Abstract
We investigate a two players zero sum differential game with incomplete information on the initial state: The first player has a private information on the initial state while the second player knows only a probability distribution on the initial state. This could be view as a generalization to differential games of the famous Aumann-Maschler framework for repeated games. In an article of the first author, the existence of the value in random strategies was obtained for a finite number of initial conditions (the probability distribution is a finite combination of Dirac measures). The main novelty of the present work consists in : first extending the result on the existence of a value in random strategies for infinite number of initial conditions and second - and mainly - proving the existence of a value in pure strategies when the initial probability distribution is regular enough (without atoms).
Highlights
We consider a two-player, zero-sum differential game with dynamics x (t) = f (x(t), u(t), v(t)) u(t) ∈ U, v(t) ∈ V x(t0) = x0 and terminal cost g : RN → R, which is evaluated at a terminal time T > 0
We investigate a two players zero sum differential game with incomplete information on the initial state: The first player has a private information on the initial state while the second player knows only a probability distribution on the initial state
The main novelty of the present work consists in : first extending the existence of a value result in random strategies for infinite number of initial conditions and second - and mainly - proving the existence of a value in pure strategies when the initial probability distribution is regular enough
Summary
- before the game starts, the initial position x0 is chosen randomly according to a probability measure μ0, - the initial state x0 is communicated to Player I but not to Player II, - the game is played on the time interval [t0, T ], - both players know the probability μ0 and observe their opponents controls Such a game with incomplete information (the first player has a private information not available for the second player) was introduced in the 1960’s in the framework of repeated games by Aumann and Maschler and are extensively studied since (see for instance [3]). The main phenomenon that appears here lies in the fact that when the initial measure μ0 has no atoms, one can built on it a “kind of randomness” which avoids the use of random strategies This is precisely this phenomenon that is explained in our main result (Theorem 4.1) of the paper. The last section contains our main result showing the existence of a value in pure strategy when μ0 has no atoms
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