The core of games on convex geometries
A cooperative game on a finite set of players \(N\) is usually defined as a function \(v\) which assigns, to every possible coalition \(S\subseteq N\), a real number \(v(S)\) which describes how much the members of \(S\) can get if they form this coalition. Some games have solutions \(x\) which are ideal for every coalition, i.e., for which \(\sum_{i\in S} x_i\geq v(S)\) for every coalition \(S\). The set of all such vectors is called a core. If not all coalitions are allowed, then it is natural to define a core as the set of all vectors \(x\) for which \(\sum_{i\in S} x_i\geq v(S)\) for every allowed coalition \(S\). The authors show that many known properties of the standard core hold for games in which the family \(\mathcal L\) of allowed coalitions forms a convex geometry. Specifically, for every family \({\mathcal L}\subseteq 2^N\) which contains \(\emptyset\) and \(N\) and which is closed under intersection, we can define, for every set \(A\subseteq N\), a closure \(\text{Co}(A)\) as the intersection of all sets from \(\mathcal L\) which contain \(A\). The family \(\mathcal L\) is called a convex geometry if it satisfies the following anti-exchange property: if \(i,j\not\in A\) and \(j\in\text{Co}(A\cup i)\), then \(i\not\in\text{Co}(A\cup j)\). This property is true for the cases when \(\mathcal L\) is the set of all convex sets in the normal geometric sense, and when \(\mathcal L\) is the family of all ``convex'' subsets \(S\) of a (partially) ordered set \(N\) (convex in the sense that if \(S\) contains two points \(a\leq b\), it must contain the entire interval \(\{c \mid a\leq c\leq b\}\)).
scientific article; zbMATH DE number 1358203