On the existence of Markovian randomized equilibria in Dynkin games of war-of-attrition-type

In optimal stopping problems, a Markov structure guarantees Markovian optimal stopping times (first exit times). Surprisingly, there is no analogous result for Markovian stopping games once randomization is required. This paper addresses this gap by proving the existence of Markov-perfect equilibria in a specific type of stopping game - a general nonzero-sum Dynkin games of the war-of-attrition type with underlying linear diffusions. Our main mathematical contribution lies in the development of appropriate topologies for Markovian randomized stopping times. This allows us to establish the existence of equilibria within a tractable and interpretable class of stopping times, paving the way for further analysis of Markovian stopping games.