Application Of Matrix Semi-tensor Product To Finite Potential Games And Mix-valued Boolean Networks

In recent years,Daizhan Cheng and his team put forward the semi-tensor prod-uct of matrices.The semi-tensor product of matrices is a novel algebraic tool,which has greatly developed Boolean network control theory and finite game theory.In this dissertation,the semi-tensor product of matrices is used to investigate finite potential games and mix-valued Boolean networks.Game theory,also called decision theo-ry,is one of the research branch of operations research,which has a wide range of applications in economics,engineering,biology,computer science and cybernetics.A game usually includes players,strategy sets and players' payoff functions defined on strategy sets.The research issues of game theory include competition,cooper-ation,optimization and equilibria.Whether there exists an equilibrium for a game is one of the core issues of game theory.A game admitting a potential function is called a potential game.Since every potential game has a pure strategy equilibrium,to find criteria of potential games has been an important issue in game theory.If different players have different strategy sets,the payoff functions can be essentially described as mix-valued Boolean logical functions.A finite game is actually a mix-valued Boolean network.The control theory of general mix-valued Boolean networks has been a research hotspot.In the following,the obtained results are stated in details.For finite potential games,an equivalent transformation matrix is proposed to analyze the potential equation,which results some new necessary and sufficient con-ditions for testing potential games.The novelty of these conditions lie in two points.One is that the interior connection between the 4-cycle condition and the potential equation based on the semi-tensor product is established.The other one is that the verification equations with the minimal number for checking potential games are pro-posed for the first time,which also reduce the complexity of the existing algorithms.For mix-valued Boolean networks,the research results lie in three points.One is that a necessary and sufficient condition for the complementation of two regular subspaces is obtained,which results a simple proof of the criterion of regular subspaces.Another one is that a new algorithm for friendly subspaces is proposed.The last one is that the general concept of invariant subspaces is introduced without the regularity assump-tion.For invariant subspaces without the regularity assumption,some necessary and sufficient conditions for testing the invariance of subspaces are put forward.