1,805 research outputs found
IST Austria Technical Report
We study finite-state two-player (zero-sum) concurrent mean-payoff games played on a graph. We focus on the important sub-class of ergodic games where all states are visited infinitely often with probability 1. The algorithmic study of ergodic games was initiated in a seminal work of Hoffman and Karp in 1966, but all basic complexity questions have remained unresolved. Our main results for ergodic games are as follows: We establish (1) an optimal exponential bound on the patience of stationary strategies (where patience of a distribution is the inverse of the smallest positive probability and represents a complexity measure of a stationary strategy); (2) the approximation problem lie in FNP; (3) the approximation problem is at least as hard as the decision problem for simple stochastic games (for which NP and coNP is the long-standing best known bound). We show that the exact value can be expressed in the existential theory of the reals, and also establish square-root sum hardness for a related class of games
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 differ by at most . The proposed new algorithm
outputs for every in finite time either a pair of stationary
strategies for the two players guaranteeing that the values from any initial
positions are within an -range, or identifies two initial positions
and and corresponding stationary strategies for the players proving
that the game values starting from and are at least
apart. In particular, the above result shows that if a stochastic game is
-ergodic, then there are stationary strategies for the players
proving -ergodicity. This result strengthens and provides a
constructive version of an existential result by Vrieze (1980) claiming that if
a stochastic game is -ergodic, then there are -optimal stationary
strategies for every . 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
Nash equilibria for non zero-sum ergodic stochastic differential games
In this paper we consider non zero-sum games where multiple players control
the drift of a process, and their payoffs depend on its ergodic behaviour. We
establish their connection with systems of Ergodic BSDEs, and prove the
existence of a Nash equilibrium under the generalised Isaac's conditions. We
also study the case of interacting players of different type
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