在這篇論文中，我們首先分析了一個艾爾法酒吧賽局的虛擬決策過程， 然後我們考慮一個廣義的艾爾法酒吧賽局， 我們提出了在廣義的艾爾法酒吧賽局中，玩家的學習過程將達到奈許平衡。之後我們提出一個學習過程是由強化學習方法和虛擬決策的混合，我們將這個混合的學習過程應用於艾爾法酒吧賽局。

In this paper we first analyze the fictitious play process of an El Farol bar game. We then consider a generalized El Farol bar game. We propose a learning procedure for the players in the generalized El Farol bar game to reach a Nash equilibrium. The proposed learning procedure is a mixture of the reinforcement learning method and the fictitious play method.

中文摘要i
Abstract ii
Acknowledgements iii
List of Figures vi
List of Tables vii
1 Introduction 1
2 Fictitious Play of an El Farol Bar Game 5
2.1 A Special Case in which c = N-1 . . . . . . . . . . . . 11
2.2 A Special Case in which c = 1 . . . . . . . . . . . . . . 19
2.3 A General Case in which c = 2: : :N-2 . . . . . . . . . 20
3 A Generalized El Farol Bar Game 25
4 Reinforcement Learning and Fictitious Play 29
5 Numerical and Simulation Results 32
5.1 Simulation settings . . . . . . . . . . . . . . . . . . . . 33
5.2 Reinforcement learning method . . . . . . . . . . . . . 34
5.3 Fictitious play method . . . . . . . . . . . . . . . . . . 36
5.4 Mixed method . . . . . . . . . . . . . . . . . . . . . . . 40
6 Conclusions 41
Bibliography 42

[1] D. Fudenberg and D. K. Levine, The theory of learning in games.Massachusetts: The MIT Press, 1999.
[2] G. W. Brown, “Iterative solution of games by fictitious play,” in Activity Analysis of Production and Allocation. New York: John
Wiley and Sons, 1951, ch. 24.
[3] E. Hopkins, “Two competing models of how people learn in
games,” Econometrica, vol. 70, no. 6, pp. 2141–2166, 2002.
[4] I. Erev and A. E. Roth, “Predicting how people play games: Reinforcement learning in experimental games with unique, mixed strategy equilibria,” The American Economic Review, vol. 88, no. 4, pp. 848–881, 1998.
[5] R. S. Sutton and A. G. Barto, Reinforcement learning: an introduction. Massachusetts: The MIT Press, 2012.
[6] W. B. Arthur, “Complexity in economic theory. inductive reasoning and bounded rationality,” American Economic Review, vol. 84, 1994.42 BIBLIOGRAPHY 43
[7] D. Easley and J. Kleinberg, Networks, crowds and markets reasoning about a highly connected world. Cambridge University Press, 2010.
[8] D. Challet, M. Marsili, and G. Ottino, “Shedding light on el farol,” Physica A: Statistical Mechanics and Its Applications, vol. 332, pp. 469–482, 2004.
[9] R. Franke, “Reinforcement learning in the el farol model,” Journal of Economic Behavior & Organization, vol. 51, pp. 367–388, 2003.
[10] D. Whitehead, “The el farol bar problem revisited: Reinforcement learning in a potential game,” in ESE Discussion Papers 186. Edinburgh School of Economics, University of Edinburgh, Tech. Rep., 2008.
[11] Y. R. Chao, “A note on “continuous mathematical induction”,” Bull. Amer. Math. Soc., vol. 26, no. 1, pp. 17–18, 1919.