On Scarf's theorem for generalized cooperative games with external relations
Mikhail V. Bludov, Oleg R. Musin
TL;DR
The paper tackles generalized NTU cooperative games with external relations, where coalitions may allocate utility beyond their members. It develops a topological framework linking the existence of a nonempty fractional core to homotopy invariants of covers, and proves a generalized Scarf theorem for balanced games: core nonemptiness holds iff the game is homotopically nontrivial. The authors introduce the triple $(U,V,r)$, define $r$-balanced sets via cover intersections, and establish equivalence between fractional core existence and topological nontriviality through KKMS-type arguments. They illustrate the theory with Scarf-type reductions, degree-k games, centrally symmetric setups, and a Hopf-fibration example, highlighting the method's reach beyond classical TU/NTU results and its potential to reveal topological structure in stability of allocations.
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.