Yuji Katsumata

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Abstract: The spatial Prisoner's Dilemma is divided into two stages: strategy selection and action (cooperation/defection) selection. This renewal allows a spatiotemporal strategy that determines the player's next action based not only on the adversary's history of actions (temporal strategy) but also on neighbors' configuration of actions. Several space-time parallelisms and dualisms would hold in this spatiotemporal generalization of strategy. Among them, this note focuses on the generosity (how many defections are tolerated). A temporal strategy involving temporal generosity, such as Tit for Tat (TFT), exhibited good performance such as noise tolerance. We report that a spatial strategy with spatial generosity can maintain a cluster of cooperators by forming a membrane that protects against defectors. The condition of membrane formation can be formulated with the spatial generosity exceeding a certain threshold determined by the number of neighborhoods.

Abstract: A spatio-temporal generalization involves not only conventional temporal strategies that determine the player's action based on the opponent past actions but also spatial strategies based on the neighbor players' current actions and configurations. This framework allows the model to be dealt as a second order cellular automaton. With this involvement of the spatial strategies, we have observed a membrane formation which protects the cooperating clusters from being corroded by defecting intruders.

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