The all-pay auction with nonmonotonic payoff

I model innovation contests as an all-pay auction in which it is possible not to achieve successful innovation despite costly R&D investments, and as a result, there is no winner. In such a case, the winning payoff turns out to be nonmonotonic in own bid. I derive the sufficient conditions for the existence of pure strategy equilibria, and fully characterize the nondegenerate mixed strategy equilibrium. In the mixed strategy equilibrium, the support of the low-value bidder is not continuous, and both the high-value and the low-value bidders place an atom in the (distinct) lower bound of their respective support. Under symmetric valuation, both bidders place an atom at zero. These results can explain why one does not observe very low quality innovation in real life, or why even symmetric firms may stay out of an innovation contest.