Characteristic sets and characteristic numbers of matrix two-person games

Abstract

Focusing on the extreme points of the solution sets of matrix two-person games, we propose the notions of characteristic sets and characteristic numbers. The characteristic sets (resp., the characteristic numbers) are the sets (resp., the numbers) of the extreme points of the solution set of the game and the optimal solution sets of the players. These concepts allow us to measure the complexity of the game. The larger the characteristic numbers, the more complex the game is. Among other things, we obtain upper bounds for the characteristic numbers and give a novel geometric construction. By the construction, we get useful descriptions of the optimal strategy set of each player of a game given by any nonsingular square matrix. Namely, the investigation of the geometry the just-mentioned sets reduces to computing or studying certain simpler sets. We also formulate several open problems.