Existence of Optimal Strategies in Zero-Sum Nonstationary Stochastic Games with Lack of Information on Both Sides

1989 ◽  
Vol 27 (2) ◽  
pp. 289-295 ◽  
Author(s):  
Andrzej S. Nowak
Keyword(s):  
Stochastic Games ◽  
Optimal Strategies ◽  
Lack Of Information ◽  
Zero Sum

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 Algorithms for uniform optimal strategies in two-player zero-sum stochastic games with perfect information

2012 ◽  
Vol 40 (1) ◽  
pp. 56-60 ◽  
Author(s):  
Konstantin Avrachenkov ◽  
Laura Cottatellucci ◽  
Lorenzo Maggi
Keyword(s):  
Stochastic Games ◽  
Optimal Strategies ◽  
Perfect Information ◽  
Zero Sum

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 Acyclic Gambling Games

2020 ◽  
Vol 45 (4) ◽  
pp. 1237-1257
Author(s):  
Rida Laraki ◽  
Jérôme Renault
Keyword(s):  
Repeated Games ◽  
Stochastic Games ◽  
Stochastic Game ◽  
Compact Metric Space ◽  
System Of Functional Equations ◽  
State Variable ◽  
Limit Value ◽  
Lack Of Information ◽  
Zero Sum ◽  
Continuous Running

We consider two-player, zero-sum stochastic games in which each player controls the player’s own state variable living in a compact metric space. The terminology comes from gambling problems in which the state of a player represents its wealth in a casino. Under standard assumptions (e.g., continuous running payoff and nonexpansive transitions), we consider for each discount factor the value vλ of the λ-discounted stochastic game and investigate its limit when λ goes to zero. We show that, under a new acyclicity condition, the limit exists and is characterized as the unique solution of a system of functional equations: the limit is the unique continuous excessive and depressive function such that each player, if the player’s opponent does not move, can reach the zone when the current payoff is at least as good as the limit value without degrading the limit value. The approach generalizes and provides a new viewpoint on the Mertens–Zamir system coming from the study of zero-sum repeated games with lack of information on both sides. A counterexample shows that under a slightly weaker notion of acyclicity, convergence of (vλ) may fail.

Zero-sum two-person semi-Markov games

1992 ◽  
Vol 29 (01) ◽  
pp. 56-72 ◽  
Author(s):  
Arbind K. Lal ◽  
Sagnik Sinha
Keyword(s):  
Stochastic Games ◽  
Optimal Strategies ◽  
Existence Of A Solution ◽  
Average Case ◽  
Optimality Equation ◽  
Markov Games ◽  
Special Cases ◽  
Ergodic Condition ◽  
Markov Decision ◽  
Zero Sum

Semi-Markov games are investigated under discounted and limiting average payoff criteria. The issue of the existence of the value and a pair of stationary optimal strategies are settled; the optimality equation is studied and under a natural ergodic condition the existence of a solution to the optimality equation is proved for the limiting average case. Semi-Markov games provide useful flexibility in constructing recursive game models. All the work on Markov/semi-Markov decision processes and Markov (stochastic) games can be viewed as special cases of the developments in this paper.

Zero-sum two-person semi-Markov games

1992 ◽  
Vol 29 (1) ◽  
pp. 56-72 ◽  
Author(s):  
Arbind K. Lal ◽  
Sagnik Sinha
Keyword(s):  
Stochastic Games ◽  
Optimal Strategies ◽  
Existence Of A Solution ◽  
Average Case ◽  
Optimality Equation ◽  
Markov Games ◽  
Special Cases ◽  
Ergodic Condition ◽  
Markov Decision ◽  
Zero Sum

Semi-Markov games are investigated under discounted and limiting average payoff criteria. The issue of the existence of the value and a pair of stationary optimal strategies are settled; the optimality equation is studied and under a natural ergodic condition the existence of a solution to the optimality equation is proved for the limiting average case. Semi-Markov games provide useful flexibility in constructing recursive game models. All the work on Markov/semi-Markov decision processes and Markov (stochastic) games can be viewed as special cases of the developments in this paper.

Optimal strategies in a class of zero-sum ergodic stochastic games

1999 ◽  
Vol 50 (3) ◽  
pp. 399-419 ◽  
Author(s):  
Andrzej S. Nowak
Keyword(s):  
Stochastic Games ◽  
Optimal Strategies ◽  
Zero Sum

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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