Balanced two-type annihilation: mean-field asymptotics

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.