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

Equilibria, Fixed Points, and Complexity Classes

Many models from a variety of areas involve the computation of an equilibrium or fixed point of some kind. Examples include Nash equilibria in games; market equilibria; computing optimal strategies and the values of competitive games (stochastic and other games); stable configurations of neural networks; analysing basic stochastic models for evolution like branching processes and for language like stochastic context-free grammars; and models that incorporate the basic primitives of probability and recursion like recursive Markov chains. It is not known whether these problems can be solved in polynomial time. There are certain common computational principles underlying different types of equilibria, which are captured by the complexity classes PLS, PPAD, and FIXP. Representative complete problems for these classes are respectively, pure Nash equilibria in games where they are guaranteed to exist, (mixed) Nash equilibria in 2-player normal form games, and (mixed) Nash equilibria in normal form games with 3 (or more) players. This paper reviews the underlying computational principles and the corresponding classes

Evolutionary games on graphs

Game theory is one of the key paradigms behind many scientific disciplines from biology to behavioral sciences to economics. In its evolutionary form and especially when the interacting agents are linked in a specific social network the underlying solution concepts and methods are very similar to those applied in non-equilibrium statistical physics. This review gives a tutorial-type overview of the field for physicists. The first three sections introduce the necessary background in classical and evolutionary game theory from the basic definitions to the most important results. The fourth section surveys the topological complications implied by non-mean-field-type social network structures in general. The last three sections discuss in detail the dynamic behavior of three prominent classes of models: the Prisoner's Dilemma, the Rock-Scissors-Paper game, and Competing Associations. The major theme of the review is in what sense and how the graph structure of interactions can modify and enrich the picture of long term behavioral patterns emerging in evolutionary games.Comment: Review, final version, 133 pages, 65 figure

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