On Balanced Games Without Side Payments

by Lloyd S. Shapley


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Presents a new proof of a basic theorem of game theory, due to Scarf, which states that every balanced game without side payments has a nonempty core. The main tool is a generalization of Sperner's topological lemma concerning triangulations of the simplex. The proof, like Scarf's, is based on a "pathfollowing" algorithm, descended from the Lemke-Howson procedure for finding equilibrium points in bimatrix games. However, it stays close to familiar ground most of the way and specializes to the game context only at the very end. An appendix details a notational scheme for iterated barycentric partitions of the [n]-simplex, suitable for use in computer programs. This paper is as self-contained as possible for the convenience of readers new to the subject of balanced sets and [n]-person games.

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