Convex covers of symmetric games

Abstract

We consider a multi-player, cooperative, transferable-utility, symmetric game (N, v) and associated convex covers, i.e., convex games (N, v) such thatv≥ v. A convex cover isefficient iffv(∅)=v(∅) andv(N)=v(N); andminimal iff there is no convex coverv ≠ v such thatv ≤ v. Efficient and minimal convex covers are closely related to the core of (N, v); in fact, extreme points of the core are shown to correspond to efficient convex covers which are minimal and extreme. A necessary and sufficient condition is provided for minimality, and another for extremity. Construction of convex covers and a form of decomposition are treated in detail, and some useful properties are identified which may be recognized in terms of visibility of points on a graph of (N, v) and other elementary concepts.