The main objective of this paper is to find structural conditions under which a stochastic game between two players with total reward functions has an $ϵ$-equilibrium. To reach this goal, the results of Markov decision processes are used to find $ϵ$-optimal strategies for each player and then the correspondence of a better answer as well as a more general version of Kakutani’s Fixed Point Theorem to obtain the $ϵ$-equilibrium mentioned. Moreover, two examples to illustrate the theory developed are presented....

A property of n-vertex graphs is called evasive if every algorithm testing this property by asking questions of the form “is there an edge between vertices u and v” requires, in the worst case, to ask about all pairs of vertices. Most “natural” graph properties are either evasive or conjectured to be such, and of the few examples of nontrivial nonevasive properties scattered in the literature the smallest one has n = 6. We exhibit a nontrivial, nonevasive property of 5-vertex graphs and show that...