  We revisit the Dynkin game problem in a general framework, improve classical
results and relax some assumptions. The criterion is expressed in terms of
families of random variables indexed by stopping times. We construct two
nonnegative supermartingales families $J$ and $J'$ whose finitness is
equivalent to the Mokobodski's condition. Under some weak right-regularity
assumption, the game is shown to be fair and $J-J'$ is shown to be the common
value function. Existence of saddle points is derived under some weak
additional assumptions. All the results are written in terms of random
variables and are proven by using only classical results of probability theory.

Probability