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

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

《經濟論文叢刊》 50卷2期 (2022/06) Pp. 217-249

https://doi.org/10.6277/TER.202206_50(2).0003

摘要

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。

關鍵字

零和賽局 ； 混合Nash均衡 ； 期許水準 ； 隨機穩定

並列摘要

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.

並列關鍵字

zero-sum game ； mixed Nash equilibrium ； aspiration ； stochastic stability

參考文獻

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