Multi-battle contests

We study equilibrium in a multistage race in which players compete
in a sequence of simultaneous move component contests. Players may
win a prize for winning each component contest, as well as a prize
for winning the overall race. Each component contest is an all-pay
auction with complete information. We characterize the unique
equilibrium analytically and demonstrate that it exhibits
endogenous uncertainty. Even a large lead by one player does not
fully discourage the other player, and each feasible state is
reached with positive probability in equilibrium (pervasiveness).
Total effort may exceed the value of the prize by a factor that is
proportional to the maximum number of stages. Important
applications are to war, sports, and R&D contests and the
results have empirical counterparts there.