Semi-tensor Product Approach To Weighted Potential Games And Networked Evolutionary Games

Game theory is the study of mathematical models of conflict and cooperation among rational decision-makers.One of the core issues in game theory is whether there exists a Nash equilibrium.Since a weighted potential game has a Nash equilibrium,it is an important topic to test weighted potential games.Moreover,in recent years,the theory of network evolutionary games has been a research hotspot.For a network evolutionary game,the players are placed in a network to play the fundamental game.With the given network and strategy updating rule,the analysis of the dynamical behavior is an typical issue of network evolutionary games.Applying the results on potential game and weighted potential games to network evolutionary games is very meaningful.The method of semitensor product of matrices for finite games is proposed in recent years,which has played an important rule in many theoretical problems of finite games.In this thesis,we use the method of semi-tensor product of matrices to investigate finite games.The obtained results include two parts.One is on weighted potential games.By using semi-tensor product,the problem of testing weighted potential games is equivalently transformed into solving linear algebraic equations,which results a new necessary and sufficient condition for weighted potential games.In the frame of linear algebra,a connection between weighted potential games and potential games is established and the criteria for potential games are generalized to those for weighted potential games.It is proved that a networked evolutionary finite game is potential if and only if its fundamental game is potential.Under the cascading myopic best response adjustment rule,it is proved that the strategies of the networked evolutionary game converge to a Nash equilibrium.Furthermore,some new results on Nash equilibrium of networked evolutionary games are obtained.