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 price in duopolistic equilibria. This...

We study implications of the choice of strategic variables, price or quantity, by firms in a duopoly with differentiated goods in which each firm maximizes its relative profit. We consider general demand and cost functions, and show that the choice of strategic variables is irrelevant in the...

We will show that Sion's minimax theorem is equivalent to the existence of Nash equilibrium in a symmetric multi-person zero-sum game. If a zero-sum game is asymmetric, maximin strategies and minimax strategies of players do not correspond to Nash equilibrium strategies. However, if it is...

About a symmetric multi-person zero-sum game we will show the following results. 1. The existence of a symmetric Nash equilibrium is proved by the Glicksberg fixed point theorem. 2. Sion's minimax theorem and the coincidence of the maximin strategy and the minimax strategy are proved by the...

We consider the relation between Sion's minimax theorem for a continuous function and a Nash equilibrium in an asymmetric three-players zero-sum game with two groups. Two players are in Group A, and they have the same payoff function and strategy space. One player, Player C, is in Group C. Then,...

We consider the relation between Sion's minimax theorem for a continuous function and a Nash equilibrium in an asymmetric three-players zero-sum game with two groups. Two players are in Group A, and they have the same payoff function and strategy space. One player, Player C, is in Group C. Then,...

We consider a formulation of a non zero-sum n players game by an n+1 players zero-sum game. We suppose the existence of the n+1-th player in addition to n players in the main game, and virtual subsidies to the n players which is provided by the n+1-th player. Its strategic variable affects only...

We consider the relation between Sion's minimax theorem for a continuous function and Nash equilibrium in a multi-players game with two groups which is zero-sum and symmetric in each group. We will show the following results. 1. The existence of Nash equilibrium which is symmetric in each group...

We consider a partially asymmetric multi-players zero-sum game with two strategic variables. All but one players have the same payoff functions, and one player (Player $n$) does not. Two strategic variables are t_i's and s_i's for each player i. Mainly we will show the following results. 1) The...

We consider a two-person zero-sum game with two sets of strategic variables which are related by invertible functions. They are denoted by (sA, sB) and (tA, tB) for players A and B. We will show that the following four patterns of competition are equivalent, that is, they yield the same outcome....