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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
Differential games with asymmetric and correlated information
Differential games with asymmetric information were introduced by
Cardaliaguet (2007). As in repeated games with lack of information on both
sides (Aumann and Maschler (1995)), each player receives a private signal (his
type) before the game starts and has a prior belief about his opponent's type.
Then, a differential game is played in which the dynamic and the payoff
function depend on both types: each player is thus partially informed about the
differential game that is played. The existence of the value function and some
characterizations have been obtained under the assumption that the signals are
drawn independently. In this paper, we drop this assumption and extend these
two results to the general case of correlated types. This result is then
applied to repeated games with incomplete information: the characterization of
the asymptotic value obtained by Rosenberg and Sorin (2001) and Laraki (2001)
for the independent case is extended to the general case.Comment: 22 page
Differential games with asymmetric information and without Isaacs condition
We 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
Decomposition of Differential Games
This paper provides a decomposition technique for the purpose of simplifying
the solution of certain zero-sum differential games. The games considered
terminate when the state reaches a target, which can be expressed as the union
of a collection of target subsets; the decomposition consists of replacing the
original target by each of the target subsets. The value of the original game
is then obtained as the lower envelope of the values of the collection of games
resulting from the decomposition, which can be much easier to solve than the
original game. Criteria are given for the validity of the decomposition. The
paper includes examples, illustrating the application of the technique to
pursuit/evasion games, where the decomposition arises from considering the
interaction of individual pursuer/evader pairs.Comment: 10 pages, 2 figure
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
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