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Complex dynamics in learning complicated games

Game theory is the standard tool used to model strategic interactions in evolutionary biology and social science. Traditional game theory studies the equilibria of simple games. But is traditional game theory applicable if the game is complicated, and if not, what is? We investigate this question here, defining a complicated game as one with many possible moves, and therefore many possible payoffs conditional on those moves. We investigate two-person games in which the players learn based on experience. By generating games at random we show that under some circumstances the strategies of the two players converge to fixed points, but under others they follow limit cycles or chaotic attractors. The dimension of the chaotic attractors can be very high, implying that the dynamics of the strategies are effectively random. In the chaotic regime the payoffs fluctuate intermittently, showing bursts of rapid change punctuated by periods of quiescence, similar to what is observed in fluid turbulence and financial markets. Our results suggest that such intermittency is a highly generic phenomenon, and that there is a large parameter regime for which complicated strategic interactions generate inherently unpredictable behavior that is best described in the language of dynamical systems theory