On Balanced Games with Infinitely Many Players: Revisiting Schmeidler's Result

TL;DR

This paper extends Schmeidler's classic result on the non-emptiness of the core in infinite-player cooperative games by relaxing the non-negativity assumption to boundedness below, and explores core properties under various conditions.

Contribution

It generalizes Schmeidler's theorem to bounded below games and analyzes core existence when the game is not bounded below, including restricted cooperation scenarios.

Findings

Core is non-empty iff the game is balanced for bounded below games.

Core may be empty even if the game is balanced when not bounded below.

Generalization to restricted cooperation scenarios.

Abstract

We consider transferable utility cooperative games with infinitely many players and the core understood in the space of bounded additive set functions. We show that, if a game is bounded below, then its core is non-empty if and only if the game is balanced. This finding is a generalization of Schmeidler's (1967) original result ``On Balanced Games with Infinitely Many Players'', where the game is assumed to be non-negative. We furthermore demonstrate that, if a game is not bounded below, then its core might be empty even though the game is balanced; that is, our result is tight. We also generalize Schmeidler's (1967) result to the case of restricted cooperation too.