关键词: 双矩阵博弈, Nash均衡, 公平关切, 公平参照点

Abstract: Game theory is mainly concerned with the players’ optimal strategies, and has become a powerful tool for analyzing interactions among different players. It has applications in diverse areas including economics, policy, psychology, environment and logistics, and so on. Bimatrix games, as an important part of non-cooperative games, play a vital role since the very beginning of game theory. A classic bimatrix game involves two different rational players who make corresponding decisions to maximize their own interests, following the assumption of “economic man”. However, this assumption limits the application of bimatrix games in reality. Due to the complex and changeable environment, cognitive limitations, emotions and preferences, people tend to show the behavior characteristics of limited rationality, which makes their choices deviate from the prediction of classical theory. Recently, a large number of behavioral and economic researches and practical cases have shown that people are concerned about fairness, and will give up part of their own interests to achieve fairness when they are treated unfairly. Namely, people show the characteristics of the bounded rationality with fairness concerns. Therefore, in order to make the theoretical prediction result more realistic, this paper incorporates fairness concerns into the bimatrix game to analyze the influence of the players’ fairness concern on the Nash equilibrium of the bimatrix game.
In order to accurately describe the players’ fairness concern, some scholars put forward several models. The model (namely F-S model) proposed by Fehr and Schmidt, one of the most famous, focuses on the issue of the fair distribution of the benefits. The F-S model points out that players show great concern for their own and others’ benefits and the basis for judging fairness is mainly to compare their own benefits with those of others. Namely, the fair reference point of a player is the opponent’s payoff. However, this point may not be fully applicable in practice, because a player would not like to get the same benefit as his opponent, if his competitive power or contribution is comparatively larger, and vice versa. Therefore, Ewerhart modified the F-S model to stress the fairness but not the outright altruism, in which a common agreement reached by the two players is regarded as their own fairness reference level. Therefore, in this paper, the model proposed by Ewerhart is used to describe the players’ fairness concern behavior in the bimatrix game. By incorporating exogenous and fixed fairness reference points, this paper constructs a model of bimatrix game with fairness concern and explores the existence of its equilibrium.
At present, some scholars have carried out research on the problem of the bimatrix game with limited rational behavior. However, these researches do not consider the strategy choice of fair-minded players. Other scholars also have discussed the influence of fairness concern on game equilibrium, which mainly focuses on bargaining game, Betrand game, prisoner’s dilemma game and so on. But, none of them consider the bimatrix games. Therefore, it is meaningful to analyze bimatrix games with fairness concerns. In view of this, in the bimatrix game, this paper considers the situation where only one player is fair-minded and his fairness reference point is exogenous and fixed. A bilinear programming method is applied to solve the Nash equilibrium of this game. Then, a complete analysis of the 2×2 bimatrix games with fairness concerns is conducted. Specifically, three scenarios are considered to explore the influence of fairness concerns on the Nash equilibrium when the fairness reference points are in different value ranges. Finally, a simple case is used for analyzing and verifying.
The results show that if a player has the characteristics of fairness concern behavior, the equilibrium strategy of his opponent and this player’s expected payoff are not only affected by his own fairness concern behavior, but also related to his fairness reference point. To be specific, when the player’s fairness reference point is in a different value range, the opponent’s equilibrium strategy and his own expected payoff will also change differently with an increase in fairness concern degree. In other words, the player’s fairness concerns may be beneficial or hurtful to himself, or may not have an effect on himself.

Key words: bimatrix games, Nash equilibrium, fairness concerns, fairness reference point