Cooperative behavior and phase transitions in co-evolving stag hunt game

Cooperative behavior and different phases in a co-evolving network dynamics based on the stag hunt game is studied. The dynamical processes are parameterized by a payoff r that tends to promote non-cooperative behavior and a probability q for a rewiring attempt that could isolate the non-cooperators. The interplay between the parameters leads to different phases. Detailed simulations and a mean field theory are employed to reveal the properties of different phases. For small r, the cooperators are the majority and form a connected cluster while the non-cooperators increase with q but remain isolated over the whole range of q, and it is a static phase. For sufficiently large r, cooperators disappear in an intermediate range qL≤q≤qU and a dynamical all-non-cooperators phase results. For q>qU, a static phase results again. A mean field theory based on how the link densities change in time by the co-evolving dynamics is constructed. The theory gives a phase diagram in the q-r parameter space that is qualitatively in agreement with simulation results. The sources of discrepancies between theory and simulations are discussed.