John M. Rulnick (Department of Electrical and Computer Engineering, Worcester Polytechnic Institute, Worcester, MA 01609, USA) Lloyd S. Shapley (Department of Mathematics, University of California, Los Angeles, CA 90095, USA)
Abstract
We consider a multi-player, cooperative, transferable-utility, symmetric game (N, ) and associated convex covers, i.e., convex games (N, ~) such that ~ \geq . A convex cover is efficient iff ~(∅) = (∅) and ~(N) = (N); and minimal iff there is no convex cover ~ \neq ~ such that ~ \leq ~. Efficient and minimal convex covers are closely related to the core of (N, ); 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, ) and other elementary concepts.
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