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  • A Coalitional Game of Four players with the Characteristic Function
  • トクセイ カンスウ オ コウリョ シタ 4ニン キョウリョク ゲーム

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Game theory is the study of mathematical models of competition and cooperation between rational decision-makers pioneered by Princeton mathematician John Von Neumann. His paper was followed by his 1944 book “Theory of Games and Economic Behavior” with Oskar Morgenstern. In this research, based on the Shapley value introduced by 2012 Nobel Economics prize-winner L.S. Shapley, we discuss how to divide the reward in a coalitional game depending on the relationships between the players. 　In this paper, we identify the relationship between two players, and define the value of the characteristic function depending on the state of that relationship. If the relationship is good, the value of the characteristic function is high. If the relationship is bad, that value goes down. We define a characteristic function to have these properties and discuss the strategy of each player. By applying the characteristic function, which depends on the relationship, we prove some properties that describe when a specific player’s reward will be the maximum or the minimum. In particular, the number who participate in this game is extended to four persons from three. 　When the number of participants increases by one, different situations develop with each player’s Shapley value changing from 33=27 to 36=729. We can discuss the strategy of each player by computation simulations with VBA （Microsoft Excel）.

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