🤖 AI Summary
Existing theoretical frameworks fail to unify the evolutionary dynamics of arbitrary multi-player, multi-strategy games on regular graphs, hindering our understanding of cooperation emergence in social and natural systems. To address this, we develop the first universal theoretical framework: leveraging a “balls-in-boxes” model to map local interaction configurations onto strategy distribution problems, and integrating weak-selection approximation with graph-structured replicator dynamics, we derive—analytically for the first time—the evolutionary equations for n-strategy multiplayer games on regular graphs. We introduce the Local Configuration Equivalence Principle, enabling derivation of a universal replicator equation solvable in polynomial time. Our theory precisely determines the critical threshold for punishment efficacy, uncovers novel phase-transition and cyclic mechanisms underlying second-order free-riding in structured populations, and successfully reproduces and extends phase behaviors under strong selection.
📝 Abstract
Multiplayer games on graphs are at the heart of theoretical descriptions of key evolutionary processes that govern vital social and natural systems. However, a comprehensive theoretical framework for solving multiplayer games with an arbitrary number of strategies on graphs is still missing. Here, we solve this by drawing an analogy with the Balls-and-Boxes problem, based on which we show that the local configuration of multiplayer games on graphs is equivalent to distributing k identical co-players among n distinct strategies. We use this to derive the replicator equation for any n-strategy multiplayer game under weak selection, which can be solved in polynomial time. As an example, we revisit the second-order free-riding problem, where costly punishment cannot truly resolve social dilemmas in a well-mixed population. Yet, in structured populations, we derive an accurate threshold for the punishment strength, beyond which punishment can either lead to the extinction of defection or transform the system into a rock-paper-scissors-like cycle. The analytical solution also qualitatively agrees with the phase diagrams that were previously obtained for non-marginal selection strengths. Our framework thus allows an exploration of any multi-strategy multiplayer game on regular graphs.