CHAOS IN COOPERATION: CONTINUOUS-VALUED PRISONER’S DILEMMAS IN INFINITE-VALUED LOGIC

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Abstract
The Prisoner’s Dilemma (PD) has become a paradigm of the evolution of cooperation. The PD can be generalized to the continuous-valued case in which players are allowed to choose intermediate levels of cooperation. When continuous-valued PDs are played in the spatial context of cellular automata, generous strategies are favored. Continuous-valued PDs are naturally represented in infinite-valued logic. The infinite-valued logic allows us to prove that cooperative interactions between continuous-valued strategies are paradigmatically chaotic. Escape-time diagrams using a given level of mutual cooperation as a threshold produce fractal images. The sensitive dependence of chaotic dynamical systems models a practical unpredictability within the PD (even though only deterministic nonstochastic strategies are involved) that is characteristic of many real life choice situations.

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