Keyword Auctions as Weighted Unit-Price Auctions

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3 (WUPAs). By allowing w to take different values, we can accommodate the following stylized auction formats. When w = 1, the winners are determined solely by bid prices. One example is the auction format adopted by Yahoo!. When w < 1, bid prices from low-yield bidders are weighted less than those from high-yield bidders. Google s auction is among this category. A WUPA may be first-score or second-score depending on whether the winning bidders are required to fulfill their own scores or the next highest score. We can show that the revenue equivalence holds for the two formats. In light of this, we will derive our main results using the first-score variation, which is simpler to work with. All bidders are risk-neutral, and their payoff functions are additive in their total valuation and the payment. In particular, in first-score WUPAs, if the probability of winning asset δ j with a score s is P j (s), bidder i s expected payoff is given by u = E [q θ i ] (v i b i ) δ j P j (s i ) (2) Synopsis of Findings. We focus on symmetric, pure-strategy Bayesian-Nash equilibria. By symmetric, we mean that bidders with the same unit-valuation and yield-signal will bid the same. We denote their equilibrium probabilities of winning the first asset as ρ 1 H (v) and ρ1 L (v) respectively. 1 Let minimum bids for high-yield and low-yield bidders be b L and b H respectively. Assuming b L b H /w 1 and w 1, 2 we have ρ 1 L (v) = ρ 1 H (v) = { [αfh (b H ) + (1 α) F L (v)] n 1 for v [b L, v 0 ] [αf H (wv) + (1 α) F L (v)] n 1 for v [v 0, 1] n 1 [αf H (v) + (1 α) F L (v 0 )] for v [b H, wv 0 ] [ ( αfh (v) + (1 α) F v )] n 1 L w for v [wv 0, w] [αf H (v) + (1 α)] n 1 for v [w, 1] (3) (4) where v 0 is the point at which lower-yield bidders jump their bids and is given by v0 w b L ρ j L (t)δ jdt = wv0 b H ρ j H (t)δ jdt. (5) Given these equilibrium winning probabilities, we can easily evaluate the equilibrium bidding functions, efficiency, and the auctioneer s revenue. Our analysis generates the following novel findings: Unlike in standard auction settings, there may be kinks in bidding functions in WUPAs. The kinks occur when one type s bids rise (or drop) to a point the other type of bidders are no longer able to match. Figure 1 shows that the bidding function for high-yield bidders has a kink at v = w when w < 1. With minimum bids, there may be a jump in one type s bidding function. The reason is that when two types of bidders face different minimum bids, the type who faces the more constraining minimum bid will have some of its bidders clustering near the minimum bid (zone-i high-yield bidders). As a result, it is rational for the other type to either leap-frog the 1 The equilibrium winning probabilities of winning other assets, ρ j L and ρj H, j = 2,..., m, can be derived analogously using different order statistics. 2 The case with w > 1 and/or b L > b H /w can be analyzed in a similar way. 3

4 (a) low-yield bidder (b) high-yield bidder Figure 1: equilibrium bidding functions without minimum bids entire cluster or to bid lower than them. The jump does not occur only when minimum bids are equally constraining, i.e., when the minimum score is the same across bidder types. For example, in Figure 2, all zone-i low-yield bidders score lower than zone-i high-yield bidders (the cluster). There is a jump in low-yield bidders bidding function at the border of zone-i and zone-ii. All Zone-II low-yield bidders score higher than zone-i high-yield bidders. No low-yield bidders can compete with Zone-III high-yield bidders. (a) low-yield bidder (b) high-yield bidder Figure 2: equilibrium bidding functions with minimum bids Remarkably, although bidders generally do not bid their true valuation (except in the case of single-object WUPA with the second-score rule), the efficient weighting scheme is simple and the same in both the first-score and the second-score WUPAs: each unit-price bid is weighted by the respective expected yield, as if bidders are bidding their true unit-valuation. This insight is due to a result central to our findings: in equilibrium, the ratio of bid prices from the high-yield and the low-yield bidders with the same score is the same as the ratio of their true unit-valuation. It is weakly efficient (i.e., the allocation is efficient among participating bidders) to impose the same minimum score across bidder types, provided the auctioneer also uses the efficient 4

5 weighting factor. As we have suggested, setting different minimum scores for different bidder types causes clustering among the more-constrained type and jump-bidding among the lessconstrained the type, which distorts allocation efficiency locally. The revenue-maximizing weighting scheme is generally different from the efficient one. The revenue-maximizing weighting scheme may favor high-yield or low-yield bidders in comparison with the efficient weighting scheme. When high-yield bidders and low-yield bidders differ only in their expected yield, the optimal weighting scheme always favor low-yield bidders. But as the unit-valuation of low-yield bidders increases by the hazard-rate order, the revenuemaximizing weighting scheme will favor low-yield bidders less and may even favor the highyield bidders. This finding is due to that when bidders differ both in the expected yield and in the unit-valuation distribution, the high-yield bidders are not necessarily stronger. Combining the previous results with the finding that efficient WUPAs are revenue-equivalent to standard auctions in which bidders bid total willingness-to-pay, we conclude that WUPAs may generate higher revenue than standard auctions. The abnormal revenue arises precisely because the interim information on bidders future-yield allows auctioneers to further discriminate among bidders, often at the cost of allocation efficiency. An optimal minimum bids policy is generally different from what we see in the optimal mechanism design literature. The reason is that besides the effect of excluding bidders of certain valuation, the minimum bids also affect the resource allocation among the participating bidders, particularly through its impact on the jump point and the size of the cluster. Furthermore, the optimal minimum-bids are generally not a uniform minimum score either, thus inefficient. References [1] Feng, Juan, H. Bhargava, and D. Pennock (2005). Implementing paid placement in web search engines: Computational evaluation of alternative mechanisms. INFORMS Journal on Computing, forthcoming. [2] Laffont, J. J. and J. Tirole (1988). The Dynamics of Incentive Contracts. Econometrica 56(5): [3] Lahaie, S. (2006). An Analysis of Alternative Slot Auction Designs for Sponsored Search. Proceedings of the 7th ACM Conference on Electronic Commerce, Ann Arbor, Michigan, ACM Press. [4] Liu, D. and J. Chen (2005). Designing online auctions with performance information. Decision Support Systems, forthcoming. [5] McAfee, R. P. and J. McMillan (1986). Bidding for Contracts - a Principal-Agent Analysis. Rand Journal of Economics 17(3): [6] Riley, J. G. (1988). Ex-Post Information in Auctions. Review of Economic Studies 55(3): [7] Samuelson, W. (1986). Bidding for Contracts. Management Science 32(12):

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