Approachability of Convex Sets in Games with Partial Monitoring

TL;DR

This paper extends Blackwell's approachability criterion to games with partial monitoring, providing necessary and sufficient conditions, explicit strategies, and applications to zero-sum games with incomplete information.

Contribution

It introduces a new approachability condition for partial monitoring games, extending classical results and constructing explicit strategies under this framework.

Findings

Established a necessary and sufficient condition for approachability with partial monitoring.

Constructed explicit approachability strategies based on auxiliary game strategies.

Identified a convex set that is neither approachable nor excludable, unlike in full monitoring cases.

Abstract

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 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.