Balanced Games

Characteristic functions of transferable utility games (abbreviated by “games” for simplicity) are set functions defined on a finite set N and vanishing on the empty set. They are of wide use in cooperative game theory, but also in decision theory (capacities are monotone games), combinatorial optimization and discrete mathematics, and are also closely related to imprecise probabilities. Besides, the core of a game is the set of additive games dominating this game and coinciding on N. When the game is a capacity, the core is the set of compatible probability measures. Games with a nonempty core are called balanced.
    The purpose of the talk is to give a survey on the known properties of balanced games and related notions. First, I will recall the classical Bondareva-Shapley result characterizing balanced games, together with the notion of (minimal) balanced collections. Then, I will address the problem of generating and enumerating minimal balanced collections, and show several applications of minimal balanced collections in cooperative game theory, including core stability, i.e., under which conditions the core of a game is a stable set in the sense of von Neumann and Morgenstern. In the last part, I will study the geometric properties of the set of balanced games, which is a cone, and find its linearity space and extreme rays.