X Revenue Optimization in the Generalized Second-Price Auction

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1 X Optimization in the Generalized Second-Price Auction David R. M. Thompson, University of British Columbia and Kevin Leyton-Brown, University of British Columbia In this paper, we consider the question of how to optimize revenue in advertising auctions without changing away from the generalized second-price paradigm that has become the de facto standard. We consider several different GSP variants (including squashing and different types of reserve prices), and how to set their parameters optimally. Our main finding is that unweighted reserve prices (i.e., where each advertiser has the same per-click reserve price) are dramatically better than the quality-weighted reserve prices that have become common practice in the last few years. This result is extremely robust, appearing in theoretical analysis as well as multiple computational experiments. Our paper also includes one of the first studies of how squashing and reserve prices interact, and of how equilibrium selection affects the revenue of GSP when features such as reserves or squashing are applied. Categories and Subject Descriptors: J. [Computer Applications]: Social and Behavioral Sciences - Economics; I..11 [Distributed Artificial Intelligence]: Multiagent Systems General Terms: Economics, Experiments, Theory Additional Key Words and Phrases: Auctions, game theory ACM Reference Format: David R. M. Thompson and Kevin Leyton-Brown, 13. optimization in the generalized secondprice auction. ACM X, X, Article X (February 13), 15 pages. DOI: 1. INTRODUCTION Advertising on search result pages is a multi-billion dollar industry and the main source of revenue for major search engines. This ad space is sold by auction on a perkeyword basis. The auction designs used for this evolved in an ad hoc way, but all the major search engines converged on the same design the weighted generalized secondprice auction (GSP). Notably, this auction design is not incentive compatible, i.e., it is typically not in an advertisers best interests to truthfully report how much she would be willing to spend per click. During this time, the research community has proposed several incentive compatible designs, but no search engine has been willing to change away from GSP. Ordinarily, the problem of revenue optimization is typically framed as a question of finding the revenue-maximizing incentive compatible auction. Given that no search engines have switched to a non-gsp design, we instead study the question how can we optimize revenue while retaining the basic GSP design? A number of variants (with adjustable parameters) have been used in practice, and we explore this space of parametrized mechanisms, searching for the optimal designs. Contribution: Although this paper makes a number of contributions, there is one exceptionally striking one. We found that the new reserve-price scheme used by major Author s addresses: D. Thompson: K. Leyton-Brown: Computer Science Department, Vancouver, Canada. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies show this notice on the first page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works requires prior specific permission and/or a fee. Permissions may be requested from Publications Dept., ACM, Inc., Penn Plaza, Suite 71, New York, NY USA, fax +1 (1) 9-1, or permissions@acm.org. c 13 ACM -/13/-ARTX $15. DOI: EC 13, June 1, 13, Philadelphia, PA, Vol. X, No. X, Article X, Publication date: February 13.

2 X: Thompson and Leyton-Brown search engines (where low-quality advertisers must pay more per click) is not optimal. In fact, the old scheme (where all advertisers have the same per-click reserve price) is substantially better. This finding is consistent across three different analysis methods and two different valuation distributions. Also, we identify new GSP variant that is provably optimal in some restricted settings, and very good in settings where it is not optimal. Finally, we also do the first systematic investigation of the interaction between reserve prices and squashing (another revenue optimization technique). Interestingly, we find that squashing can substantially improve the performance of qualityweighted reserve prices, but this effect is almost entirely due to squashing undoing the quality-weighting. We also do the first systematic investigation of how equilibrium selection interacts with revenue optimization techniques like reserve prices. Again, unweighted reserve prices prove to be superior, especially when trying to optimize the revenue of worst-case equilibria. This paper proceeds as follows. Section two describes the GSP variants and how they are used in practice, the model of advertiser preferences, and finally a review of related work. Section three covers our first analysis where we consider the problem of selling only a single position. The simplicity of this setting allowed us to identify properties of the optimal auction, and to contrast them with the GSP variants. Section two covers our first computational analysis of auctions involving multiple ad positions. Here, we did very large scale experiments, but only for a restricted class of Nash equilibria. Section five covers our other computational analysis. Using efficient game representations, we were able to actually enumerate the entire set of equilibria. Finally, in section six we discuss the implications of our results and directions for future work.. BACKGROUND The weighted generalized second-price auction (GSP) works as follows. Each advertiser i is assigned a quality score q i by the search engine. (In standard models, this quality score is proportional to the probability that the advertiser s ad would be clicked on, if it were shown in the top position.) Each advertiser submits a bid b i that specifies the maximum per-click price that she would be willing to pay. Adds are shown ordered by b i q i from highest to lowest. Any time a user clicks on an ad, the advertiser must pay the minimum amount she could bid without changing her position. Three simple variants of GSP have been proposed (and used in practice) to improve on GSP s revenue: (1) Squashing changes GSP by changing how bidders are ranked, according to a realvalued parameter s. Bidders are ranked by b i qi s. Thus, when s = 1 squashing GSP behaves identically to regular GSP. When s = squashing GSP throws away all quality information and ranks advertisers by their bids, promoting lower-quality advertisers and forcing higher-quality advertisers to pay more to retain the same position. s (, 1) gives a smooth control between these two extremes [Lahaie and Pennock 7]. () An unweighted reserve price (UWR) specifies a minimum per-click price r that an advertiser must bid (and pay) in order for her ad to appear on the page 1. Unweighted means that the reserve price per-click is constant across all the advertisers, regardless of their qualities. 1 Note: we assume that advertisers who do not bid above their minimum bid do not affect the outcome in any way. Particularly, this means that an advertiser who bids below her minimum bid does not affect the prices that other agents must pay.) EC 13, June 1, 13, Philadelphia, PA, Vol. X, No. X, Article X, Publication date: February 13.

3 Optimization in the Generalized Second-Price Auction X:3 (3) An quality reserve price (QWR) also specifies a minimum price for each advertiser, but in this case the price can vary across advertisers. Specifically, each advertiser i must pay r/q i per click. For this paper, we also explore three richer GSP variants. The first two rich variants, UWR+Sq and QWR+Sq, involve combining squashing with reserve prices. The third, anchoring, retains the back pay-to-keep-position structure of GSP and also uses unweighted reserve prices. However, unlike UWR advertisers are ranked by how much they bid above the reserve price (b i r)q i. We believe anchoring is an important design to study because, as we show in Section 3, it corresponds exactly to the optimal auction for some settings. We also found that anchoring performed well in settings where it was not provably optimal. (When GSP has none of these extensions, we refer to it as vanilla GSP. ) Squashing and both kinds of reserve prices have been used in practice (and in combination). Google initially used a UWR of $.7 per-click across all keywords, but has since switched to QWR. Yahoo! also had a UWR ($.1), but in a well-documented field experiment [Ostrovsky and Schwarz 9] switched to QWR while dramatically increasing reserve prices (and tailoring them to specific keywords). As recently as 1, Microsoft adcenter used a UWR ($.5). Because the exact formulas for quality scores are kept as trade-secrets by all major search engines, it is not immediately clear whether any search engine is employing squashing. However, Yahoo! researchers have publicly confirmed that its ad auctions do use squashing [Metz 1]..1. Model To demonstrate that our findings are not the by-product of an unusual model, we focus on one of the most widely used models of advertiser preferences, first introduced by Varian [7]. In this model, a setting is specified by a -tuple N, v, q, α ; N is the set of agents (numbered 1,, n); v i specifies agent i s value for a single click; q i specifies agent i s quality score, which is proportional to the probability of i s ad being clicked on (q i > ); and α j is the probability that an agent with quality of 1. will receive a click in position j (and α is decreasing, so higher positions get more clicks). For notational simplicity, we assume that α 1 = 1.. Thus, if agent i s ad is shown in position j with a price of p per click, then her payoff will be α j q i (v i p). As in most literature on GSP (e.g., [Edelman et al. 7; Varian 7]), we assume that the setting is common knowledge to all the advertisers in practice, advertisers can learn a great deal about each other through repeated interactions. We assume that the search engine is able to directly observe and condition on the number of bidders and their quality scores. As is common in literature on revenue optimization (going back to [Myerson 191] and extending up to efforts related to GSP [Ostrovsky and Schwarz 9; Sun et al. 11]), we further assume that these settings are drawn from some distribution P, which is known to the seller. Thus while the search engine is unable to choose a reserve price conditional on the advertisers valuations, it is able to choose a reserve price conditional on the distribution those values are drawn from... Related Work has been a major consideration since the earliest equilibrium analysis of vanilla GSP. Varian [7] and Edelman, et al, each analyzed a specific equilibrium refinement, where they found that GSP was equivalent to VCG. Subsequent work has shown that VCG (and therefore GSP) are good approximators of the optimal auction [Dughmi et al. 9] though in practice, even a good approximation could involve search engines leaving billions of dollars on the table. Other researchers have looked into general Nash equilibria (without the refinements of Varian or EOS) and found EC 13, June 1, 13, Philadelphia, PA, Vol. X, No. X, Article X, Publication date: February 13.

4 X: Thompson and Leyton-Brown that GSP has many equilibria, ranging from much worse than VCG [Thompson and Leyton-Brown 9] to substantially better [Lucier et al. 1]. Lahaie and Pennock [7] first introduced the concept of squashing. While squashing behaves similarly to the virtual values of Myerson, in that it tends to promote weak bidders, they proved that no squashing (or other manipulation of quality scores) can produce the optimal auctions. (Our Figure 1 gives some visual intuition as to why). They also did some substantial simulation experiments, demonstrating the effectiveness of squashing as a means of sacrificing efficiency for revenue. Recently, it has been shown that squashing can actually improve efficiency when quality scores are noisy [Lahaie and McAfee 11], though this paper will not consider noisy-quality models. Reserve prices have been in place for years, but our theoretical understanding of them is still relatively limited. Some researchers (e.g., [Varian 7]) have argued that as GSP is equivalent to VCG, it follows that GSP with the optimal reserve is equivalent to the optimal auction of [Myerson 191]. Others have argued that, while this equivalence holds for the case of GSP without quality scores, GSP with weighted reserves are not quite equivalent to the optimal auction in the case with quality scores [Ostrovsky and Schwarz 9]. Recently, [Sun et al. 11] showed that GSP with weighted reserves is optimal when the quality score is part of the advertisers private information. However, the question of how to optimize revenue when quality scores are known to the auctioneer remains open. Despite the fact that squashing and reserve prices have been used together in practice, there have been no significant studies of how they interact. Further, little is known about how equilibrium selection affects the revenue of GSP with reserves or squashing. Note that there are many richer models, introducing considerations such as budgets, externalities and uncertainty. While there are many important open problems relating to revenue optimization in all of these models, we chose to stick with the a simple standard model. 3. FIRST ANALYSIS: RESTRICTED CASE For our first analysis of the problem of revenue optimization in GSP, we consider an extremely restricted case selling a single slot to advertisers with independent, identically distributed per-click valuations. We do this because the problem is analytically straightforward and we can directly rely on Myerson to identify the optimal auction. Also, richer models (cascade, position preferences, etc) all collapse to the same thing in this case. For the case of two goods, it s possible to give a visual representation of the different allocation functions. Another advantage is that it s possible to calculate expected revenue analytically, rather than by sampling (as we do in subsequent sections). First we consider a simple value distribution, the uniform one. Because the uniform valuation has such a simple functional form, it is easy to identify the optimal auction for such bidders. In fact, this auction allocates identically to the anchoring rule described earlier. THEOREM 3.1. Consider any one-position setting where all the agents per-click valuations (v) are independently drawn from a uniform distribution on [, v]. For this setting, the optimal auction s allocation function is identical to the allocation of the anchoring GSP auction. PROOF. The first thing to note (and the key insight behind this proof) is that although the per-click valuations are identical distribution, the agents per-impression valuations are not identically distributed. If a agent i s valuation per-click is drawn from uniform [, v], then her per-impression valuation (given q i ) is drawn from uni- EC 13, June 1, 13, Philadelphia, PA, Vol. X, No. X, Article X, Publication date: February 13.

5 Optimization in the Generalized Second-Price Auction X:5 Table I. Comparing the different GSP variants for an extremely restricted case with two bidders (q 1 = 1, q = 1/, and v1, v uniform (, 1)) Auction Parameters VCG/GSP. Squashing.55 s =.19 UWR.31 r =.59 QWR.79 r =.375 Anchoring.33 r =.5 UWR+Sq.3 r =.55, s =.3 QWR+Sq.31 r =.7, s =. form [, q i v]. Because the auctioneer is ultimately selling impressions, per-impression valuations are what concerns us here. As was shown by Myerson [191], the optimal auction allocates by virtual values ψ: ψ i (V i ) = V i 1 F i(v i ) f i (V i ) where V i is the per-impression valuation (i.e., V i = v i q i ), f(v i ) is the probability density function for V i and F i (V i ) is the cumulative distribution function. For uniform [, x] distributions, f(v) = 1/x and F (v) = v/x. Substituting into (1) gives ψ i (q i v i ) = q i v i 1 q iv i /(q i v) 1/(q i v) (1) = q i (v i v) () That the optimal per-click reserve price ri occurs at ψ i(q i r i ) = which happens when r i = v/, regardless of q i. Thus, like in the anchoring auction, the optimal per-click reserve price is independent of advertiser quality. In the optimal auction, advertisers are ranked by ψ(q i v i ) = q i (v i v) q i (v i ri ), as in the anchoring auction. A nearly identical proof works for the exponential distribution. The uniform distribution also makes it easy to analytically calculate how much revenue an auction will make in expectation. For the case of two bidders, one with high quality (q 1 = 1) and one with lower quality (q = 1/), we calculated the optimal parameter settings for each GSP variant under consideration. (See Table I.) Although anchoring provably generates the most revenue, other mechanisms were very nearly optimal: reserves and squashing used together were 99% optimal, UWR was 9% optimal. Squashing and QWR were far behind ( 79% 3%, respectively). Visualizing the auctions allocation functions can shed some insight on their differences and similarities. In each of these figures, the x and y axes correspond to the per-click values of agent 1 and respectively. The green region indicates regions where agent wins, yellow indicates agent and white indicates that neither agent wins. The dashed line indicates the dividing line of the efficient allocation. (See Figures 1 3.) Unfortunately, not all value distributions have optimal auctions with simple (e.g., linear) forms. In fact, the log-normal distribution, which is believed to be a very good fit of real-world bidder valuations [Lahaie and Pennock 7; Ostrovsky and Schwarz 9], is one such distribution. (See Figure.) The optimal auction for log-normal distributions uses unweighted reserve prices, and behaves similarly to anchoring when bids are close to the reserve. However, far from the reserves, the optimal auction s allocation more closely resembles vanilla GSP. Even so, it is worth noting that nearly all the probability mass of these distributions is concentrated near the reserve prices, so the optimal auction and the GSP behave differently with relatively little probability. Even so, it is possible to partly recover some distribution properties of the optimal auction. Specifically, we found that for any distribution where Myerson s auction is optimal, it uses unweighted per-click reserves. EC 13, June 1, 13, Philadelphia, PA, Vol. X, No. X, Article X, Publication date: February 13.

6 X: Thompson and Leyton-Brown v v v Fig. 1. Allocation functions visualized (Green signifies regions where agent 1 wins and yellow signifies regions where agent wins.) Left: the efficient (VCG) auction; middle: the revenue-optimal auction; right: the squashing auction (with revenue optimal s). Note that squashing changes the slope of the dividing line, while the optimal auction transposes it without changing the slope. (Specifically, the optimal auction anchors the dividing line to the point where every agent bids his reserve price.) Thus, as shown in [Lahaie and Pennock 7] no combination of squashing and reserve prices can do what the optimal auction does v v Fig.. Left: the UWR auction; right: the QWR auction. Note that the optimal unweighted reserve is higher than the reserve used by the optimal auction (.5 rather than.5), and that agent only wins when agent 1 does not meet his reserve, because 1 s quality is much higher. Also, note the compromise involved in quality-weighted reserve prices: because the reserve prices must correspond to a point on the efficient dividing line, getting a reasonable reserve for agent 1 (relative to the optimal auction) results in a much-toohigh reserve for agent. Because of this compromise, QWR generates 13.3% less revenue than UWR. Note that in UWR whenever agent 1 exceeds his reserve price, he wins regardless of agent s bid. This property has a potentially undesirable side-effect in multi-slot UWR auctions: a high-quality advertiser might find it impossible to win the second position. THEOREM 3.. Consider any one-position setting where all the agents per-click valuations (v) are independently drawn from a common distribution g. If g is regular, then the optimal auction uses the same per-click reserve price for all bidders. PROOF. Because g is regular, we can use the Myerson optimal auction to maximize revenue. For any per-click valuation distribution g, we can identify the per-impression valuation distribution for an agent with quality q i : f(q i v i ) = g(v i )/q i and F (g i v i ) = G(v i ). Substituting these into (1) gives: ψ i (q i v i ) = q i v i 1 G i(v i ) g i (v i )/q i = q i ( v i 1 G ) i(v i ) g i (v i ) (3) EC 13, June 1, 13, Philadelphia, PA, Vol. X, No. X, Article X, Publication date: February 13.

7 Optimization in the Generalized Second-Price Auction X: v v Fig. 3. Left: the UWR+Sq auction; right: the QWR+Sq auction. Both have reserves that are much closer to the reserves of the optimal auction (and both are within 1% of revenue-optimal), but both use substantial squashing (. and.3 respectively) v Fig.. Unfortunately, for some value distributions, the optimal auction doesnt have a simple linear form like anchoring. For lognormal valuations (µ =, σ =.5 in this example), the optimal auction resembles anchoring when the values are below or near to the reserve price. As values get further from the reserve, the allocation tends towards the efficient auction. Note that this figure showns valuations up to the 99.9 th % quantile. As in the uniform case, the value v i that makes this expression equal to zero is independent of q i, so the optimal per-click reserve is independent of q i. Although this section concerns only the special case of auctions selling a single good (the top position), the theoretical results generalize beyond that to the full multiposition case. This is because, as Theorem.3 of [Hartline 1] shows, revenue optimization in position environments reduces to revenue optimization in multi-unit auctions (where selling by optimizing virtual surplus is optimal). In summary, the main findings of this section are: EC 13, June 1, 13, Philadelphia, PA, Vol. X, No. X, Article X, Publication date: February 13.

8 X: Thompson and Leyton-Brown 1.. p(click) v i Fig. 5. Incentive compatible pricing for position auctions is easy to compute. For every marginal increase in click probability that an agent gains by moving up a position, she must pay that probability times the minimum she must bid to be shown that position. Thus, her payment corresponds is equal to the shaded area. (1) a revenue ranking of the different GSP variants when selling a single slot: the richer mechansisms (anchoring, QWR+sq, and UWR+sq) are approximately equal, and are better than UWR which is better than QWR and squashing; () identifying properties of the optimal auction: for some distributions, it has a simple linear form (anchoring) while for others it is more complicated. Regardless, when per-click valuations are i.i.d. the optimal auction always uses unweighted reserve prices; and (3) the Myerson reserve price is not necessarily the best reserve price for auctions that differ from Myerson s auction.. SECOND ANALYSIS: NCC EQUILIBRIUM For our second analysis, we considered settings where multiple slots are available to sell. In that cases, GSP is no longer equivalent to VCG and is no longer incentive compatible. This makes the analysis more difficult because we need to account for the fact that agents may be misreporting their valuations in equilibrium. Also, GSP is known to have many Nash equilibria that can differ substantially in revenue and efficiency, so it is important to choose carefully among those equilibria. For this section, we use the non-contradiction criterion (NCC) of Edelman and Schwarz [1]. An equilibrium satisfies the NCC iff the outcome of that equilibrium implementable by some incentive compatible mechanism. NCC was proposed to rule out unrealistic, revenue-optimistic equilibria, but it also has the advantage that it cleanly generalizes the other major equilibrium refinements used when studying GSP. Using NCC also makes equilibrium computation extremely quick. Rather than having to do full Nash equilibrium computation, we can simply compute what would happen in an equivalent incentive compatible mechanism. Because every auction variant we consider allocates monotonically (i.e., increasing an agents bid weakly increases his position and therefore his expected number of clicks), such an incentive compatible mechanism always exists. It is possible to compute payments efficiently using the algorithm of [Aggarwal et al. ]. (See Figure 5.) For this section experiments, we considered two distributions: Uniform, where each agent s valuation is drawn from a uniform [,5] distribution and each agent s quality score is drawn from a uniform [,1] distribution; and These other refinements include bidder-optimal locally envy-free equilibrium [Edelman et al. 7], symmetric equilibrium [Varian 7] and impersonation-proof equilibrium [Kash and Parkes 1]. EC 13, June 1, 13, Philadelphia, PA, Vol. X, No. X, Article X, Publication date: February 13.

9 Optimization in the Generalized Second-Price Auction X: Squashing Power Squashing Power Fig.. The optimal squashing power is zero or close to it. Squashing offers modest revenue gains given a uniform value distribution (left) and substantial gains given a lognormal value distribution (right). Lognormal, where each agent s valuation and quality are drawn from log-normal distributions. These distributions are correlated using a Gaussian copula [Nelsen ]. The exact parameters of these distributions are derived from confidential data, but roughly resemble the work of Lahaie and Pennock [Lahaie and Pennock 7]. From the uniform distribution, we sampled 1 5-bidder settings, while for the lognormal we sampled 1 5-bidder settings. 3 To explore auction parameters, we used a simple grid search. Reserve prices were varied between and 3 in steps of, between 3 and 1 in 1s and 1 to 1 in 1s and from 1 to 1 in 1s. Squashing power was varied between and 1 in steps of.5. First, we investigated what the optimal parameter settings for each mechanism were. For squashing, the optimal squashing power was (as in previous work [Lahaie and Pennock 7]) for lognormal distributions, but greater than zero for uniform distributions where squashing was also somewhat less effective. (See Figure.) UWR and anchoring had similar optimal reserve prices, and which were dramatically different from QWR s optimal reserve. (See Figure 7.) Adding squashing to UWR produced some improvements and had little effect on the optimal reserve price. (See Figure.) QWR greatly benefited from squashing, but the optimal reserve price is extremely sensitive to the squashing parameter. (See Figure 9.) We then compared across GSP-variants (summarized in Table II). We found that among simple variants UWR was clearly superior to squashing or QWR. The richer variants anchoring, QWR+Sq and UWR+Sq all performed comparably well (within % of each other), though QWR+Sq was consistently the worst. Interestingly, QWR+Sq is only competitive with the other top mechanisms when squashing power is close to zero. Squashing has two effects when added to QWR: it changes the ranking among bidders who exceed their reserve prices, but it also changes the reserve prices. As squashing power gets closer to zero, reserve prices tend towards UWR. We hypothesized that QWR+Sq was only performing as well as it did because of this second effect. To tease apart these two properties, we tested them in isolation. Specifically, we tested (i) a GSP variant where reserve prices are weighted by squashed quality, but ranking among reserve-exceeding bidders is done according to their true 3 A substantial portion of the expected revenue in lognormal settings comes from rare, high valuation bidders. Thus, the expected revenue was higher variance and more samples were needed to reduce noise. EC 13, June 1, 13, Philadelphia, PA, Vol. X, No. X, Article X, Publication date: February 13.

10 X:1 Thompson and Leyton-Brown 1 1 anchor ures wres 9 anchor ures wres Fig. 7. All three reserve-based variants (anchoring, QRW and UWR) provide substantial revenue games. Anchoring is slightly better than UWR, and both are substantially better than QWR Fig.. Adding squashing to UWR can provides modest marginal improvements and doesn t substantially affect the optimal reserve price. Table II. Comparing the various auction variants given their optimal parameter settings (Uniform distribution to the left and lognormal distribution to the right) Auction Parameters VCG Squashing 9.13 s =.5 UWR 1. r = 1. QWR 1.59 r =. Anchoring 1.79 r = 1. UWR+Sq 1. r = 1., s =.5 QWR+Sq 1. r = 1., s =.5 Auction Parameters VCG.5 Squashing s =. UWR.5 r =. QWR.71 r =. Anchoring.15 r =. UWR+Sq 1.9 r =., s =.5 QWR+Sq 79. r =., s =. qualities, and () a GSP variant where reserve prices are weighted by true qualities, but rankings are done using squashed qualities. These experiments confirmed our hypothesis: the first variant (Figure 1), where squashing is used to make the mechanism more UWR-like, was much more effective at increasing revenue than the second (Figure 11). Next, we investigated what happens as the number of bidders varies. Broadly we found that the ranking among auctions remained consistent. (See Figure 1.) Interestingly, the optimal reserve price tended to increase with the number of bidders, par- EC 13, June 1, 13, Philadelphia, PA, Vol. X, No. X, Article X, Publication date: February 13.

11 Optimization in the Generalized Second-Price Auction X: Fig. 9. Adding squashing to QWR can provides dramatic improvements. However, the higher the squashing power, the less the reserve prices are actually weighted by quality. In the case of a lognormal value distribution, the optimal parameter setting (s =.) removes quality scores entirely and is thus equivalent to UWR. Note that different values of squashing power lead to dramatically different optimal reserve prices Fig. 1. When squashing is only applied to reserve prices, it can dramatically increase QWR s revenue. However, there has to be a lot of squashing (i.e., s close to ), and the optimal reserve price is very dependent on the squashing power. In fact, for both distributions, the optimal parameters set s =, in which case the mechanism is identical to UWR. ticularly with UWR. This is especially interesting, because a common intuition in the literature is to take Myerson s optimal reserve prices and add them to GSP (e.g. [Ostrovsky and Schwarz 9]). However, as our results show, Myerson s finding that the optimal reserve price does not vary with the number of bidders is only a property of the optimal auction, not of other auctions such as these GSP variants. In summary, the main findings of this section are: (1) when selling multiple positions, the revenue ranking is the same as the earier analysis (and this ranking is robust as the number of agents varies); and () QWR+sq only performs well because the squashing makes its reserve prices behave more like UWR. 5. THIRD ANALYSIS: ALL PURE NASH EQUILIBRIA As mentioned earlier, GSP is known to have multiple Nash equilibria, and that different equilibria can lead to substantial differences in both revenue and social welfare. EC 13, June 1, 13, Philadelphia, PA, Vol. X, No. X, Article X, Publication date: February 13.

12 X:1 Thompson and Leyton-Brown Fig. 11. When squashing is only applied to ranking, but not to the reserve prices, the marginal gains from squashing over QWR (with the optimal reserve) are very small. 5 VCG Squashing UWR QWR Anchoring UWR+Sq QWR+Sq VCG Squashing UWR QWR Anchoring UWR+Sq QWR+Sq Agents Agents Fig. 1. As the number of agents varies, the relative ranking of GSP variants remains mostly unchanged. The one notable exception to this is squashing and QWR, their ranking can be reversed depending on the distribution and number of bidders. For our third analysis, we investigated how equilibrium selection affects GSP and the six GSP variants, by directly calculating pure strategy Nash equilibrium. To do equilibrium computation, we use the action-graph game (AGG) representation [Jiang et al. 11]. Using an appropriate encoding, it is possible to efficiently encode GSP games as polynomial-sized, computationally usable AGGs [Thompson and Leyton-Brown 9]. Further, using support enumeration, it is possible to enumerate Actually, our analyses are presented in reverse-chronological order: we began with the Nash equilibrium enumeration of this section. However, one of our findings the search engines appear to be switching to less profitable auction designs was so surprising that we moved to progressively simpler models, looking for intuition about this effect. EC 13, June 1, 13, Philadelphia, PA, Vol. X, No. X, Article X, Publication date: February 13.

13 Optimization in the Generalized Second-Price Auction X:13 Table III. Comparing the various auction variants given their optimal parameter settings (Left: Worst-case equilibrium; Right: Best-case equilibrium) Auction Parameters Vanilla GSP 3.1 Squashing.7 s =. UWR 11. r = 15. QWR 9.39 r = 9. Anchoring 1.1 r = 13. UWR+Sq 11.3 r = 15., s =. QWR+Sq 1.17 r = 15., s =. Auction Parameters Vanilla GSP Squashing s =. UWR 11. r = 11. QWR 1. r = 5. Anchoring 1. r = 11. UWR+Sq 1.75 r = 9., s =. QWR+Sq 1.7 r = 7., s =. all 5 the pure strategy Nash equilibria of an AGG [Porter et al. ; Thompson et al. 11]. For this set of experiments, we used uniform distribution as in the NCC experiments. We generated 1 5-bidder settings, and again used grid-search to explore the space of possible parameter settings. Every auction has a built in reserve of 1 increment per click, with ties broken randomly and prices rounded up to the next whole increment. Broadly, we found that while squashing did not help at all with the problem of equilibrium selection (Figure 13, top left), any reserve price scheme dramatically improved worst-case revenue, but especially UWR (top right). Comparing the mechanisms (Table III), we found that UWR and UWR+Sq were among the best and were dramatically better than any other mechanism in worst-case equilibria. In fact, averaging samples of advertiser valuations, we found that UWR s worst-case equilibria were better than squashing and QWR s best-case equilibria. Also, we noticed that optimizing for worstcase equilibria consistently required higher reserve prices than optimizing for bestcase equilibria. In summary, the main findings of this section are: (1) There is a huge gap between best- and worst-case equilibria (over.5 for vanilla GSP). Squashing does not help to close this gap, but reserve prices do. () When considering the best-case equilibria, the revenue ranking remains roughly the same as the earlier analyses. When considering the worst-case equilibria, UWR (with or without squashing) is clearly superior to other mechanisms. (3) The optimal reserve price is much higher (for any GSP variant) when optimizing worst-case revenue, than when optimizing best-case revenue.. DISCUSSION This paper set out to find a way to maximize expected revenue, while sticking to the generalized second-price auction framework. Working the most widely-studied model, and multiple analysis techniques, we consistently found that mechanisms using unweighted per-click reserve prices (or at least, reserve prices that, due to squashing, are very nearly unweighted). However, search engines have been moving from unweighted towards weighted reserve prices. One interpretation of this discrepancy is that search engines are not optimizing their short-term expected revenue, because they are optimizing some other objective 5 As is common in worst-case analysis of equilibria of auctions, (e.g., [Roughgarden and Tardos 1], [Caragiannis, Kaklamanis, Kanellopoulos, Kyropoulou, Lucier, Paes Leme, and Tardos Caragiannis et al.]) we assume that bidders are conservative, i.e., no bidder ever follows the weakly dominated strategy of bidding more than his valuation. Without this assumption, many implausible equilibria are possible, e.g., even single-good Vickrey auctions have equilibria that are unboundedly far from efficient, and unboundedly far from the revenue of truthful bidding. Unfortunately, the log-normal distribution is too high variance to get much signal from a modest number of samples. EC 13, June 1, 13, Philadelphia, PA, Vol. X, No. X, Article X, Publication date: February 13.

14 X:1 Thompson and Leyton-Brown 1 Worst NE Best NE 1 Worst NE Best NE s r 1 Worst NE Best NE 1 Worst NE Best NE r r Fig. 13. Depending on parameter settings, revenue can vary dramatically between worst- and best-case equilibria. (Top left: squashing; top right: UWR; bottom left: QWR; bottom right: anchoring) (e.g., user satisfaction, which could lead to greater market share and therefore longterm revenue), because there are other business considerations 7, or simply because the search engines are acting in error. However, another interpretation is that there is some fundamental shortcoming in the standard position auction model, which is exposed by our findings. We believe that there is an element of truth to both interpretations. For example, in the field experiment of [Ostrovsky and Schwarz 9], where Yahoo! increased revenue by increasing reserve prices and simultaneously switching to QWR, we conjecture that the short-term revenue increases would have been even greater if they had retained UWR. However, over the longer term, advertisers may have had an incentive to change not only their bids, but also the text of their ads. This extra dimension, where advertisers can change the ad text (which will influence quality score, click probability and 7 Ostrovsky and Schwarz were explicitly instructed to use weighted reserve prices in their experiments, because they are more consistent with the weighted GSP paradigm, and are perceived to be more fair by bidders [9]. For example, consider the problem of an advertiser who has two choices of ad text. One choice will yield 1 clicks per hour, leading to 11 sales per hour. The other choice yields 1 clicks per hour, but every click produces a sale. With weighted reserve prices (and no squashing) the advertiser will always choose the first text, since it produces more sales per hour for the same price. With quality-weighted reserve prices (or squashing), the advertiser would likely chose the second, which generates nearly as many sales, and requires him to pay the reserve price far less often. Its not immediately clear which text the search engine EC 13, June 1, 13, Philadelphia, PA, Vol. X, No. X, Article X, Publication date: February 13.

15 Optimization in the Generalized Second-Price Auction X:15 possibly click value) is absent from all major models of sponsored search. Thus, we believe that the most pressing open problem stemming from this work is to invent and analyze a richer model, where quality and click probability, rather than being exogenous, are determined by the advertiser s choice of ad text (which in equilibrium must be a best response to the rules of the auction and the choices of the other agents). REFERENCES AGGARWAL, G., GOEL, A., AND MOTWANI, R.. Truthful auctions for pricing search keywords. In EC: Proceedings of the ACM Conference on Electronic Commerce. ACM, New York, NY, USA, 1 7. CARAGIANNIS, I., KAKLAMANIS, C., KANELLOPOULOS, P., KYROPOULOU, M., LUCIER, B., PAES LEME, R., AND TARDOS, E. On the efficiency of equilibria in generalized second price auctions. Invited to the Journal of Economic Theory (JET). DUGHMI, S., ROUGHGARDEN, T., AND SUNDARARAJAN, M. 9. submodularity. In ACM-EC. EDELMAN, B., OSTROVSKY, M., AND SCHWARZ, M. 7. Internet advertising and the generalized second price auction: Selling billions of dollars worth of keywords. American Economic Review 97, 1, 59. EDELMAN, B. AND SCHWARZ, M. 1. Optimal auction design and equilibrium selection in sponsored search auctions. HBS Working Paper. HARTLINE, J. D. 1. Approximation in economic design. JIANG, A. X., LEYTON-BROWN, K., AND BHAT, N. A. R. 11. Action-graph games. GEB 71, KASH, I. A. AND PARKES, D. C. 1. Impersonation strategies in auctions. In WINE. LAHAIE, S. AND MCAFEE, P. 11. Efficient ranking in sponsored search. In WINE. LAHAIE, S. AND PENNOCK, D. M. 7. analysis of a family of ranking rules for keyword auctions. In EC: Proceedings of the ACM Conference on Electronic Commerce. LUCIER, B., PAES LEME, R., AND TARDOS, E. 1. On revenue in the generalized second price auction. In WWW. METZ, C. 1. Yahoo! handicaps its search ad auctions. does google squash too? MYERSON, R Optimal auction design. Mathematics of Operations Research, 1. NELSEN, R. B.. An Introduction to Copulas. Springer. OSTROVSKY, M. AND SCHWARZ, M. 9. Reserve prices in internet advertising auctions: A field experiment. Working Paper. PORTER, R., NUDELMAN, E., AND SHOHAM, Y.. Simple search methods for finding a nash equilibrium. GEB 3,. ROUGHGARDEN, T. AND TARDOS, E. 1. Do externalities degrade GSP s efficiency? In Workshop on Advertising Auctions. SUN, Y., ZHOU, Y., AND DENG, X. 11. Optimal reserve prices in weighted gsp auctions: Theory and experimental methodology. In Workshop on Ad Auctions. THOMPSON, D. R. M., LEUNG, S., AND LEYTON-BROWN, K. 11. Computing Nash equilibria of action-graph games via support enumeration. In WINE THOMPSON, D. R. M. AND LEYTON-BROWN, K. 9. Computational analysis of perfect-information position auctions. In EC: Proceedings of the ACM Conference on Electronic Commerce. 51. VARIAN, H. 7. Position auctions. International Journal of Industrial Organization 5,, should prefer, the first satisfies more users, but also wastes the time of many users who click through but do not buy. EC 13, June 1, 13, Philadelphia, PA, Vol. X, No. X, Article X, Publication date: February 13.