25,467 research outputs found

    N-player games and mean-field games with smooth dependence on past absorptions.

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    Mean-field games with absorption is a class of games that has been introduced in Campi and Fischer (2018) and that can be viewed as natural limits of symmetric stochastic differential games with a large number of players who, interacting through a mean-field, leave the game as soon as their private states hit some given boundary. In this paper, we push the study of such games further, extending their scope along two main directions. First, we allow the state dynamics and the costs to have a very general, possibly innite-dimensional, dependence on the (non-normalized) empirical sub- probability measure of the survivors' states. This includes the particularly relevant case where the mean-eld interaction among the players is done through the empirical measure of the survivors together with the fraction of absorbed players over time. Second, the boundedness of coefficients and costs has been considerably relaxed including drift and costs with linear growth in the state variables, hence allowing for more realistic dynamics for players' private states. We prove the existence of solutions of the MFG in strict as well as relaxed feedback form, and we establish uniqueness of the MFG solutions under monotonicity conditions of Lasry-Lions type. Finally, we show in a setting with finite-dimensional interaction that such solutions induce approximate Nash equilibria for the N-player game with vanishing error as N tends to infinity.Mean-field games with absorption is a class of games that has been introduced in (Ann. Appl. Probab. 28 (2018) 2188–2242) and that can be viewed as natural limits of symmetric stochastic differential games with a large number of players who, interacting through a mean-field, leave the game as soon as their private states hit some given boundary. In this paper, we push the study of such games further, extending their scope along two main directions. First, we allow the state dynamics and the costs to have a very general, possibly infinite-dimensional, dependence on the (non-normalized) empirical sub-probability measure of the survivors’ states. This includes the particularly relevant case where the mean-field interaction among the players is done through the empirical measure of the survivors together with the fraction of absorbed players over time. Second, the boundedness of coefficients and costs has been considerably relaxed including drift and costs with linear growth in the state variables, hence allowing for more realistic dynamics for players’ private states. We prove the existence of solutions of the MFG in strict as well as relaxed feedback form, and we establish uniqueness of the MFG solutions under monotonicity conditions of Lasry–Lions type. Finally, we show in a setting with finite-dimensional interaction that such solutions induce approximate Nash equilibria for the N-player game with vanishing error as N → ∞

    Mean-Field games with absorption and singular controls

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    The first part of the work is devoted to mean-field games with absorption, a class of games that can be viewed as natural limits of symmetric stochastic differential games with a large number of players who, interacting through a mean-field, leave the game as soon as their private states hit a given boundary. In most of the literature on mean-field games, all players stay in the game until the end of the period, while in many applications, especially in economics and finance, it is natural to have a mechanism deciding when a player has to leave. Such a mechanism can be modelled by introducing an absorbing boundary for the state space. The second part of the thesis, deals with mean-field games of finite-fuel capacity expansion with singular controls. While singular control problems with finite (and infinite) fuel find numerous applications in the economic literature and originated from the engineering literature in the late 60\u2019s, many-player game versions of these problems have only very recently been introduced. They are a natural extension of the single agent set-up and allow to model numerous applied situations. In our work in particular, we make assumptions on the structure of the interaction across players that are suitable to model the so-called goodwill problem. Altogether, the original contribution to the mean-field games literature of the present work is threefold. First, it contributes to the development of mean-field games with absorption, continuing the work of Campi and Fischer (2018) and considerably generalizing the original model by relaxing the assumptions and setting it into a more abstract, infinite-dimensional, framework. Second, it introduces a new set of tools to deal with mean-field games with singular controls, extending the well-known connection between singular stochastic control and optimal stopping to mean-field games. Finally, it also contributes to the numerical literature on mean-field games, by proposing a numerical scheme to approximate the solutions of mean-field games with singular controls with a constructive approach. Overall, this thesis focuses on newly introduced branches of the theory of meanfield games that display a high potential for economic and financial applications, contributing to the literature not only by further developing the existing theory but also by working in directions that make the these models more suitable to application

    Strategic Exploitation of a Common-Property Resource under Uncertainty

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    We construct a game of noncooperative common-resource exploitation which delivers analytical solutions for its symmetric Markov-perfect Nash equilibrium. We examine how introducing uncertainty to the natural law of resource reproduction affects strategic exploitation. We show that the commons problem is always present in our example and we identify cases in which increases in risk amplify or mitigate the commons problem. For a specific class of games which imply Markov-perfect strategies that are linear in the resource stock (our example belongs to this class), we provide general results on how payoff-function features affect the responsiveness of exploitation strategies to changes in riskiness. These broader characterizations of games which imply linear strategies (appearing in an Online Appendix) can be useful in future work, given the technical difficulties that may arise by the possible nonlinearity of Markov-perfect strategies in more general settings

    Riemannian game dynamics

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    We study a class of evolutionary game dynamics defined by balancing a gain determined by the game's payoffs against a cost of motion that captures the difficulty with which the population moves between states. Costs of motion are represented by a Riemannian metric, i.e., a state-dependent inner product on the set of population states. The replicator dynamics and the (Euclidean) projection dynamics are the archetypal examples of the class we study. Like these representative dynamics, all Riemannian game dynamics satisfy certain basic desiderata, including positive correlation and global convergence in potential games. Moreover, when the underlying Riemannian metric satisfies a Hessian integrability condition, the resulting dynamics preserve many further properties of the replicator and projection dynamics. We examine the close connections between Hessian game dynamics and reinforcement learning in normal form games, extending and elucidating a well-known link between the replicator dynamics and exponential reinforcement learning.Comment: 47 pages, 12 figures; added figures and further simplified the derivation of the dynamic

    Approximate Equilibrium and Incentivizing Social Coordination

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    We study techniques to incentivize self-interested agents to form socially desirable solutions in scenarios where they benefit from mutual coordination. Towards this end, we consider coordination games where agents have different intrinsic preferences but they stand to gain if others choose the same strategy as them. For non-trivial versions of our game, stable solutions like Nash Equilibrium may not exist, or may be socially inefficient even when they do exist. This motivates us to focus on designing efficient algorithms to compute (almost) stable solutions like Approximate Equilibrium that can be realized if agents are provided some additional incentives. Our results apply in many settings like adoption of new products, project selection, and group formation, where a central authority can direct agents towards a strategy but agents may defect if they have better alternatives. We show that for any given instance, we can either compute a high quality approximate equilibrium or a near-optimal solution that can be stabilized by providing small payments to some players. We then generalize our model to encompass situations where player relationships may exhibit complementarities and present an algorithm to compute an Approximate Equilibrium whose stability factor is linear in the degree of complementarity. Our results imply that a little influence is necessary in order to ensure that selfish players coordinate and form socially efficient solutions.Comment: A preliminary version of this work will appear in AAAI-14: Twenty-Eighth Conference on Artificial Intelligenc
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