Introspection dynamics: a simple model of counterfactual learning in asymmetric games

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, 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.