More about Scarf and Sperner OIKs

The Lemke-Howson exchange algorithm for finding a Nash equilibrium in bimatrix games, as well as the classical algorithm for finding the properly colored facet in Sperner’s Lemma generalize and abstract to pure combinatorics. In particular, the idea of Lemke pivoting is extended to an arbitrary family of oiks (Euler complexes). Given a room-partition, the corresponding algorithm finds another (distinct) room-partition by traversing an 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 application of Sperner’s Lemma in combinatorial topology are well known. We also formulate the famous Scarf Lemma in terms of oiks. This Lemma has two fundamental applications in game and graph theories. In 1967, Scarf derived from it core-solvability of balanced cooperative games. In 1996, it was shown that kernel-solvability of perfect graphs also results from this Lemma. We show that Scarf’s combinatorially defined oiks are in fact realized by polytopes. We also demonstrate that the pivoting path between room-partitions can be exponentially long in d already for two equal d-dimensional Scarf oiks on 2d vertices. A similar example is constructed for a pair of d-dimensional Scarf and Sperner oiks.