  We study Dynkin games governed by a nonlinear $\mathbb E^f$-expectation on a
finite interval $[0,T]$, with payoff c\`adl\`ag processes $L,U$ of class (D)
which are not imposed to satisfy (weak) Mokobodzki's condition - the existence
of a c\`adl\`ag semimartingale between the barriers. For that purpose we
introduce the notion of Mokobodzki's stochastic intervals $\mathscr M(\theta)$
(roughly speaking, maximal stochastic interval on which Mokobodzki's condition
is satisfied when starting from the stopping time $\theta$) and the notion of
reflected BSDEs without Mokobodzki's condition (this is a generalization and
modification of the notion introduced by Hamad\'ene and Hassani (2005)). We
prove an existence and uniqueness result for RBSDEs with driver $f$ that is
non-increasing with respect to the value variable (no restrictions on the
growth) and Lipschitz continuous with respect to the control variable, and with
data in $L^1$ spaces. Next, by using RBSDEs, we show numerous results on Dynkin
games: existence of the value process, saddle points, and convergence of the
penalty scheme. We also show that the game is not played beyond $\mathscr
M(\theta)$, when starting from $\theta$.
