  We consider an interacting particle system where equal-sized populations of
two types of particles move by random walk steps on a graph, the two types may
have different speeds, and meetings of opposite-type particles result in
annihilation. The key quantity of interest is the expected extinction time.
Even for the mean-field setting of complete graphs, the correct order of
magnitude was not previously known. Under essentially optimal assumptions on
the starting configuration, we determine not only the order of magnitude but
also the asymptotics: the expected extinction time on $K_{2n}$ is
$(2+o(1))n\log n$, independently of the relative speeds of the two types.
