Online Supplement to “Keyword Auctions, Unit-Price Contracts, and the Role of Commitment”

Proof of Lemma 1. Consider a low-performance bidder with unit valuation v who bids b and a high-performance bidder with unit valuation wv who bids wb. Both bidders get a score wb, and their payoff functions are U (y L , v, b) = y L (v − b)Pr(wb is the highest score) (W1) and U (y H , wv, wb) = y H (wv − wb)Pr(wb is the highest score). (W2) It is easy to establish that U (y H , wv, wb) = wy H y L U (y L , v, b). (W3) For b L (v) and b H (v) to be equilibrium bidding functions, at any v, b = b L (v) must maximize U (y L , v, b) and b = b H (v) must maximize U (y H , v, b). So (W3) suggests that if bidding b is the best choice for a low-performance bidder with unit valuation v, bidding wb must be the best choice for a high-performance bidder with unit valuation wv, which implies b H (wv) = wb L (v).