Structure?Evolutionary Stability And Control Of Finite Games Via The Semi-tensor Product Method

Evolutionary game theory is a new field in the study of game theory,which has been developed based on classical game theory.Evolutionary game theory s-tudies the dynamic strategy updating process over time in repeated games played by boundedly rational agents,who aim at maximizing their payoff values by us-ing certain learning rule.Currently,evolutionary game theory has been applied to many practical fields such as biology,economics,social physics,engineering sci-ence.Therefore,it has extremely important theoretical value and research signifi-cance.In evolutionary game theory,when the set of players is finite and each player has finite strategies to choose,the evolutionary process is actually a finitely logic dynamic process.Then the finite evolutionary game can be expressed as a finitely logic dynamic system.After successful applications to the analysis and control of logical dynamic systems,the semi-tensor product method can also be used to study finite evolutionary games.By using this method,the paper firstly investigates the weighted subspace decomposition of finite noncooperative games.Then,stability and stabilization of Markov-type evolutionary games are studied.Finally,modeling and stability of Markov-type evolutionary games with time delays in strategies are investigated.The main contents of this paper are listed as follows:1.The weighted subspace decomposition of a finite noncooperative game is studied.First,by analyzing the properties of a finite noncooperative game,its vector space structure is determined and it can be decomposed into three orthogonal sub-spaces:the weighted pure potential game subspace,the non-strategic game subspace and the weighted pure harmonic game subspace.Then these subspaces are described in detail by constructing their bases,which provide a convenient tool for investigat-ing their properties.Finally,using these bases,we construct a weighted basis matrix,so the computational formulas and some properties of these subspaces are obtained.In addition,the optimal weight problem is considered.2.Stability and stabilization of a class of finite evolutionary games,called the Markov-type evolutionary games,are studied by using the Lyapunov-based tech-nique.First,by defining a Lyapunov function of a general k-valued logical dynamic system,an easily verifiable necessary and sufficient condition is obtained for the global stability of Markov-type evolutionary games.At the same time,an algorith-in is presented to construct the Lyapunov function via solving a set of inequalities.Then the evolutionary games with time-varying payoffs are considered and a com-mon Lyapunov function for its global stability is explored.Moreover,using the potential equation,a potential-based formula is proposed to construct a common Lyapunov function for an evolutionary game based on a near potential game and the corresponding stable condition is obtained.Finally,the stabilization problem of Markov-type controlled evolutionary games is investigated and the respective con-ditions are obtained to assure the global stabilization under fixed and time-varying payoffs.3.The effect of time delay on evolutionary stability is investigated.First,us-ing the semi-tensor product method,the evolutionary games with two kinds of time delays are expressed in the algebraic formulations respectively,and the delayed evo-lutionary dynamics can be converted into the standard discrete dynamic systems without time delays by certain transformations.Then we consider the delayed evo-lutionary dynamics based on potential games.By designing a specific strategy up-dating rule,a sufficient condition is obtained to assure the convergence of delayed evolutionary games to a pure Nash equilibrium.Finally,stability of networked evo-lutionary games with time delay is investigated.