Optimal Stopping Games and Nash Equilibrium

We show that the value function of the optimal stopping game for a right-continuous strong Markov process can be identified via equality between the smallest superharmonic and the largest subharmonic function lying between the gain functions (semiharmonic characterization) if and only if the Nash equilibrium holds (i.e., there exists a saddle point of optimal stopping times). When specialized to optimal stopping problems it is seen that the former identification reduces to the classic characterization of the value function in terms of superharmonic or subharmonic functions. The equivalence itself shows that finding the value function by “pulling a rope” between “two obstacles” is the same as establishing a Nash equilibrium. Further properties of the value function and the optimal stopping times are exhibited in the proof.