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 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 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 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.

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.

PIVOTING ALGORITHMS FOR SOME CLASSES OF STOCHASTIC GAMES: A SURVEY

2001 ◽  
Vol 03 (02n03) ◽  
pp. 253-281 ◽  
Author(s):  
S. R. MOHAN ◽  
S. K. NEOGY ◽  
T. PARTHASARATHY
Keyword(s):  
Linear Complementarity Problem ◽  
Complementarity Problem ◽  
Stochastic Games ◽  
Optimal Strategies ◽  
Linear Complementarity ◽  
Dual Simplex Method ◽  
Special Cases ◽  
Dual Simplex ◽  
Finite Step ◽  
Zero Sum

In this paper, we survey the recent literature on computing the value vector and the associated optimal strategies of the players for special cases of zero-sum stochastic games, or in computing a Nash equilibrium point and the corresponding stationary strategies of the players for special cases of nonzero-sum stochastic games, using finite-step algorithms based on pivoting. Examples of finite-step pivoting algorithms are the various simplex-type algorithms, such as the primal simplex or dual simplex method for solving the linear programming problem or Lemke's or Lemke-Howson's algorithm for solving the linear complementarity problem. Also included are Lemke-type algorithms for solving various generalisations of the linear complementarity problem. The survey also includes a few new results and observations.

scholarly journals On the existence of stationary Nash equilibria in average stochastic games with finite state and action spaces

2020 ◽  
Vol 13 ◽  
pp. 304-323
Author(s):  
Dmitrii Lozovanu ◽  
◽  
Stefan Pickl ◽  
Keyword(s):  
Nash Equilibria ◽  
Stochastic Games ◽  
Optimal Strategies ◽  
Stationary Strategies ◽  
Finite State ◽  
Optimal Stationary Strategies ◽  
Action Spaces

We consider infinite n-person stochastic games with limiting average payoffs criteria for the players. The main results of the paper are concerned with the existence of stationary Nash equilibria and determining the optimal strategies of the players in the games with finite state and action spaces. We present conditions for the existence of stationary Nash equilibria in the considered games and propose an approach for determining the optimal stationary strategies of the players if such strategies exist.

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