We show that the history of play in a population game contains exploitable information that can be successfully used by sophisticated strategies to defeat memory-one opponents, including zero determinant strategies. The history allows a player to label opponents by their strategies, enabling a player to determine the population distribution and to act differentially based on the opponent’s strategy in each pairwise interaction. For the Prisoner’s Dilemma, these advantages lead to the natural formation of cooperative coalitions among similarly behaving players and eventually to unilateral defection against opposing player types. We show analytically and empirically that optimal play in population games depends strongly on the population distribution. For example, the optimal strategy for a minority player type against a resident TFT population is ALLC, while for a majority player type the optimal strategy versus TFT players is ALLD. Such behaviors are not accessible to memory-one strategies. Drawing inspiration from Sun Tzu’s the Art of War, we implemented a non-memory-one strategy for population games based on techniques from machine learning and statistical inference that can exploit the history of play in this manner. Via simulation we find that this strategy is essentially uninvadable and can successfully invade (significantly more likely than a neutral mutant) essentially all known memory-one strategies for the Prisoner’s Dilemma, including ALLC (always cooperate), ALLD (always defect), tit-for-tat (TFT), win-stay-lose-shift (WSLS), and zero determinant (ZD) strategies, including extortionate and generous strategies.

We combine incentive, adaptive, and time-scale dynamics to study multipopulation dynamics on the simplex equipped with a large class of Riemannian metrics, simultaneously generalizing and extending many dynamics commonly studied in dynamic game theory and evolutionary dynamics. Each population has its own geometry, method of adaptation (incentive), and time-scale (discrete, continuous, and others). Using information-theoretic measures of distance we give a widely-applicable Lyapunov result for the dynamics.

Abstract:We demonstrate a vast expansion of the theory of evolutionary stability to finite populations with mutation, connecting the theory of the stationary distribution of the Moran process with the Lyapunov theory of evolutionary stability. We define the notion of stationary stability for the Moran process with mutation and generalizations, as well as a generalized notion of evolutionary stability that includes mutation called an incentive stable state (ISS) candidate. For sufficiently large populations, extrema of the stationary distribution are ISS candidates and we give a family of Lyapunov quantities that are locally minimized at the stationary extrema and at ISS candidates. In various examples, including for the Moran and Wright-Fisher processes, we show that the local maxima of the stationary distribution capture the traditionally-defined evolutionarily stable states. The classical stability theory of the replicator dynamic is recovered in the large population limit. Finally we include descriptions of possible extensions to populations of variable size and populations evolving on graphs.

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Populations of replicating entities frequently experience sudden or cyclical changes in environment. We explore the implications of this phenomenon via a environmental switching parameter in several common evolutionary dynamics models including the replicator dynamic for linear symmetric and asymmetric landscapes, the Moran process, and incentive dynamics. We give a simple relationship between the probability of environmental switching, the relative fitness gain, and the effect on long term behavior in terms of fixation probabilities and long term outcomes for deterministic dynamics. We also discuss cases where the dynamic changes, for instance a population evolving under a replicator dynamic switching to a best-reply dynamic and vice-versa, giving Lyapunov stability results.