2 \. Finally, suppose that (1,1)(1,1) is super-dominant for both populations. Then, Lemma 2 implies xβ21,∗<12,xβ12,∗<12.x_β^2^1,*< 12, x_β^1^2,*< 12. Since x¯p→xβ−p,∗ x^p→ x_β^-p^p,* and x¯p→xβ−p,∗ x^p→ x_β^-p^p,* as Np→∞N^p→∞, there exists N0(β1,β2)N_0(β^1,β^2) such that, for all N1,N2≥N0(β1,β2)N^1,N^2≥ N_0(β^1,β^2), x¯p<12<1−x¯p,p∈1,2. x^p< 12<1- x^p, p∈\1,2\. Consequently, Npx¯p<Np(1−x¯p),p∈1,2.N^p x^p<N^p(1- x^p), p∈\1,2\. Taking minima over p gives γ11N1,N2<γ00N1,N2. _11^N^1,N^2< _00^N^1,N^2. By Theorem 3, this implies that (1,1)(1,1) is uniquely stochastically stable. The proof for (0,0)(0,0) is symmetric. ∎ Theorem 8 shows that, under 2Pop-RS-BRM, equilibrium selection is determined by the relative magnitudes of the two escape costs γ00N1,N2 _00^N^1,N^2 and γ11N1,N2 _11^N^1,N^2. These escape costs depend on the ERM thresholds xβ21,∗x_β^2^1,* and xβ12,∗x_β^1^2,*, and hence on the risk-sensitivity parameters (β1,β2)(β^1,β^2). Therefore, changing agents’ risk attitudes can change the relative difficulty of escaping the two coordinated equilibria, and may consequently alter the stochastically stable equilibrium. We illustrate this risk-attitude tuning effect on equilibrium selection in the numerical example in Section 5.1. Theorem 9 (Stochastic stability under 2Pop-RS-logit). Consider an asymmetric two-population 2×22× 2 coordination game under 2Pop-RS-logit. Suppose that (1,1)(1,1) is super-dominant. Then, for every fixed pair of risk parameters (β1,β2)∈ℝ2(β^1,β^2) ^2, there exists N0(β1,β2)N_0(β^1,β^2) such that (1,1)(1,1) is uniquely stochastically stable for all N1,N2≥N0(β1,β2)N^1,N^2≥ N_0(β^1,β^2). The analogous statement holds for (0,0)(0,0). Proof. We prove the claim for (1,1)(1,1); the proof for (0,0)(0,0) is symmetric. Fix (β1,β2)∈ℝ2(β^1,β^2) ^2, and write A:=(0,0)A:=(0,0), B:=(1,1)B:=(1,1), and X:=XN1,N2X:=X^N^1,N^2 for notational convenience. Since (1,1)(1,1) is super-dominant for each population, Lemma 2 gives xβ21,∗<12,xβ12,∗<12.x_β^2^1,*< 12, x_β^1^2,*< 12. Therefore, for all sufficiently large N1,N2N^1,N^2, we have x¯p,+<1−x¯p,−,p∈1,2. x^p,+<1- x^p,-, p∈\1,2\. Fix such N1,N2N^1,N^2. Recall from Proposition 7 that ℬ0N1,N2(B)=([x¯1,+,1]×[x¯2,+,1])∩X.B_0^N^1,N^2(B)= ([ x^1,+,1]×[ x^2,+,1] )∩ X. Inside this stability set, define the rectangle ℛ:=([1−x¯1,−,1]×[1−x¯2,−,1])∩X.R:= ([1- x^1,-,1]×[1- x^2,-,1] )∩ X. By the preceding inequalities, ℛ⊆ℬ0N1,N2(B)R _0^N^1,N^2(B). Define the south and west exit layer of ℛR by ∂ℛ:=[(1−x¯1×[1−x¯2,−,1])∪([1−x¯1,−,1]×1−x¯2)]∩X. := [ (\1- x^1\×[1- x^2,-,1] )∪ ([1- x^1,-,1]×\1- x^2\ ) ]∩ X. The rectangle ℛR and its exit layer ∂ℛ are shown in Fig. 1. Every path from B∈ℛB to ∂outN1,N2(B) _out^N^1,N^2(B) must leave ℛR, because ∂outN1,N2(B)∩ℛ=∅ _out^N^1,N^2(B) = . If such a path is truncated at its first exit from ℛR, then the terminal state of the truncated path lies in ∂ℛ . Since all one-step costs are nonnegative, truncation cannot increase the cost. Hence, γ00N1,N2=CN1,N2(B,∂outN1,N2(B))≥CN1,N2(B,∂ℛ). _00^N^1,N^2=C^N^1,N^2 (B, _out^N^1,N^2(B) )≥ C^N^1,N^2 (B, ). Now define the reflection map T:X→XT:X→ X by T(x1,x2):=(1−x1,1−x2).T(x^1,x^2):=(1-x^1,1-x^2). Then, we have T(ℛ)=([0,x¯1,−]×[0,x¯2,−])∩X=ℬ0N1,N2(A),T(R)= ([0, x^1,-]×[0, x^2,-] )∩ X=B_0^N^1,N^2(A), and T(∂ℛ)=[(x¯1×[0,x¯2,−])∪([0,x¯1,−]×x¯2)]∩X=∂outN1,N2(A).T( )= [ (\ x^1\×[0, x^2,-] )∪ ([0, x^1,-]×\ x^2\ ) ]∩ X= _out^N^1,N^2(A). Let ψ=(ψ0,…,ψK)ψ=( _0,…, _K) be a minimum-cost path from (1,1)(1,1) to ∂ℛ , stopped at its first visit to this layer. Hence, ψi∈ℛ _i for every i<Ki<K. Define the reflected path ϕ=(ϕ0,…,ϕK)φ=( _0,…, _K) by ϕi:=T(ψi),i=0,…,K. _i:=T( _i),i=0,…,K. Then ϕφ is a feasible path from A to ∂outN1,N2(A) _out^N^1,N^2(A), and ϕi∈ℬ0N1,N2(A) _i _0^N^1,N^2(A) for every i<Ki<K. We compare the two path costs step by step. Suppose first that a step of ψ moves left from (u,v)∈ℛ(u,v) . Then v≥1−x¯2,−>xβ12,∗v≥ 1- x^2,->x_β^1^2,* and 1−v≤x¯2,−<xβ12,∗.1-v≤ x^2,-<x_β^1^2,*. Hence, the original left step has cost Δβ11(v) _β^1^1(v), whereas its reflected right step has cost −Δβ11(1−v)- _β^1^1(1-v). By Lemma 13, Δβ11(v)>−Δβ11(1−v). _β^1^1(v)>- _β^1^1(1-v). Similarly, a down step from (u,v)∈ℛ(u,v) has cost Δβ22(u) _β^2^2(u), whereas its reflected up step has cost −Δβ22(1−u)- _β^2^2(1-u), and Δβ22(u)>−Δβ22(1−u). _β^2^2(u)>- _β^2^2(1-u). If a step of ψ instead moves right or up, it has zero cost because its starting state lies in ℛR. Its reflected left or down step also has zero cost because its starting state lies in T(ℛ)=ℬ0N1,N2(A)T(R)=B_0^N^1,N^2(A). Thus, reflection strictly reduces the cost of every left or down step and preserves the zero cost of every right or up step. Since ψ must contain at least one left or down step to reach ∂ℛ , we obtain CN1,N2(ϕ)<CN1,N2(ψ)=CN1,N2((1,1),∂ℛ).C^N^1,N^2(φ)<C^N^1,N^2(ψ)=C^N^1,N^2 ((1,1), ). Since ϕφ is a path from A to ∂outN1,N2(A) _out^N^1,N^2(A), γ11N1,N2≤CN1,N2(ϕ)<CN1,N2((1,1),∂ℛ)≤γ00N1,N2. _11^N^1,N^2≤ C^N^1,N^2(φ)<C^N^1,N^2 ((1,1), )≤ _00^N^1,N^2. Thus, by Theorem 3, (1,1)(1,1) is uniquely stochastically stable. ∎ Theorems 8 and 9 show that the robustness result from the single-population setting extends to the two-population setting. Under both revision protocols, a super-dominant equilibrium remains uniquely stochastically stable for all fixed risk-attitude profiles (β1,β2)(β^1,β^2), provided the two population sizes are sufficiently large. Thus, even when the two populations have heterogeneous risk preferences, long-run equilibrium selection is pinned down by the payoff structure rather than by the particular risk attitudes entering the learning rule. The robustness of super-dominant equilibria is natural and useful because super-dominance aligns payoff dominance with downside protection. Such an equilibrium is attractive both to risk-seeking agents, who emphasize high-payoff outcomes, and to risk-averse agents, who emphasize worst-case guarantees. Therefore, heterogeneity in the risk parameters does not change the direction of long-run selection: the payoff structure itself keeps the dynamics anchored at the super-dominant equilibrium. 5.1 Numerical Illustration Figure 2: 2Pop-RS-BRM as β1β^1 varies, for three fixed values of β2β^2. Top: finite-time coordination frequencies with pointwise 95%95\% Wilson intervals. Bottom: finite-population escape costs; shading marks equal-cost regions. We illustrate the dependence of finite-time coordination frequencies and finite-population escape costs on risk attitudes under 2Pop-RS-BRM. Consider the asymmetric coordination game M1=(7144),M2=(5416).M^1= pmatrix7&1\\ 4&4 pmatrix, M^2= pmatrix5&4\\ 1&6 pmatrix. This is a Battle-of-the-Sexes-type game Osborne and Rubinstein (1994): population 1 receives a higher coordinated payoff at (1,1)(1,1), whereas population 2 receives a higher coordinated payoff at (0,0)(0,0). Thus, the two populations have conflicting payoff preferences over the two coordinated equilibria. We fix N1=N2=100N^1=N^2=100, η=0.1η=0.1, and the initial state x01=x02=0.5x_0^1=x_0^2=0.5. For each β2∈−0.5,0,1β^2∈\-0.5,0,1\, we use βj1=−2+4j/29 _j^1=-2+4j/29, j=0,…,29j=0,…,29, and run 250250 independent Monte Carlo simulations per parameter pair. Each run lasts exactly T=1500T=1500 individual revision opportunities. We classify a terminal state as near (1,1)(1,1) if both coordinates are at least 0.950.95, as near (0,0)(0,0) if both are at most 0.050.05, and otherwise as “other.” The top panel of Fig. 2 plots p^11,T=1250∑r=1250xT,r1≥0.95,xT,r2≥0.95. p_11,T= 1250 _r=1^2501_\x_T,r^1≥ 0.95,\;x_T,r^2≥ 0.95\. Error bars are pointwise 95%95\% Wilson score intervals for this terminal-event probability; near a frequency of 0.50.5, their half-width is approximately 0.0620.062. The bottom panel plots the integer escape costs γ11N1,N2 _11^N^1,N^2 and γ00N1,N2 _00^N^1,N^2 from Theorem 8. These costs describe stochastic stability in the vanishing-noise limit at fixed population sizes: (1,1)(1,1) is uniquely stochastically stable when γ11N1,N2<γ00N1,N2 _11^N^1,N^2< _00^N^1,N^2, and (0,0)(0,0) is uniquely stochastically stable under the reverse strict inequality. The terminal frequencies show how risk attitudes affect coordination from the specified initial state over this fixed horizon. For each fixed value of β2β^2, the observed transition occurs near the parameter interval where the two escape costs tie. The direction of this transition also has a clear payoff interpretation for population 1. As β1β^1 increases, population 1 becomes more risk-seeking, and more runs end near (1,1)(1,1), the coordinated equilibrium with the higher payoff for population 1. Conversely, as β1β^1 decreases, population 1 becomes more risk-averse, and more runs end near (0,0)(0,0), which corresponds to its maximin choice. Thus, changing only population 1’s risk attitude can change the relative difficulty of escaping the two coordinated equilibria and shift finite-time coordination from its maximin equilibrium toward the equilibrium with the higher payoff. Moreover, the location of this transition depends on population 2’s risk attitude, showing that finite-time coordination is shaped jointly by the risk attitudes of both populations. 6 Extension: k-Action Setting We now extend the single-population analysis from two actions to symmetric games with an arbitrary finite number of actions. 6.1 Model and Setup Let S=1,…,kS=\1,…,k\, with k≥2k≥ 2, denote the common action set, and let A=(aij)i,j∈SA=(a_ij)_i,j∈ S be the payoff matrix, where aija_ij is the payoff to an agent playing action i against an opponent playing action j. For a population of size N, the finite state space is ΔkN:=x∈1Nℤ+k:∑i=1kxi=1, _k^N:= \x∈ 1NZ_+^k: _i=1^kx_i=1 \, where xix_i denotes the fraction of agents currently playing action i. In a symmetric k-action game, a diagonal action profile (i,i)(i,i) is a strict Nash equilibrium if action i is the unique best response to itself, that is, aii>aji,∀j∈S∖i.a_i>a_ji, ∀ j∈ S \i\. The corresponding population state to this equilibrium is the vertex ei∈ΔkNe_i∈ _k^N, where all agents play action i. Definition 12 (k-action coordination games). A symmetric k-action game is a k-action coordination game if every diagonal action profile (i,i)(i,i), i∈Si∈ S, is a strict Nash equilibrium; equivalently, aii>aji,∀i∈S,∀j∈S∖i.a_i>a_ji, ∀ i∈ S,\ ∀ j∈ S \i\. Before proceeding to the stochastic-stability analysis, we introduce two refinement notions for diagonal Nash equilibria. Definition 13 (Payoff dominance). In a symmetric k-action coordination game, a strict coordinated equilibrium (i,i)(i,i) is payoff-dominant if its coordinated payoff is strictly larger than the coordinated payoff of every other strict coordinated equilibrium: aii>ass,∀s∈S∖i.a_i>a_s, ∀ s∈ S \i\. Definition 14 (Strong payoff dominance). In a symmetric k-action game, a diagonal profile (i,i)(i,i) is strongly payoff-dominant if aii>maxs∈S∖imaxr∈Sasr.a_i> _s∈ S \i\ _r∈ Sa_sr. In particular, any strongly payoff-dominant diagonal profile is a strict Nash equilibrium, because the above condition implies aii>aji,∀j∈S∖i.a_i>a_ji, ∀ j∈ S \i\. Strong payoff dominance requires the diagonal payoff aiia_i to exceed not only the diagonal payoffs assa_s of all other actions s≠is≠ i, but also all payoffs obtainable by any action s≠is≠ i against any opponent action r∈Sr∈ S. In a k-action coordination game, this condition is equivalent to payoff dominance. Indeed, suppose (i,i)(i,i) is payoff-dominant, so that aii>arra_i>a_r for every r≠ir≠ i. Since the game is a coordination game, each diagonal profile (r,r)(r,r) is a strict Nash equilibrium. Hence, we have arr>asr,∀s≠r.a_r>a_sr, ∀ s≠ r. Therefore, for every s≠is≠ i and every r∈Sr∈ S, we have aii>asra_i>a_sr: if r=ir=i, this follows from the strictness of (i,i)(i,i); if r≠ir≠ i, it follows from aii>arr>asra_i>a_r>a_sr when s≠rs≠ r and aii>arr=asra_i>a_r=a_sr when s=rs=r. Thus, payoff dominance implies strong payoff dominance. The converse is immediate by taking r=sr=s. Hence, in k-action coordination games, payoff dominance and strong payoff dominance coincide. The strong formulation is nevertheless useful because it is defined directly in general k-action symmetric games, and can be applied beyond the coordination-game structure. We also introduce two maximin-type refinements that will be used later. Definition 15 (Strict maximin). In a symmetric k-action game, action i∈Si∈ S is strictly maximin if it strictly maximizes the worst-case payoff: minr∈Sair>minr∈Sasr,∀s∈S∖i. _r∈ Sa_ir> _r∈ Sa_sr, ∀ s∈ S \i\. If, in addition, (i,i)(i,i) is a strict Nash equilibrium, then we call the diagonal equilibrium (i,i)(i,i) strictly maximin. Definition 16 (Strong maximin). In a symmetric k-action game, action i∈Si∈ S is strongly maximin if minr∈Sair>maxs∈S∖iasi. _r∈ Sa_ir> _s∈ S \i\a_si. In this case, the diagonal profile (i,i)(i,i) is a strict Nash equilibrium, because aii≥minr∈Sair>maxs∈S∖iasi≥aji,∀j∈S∖i.a_i≥ _r∈ Sa_ir> _s∈ S \i\a_si≥ a_ji, ∀ j∈ S \i\. Strong maximin is a stronger condition than strict maximin. To see this, suppose action i is strongly maximin. Then, for every s≠is≠ i, minr∈Sasr≤asi≤maxs′∈S∖ias′i<minr∈Sair. _r∈ Sa_sr≤ a_si≤ _s ∈ S \i\a_s i< _r∈ Sa_ir. Hence, action i strictly maximizes the worst-case payoff, and is thus strictly maximin. The stronger formulation is useful as it compares the worst-case payoff of action i with the best payoff that any alternative action can obtain when the opponent plays i. 6.2 Risk-Sensitive Evolutionary Dynamics We now introduce the risk-sensitive evolutionary dynamics for the single-population k-action setting. To keep the analysis focused, we consider the best-response-with-mutations protocol, and refer to the resulting dynamics as RS-BRM, consistent with the two-action case. Given a state x∈ΔkNx∈ _k^N, the ERM-adjusted payoff of action i∈Si∈ S is Fβ,i(x)=1βlog(∑ℓ=1kxℓeβaiℓ),β≠0,F_β,i(x)= 1β \! ( _ =1^kx_ e^β a_i ), β≠ 0, with the continuous risk-neutral extension F0,i(x)=∑ℓ=1kxℓaiℓ.F_0,i(x)= _ =1^kx_ a_i . We write Fβ(x):=(Fβ,1(x),…,Fβ,k(x)) F_β(x):= (F_β,1(x),…,F_β,k(x) ) for the vector of ERM-adjusted payoffs at state x. The corresponding ERM-adjusted best-response set is BRβ(x):=argmaxj∈SFβ,j(x). _β(x):= _j∈ SF_β,j(x). At each time step, one agent is selected uniformly at random and receives a revision opportunity. Under RS-BRM, the revising agent chooses an ERM-adjusted best response with probability 1−e−1/η1-e^-1/η, and chooses a non-best-response action with probability e−1/ηe^-1/η, where η>0η>0 is the noise level. More precisely, if BRβ(x)≠SBR_β(x)≠ S, then the probability of choosing action j∈Sj∈ S is qjη(x)=1−e−1/η|BRβ(x)|,j∈BRβ(x),e−1/ηk−|BRβ(x)|,j∉BRβ(x).q_j^η(x)= cases 1-e^-1/η|BR_β(x)|,&j _β(x),\\[5.16663pt] e^-1/ηk-|BR_β(x)|,&j _β(x). cases If all actions are tied, i.e., BRβ(x)=SBR_β(x)=S, we set qjη(x)=1k,j∈S.q_j^η(x)= 1k, j∈ S. Thus, in the vanishing-noise limit, only ERM-adjusted best responses are chosen with non-exponentially small probability. Since only one agent revises at a time, the feasible nontrivial one-step transitions are x⟶x+1N(ej−ei),i,j∈S,i≠j,xi>0,x x+ 1N(e_j-e_i), i,j∈ S, i≠ j, x_i>0, where an agent currently playing action i switches to action j. The corresponding one-step transition probability is PN,η(x,x+1N(ej−ei))=xiqjη(x).P^N,η\! (x,x+ 1N(e_j-e_i) )=x_iq_j^η(x). Here, xix_i is the probability that the selected revising agent currently plays action i, and qjη(x)q_j^η(x) is the probability that this agent chooses action j after revision. Because xi>0x_i>0 is independent of η, it does not contribute to the small-noise exponential cost. Therefore, the cost of a feasible nontrivial transition is κN(x,x+1N(ej−ei))=0,j∈BRβ(x),1,j∉BRβ(x). κ^N\! (x,x+ 1N(e_j-e_i) )= cases0,&j _β(x),\\ 1,&j _β(x). cases (14) Thus, under RS-BRM, transitions to ERM-adjusted best-response actions have zero cost, whereas transitions to actions outside the ERM-adjusted best-response set have unit cost. The associated zero-noise dynamics are defined by the directed graph on ΔkN _k^N whose edges are precisely the zero-cost transitions in (14); equivalently, these are the one-agent revisions to ERM-adjusted best responses. 6.3 Analyses We now analyze long-run equilibrium selection under RS-BRM in the single-population k-action setting. In this section, the finite state space in Definition 11 is XN=ΔkNX^N= _k^N, and the zero-noise dynamics are the directed graph generated by zero-cost RS-BRM transitions, equivalently, by one-agent revisions to ERM-adjusted best responses. For any attracting state A∈ΔkNA∈ _k^N, we use ℬ0N(A)B_0^N(A) and ∂outN(A) _out^N(A) as defined in Definition 11. Definition 17 (Radius and coradius Ellison (2000)). Let A∈ΔkNA∈ _k^N be an attracting state of the zero-noise RS-BRM dynamics. The radius of A is RN(A):=CN(A,∂outN(A)),R^N(A):=C^N\! (A, _out^N(A) ), and the coradius of A is CRN(A):=maxx∈ΔkN∖ℬ0N(A)CN(x,ℬ0N(A)).CR^N(A):= _x∈ _k^N _0^N(A)C^N\! (x,B_0^N(A) ). We use the following radius–coradius criterion to identify stochastically stable states for the risk-sensitive RS-BRM dynamics. Theorem 10 (Radius–coradius criterion for RS-BRM). Fix N, and consider the Markov chain XN,ηX^N,η induced by RS-BRM on the finite state space ΔkN _k^N. Let A∈ΔkNA∈ _k^N be an attracting state of the zero-noise RS-BRM dynamics. If RN(A)>CRN(A),R^N(A)>CR^N(A), then A is the unique stochastically stable state. Proof. For every η>0η>0, the RS-BRM learning dynamics define an irreducible finite-state Markov chain. As η→0η→ 0, this family of Markov chains is a regular perturbation of the zero-noise RS-BRM dynamics. Indeed, for a feasible transition x→x+1N(ej−ei),i≠j,xi>0,x→ x+ 1N(e_j-e_i), i≠ j, x_i>0, the transition probability is xiqjη(x)x_iq_j^η(x). Since xi>0x_i>0 is independent of η, it does not affect the exponential rate. Hence, the resistance of the transition is 00 if j∈BRβ(x)j _β(x), and is 11 if j∉BRβ(x)j _β(x). In particular, moves to ERM-adjusted best responses, including moves along payoff ties, have zero resistance, while moves to non-best-response actions have unit resistance. Thus, the unperturbed dynamics in Ellison’s framework coincide with the zero-noise RS-BRM dynamics defined above. Taking Ω=A =\A\, Ellison’s basin, radius, and coradius coincide respectively with ℬ0N(A)B_0^N(A), RN(A)R^N(A), and CRN(A)CR^N(A). The claim then follows from the radius–coradius theorem of Ellison (Ellison, 2000, Theorem 1). ∎ Theorem 11. Consider the single-population risk-sensitive evolutionary dynamics under RS-BRM in a finite symmetric k-action game with action set S=1,…,kS=\1,…,k\. Let ei∈ΔkNe_i∈ _k^N denote the state in which all agents play action i∈Si∈ S. Suppose first that the diagonal profile (i,i)(i,i) is strongly payoff-dominant, and define δi:=aii−maxs∈S∖imaxr∈Sasr>0. _i:=a_i- _s∈ S \i\ _r∈ Sa_sr>0. Let β¯i:=log2δi β_i:= 2 _i. Then, for every β>β¯iβ> β_i, there exists N0(β)N_0(β) such that, for every N≥N0(β)N≥ N_0(β), the state eie_i is uniquely stochastically stable under RS-BRM. Similarly, suppose that the diagonal profile (i,i)(i,i) is strongly maximin, and define δimm:=minr∈Sair−maxs∈S∖iasi>0. _i^m:= _r∈ Sa_ir- _s∈ S \i\a_si>0. Let β¯i:=−log2δimm. β_i:=- 2 _i^m. Then, for every β<β¯iβ< β_i, there exists N0(β)N_0(β) such that, for every N≥N0(β)N≥ N_0(β), the state eie_i is uniquely stochastically stable under RS-BRM. Proof. We prove the two claims using the same argument. Let Δk:=x∈ℝ+k:∑r=1kxr=1 _k:= \x _+^k: _r=1^kx_r=1 \ denote the continuous simplex. We first show that, in either case, there exists q<1/2q<1/2 such that action i is the unique ERM-adjusted best response at every state x∈Δkx∈ _k satisfying xi≥qx_i≥ q. The desired statement then applies in particular to all finite-population states x∈ΔkN⊂Δkx∈ _k^N⊂ _k. First, consider the strong payoff-dominance case and fix β>β¯iβ> β_i. Then, we have e−βδi<12.e^-β _i< 12. Choose q such that e−βδi<q<12.e^-β _i<q< 12. Equivalently, aii+1βlogq>aii−δi.a_i+ 1β q>a_i- _i. Now take any x∈Δkx∈ _k with xi≥qx_i≥ q. Since β>0β>0, Fβ,i(x)=1βlog(∑r∈Sxreβair)≥1βlog(xieβaii)=aii+1βlogxi≥aii+1βlogq.F_β,i(x)= 1β \! ( _r∈ Sx_re^β a_ir )≥ 1β \! (x_ie^β a_i )=a_i+ 1β x_i≥ a_i+ 1β q. Therefore, Fβ,i(x)>aii−δi.F_β,i(x)>a_i- _i. On the other hand, for every s≠is≠ i, ∑r∈Sxreβasr≤∑r∈Sxreβmaxr′∈Sasr′=eβmaxr′∈Sasr′. _r∈ Sx_re^β a_sr≤ _r∈ Sx_re^β _r ∈ Sa_sr =e^β _r ∈ Sa_sr . Thus, Fβ,s(x)≤maxr∈Sasr≤aii−δi.F_β,s(x)≤ _r∈ Sa_sr≤ a_i- _i. Hence, Fβ,i(x)>Fβ,s(x),∀s≠i.F_β,i(x)>F_β,s(x), ∀ s≠ i. So action i is the unique ERM-adjusted best response whenever xi≥qx_i≥ q. Now consider the strong maximin case and fix β<β¯iβ< β_i. Then, we have eβδimm<12.e^β _i^m< 12. Choose q such that eβδimm<q<12.e^β _i^m<q< 12. Since β<0β<0, this implies that 1βlogq<δimm. 1β q< _i^m. Let mi:=minr∈Sair.m_i:= _r∈ Sa_ir. For any x∈Δkx∈ _k, since air≥mia_ir≥ m_i for all r∈Sr∈ S and β<0β<0, we have eβair≤eβmi,∀r∈S.e^β a_ir≤ e^β m_i, ∀ r∈ S. Therefore, ∑r∈Sxreβair≤eβmi. _r∈ Sx_re^β a_ir≤ e^β m_i. Taking logarithms and then dividing by the negative number β reverses the inequality, yielding Fβ,i(x)≥mi.F_β,i(x)≥ m_i. Now fix any s≠is≠ i. Since all terms in the sum are nonnegative, ∑r∈Sxreβasr≥xieβasi. _r∈ Sx_re^β a_sr≥ x_ie^β a_si. Taking logarithms and dividing by β<0β<0, we get Fβ,s(x)≤asi+1βlogxi.F_β,s(x)≤ a_si+ 1β x_i. Since xi≥qx_i≥ q and β<0β<0, 1βlogxi≤1βlogq. 1β x_i≤ 1β q. Moreover, by the definition of δimm _i^m, asi≤maxs′≠ias′i=mi−δimm.a_si≤ _s ≠ ia_s i=m_i- _i^m. Combining these inequalities gives Fβ,s(x)≤mi−δimm+1βlogq<mi≤Fβ,i(x).F_β,s(x)≤ m_i- _i^m+ 1β q<m_i≤ F_β,i(x). Therefore, Fβ,i(x)>Fβ,s(x),∀s≠i.F_β,i(x)>F_β,s(x), ∀ s≠ i. Thus, in both cases, there exists q<1/2q<1/2 such that action i is the unique ERM-adjusted best response whenever xi≥qx_i≥ q. Now define HqN:=x∈ΔkN:xi≥q. H_q^N:=\x∈ _k^N:x_i≥ q\. Since action i is the unique ERM-adjusted best response at every state in HqNH_q^N, any one-agent revision from an action s≠is≠ i to action i has zero cost. Repeating such revisions gives a zero-cost path from every state in HqNH_q^N to eie_i. Moreover, no zero-cost path can leave HqNH_q^N. Indeed, leaving HqNH_q^N requires a transition that decreases xix_i from a state with xi≥qx_i≥ q to a state with xi<qx_i<q. Such a transition switches an agent away from action i, even though action i is the unique ERM-adjusted best response, and hence has cost 11 under RS-BRM. Therefore, HqN⊆ℬ0N(ei). H_q^N _0^N(e_i). In particular, eie_i is an attracting state of the zero-noise RS-BRM dynamics. If ℬ0N(ei)=ΔkNB_0^N(e_i)= _k^N, then eie_i is the unique recurrent class of the zero-noise RS-BRM dynamics, and the conclusion follows directly. Otherwise, since HqN⊆ℬ0N(ei)H_q^N _0^N(e_i), any path from eie_i to ∂outN(ei) _out^N(e_i) must first leave HqNH_q^N. Starting from eie_i, this requires decreasing xix_i from 11 to a value strictly below q. Hence, the path must contain at least N−⌈qN⌉+1 N- qN +1 unit-cost revisions away from action i. Therefore, RN(ei)≥N−⌈qN⌉+1. R^N(e_i)≥ N- qN +1. Conversely, from any state outside ℬ0N(ei)B_0^N(e_i), one can reach HqN⊆ℬ0N(ei)H_q^N _0^N(e_i) by revising agents to action i one at a time. Starting from any state, at most ⌈qN⌉ qN such revisions are needed to make xi≥qx_i≥ q. Each such revision has cost at most 11 under RS-BRM. Hence, CRN(ei)≤⌈qN⌉.CR^N(e_i)≤ qN . Since q<1/2q<1/2, there exists N0(β)N_0(β) such that, for every N≥N0(β)N≥ N_0(β), N−⌈qN⌉+1>⌈qN⌉.N- qN +1> qN . For such N, we have RN(ei)>CRN(ei).R^N(e_i)>CR^N(e_i). By Theorem 10, eie_i is the unique stochastically stable state. This completes the proof. ∎ We next provide a numerical illustration of how risk attitudes can affect the long-run equilibrium selection in the k-action setting. 6.4 Numerical Illustration We illustrate Theorem 11 using the following symmetric four-action game: A=(−10−10−100−1010−10044702225).A= pmatrix-10&-10&-10&0\\ -10&10&-10&0\\ 4&4&7&0\\ 2&2&2&5 pmatrix. This example is chosen to highlight the two opposite risk-sensitive selection effects in Theorem 11. In particular, one diagonal equilibrium is strongly payoff-dominant and is therefore selected under sufficiently risk-seeking behavior, while another diagonal equilibrium is strongly maximin and is selected under sufficiently risk-averse behavior. This game is a symmetric four-action game, but it is not a four-action coordination game in the sense of Definition 12, because (1,1)(1,1) is not a strict Nash equilibrium. By contrast, (2,2)(2,2), (3,3)(3,3), and (4,4)(4,4) are strict diagonal equilibria. As before, ei∈Δ4Ne_i∈ _4^N denotes the state in which all agents play action i. The diagonal equilibrium (2,2)(2,2) is strongly payoff-dominant. Its payoff is a22=10a_22=10, while the largest payoff attainable by any action other than action 22 is maxs≠2maxr∈Sasr=max0,7,5=7. _s≠ 2 _r∈ Sa_sr= \0,7,5\=7. Therefore, δ2=a22−maxs≠2maxr∈Sasr=10−7=3>0. _2=a_22- _s≠ 2 _r∈ Sa_sr=10-7=3>0. The corresponding high-risk-seeking threshold is β¯2=log2δ2=log23≈0.231. β_2= 2 _2= 23≈ 0.231. Hence, for β>β¯2β> β_2, Theorem 11 predicts that e2e_2 is the unique stochastically stable state for all sufficiently large N. The diagonal equilibrium (4,4)(4,4) is strongly maximin. Action 44 guarantees the payoff minr∈Sa4r=min2,2,2,5=2, _r∈ Sa_4r= \2,2,2,5\=2, whereas the largest payoff obtained by any other action against an opponent playing action 44 is maxs≠4as4=max0,0,0=0. _s≠ 4a_s4= \0,0,0\=0. Thus, δ4m=minr∈Sa4r−maxs≠4as4=2−0=2>0. _4^m= _r∈ Sa_4r- _s≠ 4a_s4=2-0=2>0. The corresponding low-risk threshold is β¯4=−log2δ4m=−log22≈−0.347. β_4=- 2 _4^m=- 22≈-0.347. Therefore, for β<β¯4β< β_4, Theorem 11 predicts that e4e_4 is the unique stochastically stable state for all sufficiently large N. Figure 3: Stationary mass near each pure population state under RS-BRM, with N=20N=20, η=0.3η=0.3, and threshold xi≥0.9x_i≥ 0.9. We compute the exact stationary distribution of the finite Markov chain induced by RS-BRM. The experiment uses N=20N=20, η=0.3η=0.3, and a grid of 1212 evenly spaced values of β∈[−1,1]β∈[-1,1]. Since the stationary distribution at a nonzero noise level may place substantial mass on states that are close to, but not exactly equal to, an attracting state, we also report an aggregate concentration measure. Specifically, Fig. 3 plots μN,η(xi≥0.9)μ^N,η (x_i≥ 0.9 ), the stationary probability that at least 90%90\% of the population plays action i. This statistic gives a more robust finite-noise visualization of how the stationary distribution concentrates around each attracting state. This example illustrates how the risk-sensitivity parameter shapes long-run selection. When β>0β>0, the ERM-adjusted payoff places greater emphasis on high-payoff realizations. As β increases, the high diagonal payoff a22=10a_22=10 makes action 22 increasingly attractive, leading to selection of the strongly payoff-dominant equilibrium (2,2)(2,2). When β<0β<0, the ERM-adjusted payoff places greater emphasis on low-payoff realizations, so selection is driven more by downside protection. In this example, action 44 guarantees a payoff of at least 22 against every opponent action, while every other action obtains at most 00 against an opponent playing action 44. Thus, stronger risk aversion enables the evolutionary dynamics to favor the more robust alternative against adverse opponent behavior, namely the strongly maximin equilibrium (4,4)(4,4). 7 Concluding Remarks We studied risk-sensitive evolutionary learning dynamics in coordination games. By incorporating the entropic risk measure into noisy best-response protocols, we showed that agents’ risk attitudes could change the stochastically stable outcome. In single-population symmetric 2×22× 2 coordination games, we showed that greater risk-seeking behavior favors the payoff-dominant equilibrium, whereas greater risk-averse behavior favors the maximin (and thus worst-case robust) equilibrium whenever these two equilibria differ. We also showed that this selection pattern holds under both protocols of best response with mutations and logit choice. In both the single-population and two-population settings, we also identified a robust regime in which super-dominant equilibria remain stochastically stable for all risk-sensitivity parameters. Thus, when payoff dominance and worst-case robustness are aligned, long-run selection can be insensitive to agents’ risk attitudes and to the specific noisy best-response protocol. Finally, we extended the single-population analysis to symmetric k-action games, and showed that sufficiently risk-seeking agents select strongly payoff-dominant equilibria, while sufficiently risk-averse agents select strongly maximin equilibria, whenever such equilibria exist. These results together showed that entropic risk sensitivity may serve as a systematic mechanism for steering equilibrium selection beyond the classical risk-neutral benchmark. Several directions remain open. First, our analysis focused on homogeneous risk attitudes within each population. Relaxing this assumption and allowing risk sensitivity to vary across agents or over time could provide a broader understanding of how risk preferences can shape the long-run equilibrium selection. Second, we studied entropic risk sensitivity because it provided a natural and analytically tractable way to incorporate payoff uncertainty into evolutionary learning. Other risk-sensitive criteria may induce different effective payoff comparisons and, consequently, different stochastic-stability patterns, which are worth further investigation. Finally, the present analysis focused on coordination-game environments. Extending the framework to richer population structures and other payoff structures may further illustrate how risk attitudes can shape the collective behaviors in multi-agent interactions. Acknowledgement The authors acknowledge the support from the Army Research Office (ARO) grant W911NF-24-1-0085, the NSF CAREER Award 2443704, the AFOSR YIP Award FA9550-25-1-0258, an AI Safety Research Award from Coefficient Giving, a Cisco Faculty Research Award, and a JP Morgan Faculty Research Award. The authors also acknowledge the conversations with Tamer Başar, Xiangyu Liu, and Eric Mazumdar, and the valuable feedback from the anonymous reviewers of the IEEE Conference on Decision and Control (CDC) 2026. 8 Deferred Proofs 8.1 Proof of Lemma 1 Proof. Suppress the population superscript p, and write β,Δ,a,b,c,dβ, ,a,b,c,d. Fix z∈(0,1)z∈(0,1). For β≠0β≠ 0, Δ′(z)=1β(eβa−eβbzeβa+(1−z)eβb−eβc−eβdzeβc+(1−z)eβd). (z)= 1β ( e^β a-e^β bze^β a+(1-z)e^β b- e^β c-e^β dze^β c+(1-z)e^β d ). Let α1=eβ(a−b),α2=eβ(c−d),ϕ(α):=α−1zα+1−z. _1=e^β(a-b), _2=e^β(c-d), φ(α):= α-1zα+1-z. Then Δ′(z)=1β[ϕ(α1)−ϕ(α2)],ϕ′(α)=1(zα+1−z)2>0. (z)= 1β [φ( _1)-φ( _2) ], φ (α)= 1(zα+1-z)^2>0. Hence, ϕφ is strictly increasing in α. Therefore, sgnΔ′(z)=sgn(β)sgn(α1−α2).sgn (z)=sgn(β)sgn( _1- _2). If β>0β>0, then sgn(α1−α2)=sgn((a−b)−(c−d)),sgn( _1- _2)=sgn ((a-b)-(c-d) ), whereas if β<0β<0, this sign is reversed. This reversal is exactly offset by the negative factor 1/β1/β. Thus, in both cases, sgnΔ′(z)=sgn((a−b)−(c−d)).sgn (z)=sgn ((a-b)-(c-d) ). Since a>ca>c and d>bd>b in a coordination game, we have (a−b)−(c−d)=(a−c)+(d−b)>0.(a-b)-(c-d)=(a-c)+(d-b)>0. Hence, Δ′(z)>0 (z)>0 for all z∈(0,1)z∈(0,1), so Δ is strictly increasing on (0,1)(0,1). For β=0β=0, the expected-payoff expression gives Δ0(z)=z(a−c)+(1−z)(b−d), _0(z)=z(a-c)+(1-z)(b-d), whose derivative is Δ0′(z)=(a−c)+(d−b)>0. _0 (z)=(a-c)+(d-b)>0. Thus, strict monotonicity also holds at β=0β=0. Moreover, Δ(0)=b−d<0,Δ(1)=a−c>0. (0)=b-d<0, (1)=a-c>0. By continuity, Δ has a root in (0,1)(0,1), and strict monotonicity makes this root unique. For β≠0β≠ 0, solving Δ(z)=0 (z)=0 yields zeβa+(1−z)eβb=zeβc+(1−z)eβd,ze^β a+(1-z)e^β b=ze^β c+(1-z)e^β d, and hence (4). The expression at β=0β=0 follows by taking the continuous extension, or equivalently by solving Δ0(z)=0 _0(z)=0. The stated sign characterization follows immediately from strict monotonicity. ∎ 8.2 Proof of Lemma 2 Proof. The argument is identical for every population, so we suppress the superscript p and write xβ∗=eβd−eβb(eβa−eβc)+(eβd−eβb).x_β^*= e^β d-e^β b (e^β a-e^β c )+ (e^β d-e^β b ). For β≠0β≠ 0, the denominator is nonzero because eβa−eβce^β a-e^β c and eβd−eβbe^β d-e^β b have the same sign. Hence, β↦xβ∗β x_β^* is continuous on ℝ∖0R \0\. At β=0β=0, using eβu=1+βu+o(β)e^β u=1+β u+o(β), we obtain xβ∗→d−b(a−c)+(d−b). x_β^*→ d-b(a-c)+(d-b). Therefore, β↦xβ∗β x_β^* is continuous on ℝR. For β≠0β≠ 0, differentiating gives dxβ∗dβ=xβ∗(1−xβ∗)[w2(β)−w1(β)], dx^*_βdβ=x^*_β (1-x^*_β ) [w_2(β)-w_1(β) ], (15) where w1(β):=aeβa−ceβceβa−eβc,w2(β):=deβd−beβbeβd−eβb.w_1(β):= ae^β a-ce^β ce^β a-e^β c, w_2(β):= de^β d-be^β be^β d-e^β b. Let F(β):=w2(β)−w1(β),s:=a−c>0,t:=d−b>0. F(β):=w_2(β)-w_1(β), s:=a-c>0, t:=d-b>0. A direct differentiation shows that F′(β)=g(s)−g(t),g(z):=z2eβz(eβz−1)2,z>0.F (β)=g(s)-g(t), g(z):= z^2e^β z(e^β z-1)^2, z>0. For fixed β≠0β≠ 0, differentiating g with respect to z gives g′(z)=−zeβz(eβz−1)3h(βz),h(r):=(r−2)er+r+2.g (z)=- ze^β z(e^β z-1)^3\,h(β z), h(r):=(r-2)e^r+r+2. We now show that g′(z)<0g (z)<0 for all z>0z>0. Since h(0)=0,h′(r)=(r−1)er+1,h(0)=0, h (r)=(r-1)e^r+1, and h′(r)=rer,h (r)=re^r, the function h′h decreases on (−∞,0](-∞,0], increases on [0,∞)[0,∞), and attains its minimum h′(0)=0h (0)=0. Hence, h′(r)≥0h (r)≥ 0 for all r, with strict inequality for r≠0r≠ 0. Thus, h is strictly increasing and, since h(0)=0h(0)=0, h(r)h(r) has the same sign as r. The term (er−1)3(e^r-1)^3 also has the same sign as r, so g′(z)<0g (z)<0 for all z>0z>0. It follows that F is constant when s=ts=t, and otherwise is strictly monotone on (−∞,0)(-∞,0) and on (0,∞)(0,∞), with the same monotonicity direction on both intervals. As β→0β→ 0, w1(β)=1β+a+c2+O(β),w2(β)=1β+b+d2+O(β),w_1(β)= 1β+ a+c2+O(β), w_2(β)= 1β+ b+d2+O(β), so the poles cancel and F extends continuously to β=0β=0, with F(0)=b+d−a−c2F(0)= b+d-a-c2. Therefore, F is monotone on all of ℝR. Using a>ca>c and d>bd>b, we have limβ→∞w1(β)=a,limβ→∞w2(β)=d, _β→∞w_1(β)=a, _β→∞w_2(β)=d, and limβ→−∞w1(β)=c,limβ→−∞w2(β)=b. _β→-∞w_1(β)=c, _β→-∞w_2(β)=b. Hence, limβ→∞F(β)=d−a,limβ→−∞F(β)=b−c. _β→∞F(β)=d-a, _β→-∞F(β)=b-c. Moreover, limβ→∞xβ∗=1,d>a,0,d<a,limβ→−∞xβ∗=1,b<c,0,b>c. _β→∞x_β^*= cases1,&d>a,\\ 0,&d<a, cases _β→-∞x_β^*= cases1,&b<c,\\ 0,&b>c. cases Since xβ∗(1−xβ∗)>0x_β^*(1-x_β^*)>0, (15) implies sgn(dxβ∗dβ)=sgn(F(β)).sgn\! ( dx_β^*dβ )=sgn (F(β) ). If b>cb>c and d>ad>a, then both endpoint limits of F are positive. Since F is monotone, F(β)>0F(β)>0 for all β, so xβ∗x_β^* is strictly increasing. The stated limits give case (I). Similarly, if b<cb<c and d<ad<a, then both endpoint limits of F are negative. Hence, F(β)<0F(β)<0 for all β, so xβ∗x_β^* is strictly decreasing, and the stated limits give case (I). It remains to prove cases (I) and (IV). Since a>ca>c and d>bd>b, the quantities eβa−eβce^β a-e^β c and eβd−eβbe^β d-e^β b have the same sign as β. Hence, for β≠0β≠ 0, the expression in (4) can be written as xβ∗=BβAβ+Bβ,x_β^*= B_βA_β+B_β, where Aβ:=|eβa−eβc|,Bβ:=|eβd−eβb|.A_β:= |e^β a-e^β c |, B_β:= |e^β d-e^β b |. Then xβ∗<12x_β^*< 12 if and only if Bβ<AβB_β<A_β, and xβ∗>12x_β^*> 12 if and only if Bβ>AβB_β>A_β. For any u<vu<v, |eβv−eβu|=|β|∫uveβrr. |e^β v-e^β u |=|β| _u^ve^β r\,dr. If b≥cb≥ c and d≤ad≤ a, with at least one strict inequality, then [b,d][b,d] is a proper subinterval of [c,a][c,a]. Therefore, Bβ=|β|∫bdeβrr<|β|∫caeβrr=Aβ.B_β=|β| _b^de^β r\,dr<|β| _c^ae^β r\,dr=A_β. Thus, xβ∗<12x_β^*< 12 for all β≠0β≠ 0. At β=0β=0, a−c>d−ba-c>d-b gives x0∗=d−b(a−c)+(d−b)<12x_0^*= d-b(a-c)+(d-b)< 12. This proves (I). Similarly, if b≤cb≤ c and d≥ad≥ a, with at least one strict inequality, then [c,a][c,a] is a proper subinterval of [b,d][b,d]. Hence, Aβ<BβA_β<B_β, so xβ∗>12x_β^*> 12 for all β≠0β≠ 0. At β=0β=0, d−b>a−cd-b>a-c gives x0∗>12x_0^*> 12. This proves (IV). Finally, if a=da=d and b=cb=c, then (4) simplifies to xβ∗≡12x_β^*≡ 12. ∎ 8.3 Proof of Theorem 3 Proof. Let PN,ηP^N,η be the transition matrix of XN,ηX^N,η. By the Markov-chain tree representation for finite irreducible Markov chains, also used in Freidlin–Wentzell stochastic-stability theory Freidlin and Wentzell (1998); Kandori et al. (1993); Young (1993), the stationary distribution satisfies μN,η(x)=τxN,η∑z∈XNτzN,η,μ^N,η(x)= _x^N,η _z∈ X^N _z^N,η, where τxN,η:=∑T∈x∏(u→v)∈TPN,η(u,v), _x^N,η:= _T _x _(u→ v)∈ TP^N,η(u,v), and xT_x denotes the set of directed spanning trees rooted at x, i.e., each state y≠xy≠ x has a unique directed path to x along the edges of T. For each tree T∈xT _x, (5) gives −ηlog(∏(u→v)∈TPN,η(u,v))⟶∑(u→v)∈TκN(u,v),η→0.-η ( _(u→ v)∈ TP^N,η(u,v) ) _(u→ v)∈ Tκ^N(u,v), η→ 0. Since XNX^N is finite, the set xT_x is finite. Hence, the sum defining τxN,η _x^N,η is exponentially dominated by the minimum-cost tree. Define the stochastic potential VN(x):=min∑(u→v)∈T∈xκN(u,v).V^N(x):= _T _x _(u→ v)∈ Tκ^N(u,v). Then −ηlogτxN,η⟶VN(x),η→0.-η _x^N,η V^N(x), η→ 0. Therefore, ηlogμN,η(B)μN,η(A)=ηlogτBN,ητAN,η⟶VN(A)−VN(B).η μ^N,η(B)μ^N,η(A)=η _B^N,η _A^N,η V^N(A)-V^N(B). It remains to identify VN(A)V^N(A) and VN(B)V^N(B) with the corresponding escape costs. We first show that VN(A)=γAN:=CN(B,∂outN(B)). V^N(A)= _A^N:=C^N\! (B, _out^N(B) ). The proof for VN(B)=γBNV^N(B)= _B^N is symmetric. For the upper bound, let ϕ=(ϕ0,ϕ1,…,ϕK) φ=( _0, _1,…, _K) be a minimum-cost path from B to ∂outN(B) _out^N(B), with ϕ0=B _0=B and ϕK∈∂outN(B) _K∈ _out^N(B). By definition, CN(ϕ)=γAN. C^N(φ)= _A^N. Since ϕK∉ℬ0N(B) _K _0^N(B), Assumption 2 (I) implies that CN(ϕK,A)=0C^N( _K,A)=0. Thus, there exists a zero-cost path from ϕK _K to A. Concatenating this zero-cost path with ϕφ gives a path from B to A with total cost γAN _A^N. We now construct an A-rooted spanning tree. Include the edges of the above path from B to A. For each remaining state x∈ℬ0N(A)x _0^N(A), attach x to A along a zero-cost path, stopping when the path first reaches the already constructed tree. For each remaining state x∉ℬ0N(A)x _0^N(A), Assumption 2 (I) gives CN(x,B)=0C^N(x,B)=0; attach x along such a zero-cost path to B, again stopping when the path first reaches the already constructed tree. Since all added edges have zero cost, this produces an A-rooted spanning tree with total cost γAN _A^N. Hence, VN(A)≤γAN.V^N(A)≤ _A^N. For the lower bound, let T∈AT _A be any A-rooted spanning tree. The unique directed path in T from B to A must leave the stability set ℬ0N(B)B_0^N(B). Let z be the first state on this path outside ℬ0N(B)B_0^N(B). Then z∈∂outN(B)z∈ _out^N(B). Therefore, the cost of the portion of this path from B to z is at least CN(B,∂outN(B))=γAN. C^N\! (B, _out^N(B) )= _A^N. Since all transition costs are nonnegative, the total cost of T is at least γAN _A^N. Thus, we have VN(A)≥γAN.V^N(A)≥ _A^N. Combining the upper and lower bounds gives VN(A)=γANV^N(A)= _A^N. By the same argument, VN(B)=γBN:=CN(A,∂outN(A)).V^N(B)= _B^N:=C^N\! (A, _out^N(A) ). Substituting these identities into the potential-ratio formula yields limη→0ηlogμN,η(B)μN,η(A)=γAN−γBN, _η→ 0η μ^N,η(B)μ^N,η(A)= _A^N- _B^N, which proves (6). It remains to establish the conclusions concerning stochastic stability under Definition 8. By the assumed vanishing-noise behavior of the noisy best-response functions, every entry of PN,ηP^N,η has a limit as η→0η→ 0. Define the limiting transition matrix by PN,0(x,y):=limη→0PN,η(x,y).P^N,0(x,y):= _η→ 0P^N,η(x,y). Since XNX^N is finite, PN,η→PN,0P^N,η→ P^N,0 entrywise, and PN,0P^N,0 is a stochastic matrix. Moreover, its positive-probability off-diagonal transitions are precisely the zero-cost transitions of the zero-noise dynamics. Under Assumption 2, the singleton sets A\A\ and B\B\ are closed recurrent classes of PN,0P^N,0. They are also the only recurrent classes. Indeed, suppose that ℛR were another recurrent class and let x∈ℛx . By Assumption 2(I), there is a zero-cost path from x to either A or B. This path consists of positive-probability transitions of PN,0P^N,0, contradicting the closedness of ℛR. Thus, the only recurrent classes of PN,0P^N,0 are A\A\ and B\B\. We claim that μN,η(XN∖A,B)⟶0μ^N,η (X^N \A,B\ ) 0, as η→0η→ 0. To prove this claim, consider an arbitrary sequence ηm→0 _m→ 0. Since the probability simplex on XNX^N is compact, there is a subsequence, which we continue to denote by ηm\ _m\, such that μN,ηm⟶μ¯μ^N, _m μ for some probability distribution μ¯ μ on XNX^N. Passing to the limit in the stationarity equation μN,ηmPN,ηm=μN,ηmμ^N, _mP^N, _m=μ^N, _m gives μ¯PN,0=μ¯. μP^N,0= μ. Thus, μ¯ μ is a stationary distribution of PN,0P^N,0. Every stationary distribution of a finite Markov chain is supported on its recurrent classes. Therefore, μ¯(XN∖A,B)=0. μ (X^N \A,B\ )=0. Since this conclusion holds for every subsequential limit, μN,η(XN∖A,B)⟶0.μ^N,η (X^N \A,B\ ) 0. Equivalently, μN,η(A)+μN,η(B)⟶1.μ^N,η(A)+μ^N,η(B) 1. Suppose now that γAN<γBN. _A^N< _B^N. By equation (6), ηlogμN,η(B)μN,η(A)⟶γAN−γBN<0.η μ^N,η(B)μ^N,η(A) _A^N- _B^N<0. Hence, μN,η(B)μN,η(A)⟶0. μ^N,η(B)μ^N,η(A) 0. Together with μN,η(A)+μN,η(B)⟶1,μ^N,η(A)+μ^N,η(B) 1, this implies μN,η(A)⟶1andμN,η(B)⟶0.μ^N,η(A) 1 μ^N,η(B) 0. Therefore, A is uniquely stochastically stable according to Definition 8. The case γBN<γAN _B^N< _A^N is symmetric and gives μN,η(B)⟶1.μ^N,η(B) 1. Thus, B is uniquely stochastically stable. ∎ 9 Auxiliary Lemmas Lemma 12. The function S¯logit S_logit defined in (10) is continuous on ℝR and strictly monotone unless |a−b|=|c−d||a-b|=|c-d|, in which case it is constant. Moreover, limβ→−∞S¯logit(β)=−mina,b+minc,d, _β→-∞ S_logit(β)=- \a,b\+ \c,d\, and limβ→+∞S¯logit(β)=−maxa,b+maxc,d. _β→+∞ S_logit(β)=- \a,b\+ \c,d\. Proof. Continuity of S¯logit S_logit follows from the continuity of (β,x)↦Δβ(x)(β,x) _β(x) on ℝ×[0,1]R×[0,1]. For p≤qp≤ q, define ψp,q(β):=qeβq−peβpeβq−eβp−1β,β≠0,p<q,p+q2,β=0,p<q,p,p=q. _p,q(β):= cases qe^β q-pe^β pe^β q-e^β p- 1β,&β≠ 0,\ p<q,\\[8.61108pt] p+q2,&β=0,\ p<q,\\[5.16663pt] p,&p=q. cases The apparent singularity at β=0β=0 is removable, since for p<qp<q, qeβq−peβpeβq−eβp−1β⟶p+q2as β→0. qe^β q-pe^β pe^β q-e^β p- 1β p+q2 β→ 0. Thus, ψp,q _p,q is continuous on ℝR. For β≠0β≠ 0, integrating the logarithmic expression for Δβ _β gives S¯logit(β)=−ψmina,b,maxa,b(β)+ψminc,d,maxc,d(β). S_logit(β)=- _ \a,b\, \a,b\(β)+ _ \c,d\, \c,d\(β). By continuity, the same identity holds at β=0β=0. In particular, S¯logit(0)=−a+b2+c+d2=−a+b−c−d2. S_logit(0)=- a+b2+ c+d2=- a+b-c-d2. We now prove monotonicity. If p<qp<q, then ψp,q′(β)=1β2−(q−p)2eβ(q−p)(eβ(q−p)−1)2,β≠0.ψ _p,q(β)= 1β^2- (q-p)^2e^β(q-p)(e^β(q-p)-1)^2, β≠ 0. If p=qp=q, then ψp,p≡p _p,p≡ p, so ψp,p′=0ψ _p,p=0. Hence, for β≠0β≠ 0, S¯logit′(β)=Gβ(|a−b|)−Gβ(|c−d|), S_logit (β)=G_β(|a-b|)-G_β(|c-d|), where Gβ(z):=z2eβz(eβz−1)2,z>0,1β2,z=0.G_β(z):= cases z^2e^β z(e^β z-1)^2,&z>0,\\[8.61108pt] 1β^2,&z=0. cases The value at z=0z=0 is the continuous extension, since limz↓0z2eβz(eβz−1)2=1β2. _z 0 z^2e^β z(e^β z-1)^2= 1β^2. Thus, the degenerate cases |a−b|=0|a-b|=0 or |c−d|=0|c-d|=0 are handled. For each fixed β≠0β≠ 0, the function z↦Gβ(z)z G_β(z) is strictly decreasing on [0,∞)[0,∞). This is the same calculation used in the proof of Lemma 2. Therefore, the sign of S¯logit′(β) S_logit (β) is constant on (−∞,0)∪(0,∞)(-∞,0)∪(0,∞). Since S¯logit S_logit is continuous at β=0β=0, it follows that S¯logit S_logit is monotone on all of ℝR. It is strictly monotone unless |a−b|=|c−d||a-b|=|c-d|, in which case it is constant. Finally, if p<qp<q, then limβ→−∞ψp,q(β)=p,limβ→+∞ψp,q(β)=q, _β→-∞ _p,q(β)=p, _β→+∞ _p,q(β)=q, while ψp,p≡p _p,p≡ p. Substituting these limits into the expression for S¯logit(β) S_logit(β) gives limβ→−∞S¯logit(β)=−mina,b+minc,d, _β→-∞ S_logit(β)=- \a,b\+ \c,d\, and limβ→+∞S¯logit(β)=−maxa,b+maxc,d. _β→+∞ S_logit(β)=- \a,b\+ \c,d\. ∎ Lemma 13. Fix p∈1,2p∈\1,2\. Suppose that, for population p, ap≥dpandbp≥cp,a^p≥ d^p b^p≥ c^p, with at least one inequality strict. Then, for every βp∈ℝβ^p , if Δ(⋅)=Δβp(⋅) (·)= _β^p^p(·) denotes the ERM payoff difference in (2), we have Δ(1−z)>−Δ(z),∀z∈[0,1]. (1-z)>- (z), ∀ z∈[0,1]. Equivalently, Δ(z)+Δ(1−z)>0 (z)+ (1-z)>0 for all z∈[0,1]z∈[0,1]. Proof. Fix βp∈ℝβ^p , and define Rβp(z):=Δ(z)+Δ(1−z),z∈[0,1].R_β^p(z):= (z)+ (1-z), z∈[0,1]. We first consider the case βp≠0β^p≠ 0. Let A=eβpap,B=eβpbp,C=eβpcp,D=eβpdp.A=e^β^pa^p, B=e^β^pb^p, C=e^β^pc^p, D=e^β^pd^p. Then βpRβp(z)=log((zA+(1−z)B)((1−z)A+zB)(zC+(1−z)D)((1−z)C+zD)). β^pR_β^p(z)= \! ( (zA+(1-z)B) ((1-z)A+zB )(zC+(1-z)D) ((1-z)C+zD ) ). Writing t=z(1−z)∈[0,14]t=z(1-z)∈[0, 14], we obtain exp(βpRβp(z))=AB+t(A−B)2CD+t(C−D)2. \! (β^pR_β^p(z) )= AB+t(A-B)^2CD+t(C-D)^2. Since ap≥dpa^p≥ d^p and bp≥cpb^p≥ c^p, with at least one strict inequality, we have ap+bp>cp+dp.a^p+b^p>c^p+d^p. If βp>0β^p>0, then AB>CDAB>CD, and also A+B>C+D,A+B>C+D, because A≥DA≥ D, B≥CB≥ C, and at least one inequality is strict. Hence, the affine function t↦[AB+t(A−B)2]−[CD+t(C−D)2]t [AB+t(A-B)^2 ]- [CD+t(C-D)^2 ] is positive at t=0t=0. At t=14t= 14, it equals (A+B)2−(C+D)24, (A+B)^2-(C+D)^24, which is also positive. Therefore, it is positive for all t∈[0,14]t∈[0, 14]. Thus exp(βpRβp(z))>1, \! (β^pR_β^p(z) )>1, and since βp>0β^p>0, we get Rβp(z)>0R_β^p(z)>0. If βp<0β^p<0, the inequalities reverse: AB<CDAB<CD and A+B<C+DA+B<C+D. The same affine-function argument gives exp(βpRβp(z))<1. \! (β^pR_β^p(z) )<1. Since now βp<0β^p<0, it again follows that Rβp(z)>0R_β^p(z)>0. Finally, when βp=0β^p=0, the ERM payoffs reduce to expected payoffs, so Δ(z)=z(ap−cp)+(1−z)(bp−dp). (z)=z(a^p-c^p)+(1-z)(b^p-d^p). Hence R0(z)=Δ(z)+Δ(1−z)=(ap+bp)−(cp+dp)>0.R_0(z)= (z)+ (1-z)=(a^p+b^p)-(c^p+d^p)>0. Therefore, Rβp(z)>0R_β^p(z)>0 for every βp∈ℝβ^p and every z∈[0,1]z∈[0,1], which is equivalent to Δ(1−z)>−Δ(z), (1-z)>- (z), completing the proof. ∎ References Blume (1993) L. E. Blume The statistical mechanics of strategic interaction. Games and economic behavior 5 (3), p. 387–424. Cited by: §1, §3.1. Ellison (2000) G. Ellison Basins of attraction, long-run stochastic stability, and the speed of step-by-step evolution. The Review of Economic Studies 67 (1), p. 17–45. Cited by: §6.3, Definition 17. Fiat and Papadimitriou (2010) A. Fiat and C. 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