The existence of absolute equilibrium theorem and the algorithm of the r -game with play payoff on the graph

Based on the finite connected graph, consider the dynamic game in which the players´ strategies are related with their previous actions, i.e. the choice of the next state depends on all the states he or she has experienced other than the last state only. We show the theorem that there is always existing absolute equilibrium on the connected graph (with play payoff), and state the algorithm of absolute equilibrium. When the play payoff is obtained by adding up the state payoff, we proof the lemma that the strategies can get free of the players´ historical choice.