Geometry and equilibria in bimatrix games

This thesis studies the application of geometric concepts and methods in the analysis of strategic-form games, in particular bimatrix games. Our focus is on three geometric concepts: the index, geometric algorithms for the computation of Nash equilibria, and polytopes. The contribution of this thesis consists of three parts. First, we present an algorithm for the computation of the index in degenerate bimatrix games. For this, we define a new concept, the “lex-index” of an extreme equilibrium, which is an extension of the standard index. The index of an equilibrium component is easily computable as the sum of the lex-indices of all extreme equilibria of that component. Second, we give several new results on the linear tracing procedure, and its bimatrix game implementation, the van den Elzen-Talman (ET) algorithm. We compare the ET algorithm to two other algorithms: On the one hand, we show that the Lemke-Howson algorithm, the classic method for equilibrium computation in bimatrix games, and the ET algorithm differ substantially. On the other hand, we prove that the ET algorithm, or more generally, the linear tracing procedure, is a special case of the global Newton method, a geometric algorithm for the computation of equilibria in strategic-form games. As the main result of this part of the thesis, we show that there is a generic class of bimatrix games in which an equilibrium of positive index is not traceable by the ET algorithm. This result answers an open question regarding sustainability. The last part of this thesis studies the index in symmetric games. We use a construction of polytopes to prove a new result on the symmetric index: A symmetric equilibrium has symmetric index +1 if and only if it is “potentially unique”, in the sense that there is an extended symmetric game, with additional strategies for the players, where the given symmetric equilibrium is unique.