Chaos in fictitious play
We study the dynamics of a two player continuous time bi-matrix fictitious play. In particular, we investigate the dynamics of a one-parameter family of 3 x 3 games that includes a well-known example of Shapley's as a special case. We adopt a geometric (dynamical systems) approach and study the dynamics and bifurcations both on the strategy simplex, and projected onto S3. For the more interesting parameter values we show that the flow is essentially continuous and uniquely defined everywhere except at the interior equilibrium. We prove the existence and stability properties of three periodic orbits of the flow analytically, and study in some detail the dynamics near a co-dimension two indifference set that is crucial to understanding the global dynamics. We prove that for a range of parameter values the flow has infinitely many periodic orbits. For one parameter value we find that all trajectories tend to a neighborhood of the interior equilibrium, though this is not asymptotically stable. We illustrate the properties of the flow with numerical simulations. We note that many of the phenomena we observe and results we obtain are new, and where possible we state and prove general theorems that apply to more general games. Finally, we describe some areas where further study is warranted.
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|Date of creation:||04 Jan 2001|
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