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Direct reciprocity is a well-known mechanism that could explain howcooperation emerges and prevails in an evolving population. Numerous priorresearches have studied the emergence of cooperation in multiplayer games.However, most of them use numerical or experimental methods, not theoreticalanalysis. This lack of theoretical works on the evolution of cooperation is dueto the high complexity of calculating payoffs. In this paper, we propose a newmethod, namely, the state-clustering method to calculate the long-term payoffsin repeated games. Using this method, in an $n$-player repeated game, thecomputing complexity is reduced from $O(2^n)$ to $O(n^2)$, which makes itpossible to compute a large-scale repeated game's payoff. We explore theevolution of cooperation in both infinitely and finitely repeated public goodsgames as an example to show the effectiveness of our method. In both cases, wefind that when the synergy factor is sufficiently large, the increasing numberof participants in a game is detrimental to the evolution of cooperation. Ourwork provides a theoretical approach to study the evolution of cooperation inrepeated multiplayer games.

State-clustering method of payoff computation in repeated multiplayer games