Sion's minimax theorem and Nash equilibrium in a multiplayers game with two groups which is zerosum and symmetric in each group
Abstract
We consider the relation between Sion's minimax theorem for a continuous function and a Nash equilibrium in a multiplayers game with two groups which is zerosum 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2018
 arXiv:
 arXiv:1809.01488
 Bibcode:
 2018arXiv180901488S
 Keywords:

 Mathematics  Optimization and Control;
 Computer Science  Computer Science and Game Theory;
 91A10
 EPrint:
 14 pages