Fig. 2: The Balls-and-Boxes problem, payoff calculation, and strategy updates.

a To determine a co-player configuration, we distribute k identical co-players (balls) into n distinct strategies (boxes), corresponding to the classic Balls-and-Boxes problem. For instance, with k = 4 and n = 3, there are 15 possible co-player configurations. b An individual accumulates payoffs from 1 + k multiplayer games, organized by itself and its k neighbors. Each multiplayer game involves k + 1 players, including the organizer and its neighbors. c Strategy updates in an individual are governed by the pairwise comparison rule. A random neighbor is selected, and the focal individual either adopts the neighbor’s strategy or maintains its original strategy, based on a probability proportional to the fitness (transformed from payoffs) in the pair.