  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.
