  We study a discrete-time finite-horizon two-players nonzero-sum stopping game
where the filtration of Player 1 is richer than the filtration of Player 2. A
major difficulty which is caused by the information asymmetry is that Player 2
may not know whether Player 1 has already stopped the game or not. Furthermore,
the classical backward-induction approach is not applicable in the current
setup. This is because when the informed player decides not to stop, he reveals
information to the uninformed player and hence the decision of the uninformed
player at time $t$ may not be determined by the play after time $t$, but also
on the play before time $t$. In the current work we initially show that the
expected utility of Player 2 will remain the same even if he knows whether
Player 1 has already stopped. Then, this result is applied in order to prove
that, under appropriate conditions, a recursive construction in the style of
Hamad\'ene and Zhang (2010) converges to a pure-strategy Nash equilibrium.
