Discrete Methods in Cooperative Game Theory

Abstract

Extreme point problems and the convergence of solution concepts (the core, the “competitive equilibriuml”) are topics of game theory that are frequently dealt with in a nonatomic framework (a measure space of players endowed with a nonatomic nonadditive set function). This note as a survey presents two discrete analogues of “nonatomicity”: Nondegeneracy and homogeneity of additive and nonadditive set functions. It turns out that these concepts lead to combinatorial and number theoretical problems (e.g. MINKOWSKI’s second theorem) such that general principels of game theory (convergence problems etc.) maybe formulated in a discrete manner. Hence a statement like “the core and the competitive equilibrium coincide for large sets of players” is formulated rigorously by means of Geometric Number Theory instead of nonatomic measure theory.