Title:

Nash equilibria, gale strings, and perfect matchings

This thesis concerns the problem 2NASH of ﬁnding a Nash equilibrium of a bimatrix game, for the special class of socalled “hardtosolve” bimatrix games. The term “hardtosolve” relates to the exponential running time of the famous and often used Lemke– Howson algorithm for this class of games. The games are constructed with the help of dual cyclic polytopes, where the algorithm can be expressed combinatorially via labeled bitstrings deﬁned by the “Gale evenness condition” that characterise the vertices of these polytopes. We deﬁne the combinatorial problem “Another completely labeled Gale string” whose solutions deﬁne the Nash equilibria of any game deﬁned by cyclic polytopes, including the games where the Lemke–Howson algorithm takes exponential time. We show that “Another completely labeled Gale string” is solvable in polynomial time by a reduction to the “Perfect matching” problem in Euler graphs. We adapt the Lemke–Howson algorithm to pivot from one perfect matching to another and show that again for a certain class of graphs this leads to exponential behaviour. Furthermore, we prove that completely labeled Gale strings and perfect matchings in Euler graphs come in pairs and that the Lemke–Howson algorithm connects two strings or matchings of opposite signs. The equivalence between Nash Equilibria of bimatrix games derived from cyclic polytopes, completely labeled Gale strings, and perfect matchings in Euler Graphs implies that counting Nash equilibria is #Pcomplete. Although one Nash equilibrium can be computed in polynomial time, we have not succeeded in ﬁnding an algorithm that computes a Nash equilibrium of opposite sign. However, we solve this problem for certain special cases, for example planar graphs. We illustrate the difﬁculties concerning a general polynomialtime algorithm for this problem by means of negative results that demonstrate why a number of approaches towards such an algorithm are unlikely to be successful.
