This thesis is devoted to the study of bimatrix games, in particular examples of 3x3 bimatrix games governed by the Best Response Dynamics or coupled replicator equations.
In Chapter 2, we present an approach to Hofbauer’s conjecture, positing that convergence of bimatrix games to the unique and completely mixed Nash equilibrium under the Best Response Dynamics
occurs only for zero-sum games. We present partial results concerning this conjecture and inter alia provide some dynamical characterization of zero-sum games. In section 3.7, we provide a presumptive evidence that Hofbauer’s conjecture might be incorrect.
In Chapter 3, we consider an example of a cyclic competition bimatrix game, a Rock-Scissors-Paper game. We investigate shadowing of the non-transverse heteroclinic network naturally appearing in
this model, as well as the stability of some peculiar subcycles. Moreover, we present our numerical investigations and provide our presumptions about the way chaos might be born in this model.
In Chapter 4, we consider a toy-model motivated by the Rock-Scissors-Paper game. The toy-model consists of a 2-dimensional center manifold of a saddle-elliptic point in R4 and 3-dimensional stable/unstable manifolds (of this center manifold) which coincide. We derive the normal form of the
ODE-governed toy-model being considered near the center manifold and the normal form of the return map to the vicinity of the center manifold along the non-transverse homoclinic channel. We investigate in detail the dynamics of the return map.