On the geometric dimension of the core of a TU-game

Abstract

In this paper we state a condition which characterizes the sub-class of TU-games (games with transferable utility) having non-empty core of dimension k; for any 0 <= k <= n. It improves and generalizes the adding up (AU) property used by Brandenburger and Stuart for the case k = 0 (games with one-point core) to study biform games. It also embraces one of the two conditions stated by Zhao for the case k = n - 1 (games having core with non-empty interior, relative to the set of pre-imputations) while studying some geometric properties of the core. The condition allows us to show that all the information about the geometric dimension of the core is contained in the vector of excesses associated to the nucleolus of the game. It also allows us to get some insight about the geometric properties of the cone of balanced games as well. In particular, we prove that all the games in the relative interior of each face of the cone have a core with the same geometric dimension. This fact is illustrated for the case of three-person games. We also present a couple of examples to show how the results of the paper can be used to deal with biform games from a new perspective