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Fictitious play in 3x3 games: The transition between periodic and chaotic behaviour

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  • Sparrow, Colin
  • van Strien, Sebastian
  • Harris, Christopher

Abstract

In the 1960s Shapley provided an example of a two-player fictitious game with periodic behaviour. In this game, player A aims to copy B's behaviour and player B aims to play one ahead of player A. In this paper we generalise Shapley's example by introducing an external parameter. We show that the periodic behaviour in Shapley's example at some critical parameter value disintegrates into unpredictable (chaotic) behaviour, with players dithering a huge number of times between different strategies. At a further critical parameter the dynamics becomes periodic again, but now both players aim to play one ahead of the other. In this paper we adopt a geometric (dynamical systems) approach. Here we prove rigorous results on continuity of the dynamics and on the periodic behaviour, while in the sequel to this paper we shall describe the chaotic behaviour.

Suggested Citation

  • Sparrow, Colin & van Strien, Sebastian & Harris, Christopher, 2008. "Fictitious play in 3x3 games: The transition between periodic and chaotic behaviour," Games and Economic Behavior, Elsevier, vol. 63(1), pages 259-291, May.
  • Handle: RePEc:eee:gamebe:v:63:y:2008:i:1:p:259-291
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    References listed on IDEAS

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    Cited by:

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    3. Sandholm, William H., 2015. "Population Games and Deterministic Evolutionary Dynamics," Handbook of Game Theory with Economic Applications,, Elsevier.

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