10 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 1, NO. 1, JANUARY 2014 On Dynamics and Nash Equilibriums of Networked Games Daizhan Cheng Tingting Xu Fenghua He Hongsheng Qi Abstract—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. Index Terms—Networked non-cooperative game, local infor- mation, sub-neighborhood, fundamental evolutionary equation, Nash equilibrium. I. I NTRODUCTION T HE evolutionary game was firstly proposed by biologists about four decades ago. It was used in biology to model the evolution of species in nature. Later on, it had various applications not only to biology but also to economics, social sciences, etc. [1−2] . In an evolutionary game there are usually many players, and the players as nodes form a complete graph. That is, each player plays with all others, and repeats the game for many or even infinite times. Though it has been proved very useful, this approach has an obvious drawback, that is, the topology describing the relationship of players is ignored. Recently, the network [3−6] was borrowed to describe the topology of players in an evolutionary game. Roughly speaking, an evolutionary game with a graph using players as nodes is a networked evolution- ary game (NEG). For the last decade or so, the networked evolutionary game has become a hot topic [7−9] . Some survey papers provided a general description and overall picture for the characteristics and research topics of NEGs [10−11] . The progress of the study of NEGs is rapid and many interesting results have been reported. For instance, the evo- lution of cooperation on scale-free networks was investigated Manuscript received June 24, 2013; accepted August 29, 2013. This work was supported by National Natural Science Foundation of China (61074114, 61273013, 61104065, 61333001). Recommended by Associate Editor Jie Chen Citation: Daizhan Cheng, Tingting Xu, Fenghua He, Hongsheng Qi. On dynamics and Nash equilibriums of networked games. IEEE/CAA Journal of Automatica Sinica, 2014, 1(1): 10-18 Daizhan Cheng and Tingting Xu are with the Institute of Systems Science, Chinese Academy of Sciences, Beijing 100190, China (e-mail: dcheng@iss.ac.cn; xutingting@amss.ac.cn). Fenghua He is with the Institute of Sstronautics, Harbin Institute of Technology, Harbin 150080, China (e-mail: hefenghua@gmail.com). Hongsheng Qi is with the Institute of Systems Science, Chinese Academy of Sciences, Beijing 100190, China (e-mail: qihongsh@amss.ac.cn). in [12−13], evolution on two interdependent networks was considered in [14−15], etc. We refer to [16] for some recent developments. To the best of our knowledge, most of researches on NEGs are based on Monte Carlo and similar simulations. It can hardly be used for general, e.g., asymmetric networks. Using semi-tensor product of matrices, this paper proposes a new framework to model the networked game. Then we present an algorithm to calculate the mixed strategy Nash equilibrium of a multi-player game. By exploring the relationship between local and global Nash equilibriums, the method is applied to static network games. Finally, a sufficient condition is provided to construct the Nash equilibrium of networked evolutionary game via the Nash equilibrium of its corresponding one step static game. The paper is organized as follows. Section II introduces some notations and gives a brief survey on semi-tensor product of matrices. Section III analyzes the structure of networked evolutionary games. Then a rigorous mathematical model for NEGs is presented in Section IV. Section V proposes a method to calculate the Nash equilibrium for multi-player games. In Section VI, we first investigate the static Nash equilibrium of networked game. Then the relationship between local and global Nash equilibriums is revealed, and it is used to calculate the static Nash equilibrium of networked games. Finally, a sufficient condition is obtained to construct Nash equilibrium of NEG via its corresponding one step static Nash equilibriums. Section VII contains some concluding remarks. II. PRELIMINARIES A. Notations 1) M m×n is the set of m × n real matrices. 2) Col i (M ) is the ith column of matrix M ; Col(M ) is the set of columns of M . 3) D k = {1, 2, ··· ,k}. 4) δ i n = Col i (I n ), i.e., it is the ith column of the identity matrix. 5) Δ n = Col(I n ). 6) M ∈M m×n is called a logical matrix if Col(M ) ⊂ Δ m , the set of m × n logical matrices is denoted by L m×n . 7) Assume L ∈L m×n , and L =  δ i1 m δ i2 m ··· δ in m  , then its shorthand form is L = δ m [i 1 i 2 ··· i n ] . 8) r =(r 1 , ··· ,r k ) T ∈R k is called a probabilistic vector, if r i ≥ 0, i =1, ··· ,k, and ∑ k i=1 r i =1.