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.
This report is part of the RAND Corporation paper series. The paper was a product of the RAND Corporation from 1948 to 2003 that captured speeches, memorials, and derivative research, usually prepared on authors' own time and meant to be the scholarly or scientific contribution of individual authors to their professional fields. Papers were less formal than reports and did not require rigorous peer review.
Permission is given to duplicate this electronic document for personal use only, as long as it is unaltered and complete. Copies may not be duplicated for commercial purposes. Unauthorized posting of RAND PDFs to a non-RAND Web site is prohibited. RAND PDFs are protected under copyright law. For information on reprint and linking permissions, please visit the RAND Permissions page.
The RAND Corporation is a nonprofit institution that helps improve policy and decisionmaking through research and analysis. RAND's publications do not necessarily reflect the opinions of its research clients and sponsors.