Approachability of Convex Sets in Games with Partial Monitoring

We provide a necessary and sufficient condition under which a convex set is approachable in a game with partial monitoring, i.e. where players do not observe their opponents' moves but receive random signals. This condition is an extension of Blackwell's criterion in the full monitoring framework, where players observe at least their payoffs. When our condition is fulfilled, we construct explicitly an approachability strategy, derived from a strategy satisfying some \textsl{internal consistency} property in an auxiliary game. We also provide an example of a convex set that is neither (weakly)-approachable nor (weakly)-excludable, a situation that cannot occur in the full monitoring case. We finally apply our result to describe an epsilon-optimal strategy of the uninformed player in a zero-sum repeated game with incomplete information on one side