No abstract

Social behavior in human and animal populations can be studied as an evolutionary process. Individuals often make decisions between different strategies, and those strategies that yield a fitness advantage tend to spread. Traditionally, much work in evolutionary game theory considers symmetric games: Individuals are assumed to have access to the same set of strategies, and they experience the same payoff consequences. As a result, they can learn more profitable strategies by imitation. However, interactions are oftentimes asymmetric. In that case, imitation may be infeasible (because individuals differ in the strategies they are able to use), or it may be undesirable (because individuals differ in their incentives to use a strategy). Here, we consider an alternative learning process which applies to arbitrary asymmetric games, \textit{introspection dynamics}. According to this dynamics, individuals regularly compare their present strategy to a randomly chosen alternative strategy. If the alternative strategy yields a payoff advantage, it is more likely adopted. In this work, we formalize introspection dynamics for pairwise games. We derive simple and explicit formulas for the abundance of each strategy over time and apply these results to several well-known social dilemmas. In particular, for the volunteer's timing dilemma, we show that the player with the lowest cooperation cost learns to cooperate without delay.

One landmark application of evolutionary game theory is the study of social dilemmas. This literature explores why people cooperate even when there are strong incentives to defect. Much of this literature, however, assumes that interactions are symmetric. Individuals are assumed to have the same strategic options and the same potential pay-offs. Yet many interesting questions arise once individuals are allowed to differ. Here, we study asymmetry in simple coordination games. In our set-up, human participants need to decide how much of their endowment to contribute to a public good. If a group’s collective contribution reaches a pre-defined threshold, all group members receive a reward. To account for possible asymmetries, individuals either differ in their endowments or their productivities. According to a theoretical equilibrium analysis, such games tend to have many possible solutions. In equilibrium, group members may contribute the same amount, different amounts or nothing at all. According to our behavioural experiment, however, humans favour the equilibrium in which everyone contributes the same proportion of their endowment. We use these experimental results to highlight the non-trivial effects of inequality on cooperation, and we discuss to which extent models of evolutionary game theory can account for these effects. This article is part of the theme issue ‘Half a century of evolutionary games: a synthesis of theory, application and future directions'.

Evolutionary game theory and models of learning provide a powerful framework to describe strategic decision-making in social interactions. In the simplest case, these models describe games among two identical players. However, many interactions in everyday life are more complex. They involve more than two players who may differ in their available actions and in their incentives to choose each action. Such interactions can be captured by asymmetric multiplayer games. Recently, introspection dynamics has been introduced to explore such asymmetric games. According to this dynamics, at each time step players compare their current strategy to an alternative strategy. If the alternative strategy results in a payoff advantage, it is more likely adopted. This model provides a simple way to compute the players’ long-run probability of adopting each of their strategies. In this paper, we extend some of the previous results of introspection dynamics for 2-player asymmetric games to games with arbitrarily many players. First, we derive a formula that allows us to numerically compute the stationary distribution of introspection dynamics for any multiplayer asymmetric game. Second, we obtain explicit expressions of the stationary distribution for two special cases. These cases are additive games (where the payoff difference that a player gains by unilaterally switching to a different action is independent of the actions of their co-players), and symmetric multiplayer games with two strategies. To illustrate our results, we revisit several classical games such as the public goods game.