Landscape and flux for quantifying global stability and dynamics of game theory

Game theory has been widely applied to many research areas including economics, biology and social sciences. However, it is still challenging to quantify the global stability and global dynamics of the game theory. We developed a landscape and flux framework to quantify the global stability and global dynamics of the game theory. As an example, we investigated a model of three-strategy game: a special replicator mutator game termed as the repeated Prison Dilemma model. In this model, one stable state, two stable states and limit cycle can emerge under different parameters. The repeated Prisoner's Dilemma system has Hopf bifurcation from one stable state to limit cycle state, and then to another one stable state or two stable states, and vice versa. We quantified the global stability of the repeated Prisoner's Dilemma system and identified optimal kinetic paths between the basins of attractor. The optimal paths are irreversible due to the non-zero flux. We also quantified the interplay between Peace and War.