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

• 成果类型: Article
• 署名作者: Sparrow, Colin; van Strien, Sebastian; Harris, Christopher
• 署名单位: University of Warwick
• 刊物名称: GAMES AND ECONOMIC BEHAVIOR
• ISSN/ISSBN: 0899-8256
• DOI: 10.1016/j.geb.2007.08.005
• 发表日期: 2008
• 页码: 259-291
• 关键词: fictitious pay; Learning process; bifurcation; chaos

摘要

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. Crown Copyright (c) 2007 Published by Elsevier Inc. All rights reserved.