On convergence of evolutionary games

The set of finite games with fixed numbers of players and strategies for every player becomes a vector space. Certain equivalences are introduced to classify the elements in the vector space of finite games. Then the subspace of (exact or weighted) potential games are calculated. For the evolutionary (finite) games, certain strategy updating rules are investigated, which lead to certain profile dynamics consisting with the equivalence. The convergence to (pure) Nash equilibriums is investigated. Finally, the projection of finite games to the subspace of potential games is considered, and a simple formula is given to calculate the projection. The dynamics between a game and its projection is compared, which produces a method to verify the convergence of an evolutionary game to a Nash equilibrium or an ε-equilibrium.