Applications of Category Theory to Games and Economic Behavior

2019 Fiscal Year Final Research Report

• Project/Area Number: 17K18553
• Research Category: Grant-in-Aid for Challenging Research (Exploratory)
• Allocation Type: Multi-year Fund
• Research Field: Economics, Business Administration, and related fields
• Research Institution: University of Tsukuba
• Principal Investigator: Turnbull Stephen John 筑波大学, システム情報系, 准教授 (90240621)
• Project Period (FY): 2017-06-30 – 2020-03-31
• Keywords: 圏論 / ゲーム圏 / モナド / 混合延長 / category theory / game category / monad / mixed extension

Outline of Final Research Achievements

Game theory is a branch of mathematics useful in understanding strategic behavior in society, especially economic behavior. Category theory is a set of tools for analyzing and comparing mathematical structures, and for translating from one kind of structure to another.
In this exploratory research, we first define two categories of noncooperative games in strategic form. We use these categories to examine the structure of such games, showing in both categories that when the functor extending a game to its mixed extension is paired with its right adjoint forgetful functor (the process of treating the extended game as a game, without using the fact that its strategies are probability distributions), the forgetful functor is monadic. We show that category-theoretic isomorphism classes reproduce the traditional classification of 2x2 matrix games.

Free Research Field

理論経済学

Academic Significance and Societal Importance of the Research Achievements

混合延長関手の忘却関手はモナディックという定理の異義とは混合延長ゲームが戦略が確率分布で報酬関数が線形型である特別ゲームの種類でありながらすべてのゲームの行動をモデル化できることを証明する。重要点は社会現象をモデル化する一般のゲームに均衡が存在しない場合があるが、そのゲームの混合延長には均衡が存在する。
ランダム行動の解釈が困難の場合があるが、解釈が明らかの場合もある。だが、均衡のないゲームには解釈がないのでゲームその混合延長の関係の理解を深めることが大きな貢献と考えられる。