4,132 research outputs found
A probabilistic representation for the value of zero-sum differential games with incomplete information on both sides
We 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
Zero-sum stopping games with asymmetric information
We study a model of two-player, zero-sum, stopping games with asymmetric
information. We assume that the payoff depends on two continuous-time Markov
chains (X, Y), where X is only observed by player 1 and Y only by player 2,
implying that the players have access to stopping times with respect to
different filtrations. We show the existence of a value in mixed stopping times
and provide a variational characterization for the value as a function of the
initial distribution of the Markov chains. We also prove a verification theorem
for optimal stopping rules which allows to construct optimal stopping times.
Finally we use our results to solve explicitly two generic examples
Markov games with frequent actions and incomplete information
We 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
Advances in Zero-Sum Dynamic Games
International audienceThe survey presents recent results in the theory of two-person zero-sum repeated games and their connections with differential and continuous-time games. The emphasis is made on the following(1) A general model allows to deal simultaneously with stochastic and informational aspects.(2) All evaluations of the stage payoffs can be covered in the same framework (and not only the usual Cesà ro and Abel means).(3) The model in discrete time can be seen and analyzed as a discretization of a continuous time game. Moreover, tools and ideas from repeated games are very fruitful for continuous time games and vice versa.(4) Numerous important conjectures have been answered (some in the negative).(5) New tools and original models have been proposed. As a consequence, the field (discrete versus continuous time, stochastic versus incomplete information models) has a much more unified structure, and research is extremely active
Games with incomplete information in continuous time and for continuous types
We consider a two-player zero-sum game with integral payoff and with
incomplete information on one side, where the payoff is chosen among a
continuous set of possible payoffs. We prove that the value function of this
game is solution of an auxiliary optimization problem over a set of
measure-valued processes. Then we use this equivalent formulation to
characterize the value function as the viscosity solution of a special type of
a Hamilton-Jacobi equation. This paper generalizes the results of a previous
work of the authors, where only a finite number of possible payoffs is
considered
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