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Stochastic game

About: Stochastic game is a research topic. Over the lifetime, 9493 publications have been published within this topic receiving 202664 citations.


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Journal ArticleDOI
Quan Wen1
TL;DR: In this article, the Folk theorem for repeated sequential games is extended to the case where players do not move simultaneously in stage games, and the concept of effective minimax is introduced.
Abstract: We study repeated sequential games where players may not move simultaneously in stage games. We introduce the concept of effective minimax for sequential games and establish a Folk theorem for repeated sequential games. The Folk theorem asserts that any feasible payoff vector where every player receives more than his effective minimax value in a sequential stage game can be supported by a subgame perfect equilibrium in the corresponding repeated sequential game when players are sufficiently patient. The results of this paper generalize those of Wen (1994), and of Fudenberg and Maskin (1986). The model of repeated sequential games and the concept of effective minimax provide an alternative view to the Anti-Folk theorem of Lagunoff and Matsui (1997) for asynchronously repeated pure coordination games. It has long been recognized that players behave differently in one-shot games and in the corresponding repeated games due to players' different objectives in short-term and long-term relationships. Repeat interactions allow players to respond to others' past actions in the future and a player must consider others' reactions in the future when making a decision. When all players evaluate their future sufficiently highly, repeated interactions enable the enforcement of almost all "reasonable" outcomes. This type of result is referred to as a Folk theorem, not only for repeated games but also for other situations. Intensive research makes repeated games one of the most mature subjects in game theory.' One seminal work is the infinitely repeated game model with discounting of Fudenberg and Maskin (1986). Their Folk theorem asserts that if the discounting is low enough then any feasible and strictly individually rational payoff vector of a stage game can be supported by a subgame perfect equilibrium in the corresponding infinitely repeated game, when the stage game either has only two players or satisfies the full dimensionality condition. In Fudenberg and Maskin's and many other repeated game models,2 stage games are modelled in normal form. It is interpreted that players choose their actions simultaneously in normal form games. What happens in repeated games if players do not move simultaneously in stage games? Fudenberg and Tirole (1991) first raised this question. Unfortunately, this question did not attract lot of attention at the time since it was commonly believed that a repeated game of any form of stage game should not differ too much from the repeated normal form representation of the stage game. Inspired by Fudenberg and Tirole's question, we study a class of repeated games, called repeated sequential games, in which players may not move simultaneously in stage games. We will introduce the concept of effective minimax for sequential games and establish a Folk theorem for repeated sequential games. Since normal form games are sequential games by definition, the effective minimax values and the Folk theorem in 1. Most game theory textbooks have chapters on repeated games. For a survey, see Aumann (1981), Friedman

56 citations

Book ChapterDOI
20 Aug 2012
TL;DR: This paper gives a fully polynomial-time randomized approximation scheme (FPRAS) to compute the Shapley value to within a (1 ±e) factor in monotone supermodular games, and argues that, relative to supermodularity, monotonicity is a mild assumption, and discusses how to transform super modular games to be monotonic.
Abstract: Coalitional games allow subsets (coalitions) of players to cooperate to receive a collective payoff. This payoff is then distributed “fairly” among the members of that coalition according to some division scheme. Various solution concepts have been proposed as reasonable schemes for generating fair allocations. The Shapley value is one classic solution concept: player i’s share is precisely equal to i’s expected marginal contribution if the players join the coalition one at a time, in a uniformly random order. In this paper, we consider the class of supermodular games (sometimes called convex games), and give a fully polynomial-time randomized approximation scheme (FPRAS) to compute the Shapley value to within a (1 ±e) factor in monotone supermodular games. We show that this result is tight in several senses: no deterministic algorithm can approximate Shapley value as well, no randomized algorithm can do better, and both monotonicity and supermodularity are required for the existence of an efficient (1 ±e)-approximation algorithm. We also argue that, relative to supermodularity, monotonicity is a mild assumption, and we discuss how to transform supermodular games to be monotonic.

56 citations

Journal ArticleDOI
01 Sep 2007
TL;DR: It is shown that a Simple Stochastic Game (SSG) can be formulated as an LP-type problem, and the first strongly subexponential solution for SSGs is obtained (a stronglySubexponential algorithm has only been known for binary SSGs [L]).
Abstract: We show that a Simple Stochastic Game (SSG) can be formulated as an LP-type problem. Using this formulation, and the known algorithm of Sharir and Welzl [SW] for LP-type problems, we obtain the first strongly subexponential solution for SSGs (a strongly subexponential algorithm has only been known for binary SSGs [L]). Using known reductions between various games, we achieve the first strongly subexponential solutions for Discounted and Mean Payoff Games. We also give alternative simple proofs for the best known upper bounds for Parity Games and binary SSGs. To the best of our knowledge, the LP-type framework has been used so far only in order to yield linear or close to linear time algorithms for various problems in computational geometry and location theory. Our approach demonstrates the applicability of the LP-type framework in other fields, and for achieving subexponential algorithms.

56 citations

Proceedings ArticleDOI
27 Jun 2011
TL;DR: Simulation results show that the proposed algorithms converge effectively to Nash equilibria and that the propose CS-RA mechanism achieves better performance in terms of throughput and payoff compared to conventional mechanisms.
Abstract: Cell selection and resource allocation (CS-RA) are processes of determining cell and radio resource which provide service to mobile station (MS). Optimizing these processes is an important step towards maximizing the utilization of current and future networks. In this paper, we investigate the problem of CS-RA in heterogeneous wireless networks. Specifically, we propose a distributed cell selection and resource allocation mechanism, in which the CS-RA processes are performed by MSs independently. We formulate the problem as a two-tier game named as inter-cell game and intra-cell game, respectively. In the first tier, i.e. the inter-cell game, MSs select the best cell according to an optimal cell selection strategy derived from the expected payoff. In the second tier, i.e., the intra-cell game, MSs choose the proper radio resource in the serving cell to achieve maximum payoff. We analyze the existence of Nash equilibria of both games, the structure of which suggests the interesting property that we can achieve automatic load balance through the two-tier games. Furthermore, we propose distributed algorithms named as CS-Algorithm and RA-Algorithm to enable the independent MSs converge to Nash equilibria. Simulation results show that the proposed algorithms converge effectively to Nash equilibria and that the proposed CS-RA mechanism achieves better performance in terms of throughput and payoff compared to conventional mechanisms.

56 citations

Journal ArticleDOI
TL;DR: In this paper, an unconstrained stochastic approximation method of finding the optimal measure change (in an a priori parametric family) for Monte Carlo simulations is proposed. But this method does not consider the regularity of the density of the law without assume smoothness of the payoff.
Abstract: We propose an unconstrained stochastic approximation method of finding the optimal measure change (in an a priori parametric family) for Monte Carlo simulations. We consider different parametric families based on the Girsanov theorem and the Esscher transform (or exponential-tilting). In a multidimensional Gaussian framework, Arouna uses a projected Robbins-Monro procedure to select the parameter minimizing the variance. In our approach, the parameter (scalar or process) is selected by a classical Robbins-Monro procedure without projection or truncation. To obtain this unconstrained algorithm we intensively use the regularity of the density of the law without assume smoothness of the payoff. We prove the convergence for a large class of multidimensional distributions and diffusion processes. We illustrate the effectiveness of our algorithm via pricing a Basket payoff under a multidimensional NIG distribution, and pricing a barrier options in different markets.

56 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023364
2022738
2021462
2020512
2019460
2018483