6 research outputs found
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
A Nested Family of -total Effective Rewards for Positional Games
We consider Gillette's two-person zero-sum stochastic games with perfect
information. For each k \in \ZZ_+ we introduce an effective reward function,
called -total. For and this function is known as {\it mean
payoff} and {\it total reward}, respectively. We restrict our attention to the
deterministic case. For all , we prove the existence of a saddle point which
can be realized by uniformly optimal pure stationary strategies. We also
demonstrate that -total reward games can be embedded into -total
reward games