Ad Auction Design and User Experience

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5 2.1 VCG for Sponsored Search The VCG auction collects all bids and assigns position k to the bidder with the k th highest bid. Each bidder is charged the loss in other s efficiency created by their presence. Again assuming bidders are indexed by their positions, the VCG price charged to bidder i is K j=i+1 b j(α j 1 α j ), where K is the number of total positions available. We denote the total VCG payment for the advertiser ranked k th highest in s with t (s,α) k. Let r be a ranking function such that r(u, s) is the rank of value u if it were to be inserted into the set s. Similarly, r 1 (r, s) is the value at rank r in vector s. We prove the following property of the VCG outcome: Lemma 1. For any set of values V, argmax v V (v α r(v,s) t (s+v,α) r(v,s) ) = max v V v. In words, given a choice of values per click, an advertiser s utility in the VCG outcome is highest if they choose the highest value per click. Proof First, we show that for the VCG outcome, the higher an advertiser s value, the larger their utility. By definition, s k s k+1 0. Multiplying by α k and adding s k+1 α k+1 t s,α k+1 to both sides, we see that the utility of the advertiser with the k th largest value is larger than the utility of the advertiser with the k + 1 th largest value. Take two potential values, u > w for an advertiser i. Let x be the value of the advertiser at rank r(w, s) in the set of values s + u (this is analogous to taking the value of the advertiser ranked immediately above w in s + w). For conciseness, αk = α k α k 1. Using value u gives utility k=r(u,s) uα r(u,s) αk r 1 (k 1, s+u) xα r(w,s) αk r 1 (k 1, s+u). Since both w and u are ranked higher than any value of k in the summation, αk r 1 (k 1, s + u) = αk r 1 (k 1, s + w). Because w is ranked lower than x in the set of values s + w, xα r(w,s) αk r 1 (k 1, s+u) wα r(w,s) αk r 1 (k 1, s+w). Combining the inequalities, we see that the larger value u brings higher utility in VCG than the lower value w. We define b(s, α, u) to be ts,α r(u,s u) α r(u,s u). The function b relies on the rankings and pricing that VCG outputs (based on s and α).

6 2.2 Relating VCG to GSP The following Lemma restates the results from [11], which we will use to prove Theorem 1. Lemma 2. All bidders i bidding b(s, α, s i ) is an envy-free bidder optimal equilibrium. Furthermore, the ranking and the price charged to each bidder is equivalent to the ranking and price charged to each bidder in the VCG mechanism. Corollary 1. Since in the bidder optimal equilibrium in GSP advertisers receive the same utility as in VCG, an advertiser with a choice of values per click receives highest utility in the bidder optimal equilibrium by bidding their highest choice of value. Although there exist other equilibria, it is reasonable to expect the auction to converge to the bidder optimal equilibrium. The result in [11] describes a generalized English auction that leads to the bidder optimal equilibrium. Also, in [6], it is shown that there exists a simple greedy bidding strategy that leads to the bidder optimal equilibrium. For simplicity, we assume there is only one bidder optimal equilibrium solution, since ties can be broken according to a predetermined ordering. To show the efficiency of M, we first prove the efficiency of M. Definition 3. Mechanism M. Auction M on bids b i, i I is as follows: 1. Run the generalized second price (GSP) auction on b i. 2. i I, add h i to the ppc of bidder i. Leave the ranking from GSP unchanged. Theorem 2. All bidders i I implementing websites with quality qi and submitting bids b(s(qi ) h(q i ), α) is an equilibrium point for mechanism M. Furthermore, this equilibrium point has maximum efficiency. Proof For a fixed quality q i, the utility of advertiser i in auction M is (s(q i ) h(q i ) p i )α i, where p i and α i are the price and clicks allocated by GSP. By Lemma 2 and the definition of function b, the bid b(s(q i ) h(q i ), α) must maximize (s(q i ) h(q i ) p i )α i in the bidder optimal GSP outcome. By Lemmas 1 and 2, the value of q i that maximizes (s(q i ) h(q i ) p i )α i is qi. The bidder optimal GSP outcome, since it ranks by s(qi ) h (qi ), maximizes i α i(s(qi ) h (qi )). Corollary 2. All bidders i I implementing websites with quality qi and submitting bids b(s(qi ) h(q i ), α) + h(q i ) is an equilibrium point for mechanism M. Furthermore, this equilibrium point has maximum efficiency. Since the addition of h(q i ) is immediately subtracted in the first step of M, and the utility of the bidder is unchanged from M, the maximizing behavior of the bidder is unchanged from Theorem 2.

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