Seminar

Repeated games with incremental information on one side

Fabien Gensbittel (Toulouse School of Economics - GREMAQ)

November 12, 2012, 15:00–16:00

Toulouse

Room MF 323

Decision Mathematics Seminar

Abstract

We introduce in this work a model of repeated zero-sum games with incomplete information on one side. At the beginning of the game, a state of nature is chosen randomly, and the first player receives a sequence of informative signals about this state variable all along the play while the second player does not receive any information. The payoff of each stage depends on actions taken by both players and also on the state variable. This model extends the classical model of Aumann and Maschler where the first player observes directly the state variable to a more general structure of signals. Our asymptotic approach is not related to the usual framework of undiscounted infinitely repeated games nor to the notion of uniform value. We consider that signals correspond asymptotically to observations of a continuous-time signalling process, for example the payoff function depend on the value of some Brownian motion at time T and the first player observes the value of this brownian motion at time t/n before stage t in the game of length n. This approach is well adapted in financial games with finite time horizon where time between two rounds goes to zero (or any model which approximate or can be approximated by a continuous-time model, e.g. differential games with incomplete information on one side). We prove a generalized version of the ``Cav(u)'' Theorem of Aumann and Maschler in this model using a probabilistic method based on martingales. We obtain a general characterization of the limits of optimal processes of revelation for the informed player and show that these optimal solutions induce n^{-1/2}-optimal strategies for the informed player in any game of length n. We also focus on the particular case with a finite number of informative signals and provide an abstract representation of the limit value function based on concavification operators