  The aim of this paper is twofold. First, we extend the results of [33]
concerning the existence and uniqueness of second-order reflected 2BSDEs to the
case of two obstacles. Under some regularity assumptions on one of the
barriers, similar to the ones in [10], and when the two barriers are completely
separated, we provide a complete wellposedness theory for doubly reflected
second-order BSDEs. We also show that these objects are related to non-standard
optimal stopping games, thus generalizing the connection between DRBSDEs and
Dynkin games first proved by Cvitanic and Karatzas [11]. More precisely, we
show under a technical assumption that the second order DRBSDEs provide
solutions of what we call uncertain Dynkin games and that they also allow us to
obtain super and subhedging prices for American game options (also called
Israeli options) in financial markets with volatility uncertainty
