  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.
