scholarly journals Positive Zero-Sum Stochastic Games with Countable State and Action Spaces

2018 ◽  
Vol 82 (2) ◽  
pp. 499-516
Author(s):  
János Flesch ◽  
Arkadi Predtetchinski ◽  
William Sudderth
Keyword(s):  
Stochastic Games ◽  
Positive Zero ◽  
Countable State ◽  
Zero Sum ◽  
Action Spaces

scholarly journals Characterization and simplification of optimal strategies in positive stochastic games

2018 ◽  
Vol 55 (3) ◽  
pp. 728-741 ◽  
Author(s):  
János Flesch ◽  
Arkadi Predtetchinski ◽  
William Sudderth
Keyword(s):  
State Space ◽  
Optimal Strategy ◽  
Stochastic Games ◽  
Optimal Strategies ◽  
Action Space ◽  
Positive Zero ◽  
Countable State ◽  
Zero Sum ◽  
Action Spaces

Abstract We consider positive zero-sum stochastic games with countable state and action spaces. For each player, we provide a characterization of those strategies that are optimal in every subgame. These characterizations are used to prove two simplification results. We show that if player 2 has an optimal strategy then he/she also has a stationary optimal strategy, and prove the same for player 1 under the assumption that the state space and player 2's action space are finite.

scholarly journals Subgame Maxmin Strategies in Zero-Sum Stochastic Games with Tolerance Levels

2021 ◽  
Author(s):  
János Flesch ◽  
P. Jean-Jacques Herings ◽  
Jasmine Maes ◽  
Arkadi Predtetchinski
Keyword(s):  
Stochastic Games ◽  
Sufficient Conditions ◽  
Payoff Function ◽  
Surprising Result ◽  
Countable State ◽  
Finite Action ◽  
Zero Sum ◽  
Necessary And Sufficient ◽  
Action Spaces ◽  
Special Case

AbstractWe study subgame $$\phi $$ ϕ -maxmin strategies in two-player zero-sum stochastic games with a countable state space, finite action spaces, and a bounded and universally measurable payoff function. Here, $$\phi $$ ϕ denotes the tolerance function that assigns a nonnegative tolerated error level to every subgame. Subgame $$\phi $$ ϕ -maxmin strategies are strategies of the maximizing player that guarantee the lower value in every subgame within the subgame-dependent tolerance level as given by $$\phi $$ ϕ . First, we provide necessary and sufficient conditions for a strategy to be a subgame $$\phi $$ ϕ -maxmin strategy. As a special case, we obtain a characterization for subgame maxmin strategies, i.e., strategies that exactly guarantee the lower value at every subgame. Secondly, we present sufficient conditions for the existence of a subgame $$\phi $$ ϕ -maxmin strategy. Finally, we show the possibly surprising result that each game admits a strictly positive tolerance function $$\phi ^*$$ ϕ ∗ with the following property: if a player has a subgame $$\phi ^*$$ ϕ ∗ -maxmin strategy, then he has a subgame maxmin strategy too. As a consequence, the existence of a subgame $$\phi $$ ϕ -maxmin strategy for every positive tolerance function $$\phi $$ ϕ is equivalent to the existence of a subgame maxmin strategy.

scholarly journals On N-person stochastic games by denumerable state space

1978 ◽  
Vol 10 (2) ◽  
pp. 452-471 ◽  
Author(s):  
A. Federgruen
Keyword(s):  
State Space ◽  
Stochastic Games ◽  
Transition Probability ◽  
Discount Factor ◽  
Average Return ◽  
Countable State ◽  
Transition Probability Matrices ◽  
Probability Matrices ◽  
Factor Α ◽  
Action Spaces

This paper considers non-cooperative N-person stochastic games with a countable state space and compact metric action spaces. We concentrate upon the average return per unit time criterion for which the existence of an equilibrium policy is established under a number of recurrency conditions with respect to the transition probability matrices associated with the stationary policies. These results are obtained by establishing the existence of total discounted return equilibrium policies, for each discount factor α ∈ [0, 1) and by showing that under each one of the aforementioned recurrency conditions, average return equilibrium policies appear as limit policies of sequences of discounted return equilibrium policies, with discount factor tending to one.Finally, we review and extend the results that are known for the case where both the state space and the action spaces are finite.

scholarly journals On N-person stochastic games by denumerable state space

1978 ◽  
Vol 10 (02) ◽  
pp. 452-471 ◽  
Author(s):  
A. Federgruen
Keyword(s):  
State Space ◽  
Stochastic Games ◽  
Transition Probability ◽  
Discount Factor ◽  
Average Return ◽  
Countable State ◽  
Transition Probability Matrices ◽  
Probability Matrices ◽  
Factor Α ◽  
Action Spaces

This paper considers non-cooperative N-person stochastic games with a countable state space and compact metric action spaces. We concentrate upon the average return per unit time criterion for which the existence of an equilibrium policy is established under a number of recurrency conditions with respect to the transition probability matrices associated with the stationary policies. These results are obtained by establishing the existence of total discounted return equilibrium policies, for each discount factor α ∈ [0, 1) and by showing that under each one of the aforementioned recurrency conditions, average return equilibrium policies appear as limit policies of sequences of discounted return equilibrium policies, with discount factor tending to one. Finally, we review and extend the results that are known for the case where both the state space and the action spaces are finite.

scholarly journals Zero-sum risk-sensitive stochastic games on a countable state space

2014 ◽  
Vol 124 (1) ◽  
pp. 961-983 ◽  
Author(s):  
Arnab Basu ◽  
Mrinal Kanti Ghosh
Keyword(s):  
State Space ◽  
Stochastic Games ◽  
Countable State Space ◽  
Risk Sensitive ◽  
Countable State ◽  
Zero Sum

MARKOV STRATEGIES ARE BETTER THAN STATIONARY STRATEGIES

1999 ◽  
Vol 01 (01) ◽  
pp. 9-31 ◽  
Author(s):  
J. FLESCH ◽  
F. THUIJSMAN ◽  
O. J. VRIEZE
Keyword(s):  
Stochastic Games ◽  
Stochastic Game ◽  
Initial State ◽  
Stationary Strategies ◽  
Finite State ◽  
Zero Sum ◽  
Action Spaces ◽  
Markov Strategies ◽  
Better Than

We examine the use of stationary and Markov strategies in zero-sum stochastic games with finite state and action spaces. It is natural to evaluate a strategy for the maximising player, player 1, by the highest reward guaranteed to him against any strategy of the opponent. The highest rewards guaranteed by stationary strategies or by Markov strategies are called the stationary utility or the Markov utility, respectively. Since all stationary strategies are Markov strategies, the Markov utility is always larger or equal to the stationary utility. However, in all presently known subclasses of stochastic games, these utilities turn out to be equal. In this paper, we provide a colourful example in which the Markov utility is strictly larger than the stationary utility and we present several conditions under which the utilities are equal. We also show that each stochastic game has at least one initial state for which the two utilities are equal. Several examples clarify these issues.

EQUILIBRIUM STRATEGIES IN STOCHASTIC GAMES WITH ADDITIVE COST AND TRANSITION STRUCTURE

1999 ◽  
Vol 01 (02) ◽  
pp. 131-147 ◽  
Author(s):  
HEINZ-UWE KÜENLE
Keyword(s):  
Weight Function ◽  
Stochastic Games ◽  
Optimal Strategies ◽  
Cost Structure ◽  
Equilibrium Strategy ◽  
Transition Structure ◽  
Equilibrium Strategies ◽  
Zero Sum ◽  
Action Spaces ◽  
Treated State

Two-person stochastic games with additive transition and cost structure and the criterion of expected total costs are treated. State space and action spaces are standard Borel, and unbounded costs are allowed. For the zero-sum case, it is shown that there are stationary deterministic εη-optimal strategies for every ε>0 and a certain weight function η if some semi-continuity and compactness conditions are fulfilled. Using these results, the existence of so-called quasi-stationary deterministic εη-equilibrium strategy pairs under corresponding conditions is proven.

scholarly journals Zero-Sum Stochastic Games over the Field of Real Algebraic Numbers

2019 ◽  
Vol 9 (4) ◽  
pp. 1026-1041
Author(s):  
K. Avrachenkov ◽  
V. Ejov ◽  
J. A. Filar ◽  
A. Moghaddam
Keyword(s):  
Stochastic Games ◽  
Algebraic Numbers ◽  
Zero Sum

On the optimality equation for zero-sum ergodic stochastic games

2001 ◽  
Vol 54 (2) ◽  
pp. 291-301 ◽  
Author(s):  
Anna Jaśkiewicz ◽  
Andrzej S. Nowak
Keyword(s):  
Stochastic Games ◽  
Optimality Equation ◽  
Zero Sum
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