Generalized Third-price Auctions

We first describe the environment of sponsored search auctions that was introduced in Edelman et al. (2007) and Varian (2007). For a given keyword, let N = {1, 2, . . . , n} be the set of agents (advertisers) who are interested in displaying their ads on the result page; and advertiser i’s maximum willingness to pay for a click is denoted by vi. The number of ads that the search engine can display per search is limited to k; and let cj denote the expected number of clicks that the ad in position j receives per period of time. If advertiser i buys position j for price pj per click, then his profit is simply cj(vi − pj). Without loss of generality, advertisers and positions are labeled in descending order, i.e., v1 > v2 > . . . > vn > 0 and c1 > c2 > . . . > ck > 0. It is usually the case that the number of advertisers is much larger than the number of positions since only limited number of ads can be displayed per result page, we therefore assume n ≥ k + 2 throughout. To make the notations simpler, we equate the number of positions to the number of advertisers by adding n − k fictitious positions with zero click-through-rate each. That is, cj = 0 for all j = k + 1, . . . , n. We use N as the set of positions for convenience. An auction mechanism M in this setting specifies a pair of allocation π : N → N and payment vector p ∈ Rn for each bid profile b = (b1, b2, . . . , bn) submitted by the advertisers, where π(i) is the identity of the advertiser who wins position i for price pi. Advertiser π(i)’s total payment Pi is cipi, so the total revenue of the seller is then RM (b) = ∑n i=1 Pi. The social welfare is WM (b) = ∑n i=1 civπ(i). An allocation is called efficient if it maximizes the social welfare, i.e., π(i) = i for all i ≤ k.