Search ResultsShowing 1-14 of 14
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arXiv:2311.15048 (Published 2023-11-25)
Guessing a Random Function and Repeated Games in Continuous Time
Categories: math.OCWe study a game where one player selects a random function, and the other has to guess that function, and show that with high probability the second player can correctly guess most of the random function. We apply this analysis to continuous-time repeated games played with mixed strategies with delay, identify good responses of a player to any profile of her opponents, and show that the minmax value coincides with the minmax value in pure strategies of the one-shot game.
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arXiv:2311.05229 (Published 2023-11-09)
Mean Field Games in a Stackelberg problem with an informed major player
Categories: math.OCWe investigate a stochastic differential game in which a major player has a private information (the knowledge of a random variable), which she discloses through her control to a population of small players playing in a Nash Mean Field Game equilibrium. The major player's cost depends on the distribution of the population, while the cost of the population depends on the random variable known by the major player. We show that the game has a relaxed solution and that the optimal control of the major player is approximatively optimal in games with a large but finite number of small players.
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arXiv:2012.04369 (Published 2020-12-08)
Absorption Paths and Equilibria in Quitting Games
Categories: math.OCWe study quitting games and define the concept of absorption paths, which is an alternative definition to strategy profiles that accomodates both discrete time aspects and continuous time aspects, and is parameterized by the total probability of absorption in past play rather than by time. We then define the concept of sequentially 0perfect absorption paths, which are shown to be limits of $\epsilon$-equilibrium strategy profiles as $\epsilon$ goes to 0. We finally identify a class of quitting games that possess sequentially 0-perfect absorption paths.
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arXiv:1907.09785 (Published 2019-07-23)
An example of multiple mean field limits in ergodic differential games
Categories: math.OCWe present an example of symmetric ergodic $N$-players differential games, played in memory strategies on the position of the players, for which the limit set, as $N\to +\infty$, of Nash equilibrium payoffs is large, although the game has a single mean field game equilibrium. This example is in sharp contrast with a result by Lacker [23] for finite horizon problems.
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arXiv:1903.07439 (Published 2019-03-18)
Solving Two-State Markov Games with Incomplete Information on One Side *
Categories: math.OCWe study the optimal use of information in Markov games with incomplete information on one side and two states. We provide a finite-stage algorithm for calculating the limit value as the gap between stages goes to 0, and an optimal strategy for the informed player in the limiting game in continuous time. This limiting strategy induces an-optimal strategy for the informed player, provided the gap between stages is small. Our results demonstrate when the informed player should use his information and how.
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arXiv:1802.06637 (Published 2018-02-19)
On the (in)efficiency of MFG equilibria
Categories: math.OCMean field games (MFG) are dynamic games with infinitely many infinitesimal agents. In this context, we study the efficiency of Nash MFG equilibria: Namely, we compare the social cost of a MFG equilibrium with the minimal cost a global planner can achieve. We find a structure condition on the game under which there exists efficient MFG equilibria and, in case this condition is not fulfilled, quantify how inefficient MFG equilibria are.
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arXiv:1610.02955 (Published 2016-10-10)
A two player zerosum game where only one player observes a Brownian motion
We study a two-player zero-sum game in continuous time, where the payoff-a running cost-depends on a Brownian motion. This Brownian motion is observed in real time by one of the players. The other one observes only the actions of his opponent. We prove that the game has a value and characterize it as the largest convex subsolution of a Hamilton-Jacobi equation on the space of probability measures.
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arXiv:1602.06140 (Published 2016-02-19)
A probabilistic representation for the value of zero-sum differential games with incomplete information on both sides
Categories: math.OCWe prove that for a class of zero-sum differential games with incomplete information on both sides, the value admits a probabilistic representation as the value of a zero-sum stochastic differential game with complete information, where both players control a continuous martingale. A similar representation as a control problem over discontinuous martingales was known for games with incomplete information on one side (see Cardaliaguet-Rainer [8]), and our result is a continuous-time analog of the so called splitting-game introduced in Laraki [20] and Sorin [27] in order to analyze discrete-time models. It was proved by Cardaliaguet [4, 5] that the value of the games we consider is the unique solution of some Hamilton-Jacobi equation with convexity constraints. Our result provides therefore a new probabilistic representation for solutions of Hamilton-Jacobi equations with convexity constraints as values of stochastic differential games with unbounded control spaces and unbounded volatility.
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Differential games with asymmetric information and without Isaacs condition
Categories: math.OCWe investigate a two-player zero-sum differential game with asymmetric information on the payoff and without Isaacs condition. The dynamics is an ordinary differential equation parametrised by two controls chosen by the players. Each player has a private information on the payoff of the game, while his opponent knows only the probability distribution on the information of the other player. We show that a suitable definition of random strategies allows to prove the existence of a value in mixed strategies. Moreover, the value function can be characterised in term of the unique viscosity solution in some dual sense of a Hamilton-Jacobi-Isaacs equation. Here we do not suppose the Isaacs condition which is usually assumed in differential games.
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arXiv:1307.3365 (Published 2013-07-12)
Markov games with frequent actions and incomplete information
Categories: math.OCWe study a two-player, zero-sum, stochastic game with incomplete information on one side in which the players are allowed to play more and more frequently. The informed player observes the realization of a Markov chain on which the payoffs depend, while the non-informed player only observes his opponent's actions. We show the existence of a limit value as the time span between two consecutive stages vanishes; this value is characterized through an auxiliary optimization problem and as the solution of an Hamilton-Jacobi equation.
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arXiv:1001.5191 (Published 2010-01-28)
Hölder regularity for viscosity solutions of fully nonlinear, local or nonlocal, Hamilton-Jacobi equations with super-quadratic growth in the gradient
Journal: SIAM Journal on Control and Optimization 49, 2 (2011) 555-573Categories: math.OCKeywords: viscosity solutions, super-quadratic growth, fully nonlinear, hölder regularity, nonlocal hamilton-jacobi equationsTags: journal articleViscosity solutions of fully nonlinear, local or non local, Hamilton-Jacobi equations with a super-quadratic growth in the gradient variable are proved to be H\"older continuous, with a modulus depending only on the growth of the Hamiltonian. The proof involves some representation formula for nonlocal Hamilton-Jacobi equations in terms of controlled jump processes and a weak reverse inequality.
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Another proof for the equivalence between invariance of closed sets with respect to stochastic and deterministic systems
Comments: revised version for publication in Bulletin des Sciences MathematiquesWe provide a short and elementary proof for the recently proved result by G. da Prato and H. Frankowska that -- under minimal assumptions -- a closed set is invariant with respect to a stochastic control system if and only if it is invariant with respect to the (associated) deterministic control system.
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Stochastic control problems for systems driven by normal martingales
Comments: Published in at http://dx.doi.org/10.1214/07-AAP467 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)Journal: Annals of Applied Probability 2008, Vol. 18, No. 2, 632-663DOI: 10.1214/07-AAP467Keywords: stochastic control problems, normal martingales, systems driven, second-order partial differential/difference equation, jump size control processTags: journal articleIn this paper we study a class of stochastic control problems in which the control of the jump size is essential. Such a model is a generalized version for various applied problems ranging from optimal reinsurance selections for general insurance models to queueing theory. The main novel point of such a control problem is that by changing the jump size of the system, one essentially changes the type of the driving martingale. Such a feature does not seem to have been investigated in any existing stochastic control literature. We shall first provide a rigorous theoretical foundation for the control problem by establishing an existence result for the multidimensional structure equation on a Wiener--Poisson space, given an arbitrary bounded jump size control process; and by providing an auxiliary counterexample showing the nonuniqueness for such solutions. Based on these theoretical results, we then formulate the control problem and prove the Bellman principle, and derive the corresponding Hamilton--Jacobi--Bellman (HJB) equation, which in this case is a mixed second-order partial differential/difference equation. Finally, we prove a uniqueness result for the viscosity solution of such an HJB equation.
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arXiv:math/0703155 (Published 2007-03-06)
Stochastic differential games with asymmetric information
Categories: math.OCWe investigate a two-player zero-sum stochastic differential game in which the players have an asymmetric information on the random payoff. We prove that the game has a value and characterize this value in terms of dual solutions of some second order Hamilton-Jacobi equation.