Before revision:  We have also proved its existence under strong condition ( in the so called \emph{symmetric case ) highlighting in conclusion some interesting applications in finance and future researches .
After revision:  We then provide the basic ideas to numerically search the game equilibrium via an iterative optimal stopping approach and we show the existence of the solution for our problem under strong condition ( in the so called \emph{symmetric case ) highlighting in conclusion some interesting applications in finance and future researches .