We use perturbation analysis to study independent private-value all-pay auctions with weakly risk-averse buyers. We show that under weak risk aversion: 1) Buyers with low values bid lower and buyers with high values bid higher than they would bid in the risk neutral case. 2) Buyers with low values bid lower and buyers with high values bid higher than they would bid in a first-price auction. 3) Buyers’ expected utilities in an all-pay auction are lower than in a first-price auction. 4) The seller’s expected payoff in an all-pay auction may be either higher or lower than in the risk neutral case. 5) The seller’s expected payoff in an all-pay auction may be either higher or lower than in a first-price auction.