On Dynamics and Nash Equilibriums of Networked Games

Networked noncooperative games are investigated, where each player (or agent) plays with all other players in its neighborhood. Assume the evolution is based on the fact that each player uses its neighbors' current information to decide its next strategy. By using sub-neighborhood, the dynamics of the evolution is obtained. Then a method for calculating Nash equilibriums from mixed strategies of multi-players is proposed. The relationship between local Nash equilibriums based on individual neighborhoods and global Nash equilibriums of overall network is revealed. Then a technique is proposed to construct Nash equilibriums of an evolutionary game from its one step static Nash equilibriums. The basic tool of this approach is the semi-tensor product of matrices, which converts strategies into logical matrices and payoffs into pseudo-Boolean functions, then networked evolutionary games become discrete time dynamic systems.