Displaying similar documents to “Nash ϵ -equilibria for stochastic games with total reward functions: an approach through Markov decision processes”

Empirical approximation in Markov games under unbounded payoff: discounted and average criteria

Fernando Luque-Vásquez, J. Adolfo Minjárez-Sosa (2017)

Kybernetika

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This work deals with a class of discrete-time zero-sum Markov games whose state process x t evolves according to the equation x t + 1 = F ( x t , a t , b t , ξ t ) , where a t and b t represent the actions of player 1 and 2, respectively, and ξ t is a sequence of independent and identically distributed random variables with unknown distribution θ . Assuming possibly unbounded payoff, and using the empirical distribution to estimate θ , we introduce approximation schemes for the value of the game as well as for optimal strategies considering...

An optimal strong equilibrium solution for cooperative multi-leader-follower Stackelberg Markov chains games

Kristal K. Trejo, Julio B. Clempner, Alexander S. Poznyak (2016)

Kybernetika

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This paper presents a novel approach for computing the strong Stackelberg/Nash equilibrium for Markov chains games. For solving the cooperative n -leaders and m -followers Markov game we consider the minimization of the L p - norm that reduces the distance to the utopian point in the Euclidian space. Then, we reduce the optimization problem to find a Pareto optimal solution. We employ a bi-level programming method implemented by the extraproximal optimization approach for computing the strong...

Applications of limited information strategies in Menger's game

Steven Clontz (2017)

Commentationes Mathematicae Universitatis Carolinae

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As shown by Telgársky and Scheepers, winning strategies in the Menger game characterize σ -compactness amongst metrizable spaces. This is improved by showing that winning Markov strategies in the Menger game characterize σ -compactness amongst regular spaces, and that winning strategies may be improved to winning Markov strategies in second-countable spaces. An investigation of 2-Markov strategies introduces a new topological property between σ -compact and Menger spaces.

On the Variational Inequality and Tykhonov Well-Posedness in Game Theory

C. A. Pensavalle, G. Pieri (2010)

Bollettino dell'Unione Matematica Italiana

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Consider a M-player game in strategic form G = ( X 1 , , X M , g 1 , , g M ) where the set X i is a closed interval of real numbers and the payoff function g i is concave and differentiable with respect to the variable x i X i , for any i = 1 , , M . The aim of this paper is to find appropriate conditions on the payoff functions under the well-posedness with respect to the related variational inequality is equivalent to the formulation of the Tykhonov well-posedness in a game context. The idea of the proof is to appeal to a third equivalence,...

Systems of Bellman Equations to Stochastic Differential Games with Discount Control

Alain Bensoussan, Jens Frehse (2008)

Bollettino dell'Unione Matematica Italiana

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We consider two dimensional diagonal elliptic systems Δ u + a u = H ( x , u , u ) which arise from stochastic differential games with discount control. The Hamiltonians H have quadratic growth in u and a special structure which has notyet been covered by regularity theory. Without smallness condition on H , the existence of a regular solution is established.

Some applications of the point-open subbase game

D. Guerrero Sánchez, Vladimir Vladimirovich Tkachuk (2017)

Commentationes Mathematicae Universitatis Carolinae

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Given a subbase 𝒮 of a space X , the game P O ( 𝒮 , X ) is defined for two players P and O who respectively pick, at the n -th move, a point x n X and a set U n 𝒮 such that x n U n . The game stops after the moves { x n , U n : n ø } have been made and the player P wins if n ø U n = X ; otherwise O is the winner. Since P O ( 𝒮 , X ) is an evident modification of the well-known point-open game P O ( X ) , the primary line of research is to describe the relationship between P O ( X ) and P O ( 𝒮 , X ) for a given subbase 𝒮 . It turns out that, for any subbase 𝒮 , the player P has a winning...

Metastability in reversible diffusion processes I: Sharp asymptotics for capacities and exit times

Anton Bovier, Michael Eckhoff, Véronique Gayrard, Markus Klein (2004)

Journal of the European Mathematical Society

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We develop a potential theoretic approach to the problem of metastability for reversible diffusion processes with generators of the form ϵ Δ + F ( · ) on d or subsets of d , where F is a smooth function with finitely many local minima. In analogy to previous work on discrete Markov chains, we show that metastable exit times from the attractive domains of the minima of F can be related, up to multiplicative errors that tend to one as ϵ 0 , to the capacities of suitably constructed sets. We show that...

Covariance structure of wide-sense Markov processes of order k ≥ 1

Arkadiusz Kasprzyk, Władysław Szczotka (2006)

Applicationes Mathematicae

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A notion of a wide-sense Markov process X t of order k ≥ 1, X t W M ( k ) , is introduced as a direct generalization of Doob’s notion of wide-sense Markov process (of order k=1 in our terminology). A base for investigation of the covariance structure of X t is the k-dimensional process x t = ( X t - k + 1 , . . . , X t ) . The covariance structure of X t W M ( k ) is considered in the general case and in the periodic case. In the general case it is shown that X t W M ( k ) iff x t is a k-dimensional WM(1) process and iff the covariance function of x t has the triangular...

More on the Ehrenfeucht-Fraisse game of length ω₁

Tapani Hyttinen, Saharon Shelah, Jouko Vaananen (2002)

Fundamenta Mathematicae

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By results of [9] there are models and for which the Ehrenfeucht-Fraïssé game of length ω₁, E F G ω ( , ) , is non-determined, but it is consistent relative to the consistency of a measurable cardinal that no such models have cardinality ≤ ℵ₂. We now improve the work of [9] in two ways. Firstly, we prove that the consistency strength of the statement “CH and E F G ω ( , ) is determined for all models and of cardinality ℵ₂” is that of a weakly compact cardinal. On the other hand, we show that if 2 < 2 , T is a countable...

On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes

Nicolas Fournier (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We study a one-dimensional stochastic differential equation driven by a stable Lévy process of order α with drift and diffusion coefficients b , σ . When α ( 1 , 2 ) , we investigate pathwise uniqueness for this equation. When α ( 0 , 1 ) , we study another stochastic differential equation, which is equivalent in law, but for which pathwise uniqueness holds under much weaker conditions. We obtain various results, depending on whether α ( 0 , 1 ) or α ( 1 , 2 ) and on whether the driving stable process is symmetric or not. Our...

Topological games and product spaces

Salvador García-Ferreira, R. A. González-Silva, Artur Hideyuki Tomita (2002)

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we deal with the product of spaces which are either 𝒢 -spaces or 𝒢 p -spaces, for some p ω * . These spaces are defined in terms of a two-person infinite game over a topological space. All countably compact spaces are 𝒢 -spaces, and every 𝒢 p -space is a 𝒢 -space, for every p ω * . We prove that if { X μ : μ < ω 1 } is a set of spaces whose product X = μ < ω 1 X μ is a 𝒢 -space, then there is A [ ω 1 ] ω such that X μ is countably compact for every μ ω 1 A . As a consequence, X ω 1 is a 𝒢 -space iff X ω 1 is countably compact, and if X 2 𝔠 is a 𝒢 -space,...

Time-varying Markov decision processes with state-action-dependent discount factors and unbounded costs

Beatris A. Escobedo-Trujillo, Carmen G. Higuera-Chan (2019)

Kybernetika

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In this paper we are concerned with a class of time-varying discounted Markov decision models n with unbounded costs c n and state-action dependent discount factors. Specifically we study controlled systems whose state process evolves according to the equation x n + 1 = G n ( x n , a n , ξ n ) , n = 0 , 1 , ... , with state-action dependent discount factors of the form α n ( x n , a n ) , where a n and ξ n are the control and the random disturbance at time n , respectively. Assuming that the sequences of functions { α n } , { c n } and { G n } converge, in certain sense, to α ,...

Soft local times and decoupling of random interlacements

Serguei Popov, Augusto Teixeira (2015)

Journal of the European Mathematical Society

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In this paper we establish a decoupling feature of the random interlacement process u d at level u , d 3 . Roughly speaking, we show that observations of u restricted to two disjoint subsets A 1 and A 2 of d are approximately independent, once we add a sprinkling to the process u by slightly increasing the parameter u . Our results differ from previous ones in that we allow the mutual distance between the sets A 1 and A 2 to be much smaller than their diameters. We then provide an important application...

On the central limit theorem for some birth and death processes

Tymoteusz Chojecki (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Suppose that { X n : n 0 } is a stationary Markov chain and V is a certain function on a phase space of the chain, called an observable. We say that the observable satisfies the central limit theorem (CLT) if Y n : = N - 1 / 2 n = 0 N V ( X n ) converge in law to a normal random variable, as N + . For a stationary Markov chain with the L 2 spectral gap the theorem holds for all V such that V ( X 0 ) is centered and square integrable, see Gordin [7]. The purpose of this article is to characterize a family of observables V for which the CLT holds...