All pay auctions and group size: Grading on a curve and other applications

We model contests with a fixed proportion of prizes, such as a grading curve, as all-pay auctions where higher effort weakly increases the likelihood of a prize. We find theoretical predictions for the heterogeneous effect auction size has on effort from high- and low-types. We test our predictions in a laboratory experiment that compares behavior in two-bidder, one-prize auctions with behavior in 20-bidder, 10-prize auctions. We find a statistically significant 11.8% increase in aggregate bidding when moving from the small to large auction. The impact is heterogeneous: as the auction size increases, low-types decrease effort but high-types increase effort. Additionally, the larger auction provides a stronger rank-correlation between effort and ability, awarding more prizes to the higher-skilled and improving the efficiency of prize allocation.