We study the Bertrand equilibrium in duopoly in which two firms produce a homogeneous good under quadratic cost functions, and they seek to maximize the weighted sum of their absolute and relative profits. We show that there exists a range of the equilibrium prices in duopolistic equilibria. This range of equilibrium prices is narrower and lower than the range of the equilibrium prices in duopolistic equilibria under pure absolute profit maximization, and the larger the weight on the relative profit, the narrower and lower the range of the equilibrium prices. In this sense relative profit maximization is more aggressive than absolute profit maximization.
About a symmetric three-players zero-sum game we will show the following results. 1. A modified version of Sion's minimax theorem with the coincidence of the maximin strategy and the minimax strategy are proved by the existence of a symmetric Nash equilibrium. 2. The existence of a symmetric Nash equilibrium is proved by the modified version of Sion's minimax theorem with the coincidence of the maximin strategy and the minimax strategy. Thus, they are equivalent. If a zero-sum game is asymmetric, maximin strategies and minimax strategies of players do not correspond to Nash equilibrium strategies. If it is symmetric, the maximin strategies and the minimax strategies constitute a Nash equilibrium. However, without the coincidence of the maximin strategy and the minimax strategy there may exist an asymmetric equilibrium in a symmetric three-players zerosum game.
We consider a two-person zero-sum game with two sets of strategic variables which are related by invertible functions. They are denoted by [Formula: see text] and [Formula: see text] for players A and B. The payoff function of Player A is [Formula: see text]. Then, the payoff function of Player B is [Formula: see text]. [Formula: see text] is upper semi-continuous and quasi-concave on [Formula: see text] for each [Formula: see text] (or each [Formula: see text]), upper semi-continuous and quasi-concave on [Formula: see text] for each [Formula: see text] (or each [Formula: see text]), lower semi-continuous and quasi-convex on [Formula: see text] for each [Formula: see text] (or each [Formula: see text], and lower semi-continuous and quasi-convex on [Formula: see text] for each [Formula: see text] (or each [Formula: see text]). We will show that the following four patterns of competition are equivalent, that is, they yield the same outcome. (1) Players A and B choose [Formula: see text] and [Formula: see text] (competition by [Formula: see text]). (2) Players A and B choose [Formula: see text] and [Formula: see text] (competition by [Formula: see text]). (3) Players A and B choose [Formula: see text] and [Formula: see text] (competition by [Formula: see text]). (4) Players A and B choose [Formula: see text] and [Formula: see text] (competition by [Formula: see text]).
This study derives pure strategy Bertrand equilibria in a duopoly in which two firms produce a homogeneous good with convex cost functions and seek to maximize the weighted sum of their absolute and relative profits. The study shows that there exists a range of equilibrium prices in duopolistic equilibria. This range of equilibrium prices is narrower and lower than the range of equilibrium prices in duopolistic equilibria under pure absolute profit maximization. Moreover, the larger the weight on the relative profit, the narrower and lower the range of equilibrium prices. In this sense, relative profit maximization is more aggressive than absolute profit maximization.
JEL D43 L13
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