Revenue Equivalence of Large Asymmetric Auctions

Abstract: Using asymptotic analysis, we calculate the seller's expected revenue in large asymmetric first-price and second-price auctions, as well as in optimal auctions, with risk-neutral players. These calculations show that the revenue difference between asymmetric first-price and second-price auctions scales as ϵ²/n³, where n is the number of players and ϵ is the level of asymmetry (heterogeneity) among the cumulative distribution functions of players’ valuations. This scaling law explains previous numerical findings that the revenue difference between first-price and second-price auctions is extremely small even with as few as n = 6 bidders, and shows that bidders’ asymmetry has a negligible effect on revenue ranking of large auctions. Furthermore, our asymptotic calculations show that the revenue differences between asymmetric first- or second-prices auctions and the optimal mechanism also scale as ϵ²/n³. Hence, asymmetric first-price and second-price are asymptotically optimal.