# American Institute of Mathematical Sciences

January  2019, 6(1): 65-85. doi: 10.3934/jdg.2019005

## A differential game on Wasserstein space. Application to weak approachability with partial monitoring

 1 CMLA, ENS Paris-Saclay (UMR 8536), 61 avenue du Président Wilson, 94230 Cachan 2 Criteo AI Lab, 32 rue Blanche, 75009 Paris, France 3 Laboratoire de Mathématiques de Bretagne Atlantique (CNRS UMR 6205), 6, Avenue Victor Le Gorgeu, 29200 Brest, France

* Corresponding author: vianney.perchet@normalesup.org

Received  January 2018 Revised  December 2018 Published  January 2019

Fund Project: This research was partially by research contract AFOSR-FA9550-18-1-0254. The first author also benefited from the support of the FMJH Program Gaspard Monge in optimization and operations research (supported in part by EDF) and the CNRS through the PEPS 3IA program.

Studying continuous time counterpart of some discrete time dynamics is now a standard and fruitful technique, as some properties hold in both setups. In game theory, this is usually done by considering differential games on Euclidean spaces. This allows to infer properties on the convergence of values of a repeated game, to deal with the various concepts of approachability, etc. In this paper, we introduce a specific but quite abstract differential game defined on the Wasserstein space of probability distributions and we prove the existence of its value. Going back to the discrete time dynamics, we derive results on weak approachability with partial monitoring: we prove that any set satisfying a suitable compatibility condition is either weakly approachable or weakly excludable. We also obtain that the value for differential games with nonanticipative strategies is the same that those defined with a new concept of strategies very suitable to make links with repeated games.

Citation: Vianney Perchet, Marc Quincampoix. A differential game on Wasserstein space. Application to weak approachability with partial monitoring. Journal of Dynamics & Games, 2019, 6 (1) : 65-85. doi: 10.3934/jdg.2019005
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