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.

scholarly journals On the Existence and Determining Stationary Nash Equilibria for Switching Controller Stochastic Games

2021 ◽  
Vol 14 ◽  
pp. 290-301
Author(s):  
Dmitrii Lozovanu ◽  
◽  
Stefan Pickl ◽  
Keyword(s):  
Nash Equilibrium ◽  
Normal Form ◽  
Nash Equilibria ◽  
Stochastic Games ◽  
Stochastic Game ◽  
Static Game ◽  
Stationary Strategies ◽  
Switching Controller ◽  
Optimal Stationary Strategies ◽  
Stationary Nash Equilibrium

In this paper we consider the problem of the existence and determining stationary Nash equilibria for switching controller stochastic games with discounted and average payoffs. The set of states and the set of actions in the considered games are assumed to be finite. For a switching controller stochastic game with discounted payoffs we show that all stationary equilibria can be found by using an auxiliary continuous noncooperative static game in normal form in which the payoffs are quasi-monotonic (quasi-convex and quasi-concave) with respect to the corresponding strategies of the players. Based on this we propose an approach for determining the optimal stationary strategies of the players. In the case of average payoffs for a switching controller stochastic game we also formulate an auxiliary noncooperative static game in normal form with quasi-monotonic payoffs and show that such a game possesses a Nash equilibrium if the corresponding switching controller stochastic game has a stationary Nash equilibrium.

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.

Existence of p-Equilibrium and Optimal Stationary Strategies in Stochastic Games

1976 ◽  
Vol 60 (1) ◽  
pp. 245 ◽  
Author(s):  
C. J. Himmelberg ◽  
T. Parthasarathy ◽  
T. E. S. Raghavan ◽  
F. S. Van Vleck
Keyword(s):  
Stochastic Games ◽  
Stationary Strategies ◽  
Optimal Stationary Strategies

scholarly journals Stochastic Games with Finite State and Action Spaces.

1988 ◽  
Vol 39 (6) ◽  
pp. 612
Author(s):  
L. C. Thomas ◽  
O. J. Vrieze
Keyword(s):  
Stochastic Games ◽  
Finite State ◽  
Action Spaces

On zero-sum two-person undiscounted semi-Markov games with a multichain structure

2017 ◽  
Vol 49 (3) ◽  
pp. 826-849 ◽  
Author(s):  
Prasenjit Mondal
Keyword(s):  
Stochastic Game ◽  
Optimal Strategies ◽  
Markov Games ◽  
Stationary Strategies ◽  
Current State ◽  
Strategy Evaluation ◽  
Optimality Equations ◽  
Optimal Stationary Strategies ◽  
Zero Sum ◽  
Absorbing States

Abstract Zero-sum two-person finite undiscounted (limiting ratio average) semi-Markov games (SMGs) are considered with a general multichain structure. We derive the strategy evaluation equations for stationary strategies of the players. A relation between the payoff in the multichain SMG and that in the associated stochastic game (SG) obtained by a data-transformation is established. We prove that the multichain optimality equations (OEs) for an SMG have a solution if and only if the associated SG has optimal stationary strategies. Though the solution of the OEs may not be optimal for an SMG, we establish the significance of studying the OEs for a multichain SMG. We provide a nice example of SMGs in which one player has no optimal strategy in the stationary class but has an optimal semistationary strategy (that depends only on the initial and current state of the game). For an SMG with absorbing states, we prove that solutions in the game where all players are restricted to semistationary strategies are solutions for the unrestricted game. Finally, we prove the existence of stationary optimal strategies for unichain SMGs and conclude that the unichain condition is equivalent to require that the game satisfies some recurrence/ergodicity/weakly communicating conditions.

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 Existence of $p$-equilibrium and optimal stationary strategies in stochastic games

1976 ◽  
Vol 60 (1) ◽  
pp. 245-245
Author(s):  
C. J. Himmelberg ◽  
T. Parthasarathy ◽  
T. E. S. Raghavan ◽  
F. S. Van Vleck
Keyword(s):  
Stochastic Games ◽  
Stationary Strategies ◽  
Optimal Stationary Strategies
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