A Potential Reduction Algorithm for Two-person Zero-sum Mean Payoff Stochastic Games

We suggest a new algorithm for two-person zero-sum undiscounted stochastic games focusing on stationary strategies. Given a positive real $\epsilon$, let us call a stochastic game $\epsilon$-ergodic, if its values from any two initial positions differ by at most $\epsilon$. The proposed new algorithm outputs for every $\epsilon>0$ in finite time either a pair of stationary strategies for the two players guaranteeing that the values from any initial positions are within an $\epsilon$-range, or identifies two initial positions $u$ and $v$ and corresponding stationary strategies for the players proving that the game values starting from $u$ and $v$ are at least $\epsilon/24$ apart. In particular, the above result shows that if a stochastic game is $\epsilon$-ergodic, then there are stationary strategies for the players proving $24\epsilon$-ergodicity. This result strengthens and provides a constructive version of an existential result by Vrieze (1980) claiming that if a stochastic game is $0$-ergodic, then there are $\epsilon$-optimal stationary strategies for every $\epsilon > 0$. The suggested algorithm is based on a potential transformation technique that changes the range of local values at all positions without changing the normal form of the game

A potential reduction algorithm for two-person zero-sum mean payoff stochastic games

We suggest a new algorithm for two-person zero-sum undiscounted stochastic games focusing on stationary strategies. Given a positive real , let us call a stochastic game -ergodic, if its values from any two initial positions dier by at most . The proposed new algorithm outputs for every > 0 in nite time either a pair of stationary strategies for the two players guaranteeing that the values from any initial positions are within an -range, or identies two initial positions u and v and corresponding stationary strategies for the players proving that the game values starting from u and v are at least =24 apart. In particular, the above result shows that if a stochastic game is -ergodic, then there are stationary strategies for the players proving 24-ergodicity. This result strengthens and provides a constructive version of an existential result by Vrieze (1980) claiming that if a stochastic game is 0-ergodic, then there are -optimal stationary strategies for every > 0. The suggested algorithm is based on a potential transformation technique that changes the range of local values at all positions without changing the normal form of the game

A potential reduction algorithm for two-person zero-sum mean payoff stochastic games

We suggest a new algorithm for two-person zero-sum undiscounted stochastic games focusing on stationary strategies. Given a positive real , let us call a stochastic game -ergodic, if its values from any two initial positions dier by at most . The proposed new algorithm outputs for every > 0 in nite time either a pair of stationary strategies for the two players guaranteeing that the values from any initial positions are within an -range, or identies two initial positions u and v and corresponding stationary strategies for the players proving that the game values starting from u and v are at least =24 apart. In particular, the above result shows that if a stochastic game is -ergodic, then there are stationary strategies for the players proving 24-ergodicity. This result strengthens and provides a constructive version of an existential result by Vrieze (1980) claiming that if a stochastic game is 0-ergodic, then there are -optimal stationary strategies for every > 0. The suggested algorithm is based on a potential transformation technique that changes the range of local values at all positions without changing the normal form of the game

A potential reduction algorithm for two-person zero-sum mean payoff stochastic games

We suggest a new algorithm for two-person zero-sum undiscounted stochastic games focusing on stationary strategies. Given a positive real , let us call a stochastic game -ergodic, if its values from any two initial positions dier by at most . The proposed new algorithm outputs for every > 0 in nite time either a pair of stationary strategies for the two players guaranteeing that the values from any initial positions are within an -range, or identies two initial positions u and v and corresponding stationary strategies for the players proving that the game values starting from u and v are at least =24 apart. In particular, the above result shows that if a stochastic game is -ergodic, then there are stationary strategies for the players proving 24-ergodicity. This result strengthens and provides a constructive version of an existential result by Vrieze (1980) claiming that if a stochastic game is 0-ergodic, then there are -optimal stationary strategies for every > 0. The suggested algorithm is based on a potential transformation technique that changes the range of local values at all positions without changing the normal form of the game

A pseudo-polynomial algorithm for mean payoff stochastic games with perfect information and few random positions

We consider two-person zero-sum stochastic mean payoff games with perfect information, or BWR-games, given by a digraph G = (V;E), with local rewards r : E Z, and three types of positions: black VB, white VW, and random VR forming a partition of V . It is a long- standing open question whether a polynomial time algorithm for BWR-games exists, or not, even when |VR| = 0. In fact, a pseudo-polynomial algorithm for BWR-games would already imply their polynomial solvability. In this paper, we show that BWR-games with a constant number of random positions can be solved in pseudo-polynomial time. More precisely, in any BWR-game with |VR| = O(1), a saddle point in uniformly optimal pure stationary strategies can be found in time polynomial in |VW| + |VB|, the maximum absolute local reward, and the common denominator of the transition probabilities

Aspirations, adaptive learning and cooperation in repeated games

Game Theory;Repeated Games

A survey of random processes with reinforcement

The models surveyed include generalized P\'{o}lya urns, reinforced random walks, interacting urn models, and continuous reinforced processes. Emphasis is on methods and results, with sketches provided of some proofs. Applications are discussed in statistics, biology, economics and a number of other areas.Comment: Published at http://dx.doi.org/10.1214/07-PS094 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org