Sperner Oiks

Jack Edmonds, Stéphane Gaubert, Vladimir Gurvich

Abstract: The idea of “Lemke pivoting in a family of oiks (Euler complexes)” generalizes, and abstracts to pure combinatorics, the Lemke-Howson exchange algorithm for finding a Nash equilibrium in bimatrix games, as well as the classical algorithm for finding the properly colored room in Sperner's Lemma. Given a “room-partitioning”, this algorithm finds another (distinct) room-partitioning by traversing the exchange graph. In this paper we show that each family of k oiks O={O1,…,Ok} can be reduced to a pair of oiks O′={O1+…+Ok,O0} (one of which, O0, is a Sperner oik) such that the exchange graphs for O and O′ are isomorphic. Numerous applications of Sperner's Lemma in combinatorial topology are well known.