On networked non-cooperative games — A semi-tensor product approach

The networked competitive games are investigated, where each player (or agent) plays with all other players in his neighborhood. Assume the evolution is based on the fact that players use local strategy, that is, each player´s strategy depends on the previous information of its neighborhood players, including strategies and payoffs. Using sub-neighborhood, the dynamics of the evolution is obtained. Then Formula for calculating Nash equilibrium from mixed strategies of multi-players is proposed. The relationship between local Nash equilibriums on individual neighborhoods and global Nash equilibriums of overall network is revealed. Certain related properties are investigated. The basic tool of this approach is the semi-tensor product of matrices, which converts the strategies into logical matrices, the payoffs into pseudo-Boolean functions, and the evolutionary games become discrete time dynamic systems.