On Scarf's theorem for generalized cooperative games with external relations

TL;DR

This paper extends cooperative game theory by allowing coalitions to share utility with non-members and establishes an analogue of Scarf's theorem, linking core existence to a homotopy invariant of covers.

Contribution

It introduces a generalized framework for cooperative games with external relations and proves a new version of Scarf's theorem applicable to these games.

Findings

Core existence relates to a homotopy invariant of covers.

Generalized sharing rules include non-contributing players.

Analogue of Scarf's theorem established for external relations.

Abstract

In this paper, we consider a generalization of cooperative games to the case where a coalition can distribute the earned utility not only among its members but also to other players. In particular, we consider an example where coalitions are required to share their winnings with non-contributing players. For these generalized games, we also provide an analogue of Scarf's theorem. It turns out that in this generalization, the existence of a non-empty core is closely related to a homotopy invariant of covers defined by the cooperative game.