Continuous stochastic games

Nonzero-sum N-person stochastic games are a generalization of Shapley's two-person zero-sum terminating stochastic game. Rogers and Sobel showed that an equilibrium point exists when the sets of states, actions, and players are finite. The present paper treats discounted stochastic games when the sets of states and actions are given by metric spaces and the set of players is arbitrary. The existence of an equilibrium point is proven under assumptions of continuity and compactness.

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On a Simplified Method of Defining Characteristic Function in Stochastic Games

Mathematics ◽

10.3390/math8071135 ◽

2020 ◽

Vol 8 (7) ◽

pp. 1135

Author(s):

Elena Parilina ◽

Leon Petrosyan

Keyword(s):

Characteristic Function ◽

Stochastic Games ◽

Stochastic Game ◽

New Method ◽

Characteristic Functions ◽

Zero Sum Games ◽

Subgame Consistency ◽

The Core ◽

Simplified Method ◽

Zero Sum

In the paper, we propose a new method of constructing cooperative stochastic game in the form of characteristic function when initially non-cooperative stochastic game is given. The set of states and the set of actions for any player is finite. The construction of the characteristic function is based on a calculation of the maximin values of zero-sum games between a coalition and its anti-coalition for each state of the game. The proposed characteristic function has some advantages in comparison with previously defined characteristic functions for stochastic games. In particular, the advantages include computation simplicity and strong subgame consistency of the core calculated with the values of the new characteristic function.

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Acyclic Gambling Games

Mathematics of Operations Research ◽

10.1287/moor.2019.1030 ◽

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.

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MARKOV STRATEGIES ARE BETTER THAN STATIONARY STRATEGIES

International Game Theory Review ◽

10.1142/s0219198999000037 ◽

1999 ◽

Vol 01 (01) ◽

pp. 9-31 ◽

Cited By ~ 2

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.

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Cooperative Stochastic Games with Mean-Variance Preferences

Mathematics ◽

10.3390/math9030230 ◽

2021 ◽

Vol 9 (3) ◽

pp. 230

Author(s):

Elena Parilina ◽

Stepan Akimochkin

Keyword(s):

Standard Deviation ◽

Stochastic Games ◽

Stochastic Game ◽

Stochastic Variable ◽

Expected Payoff ◽

The Core ◽

Risk Sensitive ◽

Payoff Functions ◽

Cooperative Solutions ◽

Mean Variance

In stochastic games, the player’s payoff is a stochastic variable. In most papers, expected payoff is considered as a payoff, which means the risk neutrality of the players. However, there may exist risk-sensitive players who would take into account “risk” measuring their stochastic payoffs. In the paper, we propose a model of stochastic games with mean-variance payoff functions, which is the sum of expectation and standard deviation multiplied by a coefficient characterizing a player’s attention to risk. We construct a cooperative version of a stochastic game with mean-variance preferences by defining characteristic function using a maxmin approach. The imputation in a cooperative stochastic game with mean-variance preferences is supposed to be a random vector. We construct the core of a cooperative stochastic game with mean-variance preferences. The paper extends existing models of discrete-time stochastic games and approaches to find cooperative solutions in these games.

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