Dynamics and Stability of Potential Hyper-networked Evolutionary Games

This paper considers the modeling and convergence of hyper-networked evolutionary games (HNEGs). In an HNEG the network graph is a hypergraph, which allows the fundamental network game to be a multi-player one. Using semi-tensor product of matrices and the fundamental evolutionary equation, the dynamics of an HNEG is obtained and we extend the results about the networked evolutionary games to show whether an HNEG is potential and how to calculate the potential. Then we propose a new strategy updating rule, called the cascading myopic best response adjustment rule (MBRAR), and prove that under the cascading MBRAR the strategies of an HNEG will converge to a pure Nash equilibrium. An example is presented and discussed in detail to demonstrate the theoretical and numerical results.