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We consider Blackwell approachability, a very powerful and geometric tool ingame theory, used for example to design strategies of the uninformed player inrepeated games with incomplete information. We extend this theory to"generalized quitting games" , a class of repeated stochastic games in whicheach player may have quitting actions, such as the Big-Match. We provide threesimple geometric and strongly related conditions for the weak approachabilityof a convex target set. The first is sufficient: it guarantees that, for anyfixed horizon, a player has a strategy ensuring that the expected time-averagepayoff vector converges to the target set as horizon goes to infinity. Thethird is necessary: if it is not satisfied, the opponent can weakly exclude thetarget set. In the special case where only the approaching player can quit thegame (Big-Match of type I), the three conditions are equivalent and coincidewith Blackwell's condition. Consequently, we obtain a full characterization andprove that the game is weakly determined-every convex set is either weaklyapproachable or weakly excludable. In games where only the opponent can quit(Big-Match of type II), none of our conditions is both sufficient and necessaryfor weak approachability. We provide a continuous time sufficient conditionusing techniques coming from differential games, and show its usefulness inpractice, in the spirit of Vieille's seminal work for weakapproachability.Finally, we study uniform approachability where the strategyshould not depend on the horizon and demonstrate that, in contrast withclassical Blackwell approacha-bility for convex sets, weak approachability doesnot imply uniform approachability.

Approachability of convex sets in generalized quitting games