Asymmetric contests with head starts and nonmonotonic costs

This paper studies multiprize contests in which players costs need not be strictly increasing in their performance. Such costs accommodate various types of asymmetries, including head starts. Head starts capture incumbency advantages, prior investments, and technological differences. I provide an algorithm that constructs the unique equilibrium in which players do not choose weakly-dominated strategies, and apply it to study multiprize all-pay auctions with head starts. A comparison to the standard all-pay auction shows that the strategic effects of head starts differ substantially from those of differing valuations.