Solving zero-sum matrix games is polynomial, because it boils down to linear programming. The approximate solving is sublinear by randomized algorithms on machines with random access memory. Algorithms working separately and independently on columns and rows have been proposed, with the same performance; these versions are compliant with matrix games with stochastic reward. (Flory and Teytaud, ...

We present a novel variant of fictitious play dynamics combining classical with $Q$-learning for stochastic games and analyze its convergence properties in two-player zero-sum games. Our involves players forming beliefs on the opponent strategy their own continuation payoff ($Q$-function), playing greedy best response by using estimated payoffs. Players update from observations actions. A key p...

We consider two-player zero-sum games with countably infinite action spaces and bounded payoff functions. The players’ strategies are finitely additive probability measures, called charges. Since a strategy profile does not always induce a unique expected payoff, we distinguish two extreme attitudes of players. A player is viewed as pessimistic if he always evaluates the range of possible expec...

Definable zero-sum stochastic games involve a finite number of states and action sets, reward and transition functions that are definable in an o-minimal structure. Prominent examples of such games are finite, semi-algebraic or globally subanalytic stochastic games. We prove that the Shapley operator of any definable stochastic game with separable transition and reward functions is definable in...

Matching Pennies is a well-known example of a two player, zero-sum game. In this game, each of the players, the matcher and the mismatcher, flips a coin, and the payoffs are determined as follows. If the coins come up matching (i.e., both heads or both tails), then the matcher wins, so the mismatcher pays the matcher the sum of $1. If the coins do not match (i.e., one head and one tail), then t...

We briefly introduce and discuss two two-player non-zero-sum games, mainly to illustrate that such games are substantially more complex to investigate than zero-sum games. Actually, the key aspect of such games is that the sum of the pay-offs to the two players is not constant over all pairs of strategies, as constant sum games are equivalent to zero-sum games (think about this; one can imagine...

In this article we consider zero and non-zero sum risk-sensitive average criterion games for semi-Markov processes with a finite state space. For the zero-sum case, under suitable assumptions show that game has value. We also establish existence of stationary saddle point equilibrium. Nash