Approachability of convex sets in generalized quitting games

TL;DR

This paper extends Blackwell approachability to generalized quitting games, providing geometric conditions for weak approachability of convex sets, and explores the differences between weak and uniform approachability.

Contribution

It introduces new geometric conditions for approachability in quitting games and characterizes when convex sets are weakly approachable or excludable.

Findings

Three geometric conditions for weak approachability are identified.

Full characterization of approachability in type I Big-Match games.

Continuous time conditions are useful for approachability analysis.

Abstract

We consider Blackwell approachability, a very powerful and geometric tool in game theory, used for example to design strategies of the uninformed player in repeated games with incomplete information. We extend this theory to "generalized quitting games" , a class of repeated stochastic games in which each player may have quitting actions, such as the Big-Match. We provide three simple geometric and strongly related conditions for the weak approachability of a convex target set. The first is sufficient: it guarantees that, for any fixed horizon, a player has a strategy ensuring that the expected time-average payoff vector converges to the target set as horizon goes to infinity. The third is necessary: if it is not satisfied, the opponent can weakly exclude the target set. In the special case where only the approaching player can quit the game (Big-Match of type I), the three conditions are equivalent and coincide with Blackwell's condition. Consequently, we obtain a full characterization and prove that the game is weakly determined-every convex set is either weakly approachable or weakly excludable. In games where only the opponent can quit (Big-Match of type II), none of our conditions is both sufficient and necessary for weak approachability. We provide a continuous time sufficient condition using techniques coming from differential games, and show its usefulness in practice, in the spirit of Vieille's seminal work for weak approachability.Finally, we study uniform approachability where the strategy should not depend on the horizon and demonstrate that, in contrast with classical Blackwell approacha-bility for convex sets, weak approachability does not imply uniform approachability.