Spatial indirect reciprocation

In a variant of evolutionary games, spatially distibuted populations are considered, with each individual interacting only with the closest neighbors and updating by switching to the strategy of a random neighbor with a probability proportional to the payoff difference. Let us assume, for instance, that the players sit on an N x N-lattice, with the usual identification of opposite borders, and that the neighborhood of site (i,j) consists of the 8 sites whose coordinates differ by at most one unit (a Moore neighborhood). Since the score depends on how many games have already been played, it is important to introduce no systematic bias in the ordering of the games. A simple approach is to arrange all individuals in a random sequence and let the interactions take place in that order, with this individual as recipient, and one of the neighbors (randomly chosen) as potential donor. Individuals cannot receive help more than once per round, but they may be asked more than once to help a co-player. Not surprisingly, the spatial games lead to the evolution of cooperation for even smaller b/c-values than in the well-mixed case (see Fig. 3.6 and, for interactive experimentation, Brandt 2004). In these simulations, a small mutation probability and a probability of not being able to cooperate, due to lack of resources for example, is included. Moreover, discriminators are tempted to defect instead of helping with a small temptation rate. Every generation consists of 5 rounds played as described. Then, in the spatial case, sites are updated by comparing their payoff with that of a ran-

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