Aspiration dynamics of multi-player games in finite populations

Abstract

On studying strategy update rules in the framework of evolutionary game theory, one can differentiate between imitation processes and aspirationdriven 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.

Evolutionary dynamics of general group interactions in structured populations

“…Evolutionary dynamics for strategies in group interactions are complex even in the ideally structureless (well-mixed) populations, with outcomes which cannot be obtained from pairwise interactions [28][29][30]33,34,37,42,43]. In reality, the introduction of not merely multiplayer games but also * longwang@pku.edu.cn structured populations gives rise to polynomial as well as nonlinear fitness functions in evolutionary dynamics [28,29,33,34,38,[44][45][46][47].…”

“…Within the area of dynamics, two-player games [14][15][16] are frequently adopted to model typical pairwise interactions to understand the evolution of cooperation [17][18][19][20][21][22][23][24][25][26]. Considering the ubiquitously group interactions ranging from the natural world to human society, researchers recently generalized two-player games to their multiplayer versions [27][28][29][30][31][32][33][34][35][36][37], such as the N -person prisoner's dilemma [30,38], N-person snowdrift game [31,32], N -person stag hunt game [39], as well as the N -person ultimatum game [40]. In a typical collective action, an individual's payoff could be no longer the simple summation of many pairwise interactions [33,41], and instead it is replaced by the multiple interactive payoffs from multiplayer games, which depends on what strategies all other opponents hold in the same group.…”

Evolution of cooperation driven by incremental learning

“…Nonetheless, these imitation dynamics could give some useful inspirations for understanding the high levels of 28 cooperation characterized by the underlying dynamics. According to social psychology, individuals' cognitive processes are 29 heavily influenced by those around them and individuals renew knowledge to promote their survival probability through 30 social interactions [33,[37][38][39]. In other words, learning capabilities originate from social interactions, suggesting that social 31 sharing of information provides great evolutionary advantages and offer alternatives to solve complicated social problems.…”

“…This mechanism allows the coexistence of cooperators and defectors 7 even in the prisoner's dilemma game, and the cooperation dominant state, in some cases. 8 It is well accepted that the population dynamics in structured games, stemming from the microscopic dynamic or strategy 9 update rules that define how successful strategies spread, have a crucial impact on evolutionary outcomes [29,30]. So 10 it is important to explore the potential dynamic update rules underlying the evolutionary process, which is crucial for 11 understanding the emergence of cooperation.…”

“…Under this condition, an 18 inferior strategy is possible to replace a more profitable one, as is often seen in the society or nature. Another representative 19 strategy update rule is aspiration-based preference learning [12, 30,35], under which strategies judged to have resulted in 20 satisfactory payoffs are more likely to be selected and those leading to unsatisfactory payoffs tend to be abandoned, namely 21 win-stay-lose-shift. 22 Although extensive research has been conducted regarding strategy update rules, how strategies are actually adopted and 23 spread within the population is still unknown.…”