Aspiration dynamics of multi-player games in finite populations

On 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 pay-offs 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 games 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 favoured under imitation dynamics can be disfavoured under aspiration dynamics. This does not require any population structure, and thus highlights the intrinsic difference between imitation and aspiration dynamics.