Harmful competition in all-pay auctions

We consider an all-pay auction in a standard symmetric independent private value setting with a risk averse seller. We prove that if the distribution for the bidders' valuations attaches probability almost one to a single value, then the seller prefers that only two bidders participate in the auction because more bidders increase substantially the revenue's volatility but only slightly its expectation. Furthermore, we show that the same result holds also for a more general class of distributions if the seller is sufficiently risk averse.