  We construct subgame-perfect equilibria with mixed strategies for symmetric
stochastic timing games with arbitrary strategic incentives. The strategies are
qualitatively different for local first- or second-mover advantages, which we
analyse in turn. When there is a local second-mover advantage, the players may
conduct a war of attrition with stopping rates that we characterize in terms of
the Snell envelope from the theory of optimal stopping. This is a very general
result, but it provides a clear interpretation. When there is a local
first-mover advantage, stopping typically results from preemption and is
abrupt. Equilibria may differ in the degree of preemption, precisely when it is
triggered or not. We develop an algorithm to characterize when preemption is
inevitable and to construct corresponding payoff-maximal symmetric equilibria.
