  We prove the existence of a Markov-perfect equilibrium in randomized stopping
times for a model of the war of attrition in which the underlying state
variable follows a homogenous linear diffusion. The proof uses the fact that
the space of Markovian randomized stopping times can be topologized as a
compact absolute retract, which in turn enables us to use a powerful
fixed-point theorem by Eilenberg and Montgomery. We illustrate our results with
an example of a war of attrition that admits a mixed-strategy Markov-perfect
equilibrium but no pure-strategy Markov-perfect equilibrium.
