A: It depends on who played the Exchange Keepers card – the person with the Batcave, trading it away for some other Keeper, or another player, using Exchange Keepers to try to get the Batcave.

To refresh: the Batcave says “Other players cannot play Surprises during your turn.”

In the first case, where the person with the Batcave plays Exchange Keepers, that cannot be stopped with a Surprise, because, at the time you’d have to play the Surprise, that person has the Batcave, and you can’t play Surprises during their turn. Letting it be possible results in a paradox: If you stop the Exchange, then they have the Batcave, and you can’t stop the Exchange…

In the second case, however, the person playing Exchange Keepers does not have the Batcave at the time they play that Action, so they are not yet protected by the Batcave’s super-ultra-high-security, and are susceptible to being Surprised.