Summary
When people or agents try to coordinate their actions, different possible outcomes can be more or less stable over time. This paper looks at how attitudes toward risk—being risk-seeking or risk-averse—affect which outcomes become most likely in the long run. The authors find that risk-seeking behavior tends to favor more rewarding but uncertain outcomes, while risk-averse behavior tends to favor safer, more guaranteed options. These patterns hold in different game scenarios and help explain how a group's overall risk mindset can steer the final coordinated outcome. This offers a way to understand and potentially guide collective decision-making based on risk preferences.
coordination gamesevolutionary learningrisk sensitivityentropic risk measurestochastic stabilitybest response dynamicslogit choicerisk-dominant equilibriumpayoff-dominant equilibriummaximin equilibrium
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.