Complex behaviour in cyclic competition bimatrix games

Abstract: We consider an example of cyclic competition bimatrix game which is a Rock-Scissors-Paper game with assumption about perfect memory of the playing agents. At first we investigate the dynamics in the neighbourhood of the Nash equilibrium as well as the dynamics on the boundary of codimension 1 - that is when one of the strategies is not played by any agent. For the analysis of the asymptotic behaviour close to the boundary of state space, we provide the description of a naturally appearing heteroclinic network. Due to the symmetry, the heteroclinic network induces a quotient network. This quotient network is investigated as well. In the literature this model was already studied, but unfortunately the results stated there are invalid, as we will show in this paper. It turns out that certain types of behaviour are never possible or appear in the system only for some parameter values, contradicting what was stated before in the literature. Moreover, the parameter space (-1,1)x(-1,1) is divided into four regions where we observe either irregular behaviour, or preference to follow an itinerary consisting of strategies for which one or the other agent does not lose, or they alternate in winning. These regions in parameter space are separated by two analytical curves and lines where the game is either symmetric (system is not $C^1$ linearisable at stationary saddle points) or is zero-sum and the system is Hamiltonian. On each of these curves we observe different bifurcation scenarios: e.g. transition from order to chaos, or from one kind of stability to another kind, or just loss of one dimension of the local stable manifold of the subcycle.