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 nmnbers of players and strategies for each players. Two statical equivalences are used to reduce tile 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 c-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.