Sion's mini-max theorem and Nash equilibrium in a multi-players game with two groups which is zero-sum and symmetric in each group

We consider the relation between Sion's minimax theorem for a continuous function and a 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 implies Sion's minimax theorem with the coincidence of the maximin strategy and the minimax strategy for players in each group. %given the values of the strategic variables. 2. Sion's minimax theorem with the coincidence of the maximin strategy and the minimax strategy for players in each group implies the existence of a Nash equilibrium which is symmetric in each group. Thus, they are equivalent. An example of such a game is a relative profit maximization game in each group under oligopoly with two groups such that firms in each group have the same cost functions and maximize their relative profits in each group, and the demand functions are symmetric for the firms in each group.Comment: 14 page

Equivalence of Cournot and Bertrand equilibria in duopoly under relative profit maximization: A general analysis

Abstract. We study the relationship between Cournot equilibrium and Bertrand equilibrium in duopoly with differentiated goods in which each firm maximizes its relative profit. We show that Cournot equilibrium and Bertrand equilibrium coincide under relative profit maximization even with general demand and cost functions. This result is due to the fact that a game of relative profit maximization in duopoly is a two-person zero-sum game.Keywords. Relative profit maximization, Duopoly, Cournot equilibrium, Bertrand equilibrium.JEL. D43, L13

Relative profit maximization and equivalence of Cournot and Bertrand equilibria in asymmetric duopoly

We study the relation between a Cournot equilibrium and a Bertrand equilibrium in an \emph{asymmetric} duopoly with differentiated goods in which each firm maximizes its relative profit that is the difference between its profit and the profit of the rival firm. Both demand and cost functions are linear but asymmetric, that is, demand functions for the goods are asymmetric and the firms have different marginal cots. We will show that a Cournot equilibrium and a Bertrand equilibrium coincide even in an asymmetric duopoly

Two person zero-sum game with two sets of strategic variables

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. 1. Player A and B choose sA and sB (competition by (sA, sB)). 2. Player A and B choose tA and tB (competition by (tA, tB)). 3. Player A and B choose tA and sB (competition by (tA, sB)). 4. Player A and B choose sA and tB (competition by (sA, tB))

Choice of strategic variables under relative profit maximization in asymmetric oligopoly

We consider a simple model of the choice of strategic variables under relative profit maximization by firms in an asymmetric oligopoly with differentiated substitutable goods such that there are three firms, Firm 1, 2 and 3, demand functions are linear and symmetric, marginal costs are constant, there is no fixed cost, Firm 2 and 3 have the same cost function, but Firm 1 has a different cost function. In such a model we show that there are two pure strategy sub-game perfect equilibria. One is such that all firms choose the outputs as their strategic variables, and the other is such that Firm 2 and 3 choose the outputs as their strategic variables, and Firm 1 chooses the price as its strategic variable

Choice of strategic variables under relative profit maximization in asymmetric oligopoly

We consider a simple model of the choice of strategic variables under relative profit maximization by firms in an asymmetric oligopoly with differentiated substitutable goods such that there are three firms, Firm 1, 2 and 3, demand functions are linear and symmetric, marginal costs are constant, there is no fixed cost, Firm 2 and 3 have the same cost function, but Firm 1 has a different cost function. In such a model we show that there are two pure strategy sub-game perfect equilibria. One is such that all firms choose the outputs as their strategic variables, and the other is such that Firm 2 and 3 choose the outputs as their strategic variables, and Firm 1 chooses the price as its strategic variable