Aspiration Dynamics of Multi-player Games in Finite Populations

Studying strategy update rules in the framework of evolutionary game theory, one can differentiate between imitation processes and aspiration-driven dynamics. In the former case, individuals imitate the strategy of a more successful peer. In the latter case, individuals adjust their strategies based on a comparison of their payoffs from the evolutionary game to a value they aspire, called the level of aspiration. Unlike imitation processes of pairwise comparison, aspiration-driven updates do not require additional information about the strategic environment and can thus be interpreted as being more spontaneous. Recent work has mainly focused on understanding how aspiration dynamics alter the evolutionary outcome in structured populations. However, the baseline case for understanding strategy selection is the well-mixed population case, which is still lacking sufficient understanding. We explore how aspiration-driven strategy-update dynamics under imperfect rationality influence the average abundance of a strategy in multi-player evolutionary games with two strategies. We analytically derive a condition under which a strategy is more abundant than the other in the weak selection limiting case. This approach has a long standing history in evolutionary game and is mostly applied for its mathematical approachability. Hence, we also explore strong selection numerically, which shows that our weak selection condition is a robust predictor of the average abundance of a strategy. The condition turns out to differ from that of a wide class of imitation dynamics, as long as the game is not dyadic. Therefore a strategy favored under imitation dynamics can be disfavored under aspiration dynamics. This does not require any population structure thus highlights the intrinsic difference between imitation and aspiration dynamics

Evolutionary consequences of behavioral diversity

Iterated games provide a framework to describe social interactions among groups of individuals. Recent work stimulated by the discovery of "zero-determinant" strategies has rapidly expanded our ability to analyze such interactions. This body of work has primarily focused on games in which players face a simple binary choice, to "cooperate" or "defect". Real individuals, however, often exhibit behavioral diversity, varying their input to a social interaction both qualitatively and quantitatively. Here we explore how access to a greater diversity of behavioral choices impacts the evolution of social dynamics in finite populations. We show that, in public goods games, some two-choice strategies can nonetheless resist invasion by all possible multi-choice invaders, even while engaging in relatively little punishment. We also show that access to greater behavioral choice results in more "rugged " fitness landscapes, with populations able to stabilize cooperation at multiple levels of investment, such that choice facilitates cooperation when returns on investments are low, but hinders cooperation when returns on investments are high. Finally, we analyze iterated rock-paper-scissors games, whose non-transitive payoff structure means unilateral control is difficult and zero-determinant strategies do not exist in general. Despite this, we find that a large portion of multi-choice strategies can invade and resist invasion by strategies that lack behavioral diversity -- so that even well-mixed populations will tend to evolve behavioral diversity.Comment: 26 pages, 4 figure

Cooperation, Norms, and Revolutions: A Unified Game-Theoretical Approach

Cooperation is of utmost importance to society as a whole, but is often challenged by individual self-interests. While game theory has studied this problem extensively, there is little work on interactions within and across groups with different preferences or beliefs. Yet, people from different social or cultural backgrounds often meet and interact. This can yield conflict, since behavior that is considered cooperative by one population might be perceived as non-cooperative from the viewpoint of another. To understand the dynamics and outcome of the competitive interactions within and between groups, we study game-dynamical replicator equations for multiple populations with incompatible interests and different power (be this due to different population sizes, material resources, social capital, or other factors). These equations allow us to address various important questions: For example, can cooperation in the prisoner's dilemma be promoted, when two interacting groups have different preferences? Under what conditions can costly punishment, or other mechanisms, foster the evolution of norms? When does cooperation fail, leading to antagonistic behavior, conflict, or even revolutions? And what incentives are needed to reach peaceful agreements between groups with conflicting interests? Our detailed quantitative analysis reveals a large variety of interesting results, which are relevant for society, law and economics, and have implications for the evolution of language and culture as well