1,720,966 research outputs found
Finite State N-player and Mean Field Games
Full text linkMean field games represent limit models for symmetric non-zero sum dynamic games when the number N of players tends to infinity. In this thesis, we study mean field games and corresponding N- player games in continuous time over a finite time horizon where the position of each agent belongs to a finite state space. As opposed to previous works on finite statemean field games, we use a probabilistic representation of the system dynamics in terms of stochastic differential equations driven by Poisson random measures. Firstly, under mild assumptions, we prove existence of solutions to the mean field game in relaxed open-loop as well as relaxed feedback controls. Relying on the probabilistic representation and a coupling argument, we show that mean field game solutions provide symmetric εN- Nash equilibria for the N-player game, both in open-loop and in feedback strategies (not relaxed), with εN≤ constant √N. Under stronger assumptions, we also find solutions of the mean field game in ordinary feedback controls and prove uniqueness either in case of a small time horizon or under monotonicity. Then, assuming that players control just their transition rates from state to state, we show the convergence, as N tends to infinity, of the N-player game to a limiting dynamics given by a finite state mean field game system made of two coupled forward-backward ODEs. We exploit the so-called master equation, which in this finite-dimensional framework is a first order PDE in the simplex of probability measures. If the master equation possesses a unique regular solution, then such solution can be used to prove the convergence of the value functions of the N players and of the feedback Nash equilibria, and a propagation of chaos property for the associated optimal trajectories. A sufficient condition for the required regularity of the master equation is given by the monotonicity assumptions. Further, we employ the convergence results to establish a Central Limit Theorem and a Large Deviation Principle for the evolution of the N-player optimal empirical measures. Finally, we analyze an example with as state space and anti-monotonous cost,and show that the mean field game has exactly three solutions. The Nash equilibrium is always unique and we prove that the N-player game always admits a limit: it selects one mean field game solution, except in one critical case, so there is propagation of chaos. The value functions also converge and the limit is the entropy solution to the master equation, which for two state models can be written as a scalar conservation law. Moreover, viewing the mean field game system as the necessary conditions for optimality of a deterministic control problem, we show that the N-player game selects the optimum of this problem when it is unique
Finite state N-agent and mean field control problems
Full text linkInternational audienceWe examine mean field control problems on a finite state space, in continuous time and over a finite time horizon. We characterize the value function of the mean field control problem as the unique viscosity solution of a Hamilton-Jacobi-Bellman equation in the simplex. In absence of any convexity assumption, we exploit this characterization to prove convergence, as N grows, of the value functions of the centralized N-agent optimal control problem to the limit mean field control problem value function, with a convergence rate of order . Then, assuming convexity, we show that the limit value function is smooth and establish propagation of chaos, i.e. convergence of the N-agent optimal trajectories to the unique limiting optimal trajectory, with an explicit rate
Selection by vanishing common noise for potential finite state mean field games
No full textInternational audienceThe goal of this paper is to provide a selection principle for potential mean field games on a finite state space and, in this respect, to show that equilibria that do not minimize the corresponding mean field control problem should be ruled out. Our strategy is a tailor-made version of the vanishing viscosity method for partial differential equations. Here, the viscosity has to be understood as the square intensity of a common noise that is inserted in the mean field game or, equivalently, as the diffusivity parameter in the related parabolic version of the master equation. As established in the recent contribution (Bayraktar et al., 2021, J. Math. Pures Appl. 147:98-162), the randomly forced mean field game becomes indeed uniquely solvable for a relevant choice of a Wright-Fisher common noise, the counterpart of which in the master equation is a Kimura operator on the simplex. We here elaborate on (Bayraktar et al., 2021, J. Math. Pures Appl. 147:98-162) to make the mean field game with common noise both uniquely solvable and potential, meaning that its unique solution is in fact equal to the unique minimizer of a suitable stochastic mean field control problem. Taking the limit as the intensity of the common noise vanishes, we obtain a rigorous proof of the aforementioned selection principle. As a byproduct, we get that the classical solution to the viscous master equation associated with the mean field game with common noise converges to the gradient of the value function of the mean field control problem without common noise. We hence select a particular weak solution of the master equation of the original mean field game. Lastly, we establish an intrinsic uniqueness criterion for this solution within a suitable class of weak solutions to the master equation satisfying a weak one-sided Lipschitz inequality
Mean field games master equations: from discrete to continuous state space
No full textThis paper studies the convergence of mean field games with finite state
space to mean field games with a continuous state space. We examine a space
discretization of a diffusive dynamics, which is reminiscent of the Markov
chain approximation method in stochasctic control, but also of finite
difference numerical schemes; time remains continuous in the discretization,
and the time horizon is arbitrarily long. We are mainly interested in the
convergence of the solution of the associated master equations as the number of
states tends to infinity. We present two approaches, to treat the case without
or with common noise, both under monotonicity assumptions. The first one uses
the system of characteristics of the master equation, which is the MFG system,
to establish a convergence rate for the master equations without common noise
and the associated optimal trajectories, both in case there is a smooth
solution to the limit master equation and in case there is not. The second
approach relies on the notion of monotone solutions introduced by Bertucci. In
the presence of common noise, we show convergence of the master equations, with
a convergence rate if the limit master equation is smooth, otherwise by
compactness arguments
Convergence, fluctuations and large deviations for finite state mean field games via the Master Equation
No full textWe show the convergence of finite state symmetric N-player differential games, where players control
their transition rates from state to state, to a limiting dynamics given by a finite state Mean Field Game
system made of two coupled forward–backward ODEs. We exploit the so-called Master Equation, which
in this finite-dimensional framework is a first order PDE in the simplex of probability measures, obtaining
the convergence of the feedback Nash equilibria, the value functions and the optimal trajectories. The
convergence argument requires only the regularity of a solution to the Master Equation. Moreover, we
employ the convergence results to prove a Central Limit Theorem and a Large Deviation Principle for the
evolution of the N-player empirical measures. The well-posedness and regularity of solution to the Master
Equation are also studied, under monotonicity assumptions
Probabilistic Approach to Finite State Mean Field Games
Full text linkWe study mean field games and corresponding N-player games in continuous time over a finite time horizon where the position of each agent belongs to a finite state space. As opposed to previous works on finite state mean field games, we use a probabilistic representation of the system dynamics in terms of stochastic differential equations driven by Poisson random measures. Under mild assumptions, we prove existence of solutions to the mean field game in relaxed open-loop as well as relaxed feedback controls. Relying on the probabilistic representation and a coupling argument, we show that mean field game solutions provide symmetric ε_N-Nash equilibria for the N-player game, both in open-loop and in feedback strategies (not relaxed), with ε_N≤constant√N. Under stronger assumptions, we also find solutions of the mean field game in ordinary feedback controls and prove uniqueness either in case of a small time horizon or under monotonicity
Weak solutions to the master equation of potential mean field games
Full text linkThe purpose of this work is to introduce a notion of weak solution to the
master equation of a potential mean field game and to prove that existence and
uniqueness hold under quite general assumptions. Remarkably, this is achieved
without any monotonicity constraint on the coefficients. The key point is to
interpret the master equation in a conservative sense and then to adapt to the
infinite dimensional setting earlier arguments for hyperbolic systems deriving
from a Hamilton-Jacobi-Bellman equation. Here, the master equation is indeed
regarded as an infinite dimensional system set on the space of probability
measures and is formally written as the derivative of the
Hamilton-Jacobi-Bellman equation associated with the mean field control problem
lying above the mean field game. To make the analysis easier, we assume that
the coefficients are periodic, which allows to represent probability measures
through their Fourier coefficients. Most of the analysis then consists in
rewriting the master equation and the corresponding Hamilton-Jacobi-Bellman
equation for the mean field control problem as partial differential equations
set on the Fourier coefficients themselves. In the end, we establish existence
and uniqueness of functions that are displacement semi-concave in the measure
argument and that solve the Hamilton-Jacobi-Bellman equation in a suitable
generalized sense and, subsequently, we get existence and uniqueness of
functions that solve the master equation in an appropriate weak sense and that
satisfy a weak one-sided Lipschitz inequality. As another new result, we also
prove that the optimal trajectories of the associated mean field control
problem are unique for almost every starting point, for a suitable probability
measure on the space of probability measures
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
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