Yuanhua Wang

Vector space structure of finite evolutionary games and its application to strategy profile convergence

A vector space structure is proposed for the set of finite games with fixed numbers of players and strategies for each players. Two statical equivalences are used to reduce the dimension of finite games. Under the vector space structure the subspaces of exact and weighted potential games are investigated. Formulas are provided to calculate them. Then the finite evolutionary games (EGs) are considered. Strategy profile dynamics is obtained using different strategy updating rules (SURs). Certain SURs, which assure the convergence of trajectories to pure Nash equilibriums, are investigated. Using the vector space structure, the projection of finite games to the subspace of exact (or weighted) potential games is considered, and a simple formula is given to calculate the projection. The convergence of near potential games to an e-equilibrium is studied. Further more, the Lyapunov function of EGs is defined and its application to the convergence of EGs is presented. Finally, the near potential function for an EG is defined, and it is proved that if the near potential function of an EG is a Lyapunov function, the EG will converge to a pure Nash equilibrium. Some examples are presented to illustrate the results.

A survey on potential evolutionary game and its applications

Basic concepts about the finite potential games and the networked evolutionary games (NEGs) are introduced. Some new developments are surveyed, including (i) formulas for verifying whether a finite game is (weighted) potential and for calculating the (weighted) potential function; and (ii) the fundamental network equation and strategy profile dynamics of NEGs. Then some applications are introduced, which include: (i) convergence of NEGs; (ii) congestion control; (iii) distributed coverage of graphs.

Dynamics and stability of evolutionary games with time-invariant delay in strategies

This paper investigates the modeling and stability of finite evolutionary games (EGs) with time-invariant delay in strategies. Unlike EGs without delay, the evolutionary dynamics of a sequence of strategy profiles, named as the profile trajectory, is proposed to describe the strategy updating process of the delayed EGs. Using the semi-tensor product of matrices, the evolutionary dynamics of the delayed EGs is expressed into an algebraic model, and then some sufficient conditions are proposed to assure the convergence of profile trajectory to a pure Nash equilibrium. Finally, we apply our model to the networked evolutionary games and propose a new strategy updating rule, called the distributed sequential Myopic Best Response Adjustment Rule (MBRAR), and prove that under the distributed sequential MBRAR, a delayed networked evolutionary game will also converge to a pure Nash equilibrium. Some examples are given to illustrate the theoretical results.

A note on observability of Boolean control networks

The observability of Boolean control networks is investigated. The pairs of states are classified into three classes: (i)diagonal, (ii) h-distinguishable, and h-indistinguishable. For h-indistinguishable pairs, we construct a matrix W called the transferable matrix, which indicates the control-transferability among h-indistinguishable pairs. Modifying W yields a matrix U0, which is used as the initial matrix for an iterative algorithm. After finite iterations a stable Uk∗ is reached, which is called the observability matrix. It is proved that a Boolean control network is observable, if and only if, the last column for all rows of Uk∗ are one. Some numerical examples are presented.

Convergence of potential networked evolutionary games

This paper considers when a potential networked evolutionary game (NEG) converges to a Nash equilibrium. First, based on the fundamental evolutionary equation, the profile dynamics of an NEG is revealed. Then we show that an NEG is potential, if the fundamental network game is. Finally, a sufficient condition for an NEG to converge to a Nash equilibrium is presented. An illustrative example is included to demonstrate the theoretical and numerical results.