混合均衡的隨機穩定：期許水準於零和賽局之應用

Learning through Aspiration to Play the Mixed Equilibrium in Zero-Sum Games

10.6277/TER.202206_50(2).0003

袁國芝(Kuo-Chih Yuan)；王子豪(Tzu-Hao Wang)；莊委桐(Wei-Torng Juang)

零和賽局 ； 混合Nash均衡 ； 期許水準 ； 隨機穩定 ； zero-sum game ； mixed Nash equilibrium ； aspiration ； stochastic stability

經濟論文叢刊

50卷2期（2022 / 06 / 01）

217 - 249

繁體中文

在有唯一混合Nash均衡的兩人零和賽局中，我們導入期許水準（Aspiration）建立了一個超賽局模型。參賽者於每期選定一個策略並重複玩N次零和賽局；以此方式進行無限多期。參賽者透過檢視當前策略於前期的表現是否達到設定期許水準來決定繼續使用原策略或挑選新的策略。當策略都不能滿足時則降低期許水準。參賽者有微小的機率會進行新的嘗試或犯錯。依據Young（1993, 1998）的隨機隱定方法，我們證明「兩個玩家期許水準為零且採用混合均衡策略」為隨機穩定狀態之一。因此我們部分解決了Crawford難題。並以兩性戰爭賽局為例，說明混合均衡也可以如純粹均衡一般穩定。

This paper aims to provide a theoretical foundation for players learning to play mixed strategies and analyze the stochastic stability of the unique mixed Nash equilibrium in zero-sum games. We construct a supergame in which each player selects a strategy to play a zero-sum game for N rounds in each period. Each player then compares the average payoffs received in the N rounds with her aspiration. If the former exceeds the latter, the player is satisfied and sticks to the same strategy for the next period; otherwise she randomly selects a new strategy from the set of feasible strategies that have not yet been adopted. If all strategies have been tried and none of them can fulfill the player's current aspiration, then the player lowers her aspiration. Players also have a small probability of making mistakes when adjusting their strategies or aspiration. We apply the stochastic stability approach proposed by Young (1993), combined with the aspiration hypothesis and show that the unique mixed Nash equilibrium outcome is stochastically stable in a zero-sum game. We also use Battle of the Sexes to illustrate that mixed equilibrium could be as stable as pure ones.

1. Börgers, Tilman,Sarin, Rajiv(2000).Naïve Reinforcement Learningwith Endogenous Aspirations.International Economic Review,41(4),921-950.
2. Brown, George William(1951).Iterative Solution of Games by Fictitious Play.Activity Analysis of Production and Allocation, Cowles Commission Monography,New York:
3. Chong, Juin-Kuan,Ho, Teck-Hua,Camerer, Colin(2016).A Generalized Cognitive Hierarchy Model of Games.Games and Economic Behavior,99,257-274.
4. Conlisk, John(1993).Adaptation in Games: Two Solutions to the Crawford Puzzle.Journal of Economic Behavior and Organization,22(1),25-50.
5. Conlisk, John(1993).Adaptive Tactics in Games: Further Solutions to the Crawford Puzzle.Journal of Economic Behavior and Organization,22(1),51-68.
6. Crawford, Vincent P.(1985).Learning Behavior and Mixed-Strategy Nash Equilibria.Journal of Economic Behavior and Organization,6(1),69-78.
7. Crawford, Vincent P.(1974).Learning the Optimal Strategy in a Zero-SumGame.Econometrica,42(5),885-891.
8. Dieckmann, Tone(1998).Stochastic Learning and the Evolution of Convention.Constitutional Political Economy,9(3),187-212.
9. Gilboa, Itzhak,Schmeidler, David(1996).Case-Based Optimization.Games and Economic Behavior,15(1),1-26.
10. Juang, Wei-Torng(2002).Rule Evolution and Equilibrium Selection.Gamesand Economic Behavior,39(1),71-90.
11. Karandikar, Rajeeva,Mookherjee, Dilip,Ray, Debraj,VegaRedondo, Fernando(1998).Evolving Aspirations and Cooperation.Journal ofEconomic Theory,80(2),292-331.
12. Lambson, Val E.,Probst, Daniel A.(2004).Learning by Matching Patterns.Games and Economic Behavior,46(2),398-409.
13. Napel, Stefan(2003).Aspiration Adaptation in the Ultimatum Minigame.Games and Economic Behavior,43(1),86-106.
14. Oechssler, Jörg(2002).Cooperation as Result of Learning with AspirationLevels.Journal of Economic Behavior and Organization,49(3),405-409.
15. Pangallo, Marco,Heinrich, Torsten,Doyne Farme, J.(2019).Best Reply Structure and Equilibrium Convergence in Generic Games.ScienceAdvances,5(2)
16. Pazgal, Amit(1997).Satisficing Leads to Cooperation in Mutual InterestGames.International Journal of Game Theory,26(4),439-453.
17. Robinson, Julia(1951).An Iterative Method of Solving a Game.Annals ofMathematics,54(2),296-301.
18. Schade, Christian,Schroeder, Andreas,Krause, Kai Oliver(2010).Coordination after Gains and Losses: Is Prospect Theory’s Value FunctionPredictive for Games?.Journal of Mathematical Psychology,54(5),426-445.
19. Spiliopoulos, Leonidas(2012).Pattern Recognition and Subjective BeliefLearning in a Repeated Constant-Sum Game.Games and Economic Behavior,75(2),921-935.
20. Stahl, Dale O.(1988).On the Instability of Mixed-strategy Nash Equilibria.Journal of Economic Behavior and Organization,9(1),59-69.
21. Tucker, Albert William(ed.)(1964).Advances in Game Theory.New Jersey:Princeton Univ. Press.
22. Young, H. Peyton(1993).The Evolution of Conventions.Econometrica,61(1),57-84.
23. Young, H. Peyton(1998).Individual Strategy and Social Structure: An Evolutionary Theory of Institutions.New Jersey:Princeton University Press.
24. Young, H. Peyton,Dean Foster(1991).Cooperation in the Short andin the Long Run.Games and Economic Behavior,3(1),145-156.