Soft Floors in Auctions - UCLA Anderson Home...one of the key forces behind the resurgence of banners: unlike the banners from the 1990s, today’s banner ads are targeted to the individual

MANAGEMENT SCIENCEArticles in Advance, pp. 1–18

http://pubsonline.informs.org/journal/mnsc/ ISSN 0025-1909 (print), ISSN 1526-5501 (online)

Soft Floors in AuctionsRobert Zeithammera

aUCLA Anderson School of Management, University of California, Los Angeles, Los Angeles, CA 90095Contact: [email protected], http://orcid.org/0000-0002-8388-247X (RZ)

Received: July 6, 2017Revised: March 16, 2018; May 28, 2018Accepted: June 30, 2018Published Online in Articles in Advance:June 11, 2019

https://doi.org/101287/mnsc.2018.3164

Copyright: © 2019 INFORMS

Abstract. Several of the auction-driven exchanges that facilitate programmatic buying ofinternet display advertising have recently introduced “soft floors” in addition to standardreserve prices (called “hard floors” in the industry). A soft floor is a bid level below whicha winning bidder pays his own bid instead of paying the second-highest bid as in a second-price auctionmost ad exchanges use by default. This paper characterizes soft floors’ revenue-generating potential as a function of the distribution of bidder independent private values.When bidders are symmetric (identically distributed), soft floors have no effect on revenue,because a symmetric equilibrium always exists in strictly monotonic bidding strategies, andstandard revenue-equivalence arguments thus apply. The industry oftenmotivates soft floorsas tools for extracting additional expected revenue from an occasional high bidder, for ex-ample a bidder retargeting the consumer making the impression. Such asymmetries in thedistribution of bidder preferences do not automaticallymake soft floors profitable. This paperpresents two examples of tractable modeling assumptions about such occasional high bid-ders, with one example implying low soft floors always hurt revenues because of strategicbid-shading by the regular bidders, and the other example implying high soft floors canincrease revenues by making the regular bidders bid more aggressively.

History: Accepted by Juanjuan Zhang, marketing.

Keywords: advertising and media • marketing: pricing • microeconomics: market structure and pricing

1. IntroductionSince the world’s first banner ad in 1994 (Singel 2010),advertising dollars have followed the shift of consumerattention to digital media, reaching more than one-thirdof total U.S. advertising spending by 2016. Despitestarting with display banner ads, the lion’s share ofdigital advertising dollars was initially spent on searchads, because they offered an unparalleled level oftargeting (Goldfarb 2014). However, for the first time inthe more recent history of digital advertising, spendingon display ads surpassed spending on search ads in2016 (emarketer 2016). An improved targeting ability isone of the key forces behind the resurgence of banners:unlike the banners from the 1990s, today’s banner adsare targeted to the individual viewer one impression ata time by computer algorithms—a practice called “pro-grammatic buying.” A dominant method of allocatingand pricing the display advertising space sold pro-grammatically is real-time bidding (RTB), whereby eachavailable impression is sold to interested advertisers bya sealed-bid auction that lasts a fraction of a second.Experts estimate thatmore than $20 billion in advertisingis sold by RTB per year in the United States (emarketer2016) in more than 30 trillion unique transactions(Friedman 2015).

What are the rules of these trillions of auctions? Thevast majority of the “ad exchange” auctioneers usesecond-price sealed-bid “Vickrey” auctions—a dramatic

shift from the obscurity of the Vickrey pricing rule in pastauction-driven marketplaces documented by Rothkopfet al. (1990). However, several important players in theRTB industry have recently partially reversed this shiftby introducing “soft floors”—bid levels belowwhich theauction’s pricing rule switches from second-price to first-price, sometimes also called “high-bid.”1 The “soft” partof “soft floor” contrasts with a “hard floor”—a bid levelbelowwhich the auctioneerwill not sell the impression,also known as “reserve price” in the auction literature(Myerson 1981). This paper provides the first theo-retical treatment of soft floors and shows that theirusefulness depends on the distribution of bidderpreferences. Throughout this paper, bidders are as-sumed to have independent private values. When thebidders are symmetric (i.e., when their valuations aredrawn from the samedistribution), I show that the use ofsoft floors is misguided because they complicate biddingandhavenoeffect on expected revenue.When the biddersare asymmetric in an RTB-relevant fashion (i.e., whenhigh-valuation bidders occasionally join the auction),I showby two examples that softfloors can bothhurt andbenefit the auctioneer, depending both on themagnitudeof the soft floor and on the valuation overlap betweenthe regular bidders and the high bidders. The next fewparagraphs introduce the three different modelingassumptions used in this paper and preview the resultsthey imply. Please see Figure 1 for a representation of

1

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all the modeling variants covered in this paper, bysection.

As long as the bidders are symmetric, I show that softfloors have no impact on auction revenue. In otherwords, when the different advertisers’ valuations ofeach impression are drawn from the same distribution,soft-floor auctions are revenue-equivalent with standardauctions that have the same hard floor. The revenue-equivalence result is not a trivial extension of the well-known equivalence between first- and second-priceauctions: just because first- and second-price auc-tions yield the same expected revenue (under biddersymmetry), it does not immediately follow that theirhybrid arising from the presence of a soft floor will alsobe revenue-equivalent with the simple second-priceauction: strategic bidders may react to the introductionof a soft floor by playing mixed strategies or by pooling,thus changing the relationship between valuations andthe chance of winning. The first main result of this paper(discussed in Section 4) is a general proof that whereasbidders indeed react to the introduction of a soft floor byadjusting their bids, the resulting equilibrium is in puremonotonic strategies. The monotonicity of the biddingequilibrium in the soft-floor auction guarantees the soft-floor auction does not change any bidder’s chance ofwinning relative the second-price auction with the samehard floor, which in turn keeps the expected revenueof the auctioneer unaffected according to the revenue-equivalence result of Myerson (1981). In the extensions,analogous arguments are then used to also show thatrevenue equivalence continues to hold evenwhen biddersparticipate randomly, and also when the soft floor ishidden, as it tends to be on some exchanges.

Given the robust revenue equivalence in the sym-metric model, the rest of this paper explores the ob-vious possibility that a rationalization of the soft-floorindustry practice can arise from asymmetries amongbidders. Inspired by the industry analysts who origi-nally motivated the use of soft floors (e.g., Weatherman2013), I consider the possibility that high-value biddersmay occasionally enter the auction. For example,a “retargeting advertiser” (whose website the customerhas just visited before arriving to the publisher

auctioning off the customer’s impression) likelyvalues the impressionmuchmore than other advertiserswho bid only on demographics. If such a high-valueadvertiser were always present, there would be littlebenefit to softfloors—the seller could simply increase thehard floor; but such a high-value advertiser may notparticipate in every RTB auction, so a soft floor mightseem to be a clever adaptive mechanism that automat-ically activates a higher reserve price only when theadvertiser does appear (Weatherman 2013). In contrastto this industry intuition, I show that adding randomlyappearing asymmetrically high bidders always makeslow-enough soft floors suboptimal for the auctioneer. Onthe other hand, very high softfloors can increase revenueunder some assumptions. I now discuss both of thesecontrasting examples in turn.In the second main result of the paper (covered in

Section 5), I show that when the high bidders areguaranteed to have valuations above the regular bid-ders, soft floors low enough that all the high biddersface second-price pricing reduce expected revenues.The reason is that the strategic bid-shading by regularbidders always more than offsets the additional pricingpressure on the high bidders generated by the soft floor.The revenue loss can be derived in closed form, and it isbounded by the amount of revenue an auctioneer run-ning a second-price auction would lose from losing oneof the regular bidders. The bound echoes the classicresult of Bulow and Klemperer (1996), who show thata revenue gain from setting the hard floor optimally isless than the gain from finding one more bidder.If low soft floors hurt revenue, might higher ones

help? The analysis of higher soft floors that “kick in” forthe high bidders is not tractable, but tractability resumesin a slightly modified asymmetric model when the softfloor is so high that the auction effectively becomesa first-price sealed-bid auction. The third main result ofthis paper (covered by Section 6) provides a lowerbound of the profitability of a soft-floor auction in anRTB-relevant asymmetric setting by analyzing therevenue potential of a first-price auction when oneof two uniformly distributed potential bidders sto-chastically dominates the other, but only participatesoccasionally. This analysis extends the analysis ofequilibrium bidding by Maskin and Riley (2000) andKaplan and Zamir (2012) to random participation byone of the two bidders. The bidding equilibrium is inclosed form, but the expected revenue calculation,and hence the ultimate revenue comparison with asecond-price auction, involves an intractable integral.Approximating the integral numerically, I find thatfirst-price auctions can revenue-dominate second-priceauctions as long as the high bidder’s chance of partic-ipation is high enough to induce aggressive bidding bythe regular bidder. Because soft floor auctions weaklydominate first-price auctions, this result is an example

Figure 1. (Color online) Effects of Soft Floor on AuctionRevenue, and Assumptions They Rely on

Zeithammer: Soft Floors in Auctions2 Management Science, Articles in Advance, pp. 1–18, © 2019 INFORMS

of a situation in which soft floor can strictly increasethe auctioneer’s revenue. The exact conditions for therevenue dominance, as well as the magnitude of therevenue difference, are sensitive to whether the auc-tioneer can optimize the hard floor. Once the hard flooris optimized for the demand situation, only a smallamount of additional revenue is available from alsooptimizing the pricing rule.

2. Literature ReviewThe literature on online display advertising (for thor-ough literature reviews, see Hoban and Bucklin 2015,Choi et al. 2017, or Johnson et al. 2017) contains veryfew papers about soft floors: Yuan et al. (2013) examinedata from a large ad exchange that uses soft floors andestimate that more than half of the exchange’s revenueis transacted using the first-price rule instead of thesecond-price rule. They conclude soft floors are aneconomically important phenomenon in the RTB mar-ketplace. In contrast to the predictions of this paper,Forsch et al. (2017) analyze the profitability of soft floorsusing a large-scale field experiment and conclude thateven relatively low soft floors can increase the auction-eer’s revenue. One possible explanation for the diver-gence between this paper’s predictions and the fieldexperiment’s results is that the bidders in the experimentdid not have enough time to adjust their strategies tothe novel mechanism: unlike the rational biddersassumed herein, the bidders in Forsch et al. (2017) donot reduce their bids when soft floors are introduced.

This paper also contributes to the broader literatureon mechanism design in the online display advertisingindustry. Most work in that literature focuses on op-timizing the pricing and allocation rules of RTB ex-changes to address specific ways the bidders for onlinedisplay impressions differ from bidders assumed incanonical models. For example, Abraham et al. (2016)focus on the informational asymmetry arising frominformative “cookies” available to only some bidders ina pure common-value model, and they compare thetwo dominant auction pricing rules in terms of revenue.Arnosti et al. (2016) also study the impact of bidderasymmetries in a common-value model, with the asym-metry arising from the difference between “performanceadvertisers”whoknow their valuations of each impressionand “brandadvertisers”whodonot. They focusmainly onmarket efficiency and propose a new theoretical mecha-nism that is nearly efficient. In contrast to Abraham et al.(2016) and Arnosti et al. (2016), this paper assumes thatbidders have and know their private values, focuses on theex ante asymmetry between “regular” bidders whotend to bid low and often and “high” retargeting bidderswho bid high but rarely, and restricts attention to aprominent design implemented in the industry—thesoft-floor auction. Celis et. al (2014) take a differentapproach to analyzing the competition between regular

and retargeting bidders: instead of considering an exante asymmetric set of bidders, they assume an interim-asymmetric model of bidders drawn from a mixtureof regular bidders poorly matched with the customermaking the impression and high-valuation bidders whodo match with the customer well. They note that suchamixture distribution is irregular in the sense ofMyerson(1981), so standard auctions may not perform well, andthey propose a novel mechanism called “buy-it-now ortake-a-chance” which does better.All of the above mechanism-design papers—

including this paper—focus on the sale of a singleimpression. There is also an emerging stream of workthat addresses themultiunit nature of RTBmarketplaces.For example, Balseiro et al. (2015) use a modern fluidmean-field equilibrium notion to simplify an otherwiseintractable model of budget-constrained bidders par-ticipating in a sequence of second-price sealed-bid auc-tions. Having outlined the contribution of this paperto the literature, I now describe the main mechanismof interest—the soft-floor auction.

3. Soft-Floor Auction Definition and OtherSupply-Side Assumptions

One object (e.g., an ad impression in the RTB context) isfor sale. The auctioneer values the object at zero andsets two reserves: a hard floor h≥ 0 and a soft floor s> h.The soft-floor sealed-bid auction collects bids, sortsthem such that b(1) ≥ b(2) ≥ b(3) . . . , and determines theauction winner and the price paid as follows:

(1) When b(1) > s, the bidder who submitted b(1) winsand pays max{s, b(2)}.

(2) When s≥ b(1) ≥ h, the bidder who submitted b(1)wins and pays b(1).

(3) When h> b(1), the auctioneer keeps the object.In words, the soft floor functions as a reserve price in

a second-price sealed-bid auction (2PSB) as long as atleast one bid exceeds it (case 1). When no bids exceed s,the auction becomes a first-price sealed-bid auction(1PSB) with a reserve price equal to h (cases 2 and 3).Throughout this paper, I assume h is common knowl-

edge; that is, the auctioneer announces the reserveprice before the auction. Regarding the bidders’ in-formation about s, I first assume the auctioneer alsopreannounces s (or that, equivalently, the biddersfigure out both values through experimentation) andthen address the possibility of keeping s hidden frombidders whenever tractable.An analysis of the revenue implications of soft floors

requires a demand-side model of bidders. This paperconsiders independent private-valuation (IPV) bidders—a standard assumption in auction theory. IPV is areasonable model of bidders in the RTB context thatmotivates this paper: “valuation” of an impression isthe increase in the advertiser’s profit from winning the

Zeithammer: Soft Floors in AuctionsManagement Science, Articles in Advance, pp. 1–18, © 2019 INFORMS 3

impression, “private”means no advertiser can learnabout his own valuation of the impression from howmuch another advertiser values it, and “independent”means the values are statistically independent of eachother in the population of bidders. Given the IPV as-sumption, a population distribution of valuations com-pletes the model. This paper makes three partiallynested assumptions about the distribution, summa-rized in Figure 1 and introduced in the previous sec-tion. I turn to the symmetric case next.

4. Symmetric Bidders: Soft Floors HaveNo Impact on the Auctioneer’s Revenue

SupposeN bidders indexed by i = 1,2,. . .,N have privatevaluations vi drawn independently from a continuousdistribution F with full support on [0,M]. FollowingKrishna’s (2009) notation, let G be the distributionof the maximum from N − 1 independent and iden-tically distributed (iid) draws from F, itself denoted Y1:G(Y1) ≡ FN−1 (Y1), and let X1 be the highest of N iiddraws from F, distributed FN(X1).

This section demonstrates that when bidders aresymmetric, soft floors have no impact on the auction-eer’s expected revenue. The proof proceeds in twosteps. First, for any s> h≥ 0, I construct a monotonicallyincreasing equilibrium bidding strategy β(v) that bestresponds to s and h. Second, the fact that the biddingstrategy is monotonic means the soft-floor auction allo-cates the object to the same bidder as a standard auctionwith a hard floor of h would, and so the revenue-equivalence theorem of Myerson (1981) implies thesoft-floor auction also produces the same expectedrevenue to the auctioneer. The exact form of the biddingequilibrium depends on the bidders’ information aboutthe soft floor. The following subsection (Section 4.1)analyzes the case of the soft floor being commonknowledge among a fixed set of participating bidders.Section 4.2 then generalizes the bidding strategies tobidders participating randomly, and Section 4.3 takes upthe case of bidders uncertain about the soft floor at thetime of bidding. The main revenue-equivalence result isoutlined and discussed in Section 4.4.

4.1. Bidding Equilibrium When the Soft FloorIs Common Knowledge and AllBidders Participate

I begin the exposition of bidding in a soft-floor auctionunder the canonical assumption that allN bidders knows and participate in the auction for sure. Let βI(v) denotethe standard symmetric bidding equilibrium in a 1PSBwith N bidders and a public reserve h (for a detailedderivation, see Krishna 2009):

βI(v) � hG(h)G(v) +

1G(v)

∫ v

hxg(x)dx � E

[max{Y1,h}|Y1< v

],

(1)

where the roman numeral subscript on β indicates thefirst-price pricing rule. Then the bidding equilibriumβ(v) in the soft-floor auction can be characterized interms of βI(v) as follows:

Proposition 1. When s< βI(M), the following is a uniquesymmetric monotonic pure-strategy equilibrium of the soft-floor auction:

β(v) �{v≤ β−1I (s) : βI(v)v> β−1I (s) : v .

When s≥ βI(M), the soft-floor auction becomes a 1PSBauction, and β(v) � βI(v).Please see the appendix for detailed proofs of all

propositions in this paper. The intuition for the result isas follows: when s< βI(M) (i.e., when the highest bidderwould bid above it in a 1PSB), the equilibrium β in-volves a jump discontinuity at a valuation v* such thatβI(v*) � s. Bidders with v < v* shade their bids as if theywere in a 1PSB auction. They effectively ignore thehigher-valuation bidders because they cannot winagainst them. Bidders with v> v* bid their valuations asif they were in a second-price auction. Their bids areunaffected by the behavior of lower-valuation bidders,because bidding one’s valuation is a dominant strategyunder the 2PSB incentives. The jump discontinuity’s lo-cation in the space of valuations and themagnitude of thejump ensure no bidderwants to unilaterally deviate fromthe pricing rule “assigned” to him by his valuation.

Example (F = Uniform[0,1]). Illustrating Proposition 1on a concrete distributional example is useful. A uni-form distribution of valuations implies G(x) � xN−1, sothe 1PSB bidding strategy is

βI(v) �N − 1N

v + hN

NvN−1. (2)

Figure 2. Equilibrium Bidding Strategy with a Known SoftFloor and Guaranteed Bidder Participation

Notes. F = uniform [0,1], s = 0.6, and h = 0.5 (the h is optimal giventhe F). The dashed line indicates the 45° line; the dotted vertical linesindicate the jump discontinuities at v* for the given numbers ofbidders indicated by the numbers next to the lines.


The indifference equation βI(v*) � s becomes (N − 1) ·(v*)N − sN(v*)N−1 + hN � 0, which does not have aclosed-form solution for a general N, but does forN = 2: v*(s;N � 2) � s + ��

s2 − h2√

< 1⇔ s< 1+h22 . Figure 2

illustrates the bidding function β(v) for N = 2, 3, 4, and10. I now turn to the possibility that the bidders par-ticipate randomly.

4.2. Bidding Equilibrium When the Soft Floor IsCommon Knowledge and ParticipationIs Random

One of the apparent benefits of a soft floor is its ability toput pricing pressure on a single high-valuation bidder,who only pays the hard floor h under 2PSB rules. Whensuch a bidder’s presence is assured, the auctioneer cansimply increase the hard floor; but when the number ofbidders is uncertain, the soft floor s > h can “kick in”precisely when there happens to be just one bidder.I will show this intuition is incomplete because it doesnot consider the associated revenue decrease whenthere happen to be multiple bidders.

AssumeN symmetric potential bidders exist, and eachof them enters independently of the other with proba-bility 0≤α≤ 1. Because an entrant might face feweropponents, he should bid less aggressively comparedwith facing all potential opponents for sure. Indeed,Harstad et al. (1990) show the existence of a βα(v)<βI(v), which can be expressed as a weighted average ofthe contingent 1PSB bidding functions that would applyfor a fixed number of present bidders between 1 and N.Please see Equation (A.7) in the appendix for the βα(v)for general F when N = 2. A concrete example witha uniform distribution is again helpful:

Example (F = Uniform[0,1], N = 2, and h = 0). This ex-ample makes clear how random participation reduces1PSB bids relative to certain participation:

βα(v) �αv2

2[1 − α(1 − v)] � βI(v) −(1 − α)v

2[1 − α(1 − v)]. (3)

Not surprisingly, Proposition 1 generalizes to the sit-uation with random participation:

Corollary 1. For any s≤ βα(M), the following is a uniquesymmetric monotonic pure-strategy equilibrium of the soft-floor auction:

β(v) �{v≤ β−1α (s) : βα(v)v> β−1α (s) : v .

When s≥ βα(M), the soft-floor auction becomes a 1PSBauction, and β(v) � βα(v).

The proof is analogous to the proof of Proposition 1,with G(v)≡ (1−α)+αF(v) as the equilibrium probability

of winning. Figure 3 illustrates the bidding strategies fora range of α’s in the uniform example.

4.3. Bidding Equilibrium When the Soft Floor IsHidden at the Time of Bidding

Suppose the bidders are uncertain about the soft floorat the time of bidding, and they all summarize theirbeliefs about it by some distributionΩ on [h,M]. Optimalbidding must now average over the possibility that thesoft floor happens to be low (and 2PSB rules will thusapply) and the possibility that the soft floor happens tobe high (and the price paid will be equal to the winningbid). Unlike in the previous two subsections, charac-terizing the equilibrium in closed form is not possibleeven in the uniform example. However, the followingproposition provides weak sufficient conditions for asymmetric monotonic equilibrium to exist and boundsthe resulting bidding function below with βI(v):Proposition 2. When f (v) · FN−2(v) and Ω(v) are contin-uous on [h,M], the sealed-bid auction with a hard floor h anda hidden soft floor drawn from Ω on [h,M] has a symmetricpure-strategy equilibrium characterized by an increasingbidding function β(v)> βI(v) that satisfies

β′(v) � g(v)(v − β(v))G(v)[1 −Ω(β(v))] . (4)

The proof uses the Peano existence theorem to assure usEquation (4) has a solution. Compared with the 1PSBdifferential equation that gives rise to βI(v), Equation (4)adds the term in the square bracket. Because β(v) is thussteeper everywhere, the relative ranking of the twobidding functions follows.

Figure 3. Equilibrium Bidding Strategy with a Known SoftFloor and Two Potential Bidders Who ParticipateRandomly.

Notes. F = uniform [0,1], h = 0, s = 0.2,N = 2. The dashed line indicatesthe 45° line; the thicker dotted line indicates the 1PSB bidding strategywith two bidders, and the vertical lines indicate the jumpdiscontinuities at v* for the participation probabilities shown nextto the jumps, if any.


Intuitively, relative to 1PSB, random softfloors partiallymitigate the increased payment associated with higherbids by switching the pricing rule to 2PSB. The resulting“random discount” gives the bidders an incentive to raisebid levels, and so the β(v) exceeds equilibrium biddingin a 1PSB with the same number of bidders everywhereabove h . See Figure 4 for a concrete uniform-distributionexample.

4.4. Revenue Equivalence Under Bidder SymmetryAll of the previous subsections (Sections 4.1–4.3) finda monotonically increasing symmetric pure-strategyequilibrium of the soft-floor auction game. Under allthree potential assumptions regarding the bidders’ in-formation about the soft floor and the bidders’ partici-pation, the introduction of a soft floor therefore doesnot affect any bidder’s probability of winning. The in-troduction of a soft floor also does not affect the payoff ofthe bidder with the lowest trading valuation v = h, whomakes zero surplus both with and without the soft floor.Therefore, the revenue-equivalence result of Myerson(1981) implies the soft floor does not affect the auc-tioneer’s revenue and the bidders’ surpluses—a factI summarize in the next proposition.

Proposition 3. Suppose bidders are symmetric in theirvaluations, their participation behavior, and their beliefs aboutthe soft floor. Then, for every hard floor h, the introduction ofa soft floor s > h has no effect on the auctioneer’s revenue orany bidder’s expected surplus.

The “magic” of revenue equivalence stems from thefact that we only need to consider the allocation prob-ability (the chance of winning) for every bidder type—the revenues are then implied by incentive compatibility.Please see Myerson (1981) for the original result and

Krishna (2009) for the straightforward extension to thecase of randomly participating bidders.The case of randomly participating bidders is es-

pecially interesting to analyze deeper because it seemsto agree with a common argument in favor of softfloors. Let N = 2 to simplify the combinatorics. Bidderswith v < h have no impact on revenue. When only onebidder with v > h happens to participate, he wins forsure regardless of the pricing rule. Whereas he wouldpay only h in the 2PSB, the soft-floor auction chargeshim more, namely min(βα(v), s)> h. It seems that thisrevenue advantage of the soft-floor auction over 2PSBmight dominate its associated revenue disadvantagewhen both bidders happen to participate and they bothhave v > h because that scenario happens much lessoften. For example, when α � h � 1/2, the chance ofonly one v > h bidder participating is 3/8—three timesgreater than the chance that both bidders participate andhave v > h. Proposition 3 shows that βα(v) is calibratedsuch that the single-bidder advantage of the soft-floorauction is exactly offset by its two-bidder disadvantage.In the appendix, I demonstrate the revenue equivalenceunder random participation explicitly (i.e., without re-lying on the mechanism-design results used in the quickproof of Proposition 3) to illustrate how the advantageand the disadvantage cancel each other out in the ex-pected revenue calculation.Reflecting on the predictions of this section for bid-

ding behavior is also useful empirically. Looking atFigures 2–4, one can make the following observations:keeping the hard floor constant, adding a soft floorshould lead to bid-shading by low-valuation bidders,and so the distribution of the observed bids shouldbecome more skewed to the right. If the soft floor iscommon knowledge, the distribution of observedbids should also have a hole just above the soft floor.The data collected by Forsch et al. (2017) do not haveeither of these features, suggesting the bidders in theexperiment did not rationally adjust their biddingstrategies to the presence of the soft floor.

5. Randomly Appearing High-ValuationBidders: Low Soft Floors ReduceAuctioneer’s Revenue, and High SoftFloors Break the Monotonicity ofBidding Strategies

In a prominent explanation of soft floors, Kevin Weather-man of theMoPub platform used a stylized example ofa seller who sets a hard floor of $1 and faces occasionalbids around $2 in addition to regular bidding activityin the $0.75–$0.90 range (Weatherman 2013). Weath-erman’s argument for why such a seller would benefitfrom soft floors is that the seller can lower his hardfloor toward the bidding range of the regular bid-ders while introducing a soft floor above $1: such an

Figure 4. Bidding Strategy When Soft floor Is Hidden andBidder Participation Is Guaranteed

Notes. F = uniform [0,1], and h = 0.5 (the h is optimal given the F), ands ~ Uniform[h,1]. The dashed line indicates the 45° line. The dotted linesindicate 1PSB bidding strategies without a soft floor, and the solid linesindicate bidding strategies with a hidden soft floor. Several levels of thenumber of bidders are indicated by the numbers next to the lines.


arrangement seems to preserve the price pressure onthe high bidder (when exactly one such bidderhappens to participate—multiple high bidders putprice pressure on each other) while also collectingmore revenue from low bidders (when no highbidders happen to be present). As Weatherman puts it,“the goal is to ‘harvest’ higher bids while not compro-mising on lower bid opportunities” (Weatherman 2013).Diksha Sahni of AppLift eloquently makes the same ar-gument by pointing out that “when the gap between a bidand the second bid is significant, it may create a gapbetween potential revenues and actual revenues” (Sahni2016). Motivated by these industry experts, this sectionanalyzes the possibility of randomly present high-valuation bidders somehow making soft floors prof-itable for the auctioneer. I focus on the followingasymmetric case:2

Definition. Let a market with randomly appearing high-valuation bidders always contain N “regular” biddersdrawn from some F on [0,1] and K potential “high”bidders with valuations drawn from some Φ on [L,M]with L ≥ 1. The high bidders participate randomly andindependently of each other with probability α. Par-ticipation by competing bidders is not observable byanyone before bidding.

An analysis of 1PSB equilibrium bidding in theabove-defined market is not tractable even with α = 1(Maskin and Riley 2000), so assessing the profitabilityof a soft floor so high that nobody pays it is difficult.When the soft floor “kicks in,” the analysis remainsintractable when s> βI(1): no globally monotonicequilibrium strategy exists that would produce anincentive-compatible separation of the high bid-ders above s from the rest of the bidders. On theflipside, when s≤ βI(1) (i.e., the soft floor is lowenough that at least some regular bidders bid aboveit), the analysis of equilibrium bidding is simple inthat the soft-floor auction has the same equilib-rium (outlined in Proposition 1) as without the highbidders:

Lemma 1. When the soft floor is small enough that at leastsome regular bidders would bid above it in a 1PSB, i.e., whens≤ βI(1), the soft-floor auction in a market with randomlyappearing high-valuation bidders has the same biddingequilibrium as the soft-floor auction without the high-valuation bidders characterized in Proposition 1, namely

β(v) �{v≤ β−1I (s) : βI(v)v> β−1I (s) : v

for both bidder types.

The argument behind Lemma 1 is straightforward: ifthe high bidders indeed bid their valuations, the reg-ular bidders assume they can only win when no high

bidder is present, and so they behave the same as in anauction without high bidders. The high bidders, on theother hand, do not want to deviate from bidding theirvaluations to bidding the soft floor, because the asso-ciated lower price comes with leaving too many po-tential wins on the table. Intuitively, the low-enoughsoft floor keeps bidding tractable because it guaranteesthe bidders facing 1PSB rules are symmetric (they arejust the regular bidders with valuations up to v*) whileall the remaining bidders face 2PSB incentives that areunaffected by asymmetries (bidding one’s true valuationremains an equilibrium strategy in a 2PSB even withasymmetries because it is a dominant strategy).So can soft floors increase revenue in this market?

Because the bidding equilibrium is the same as inProposition 1 when s≤ βI(1), the auctioneer’s revenuefrom a soft-floor auction is easy to compare with thatin a 2PSB auction with the same hard floor. When nohigh bidder enters, Proposition 3 proves that the softfloor does not impact revenue.When two ormore highbidders enter, 2PSB rules with bids above L > s de-termine the price in the soft-floor auction, so the softfloor does not impact revenue. Adding the soft flooronly makes a difference when exactly one high bidderenters. The lone high bidder wins and pays either s orthe highest regular valuation X1, whichever is greater:FN(v*)s + (1 − FN(v*))E(X1 |X1 > v*) instead of payingE(max{X1, h}) without the soft floor. The revenueimpact of soft floor is therefore

ΠA(h, s) −ΠA2PSB(h) � Pr(1 high) [ FN(v*)s

+ (1 − FN(v*))E(X1 |X1 > v*) − E(max{X1, h})], (5)

where the probability of exactly one high bidder en-tering is Pr(1 high) � Kα(1 − α)K−1. Now recall that scan be expressed as a conditional expectation of orderstatistics: s � βI(v*) � E[max{Y1, h}|Y1 < v*]. Substitut-ing for s in Equation (5) yields

ΠA(h, s) −ΠA2PSB(h)

Kα(1 − α)K−1� FN(v*)︸�︷︷�︸

Pr(X1<s)E(max{Y1, h}|Y1 < v*)︸��︷︷��︸

s

+ (1 − FN(v*))E(X1 |X1 > v*)

− FN(v*)E(max{X1, h}|X1 < v*)

− (1 − FN(v*))E(max{X1, h}|X1 > v*)︷��︸︸��︷�E(X1 |X1 > v*) because v*> h

� FN(v*)[E(max{Y1, h}|Y1 < v*)−E(max{X1, h}|X1 < v*)]< 0.

(6)


I have just derived the key ingredient of the followingresult:

Proposition 4. For every hard floor h, adding a soft floorsmall enough that the highest regular bidder would bidabove it in a 1PSB reduces the expected revenue com-pared with a 2PSB with the same hard floor. The expectedrevenue reduction increases in the soft floor magnitude,and it is the same as if the auction followed 2PSB pricing,but lost one regular bidder whenever exactly one high-valuation bidder entered and all the regular biddershappened to have valuations low enough to bid below thesoft floor.

Note that soft floors reduce expected auctioneerrevenue precisely when they “kick in” to put pricingpressure on a randomly appearing high bidder, thatis, precisely in the situation discussed by soft-flooradvocates (e.g., Weatherman 2013, Sahni 2016). Theadvocates are correct in noting the soft floor addspricing pressure on the high bidder whenever he ispresent; but Proposition 4 shows the coincident bid-shading by low-valuation regular bidders more thanerodes the benefits of the added pricing pressure.

Because the revenue reduction is increasing in sand weighted by the probability of exactly one highbidder entering, the revenue reduction is boundedby the revenue loss from losing one regular bidder.Proposition 4 thus echoes the classic result of Bulowand Klemperer (1996), who show that a revenue gainfrom setting the hard floor optimally in a symmetricmodel is less than the gain from finding one morebidder. Unlike in the Bulow and Klemperer (1996)case in which reserves obviously discourage bidderentry, it is not clear whether having a soft floor mightencourage bidder entry; but if it did, Proposition 4shows that the auctioneer would always prefer add-ing a small soft floor to losing one regular bidder ina 2PSB.

6. Example of Profitable Soft Floors:First-Price Auctions CanRevenue-Dominate Second-PriceAuctions in a Market with a RandomlyAppearing StochasticallyDominant Bidder

The previous section argues that low-enough softfloors hurt expected revenue of the auctioneer. Un-fortunately, the analysis of medium soft floors inthe market with randomly appearing high-valuationbidders defined above is not tractable. However,tractability resumes for very high soft floors in a slight-ly modified but still RTB-relevant market with randomlyappearing stochastically dominant bidders, where a “veryhigh” softfloor is one that never kicks in. Such a softfloor

effectively implements 1PSB in a marketplace origi-nally designed around the 2PSB rule. The soft-floorauction obviously weakly dominates both the 1PSBauction and the 2PSB auction. So a situation in which1PSB strictly revenue-dominates 2PSB is an exampleof a market in which soft floors strictly increase theauctioneer’s expected revenue. This section providessuch an example.Throughout this section, I assume that there is only

one regular bidder (bidder 1) and one randomlyappearing bidder (bidder 2), both have uniformly dis-tributed valuations, and the randomly appearing bid-der who appears with probability α is stochasticallydominant in that v1 ~U[0,1] while v2 ~U[0,M] forsome M≥ 1. The common lower bound of the two val-uation supports simplifies analysis, and the stochasticdominance captures the idea of a “high” bidder (dif-ferently from Section 5, which assumed guaranteeddominance). To provide an example of a market inwhich soft floors increase revenue, this section exhibitsa range of (α,M) parameter values for which the 1PSBpricing rule (and hence also the soft-floor auction)revenue-dominates the 2PSB pricing rule even withrule-optimized hard floors.

6.1. The Optimal Mechanism Favors theRegular Bidder

Before comparing the revenues of the two standardpricing rules, it is useful to consider the optimalmechanism of Myerson (1981), which would allocatethe impression to the bidder with the highest positivevirtual value ψi(vi) � vi − 1−Fi(vi)

fi(vi) . The random partic-ipation of the high bidder does not change his virtualvalue, and the uniform Fi then imply ψ1(v1) � 2v1 −1> 2v2 −M � ψ2(v2). Therefore, the optimal mecha-nism would impose a higher hard floor of h2 � M/2on the high bidder than the optimal hard floor of h1 �1/2 for the regular bidder, and it would level theplaying field further by favoring the regular bidderin picking the auction’s winner. Specifically, whenψi(vi)> 0, it would award the impression to the reg-ular bidder whenever v1 > v2 − M−1

2 . The problemwith such a scheme is obvious: the high bidderwould try to obscure its identity—a behavior called“false-name bidding” by Arnosti et al. (2016). Nev-ertheless, the optimal mechanism suggests when1PSB is likely to have a revenue advantage over2PSB, namely whenever the regular bidder bids ag-gressively in the 1PSB, and thus wins the auctiondespite having a smaller valuation than the highbidder.

6.2. Equilibrium Bidding Under the First-Price RuleBidding in the 2PSB is in dominant strategies, and thederivation of the expected revenue πII and its associated


optimal hard floor h*II are relegated to the appendix.In contrast, the characterization of 1PSB bidding inthe market with a stochastically dominant randomlyappearing bidder is new to the literature, extending theanalysis of Kaplan and Zamir (2012) who derived theα = 1 special case for an arbitrary h, and themselvesextended the result of Griesmer et al. (1967), who derivedthe α = 1 & h = 0 special case. The next propositioncharacterizes the equilibrium inverse bidding functionsin closed form:

Proposition 5. For any h,α ∈ [0,1] and M≥ 1, equilibriumbidding involves both bidders submitting bids in the [h, b]interval, where

b � hM + α(h2 +M − hM)α +M

.

The inverse bidding function of the regular bidder is

v1(b) � h(A + h)A+b+[h(A + h)−(A + b)]

(b − hb − h

)θ(b + h + Ab + h + A

)1−θ,where

A � M(1 − α)α

and

θ � hA + 2h

.

The inverse bidding function of the high bidder is

v2(b) � b[v1(b) + A] − h(h + A)v1(b) − b

.

The proof generally follows the approach of Kaplanand Zamir (2012), adapted for the randomly missingbidder. Figure A.2 in the appendix illustrates the fol-lowing intuitive comparative statics of equilibriumbidding: as α decreases, the support of the bids shrinksdown to h (b approaches h), and most regular biddersbid just above h, effectively banking on the high biddernot showing up. In response, the high bidder becomesless aggressive (bids less for a given value of v) becausethe regular bidder’s behavior presents an opportunityto win very often. As M increases, the support of thebids expands to accommodate the greater gains oftrade on the table, the high bidder becomes more ag-gressive in his attempt to win the gains from trade, andthe regular bidder becomes more aggressive in re-sponse. No closed-form solution exists for the biddingfunctions shown in Figure A.2, but bidding functionsare not necessary for the computation of the expected

1PSB revenue πI , because the expected revenue followsdirectly from the inverse bidding functions vi(b):Lemma 2. E(πI |h<1)� h[1−h(1−α+αh)]+

∫ b

h1−v1(x)[1−α+αv2(x)]dx.

The integral in Lemma 2 does not have a closed formbut can be trivially approximated numerically as asum on a fine grid. Optimizing the hard floor is thenstraightforward. I now turn to comparing the revenue-generating potential of the two pricing rules.

6.3. Comparing the Expected Revenue of the First-Price Rule (1PSB) with the Second-PriceRule (2PSB)

Let the percentage revenue lift from 1PSB versus 2PSB bedefined as follows:

% revenue lift from1PSB versus 2PSB

≡ E(πI |h � h*I) − E(πII |h � h*II)E(πII |h � h*II)

.(7)

The top panel of Figure 5 plots the percentage revenuelift as a function of α, for three qualitatively differentexamples of M: 1, 2, and 3. When M = 1, the bidders’valuations are symmetric, but one bidder participatesrandomly. When M≤ 2, the optimizing seller caters toboth bidders regardless of pricing regime by setting bothhard floors below 1.WhenM= 3, the seller caters to bothbidderswhenα it low, andonly the high bidder otherwise.To aid in understanding the forces underlying the revenue

Figure 5. Revenue Comparison of the Two Pricing Rules forDifferent M, as a Function of α


results, the bottom panel plots the percentage differ-ence in the optimal hard floor, defined as %hard floor

difference ≡ h*I−h*IIh*II

on the same α axis as the top panel.Slowly unpacking the different effects illustrated in

Figure 5 is helpful, starting with the high bidder alwaysbeing present (α = 1): when M = 1, the two bidders aresymmetric, and standard revenue equivalence thus im-plies no difference in revenue. WhenM = 2, we have the“stretch case” of Maskin and Riley (2000)—a situationthey show to favor the 1PSB as long as h = 0. I find 1PSBcontinues to dominate 2PSB in the stretch case even withoptimal rule-specific hard floors: in addition to the ag-gressive bidding by the regular bidder, the optimal 1PSBis alsomore efficient (has lower reserve; see bottompanelof Figure 5). Once M increases to 3, revenue equivalenceunder α = 1 is restored because both pricing rules onlycater to the high bidder and effectively turn from auc-tioning to posted pricing.

Now consider intermediate values of α: Figure 5indicates 1PSB can revenue dominate 2PSB whenenough asymmetry exists in valuations (M > 1), thehigh bidder’s participation is likely enough to make theregular bidder bid aggressively (α high enough), andthe optimal hard floor is low enough to cater to bothbidders (α and M small enough). A goldilocks (α,M)is thus required for 1PSB to dominate 2PSB. One wayto understand the revenue-dominance of 1PSB is viaits better approximation of the optimal mechanismwhenever it induces the regular bidders to bid ag-gressively and effectively favors them in allocation.

Finally, consider very low α: when α = 0, the twopricing rules are revenue-equivalent because they againreduce to posted pricingwith a single bidder.When α issmall but positive, 2PSB outperforms 1PSB for all Mbecause the 1PSB’s b approaches h, whereas 2PSB in-volves a broader range of prices (up to 1). The optimalhard floor for 1PSB rises above that of 2PSB to try andcompensate for the low bids, but it apparently cannotcompensate fully.

To generalize from the three particular levels of Mconsidered in Figure 5, as well as to examine the role ofhard-floor optimization, please see Figure 6, whichdisplays a contour plot of percentage revenue lift from1PSB. One way to summarize the patterns in Figures 5and 6 is as follows:

Summary of Revenue Comparisons. The first-price rulerevenue dominates the second-price rule when α is highenough for the regular bidder to bid aggressively. When thehard floor is optimized, a second condition for the revenuedominance of the first-price rule is the asymmetry beingsmall enough for the seller to cater to both bidders instead ofjust making a take-it-or-leave-it offer to the high bidder.

The result illustrated by the right-hand panel ofFigure 6 extends thefindings ofMaskin andRiley (2000),

who separately analyze the two asymmetries involvedin the construction of the RTB-relevant “high” bidderconsidered here and find that (1) stretching the dis-tribution of bidder 2 relative to the bidder 1’s distri-bution favors the first-price rule, and (2) randomparticipation of bidder 2 (which Maskin and Riley call“shifting of probability weight to the bottom of sup-port”) favors the second-price rule. The right-handpanel of Figure 6 replicates both individual effects asspecial cases (α = 1, M > 1) and (α < 1, M = 1), re-spectively, and shows the effect of a combination of bothasymmetries on the relative profitability of the two pricerules. One way to interpret the joint effect is to concludethe effect of random participation is stronger than theeffect of stochastic dominance by one bidder because the2PSB rule revenue dominates the first-price rule for allα < 0.75 regardless of M, and the revenue loss frompicking the wrong pricing rule is much bigger when the1PSB rule is chosen erroneously than when the 2PSB ruleis chosen erroneously.The left-hand panel of Figure 6 then examines how

the joint effect of “stretching” and random participa-tion changes when the auctioneer optimizes the hardfloor. Two qualitative differences relative to the right-handpanel emerge, each due to the auctioneer effectivelycatering to only one bidder: when the high bidder isunlikely to show up (α low), the auctioneer caters onlyto the regular bidder. When the high bidder is likely toshow up and likely to have a much higher valuationthan the regular bidder (α and M high), the auctioneercaters only to the high bidder. In either situation, norevenue difference exists between the two pricing rules.One important difference between the left and the right

panels in Figure 6 is the magnitude of the revenue lift:when the hard floor is optimized, the absolute differencebetween the two pricing rules is only a few percentage

Figure 6. Percentage Revenue Lift of 1PSB vs. 2PSB in(α,M) Space

Notes. The shaded area indicates the region of the parameter spacefor which 1PSB generates higher expected revenue than 2PSB. Thecontours in the left plot are in 0.01 intervals, and the contours in theright plot are in 0.1 intervals.


points. By contrast, 1PSB can increase profit bymore than30%when no binding hardfloor exists andα is very high,but it can also lose the entire potential revenue when α isvery low. Once the hard floor is optimized for the (α,M)situation, only a small amount of revenue seems to beavailable from also optimizing the pricing rule.

7. DiscussionSoft floors have emerged in the RTB digital displayadvertising industry as a potential tool for increasingpublisher revenues. This paper shows soft floors are notlikely to deliver on this promise in the long runwhen thebidders are ex ante symmetric, even if the auctioneerkeeps the exact soft-floor level hidden from the bid-ders, or when bidders participate in the auction ran-domly. Adding randomly appearing “high” bidders(e.g., retargeting advertisers in the RTB context) to theauction does not automaticallymake soft floors profitableeither: the profitability of soft floors depends on theirmagnitude and on the amount of valuation overlap be-tween regular and high bidders. To illustrate the nuancedprofitability of soft floors in RTB-relevant asymmetricmarkets, this paper provides both an example in whichsoft floors reduce revenue and an example in which theyincrease it. Overall, the paper provides threemain resultsshown in Figure 1. I now discuss the three results in turn,with one paragraph devoted to each of them.

When the bidder valuations are all drawn from thesame distribution (and the bidders are thus “sym-metric” in the auction-theory parlance), low-valuationbidders shade their bids down in response to a softfloor, but the fact that low-bidding winners pay theirbids exactly compensates for the seeming loss of revenue.Soft floors are revenue neutral because the equilibriumbidding function remains monotonically increasing asin the benchmark second-price auction with the samehard floor, and the classic revenue-equivalence result ofMyerson (1981) thus applies. The monotonicity of equi-librium bidding (and hence the revenue equivalence)continues to hold even when the auctioneer hides theexact level of the soft floor before bidding, or whenbidders participate randomly.

Soft-floor advocates often point to a gap betweenthe winning bid and the second-highest bid in RTBauctions and argue the auctioneer can capture some ofthis gap as extra revenue using a softfloor. The symmetriccase discussed in the previous paragraph shows anoccasional random realization of a large gap by a set ofotherwise ex ante similar bidders is not a good argumentfor soft floors. However, the possibility remains that thegap is systematic, arising from the presence of bidderswhose valuation is known to be relatively high. For ex-ample, one of the bidders bidding on a particular im-pression may be a retargeting advertiser whose websitethe customer has just visited. A soft floor might seem to

put pricing pressure on such an “asymmetrically high”bidder while preserving revenues when he happens notto show up at the auction. The second main result of thispaper shows that this intuition is incomplete when thesoft floor is low: the soft floor does indeed add pricingpressure on the asymmetrically high bidders, but thecoincident bid-shading by low-valuation bidders alwaysmore than erodes the benefits of the added pricingpressure. The phenomenon of bid-shading shows thateven in a soft-floor auction, the pricing pressure on theasymmetrically high bidder ultimately stems from thepresence of lower-valuation bidders, more of whomshade their bids down when the soft floor increases. Asa result, soft floors can actually reduce expected auc-tioneer revenue precisely in the asymmetric situationthat motivates their use in the industry.The third main result of this paper is an example of

an asymmetric market with randomly appearing “high”bidders, in which the auctioneer can profit from a softfloor. Unlike in the second result, which considered theeffect of a soft floor that “kicks in” for at least somebidders, the third result only considers soft floors highenough to turn the auction into a 1PSB auction. Maskinand Riley (2000) provide an example of an asymmetricmarketwith one stochastically dominant bidder inwhich1PSB revenue dominates 2PSB. This paper extendsMaskinand Riley’s (2000) revenue-dominance result to the RTB-relevant situation of the stochastically dominant bidderpresent only occasionally and also shows the revenuedominance can survive hard-floor optimization by theauctioneer under some conditions: 1PSB rules can revenue-dominate 2PSB rules as long as the high bidder’s partici-pation probability is high enough and the asymmetry issmall enough for the auctioneer to cater to both bidders.However, the relative revenue advantage of the 1PSBrule is much smaller once the auctioneer optimizes hardfloors. Thus, changing the pricing rule in RTB auctionsseems likely to only lead to large revenue effects if thehard floors are difficult to optimize for some reason. Thisanalysis is orthogonal to other arguments for switchingto the 1PSB rule, such as its increased transparency asargued by Chen (2017) and Moesman (2017), who echothe earlier analysis of Rothkopf et al. (1990).Several directions of future research remain. One po-

tentially fruitful avenue would be to consider the directimpact of soft floors in the model of Section 6: as the softfloor rises toward the point when the auction turns intoa 1PSB auction for all the bidders, what happens tothe revenue advantage of the soft-floor auction over theunderlying simple second-price auction? If the revenueadvantage is decreasing, a case could be made for op-timizing soft floors; but if it increases toward the pointwhen soft floors do not kick in at all, the auctioneerwould have a much simpler choice between the twostandard pricing rules.


Soft floors are not the only proposed novel mecha-nisms in the RTB space: for example, Celis et. al (2014)propose a “buy-it-now or take-a-chance”mechanism toaddress an irregularity in the distribution of biddersarising from random matching with advertisers. Itwould be interesting to analyze how soft floors wouldperform under their assumptions because the idea ofrandom matching is similar in spirit to the randomlypresent high bidders analyzed here. Unlike in this paper,the bidders in Celis et. al (2014) are ex ante symmetric,albeit coming from a mixture distribution. So one canconjecture on the basis of Proposition 3 that soft floorswould not impact revenues under random matching aslong the mixture had a full support and implied a mono-tonic 1PSB equilibrium bidding strategy.

Another extension could address the multiunit re-ality of RTB marketplaces: throughout the paper, Ifocused on a single auction attended by several inde-pendent private-value bidders. However, advertiserslooking to purchase impressions on ad exchanges facea sequence of opportunities to show their ad, and theyoften view these opportunities as substitutes becausethey are budget constrained. Some advocates of softfloors correctly point out that bidding one’s full privatevaluation in a sequential auction for substitutes is notoptimal (e.g., Nolet 2010, Strong 2012). Instead, oneneeds to bid the valuation of winning net of the op-portunity cost of trying again later, and the opportunity-cost calculation needs to take into account equilibriumconsiderations, because the opportunity cost dependson the behavior and types of competing bidders (see,e.g., Engelbrecht-Wiggans 1994, Milgrom and Weber2000). Balseiro et al. (2015) provide a new solutionconcept called “Fluid Mean Field Equilibrium” andstationarity assumptions that together make account-ing for multiple bidding opportunities practical. How-ever, existing theory does not support the advocates’leap of faith that soft-floor or 1PSB auctions somehowresolve this issue, but rather continues to find 1PSB and2PSB are revenue-equivalent even in sequential settingsunder the symmetric model (e.g., Reiß and Schondube2010, Chattopadhyay and Chatterjee 2012). Given theserevenue-equivalence results, one can conjecture that ananalysis analogous to Section 4 of this paper would showsoft floors have no effect on auction revenue even ina sequential-auction model under bidder symmetry. Aninteresting future direction of inquirywould examine softfloors in sequential auctions with asymmetric bidders.

The managerial recommendations of the results pre-sented in this paper are clear. First, soft floors should beeliminated from ad exchanges that do not seem to haveasymmetrically high bidders. Second, low soft floors(i.e., soft floors that kick in for regular bidders) shouldbe avoided, but markets with randomly appearing asym-metrically high bidders might benefit from high soft

floors or even soft floors that never “kick in” as reserves.Third, managers should focus on setting their hard-floor levels correctly given the demand they face, in-stead of worrying about switching pricing rules. Fi-nally, managers should not worry about the “revenuegap” between the top two bids in the second-priceauction identified by Sahni (2016). They should resistthe temptation to somehow monetize the gap and resteasy knowing the winner needs to capture the entiregap as his surplus to continue bidding truthfully indominant strategies, that is, to preserve the clear bid-ding incentives that make the second-price rules de-sirable. One of the most powerful implications ofMyerson’s (1981) revenue equivalence is that this stra-tegic simplicity for bidders does not come at a cost to theauctioneer as long as the bidders are symmetric—thesecond-price auction with a correctly chosen reserve isat least as profitable as any other auction format themanager may wish to implement.

Appendix. Proofs Not Covered in the Main Text

Proof of Proposition 1. The proof proceeds in three steps,establishing the following claims in turn.

Claim 1. Any monotonically increasing equilibrium biddingfunction β(v)with β(M)> smust have a jump discontinuity atthe valuation level v* such that β(v*) � s, such that no bids inthe (s, v*) interval are submitted.

Claim 2. The proposed bidding strategy is a Nash equilib-rium strategy.

Claim 3. The proposed bidding strategy is a unique Nashequilibrium strategy.

Figure A.1. Equilibrium and Deviation Expected Surplusesof the Focal Bidder (N = 3, F=U[0,1])


Proof of Claim 1 (Jump Discontinuity). Suppose a symmetricmonotonically increasing bidding equilibrium β(v) exists suchthat β(M)> s. Let v* be the valuation level such that β(v*) � s.By construction, the v* bidder receives a positive surplus ofPr(win)(v* − s)> 0, so v*> s. Now consider bidders with v> v*,who also bid above s by monotonicity and hence face 2PSBpricing with a reserve of s. By the dominant-strategyproperties of 2PSB, these bidders bid their valuations,that is, β(v) � v for all v> v*. Therefore, the bidding func-tion β(v) must approach v* as v approaches v* from above,resulting in the jump discontinuity:limv→v*+β(v) � v*> s �β(v*). By monotonicity, bidders with v > v* bid above v* andbidders with v < v* bid below s, so no bids in the (s, v*) aresubmitted.

Proof of Claim 2 (The Proposed Bidding Strategy Is a Nash-Equilibrium Strategy). Suppose all N-1 competitors followthe bidding strategy β(v) outlined in the proposition, andconsider a focal bidder with valuation v. It is enough to showthat bidding according to β(v) is the focal bidder’s best re-sponse to those competitors. Three cases depend on themagnitude of v; please see Figure A.1 for an illustration ofthe three cases and all the relevant expected surplusesfor the N = 3 and F = Uniform[0,1] example with h = 0.5 ands = 0.6.

Case 1 (v ≤ s). The bidder can bid below s and guarantee 1PSBpricing should hewin.Winning is only possible if all competitorsalso bid below s, and such competitors follow βI by assumption.Because βI is an equilibrium bidding function in 1PSB, biddingβI(v) is the focal bidder’s best response to the relevant compe-tition. Hence, he canmake a positive expected surplus of πI(v) �G(v)(v − βI(v)) by bidding below s. There is only one nonlocaldeviation to consider: bidding more than s and triggering2PSB pricing. But triggering 2PSB pricing also triggers thesoft floor as the reserve, so the focal bidder will pay at leasts upon winning, which is in turn weakly more than hisvaluation. Therefore, the focal bidder cannot make a positivepayoff by bidding above s, and his overall best response tothe soft-floor auction incentives is to follow the proposedβ(v) and bid below s.

Case 2 (s< v< v*). The bidder can bid below s and guarantee1PSB pricing should he win. The argument presented inCase 1 shows that doing so will yield an expected surplus ofπI(v) � G(v)(v − βI(v)). Also as in Case 1, this strategy domi-nates bidding over v* because v< v*. Alternatively, the biddercan bid in the (s, v*) interval under 2PSB rules. Because nocompetitors bid in the (s, v*) interval (they follow the pro-posed β(v) by assumption), any bid there by the focal bidderwinswhenever all the competitors bidweakly below s, that is,with probability G(v*), and thus triggers the soft floor as thereserve price. The alternative payoff from bidding in the (s, v*)interval is thus π(v) � G(v*)(v − s).

I now prove πI(v) ≥π(v) for all s< v< v*; that is, β(v) � βI(v)is the best response of the focal bidder. By construction,the two payoffs are increasing in v and coincide at v*:πI(v*) � π(v*). However, πI(v) dominates π(v) on (s, v*) atlower valuations because πI(v) has a lower slope: π′

I(v) �G(v)<G(v*) � π′(v). The slope of π(v) is trivial from its formabove, and the slope of πI(v) can either be derived from

Equation (1) or obtained from the standard mechanism-design result that the slope of equilibrium expected sur-plus in a standard auction is the probability of winning.

Case 3 (v ≥ v*). The bidder has three options for obtaininga positive expected surplus:

(a) Bidding over v* triggers 2PSB pricing, and so the bestbid above v* is one’s true valuation. The expected surplus isπII(v) � G(v*)(v − s) + ∫ vv*(v − Y1)dG(Y1).

(b) As in Case 2, bidding in the (s, v*) interval results inan expected payoff of π(v). Bidding his true valuation as in(a) dominates this option because πII(v) − π(v) � ∫ vv*(v − Y1)dG (Y1)> 0.

(c) Bidding below s triggers the 1PSB pricing rule, but thebidder cannot bid the same amount as he would in a 1PSB,because βI(v)> s by construction. Instead, the bidder solvesa constrained optimization problem, finding the best valua-tionw below v* tomimic: maxw≤v*G(w)(v − βI(w)) . It is easy tosee the objective function is increasing on [0, v*]: its derivativein the mimicked type w is D(w; v) ≡ d

dw G(w)(v − βI(w)) �g(w)(v − βI(w)) − G(w)β′I(w). Because the w type is optimizing,the first-order condition D(w;w) � 0 must hold, so g(w)(w−βI(w)) � G(w)β′I(w). Substituting the first-order condition backinto D(w; v) yields D(w; v) ≡ g(w)(v − w)> 0. Therefore, theoptimal type to mimic is v*; that is, the optimal bid weaklybelow s is s, and the expected surplus from bidding it is thesame as π(v). As shown above in (b), this surplus is alsodominated by bidding true valuation.

In summary, the best response of the focal bidder withv≥ v* is to bid his true valuation, that is, to follow the pro-posed β(v).Proof of Claim 3 (Uniqueness). Now consider another mono-tonic equilibrium bidding strategy β(v). From Claim 1 weknow β(v) must have a jump discontinuity at v* such thatβ(v*) � s. Consider bidderswith valuations below v* first: frommonotonicity of β(v), the bidding strategy of any bidder withv< v* depends only on the bidding strategy of bidders withvaluations below v and hence also below v*. Therefore, allbidders with v< v* effectively face 1PSB incentives. It is wellknown that βI(v) is the unique equilibrium bidding functionof 1PSB, so β(v) must coincide with βI(v) below v* unless thebidders can profitably deviate nonlocally to bid above s.However, the above derivation of the deviation payoff π(v) inCase 2 holds for any monotonic bidding function such that nobids in the (s, v*) are submitted, so it also holds for the presentβ(v), and the above argument in Case 2 implies the nonlocaldeviation to bidding above s by bidders with v< v* cannot payoff. It follows that β(v) � βI(v) for v< s, and so v* � v* � β−1I (s).Now consider the bidders with valuations above v* � v*, whoeither bid their valuation (as implied by the properties of 2PSB)or deviate nonlocally below v*. Because we have establishedthat β(v)must coincidewith βI(v) below v*, the above argumentin Case 3 implies β(v) � v for v>v*, and so β(v) must coincidewith β(v) on the entire support of F. In other words, β(v) is theunique symmetric Nash-equilibrium bidding strategy.Q.E.D. Proposition 1.

Proof of Proposition 2. Consider one bidder with valuationv and suppose all N-1 of his competitors bid according to


some increasing bidding function β(v). The focal biddersolves

maxb

G(β−1(b))︸��︷︷��︸Pr(win)

[Ω(b)(v − E[max{s, β(Y1)}|s< b&Y1 < β−1(b)])︸��︷︷��︸

b>s→ 2PSB

+ (1 −Ω(b))(v − b)︸��︷︷��︸s>b→ 1PSB

].

(A.1)

The E[max{s, β(Y1)}|s< b&Y1 < β−1(b)] term, which cap-tures price paid whenever the bid exceeds the soft floor,seems rather complex at first but simplifies to

E[max{s, β(Y1)}|s< b&Y1 < β−1(b)]� b −

∫ b

h

∫ b

s

G(β−1(x))G(β−1(b)) dxd

Ω(s)Ω(b). (A.2)

I now prove the above simplification: write the expectedpayment as a double integral:

E[max{s,β (Y1)}|s<b&Y1<β−1(b)]

�∫ b

0

1G(β−1(b))

[∫ β−1(s)

0sdG (Y1)+

∫ β−1(b)

β−1(s)β(Y1)dG (Y1)

]dΩ(s)Ω(b).

(A.3)

The material in the square bracket simplifies as follows:∫ β−1(s)

0sdG (Y1) +

∫ β−1(b)

β−1(s)β(Y1)g(Y1)dY1

� G(β−1(s))s + G(β−1(b))b − G(β−1(s))s−∫ β−1(b)

β−1(s)G(Y1)β′ (Y1)dY1

� G(β−1(b))b −∫ b

sG(β−1(x))dx,

where the second line follows from the first line using in-tegration by parts and a subsequent change of variablesx � β(Y1). Plugging the simplified material into Equation (A.3)yields Equation (A.2). In words, Equation (A.2) shows the ex-pected payment of a winner who randomly faces a soft floorbelow his bid b involves a discount below b (the inside integral),which in turn depends on the realized s: Given Equation (A.2),Equation (A.1) simplifies to

maxb

G(β−1(b))(v − b)︸��︷︷��︸1PSB surplus

+∫ b

h

∫ b

sG(β−1(x))dxdΩ(s)︸��︷︷��︸

2PSBdiscountwhen s<b

. (A.4)

The first-order condition of the bidding problem is

g(β−1(b)) 1β′(β−1(b)) (v − b)︸��︷︷��︸

winningmore often

− G(β−1(b))︸��︷︷��︸payingmore ina simple 1PSB

+ G(β−1(b))Ω(b)︸��︷︷��︸larger discountwhen s<b

� 0,(A.5)

where the first two terms are the same as in a textbook solutionof a 1PSB problem, and the third term arises from the hiddensoft floor. In a symmetric equilibrium, it must be that b � β(v),and so the equilibrium bidding function must satisfy thedifferential equation in Equation (4). The differential equationdoes not have a closed-form solution, but the Peano exis-tence theorem implies a solution exists whenever the right-hand side (RHS) of Equation (4) is continuous in (v, β), forwhich a sufficient condition is that g(y) � (N − 1)f (y)FN−2 (y)and Ω(v) are continuous. Q.E.D. Proposition 2.

Proof of Lemma 1. First consider the incentives of a regularbidder: if the high bidders indeed bid their valuations as sug-gested by Proposition 1, the regular bidders assume they canonlywinwhen no high bidder is present, and so they behave thesame as in an auction without high bidders. A deviation by thehighest regular bidder v= 1 to a bid above L thatwould competewith the high bidders is not profitable, because it only changesthe outcome of the game when it actually beats a high bidderand so results in a payment above L thatmust exceed the regularbidder’s valuation by construction. Because the highest regularbidder does not deviate, neither do other regular bidders.

Second, consider the incentives of a high bidder whohappens to participate andwho should bid his valuation underthe proposed equilibrium. The only nontrivial deviation Ineed to analyze is bidding s or below to trigger 1PSB pricing,resulting in winning much less often but also paying less.Given the high bidder’s valuation level, bidding exactly s is thebest such deviation from all possible bids weakly below s (seeCase 3(c) in the Proof of Proposition 1 for a mathematicalargument for why s is the best deviation from all possible bidsweakly below s). This deviation is not profitable, because itforegoes both the positive surplus available by possiblybeating the other high bidders should they also participate andthe positive surplus available by beating high-valuation reg-ular bidders should the other high bidder stay out:

E[surplus | bid � v] � (1 − α)K−1︸��︷︷��︸other highbidders out

[v − E(price | s)]︸��︷︷��︸

win for sure, pay eitherX1 ifX1>v*, s otherwise

+ [1 − (1 − α)K−1]︸��︷︷��︸

at least 1otherhigh bidder in

E[surplus | other(s)in]︸��︷︷��︸

→win if present high bidder(s)have valuations below v

︷��︸︸��︷> 0 byproperties of 2PSB

> (1 − α)K−1︸��︷︷��︸other highbidders out

FN(v*)(v − s)︸��︷︷��︸X1<v*→ s kicks in 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣+ (1 − FN(v*))[v − E(X1 |X1 > v*)]︸��︷︷��︸

X1>v*→payX1

︷��︸︸��︷> 0 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦> (1 − α)K−1FN(v*)(v − s) � E

[surplus|bid � s

]. (A.6)

In words, bidding v yields an additional expected surpluscompared with bidding (and paying) s. The additional ex-pected surplus arises from winning more often, namely,when the other high bidder(s) is (are) out and the highestregular competitor is above v*, and also from the 2PSB auc-tion that results when the other high competitor(s) is (are) inbut his (their) valuation is below v. Q.E.D. Lemma 1.


Verification of Revenue Equivalence When Soft FloorIs Common Knowledge and Participation Is RandomLet N = 2 in Section 4.2. The 1PSB equilibrium biddingfunction is the following, easily derived by substitutingG(v) � (1 − α) + αF(v) in Equation (1) to account for the in-creased probability of winning:

βα(v) � h(1 − α) + αF(h)(1 − α) + αF(v) +

α

(1 − α) + αF(v)∫ v

hzdF(z). (A.7)

The auctioneer’s expected revenue depends on how many, ifany, bidders are present:

Π(h, s) � α2︸︷︷︸Pr(2 bidders)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩Pr(h≤X1 < v*)EX1

[βα(X1)|h≤X1 < v*

]︷��︸︸��︷highest bid below soft floor → 1PSBwith reserve of h

+Pr(X1 > v*)EX2

[sPr(βα(X2)< s)

+X2Pr(βα(X2)> s)|X1 > v*]︸��︷︷��︸

highest bid above soft floor → 2PSBwith reserve of s

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭+ 2α(1 − α)︸��︷︷��︸Pr(only 1 bidder)

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩Pr(h≤X< v*)EX[βα(X)|h≤X< v*

]︷��︸︸��︷the only bid below soft floor → 1PSBwith reserve of h

+ Pr(X> v*)s︸��︷︷��︸the only bid above soft floor→ price is the reserve s

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭,(A.8)

whereX1 is the highest of two valuations distributed ~ F2, andwhere X2 is the second-highest of two valuations, and itsdistribution conditional on X1 � x is just F( · )

F(x): the highestof N-1 = 1 draws from F conditional on the draws beingbelow x. Plugging these distributions of the order statisticsinto Equation (A.8), the auctioneer’s expected revenuebecomes

Π(h, s) � α2∫ v*

hβα(x)dF2(x)

+ α2∫ 1

v*

[sF(v*)F(x) + 1

F(x)∫ x

v*zdF(z)

]dF2(x)

+ 2α(1 − α)∫ v*

hβα(x)dF(x)

+ 2α(1 − α)s[1 − F(v*)].(A.9)

Collecting terms yields

Π(h, s)2α

�∫ v*

hβα(x)

[(1 − α) + αF(x)]dF(x)+∫ 1

v*

[s[(1 − α) + αF(v*)] + α

∫ x

v*zdF(z)

]dF(x).

(A.10)

Now note that βα(v*) � s, so one can substitute for s usingEquation (A.7) as follows:

s[(1 − α) + αF(v*)]� h[(1 − α) + αF(h)] + α

∫ v*

hzdF(z).

After this substitution, note the integrand in the second in-tegral (from v* to 1) is exactly the same as the integrand in thefirst(h to v*), and so Equation (A.10) simplifies to

Π(h, s) � 2α∫ 1

h

{h[(1 − α) + αF(h)] + α

∫ x

hzdF(z)

}dF(x)

� Π(h),(A.11)

where the last equality emphasizes that the s has no impact onΠ, because neither it nor v*are present. To seewhy the expectedrevenueΠ(h) is exactly the same as in the second-price auctionwith the same reserve, rearrange Equation (A.11) as follows:

Π(h) � h {2α(1 − α)[1 − F(h)]︸��︷︷��︸Pr(1 bidderwith v>h)

+ α22F(h)[1 − F(h)]︸��︷︷��︸Pr(2 bidderswith vi<h<v−i)

}+ α2︸︷︷︸

Pr(2 bidders)

∫ 1

h2z[1 − F(z)]dF(z)︸��︷︷��︸Pr(X2>h)E(X2 |X2>h)

,

where the∫ 1

h2z[1 − F(z)]dF(z) � Pr(X2 > h)E(X2 |X2 > h)

because the pdf of a minimum of two draws is 2z[1 − F(z)].This concludes the direct proof of revenue equivalencewhen two randomly participating symmetric bidders arepresent.

Proof of Proposition 4. Equation (6) derives the formula for

the difference in profits ΔΠ(v*) ≡ ΠA2PSB(h)−ΠA(h, s)Kα(1−α)K−1 . It is enough to

Figure A.2. 1PSB Bidding Strategies with One Regular andOne Randomly Present High Bidder

Note. Bidding functions implied by Proposition 5 for two values ofMand three values of α, with the x-axis corresponding to the support ofthe high bidder’s valuations, h= 0.5 throughout, and the y-axisidentical in all six subplots.


show ΔΠ is positive for all v* > h and increasing in v*.Omitting the asterisk on v for clarity, plug in the distributionsof X1 and Y1 in terms of F:

ΔΠ(v) � FN(v)⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣h F

N(h)FN(v) +

1FN(v)

∫ v

hxdFN(x) − h

FN−1(h)FN−1(v)

− 1FN−1(v)

∫ v

hxdFN−1(x)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦�∫ v

hxdFN(x) − F(v)

∫ v

hxdFN−1(x)

− hFN−1(h) ( F(v) − F(h)).Therefore, ΔΠ(h) � 0. To show ΔΠ(v)> 0 and ΔΠ′(v)> 0 forv > h, it is enough to show ΔΠ′(v)> 0 for all v > h:

ΔΠ′(v) � f (v)⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣vNFN−1(v) −

∫ v

hxdFN−1(x)

− F(v)v(N − 1)FN−2(v) − hFN−1(h)⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

� f (v)[vFN−1(v) − (FN−1(v)− FN−1(h))E(Y1 |h<Y1 < v) − hFN−1(h)]

� f (v){ [FN−1(v) − FN−1(h)][v − E (Y1 |h<Y1 < v)]︸��︷︷��︸

>0

+ FN−1(h)(v − h)}> 0,

where the last line follows from the previous expression byadding and subtracting f (v)vFN−1(h). Q.E.D. Proposition 4.

Details of Section 6:ExpectedRevenueUnder theSecond-Price Rule (2PSB) in the Market with One Regular Uni-form Bidder and One Stochastically Dominant RandomlyAppearing BidderThe expected revenue πII of a 2PSB auction is straightfor-ward to derive because the bidders bid their valuations asa dominant strategy. As a function of h, the expected reve-nue function is not necessarily globally concave, because anyh > 1 excludes the regular low bidder, effectively acting asa posted price for the high bidder. The expected revenuefrom such a high hard floor is obviously E(πII | h> 1) � αh(M−h)

M ,so the optimal high hard floor is hHII ≡ M

2 , which exceeds 1whenever M≥ 2 and yields the expected revenue ofE(πII | hHII ) � αM

4 . Alternatively, the seller can select an h≤ 1,which engages the regular bidder. The resulting ex-pected revenue is as follows:

Claim 4.

E(πII |h≤1) � α

6M(3M− 1+ 3h2(1+M) − 8h3) + (1−α)h(1− h).

(A.12)

Proof of Claim 4. When the high bidder does not enter, therevenue is h(1 − h) because the low bidder wins and pays the

reserve price whenever his valuation is above the reserveprice—a probability of 1 − h. When the high bidder doesenter, four distinct revenue regions of the (v1, v2) space exist:(1) With probability h2

M, maxvi < h, and so the revenue is zero.

(2) With probability h(M−h)+h(1−h)M , vi < h< v−i, and the reve-

nue is thus h.(3) With probability (M−1)(1−h)

M , h< v1 < 1< v2, so the highbidder wins for sure and pays the expected conditionalvaluation of the low bidder E(v1 |v1 > h) � 1+h

2 .(4) With probability (1−h)2

M , h< vi < 1, so the bidding com-petition is as if two iid bidders from Uniform[h,1] exist, andthe relevant revenue is thus E(minvi |h< vi > 1) � 1+2h

3 .Combining the above four cases with the possibility of the

high bidder staying out yields Equation (A.12). This con-cludes the proof of the Claim.

The optimal hard floor below 1, denoted hLII , optimizes theexpected revenue in Equation (A.12). The solution is in closedform because the first-order condition of max

h≤1E(πII |h≤ 1)

is quadratic. It is possible to show that hLII < 1⇔α< M2M− 3.

Therefore, α> M2M−3 is a sufficient condition for E(πII |hHII )>

E(πII |hLII) because it implies E(πII |h< 1) is increasing in h onthe [0,1] interval. However, it is clearly possible forE(πII |hHII )>E(πII |hLII) even when hLII < 1—the hLII may only bea local minimum.

The globally optimal hard floor h*II is clearly hLII or hHII ,depending on which leads to higher expected revenue. I nowshow h*II � hLII for all α as long asM is small, namely as long asM3 − 21M2 + 51M − 15> 0⇔M<≈ 2.4, above which pointhHII becomes globally optimal for high-enough α. To see thecondition, note first that hLII > 1 /2= the hard floor optimalwithout the high bidder. For any 1

2< h< 1, the differenceE(πII |h< 1) − αM

4 decreases in α. Therefore, for a given M, theoptimal hard floor is below 1 for all α as long as E(πII |hLII ,α � 1)>M

4 . When α � 1, hLII � 1+M4 , the associated revenue is

M3+3M2+51M−1596M , which exceeds M

4 whenever M3 − 21M2+51M − 15> 0.

Proof of Proposition 5. Following Kaplan and Zamir (2012),let vi(b) be the inverse bidding function of bidder i and lookfor a pure-strategy equilibrium whence both bidders bidinside an interval [h, b], with initial conditions vi(h) � h. Thetwo bidders solve the following optimization problems,respectively:

maxb≥h

[α

(v2(b)M

)+ 1 − α

](v1 − b)

maxb≥h

[v1(b)](v2 − b),(A.13)

where the square brackets are the probabilities of winningwith a bid of b. For vi > h, the b≥ h constraints are not binding,and so first-order conditions are necessary for optimality.The first-order conditions imply the two inverse biddingfunctions must satisfy the following system of differentialequations:

αv′2(b)(v1(b) − b) � αv2(b) +M(1 − α)v′1(b)(v2(b) − b) � v1(b).

(A.14)


Multiplying both sides of the second equation by α andadding to the first equation yields

v′2(b)v1(b) + v′1(b)v2(b)︸��︷︷��︸[v2(b)v1(b)]′

� bv′1(b) + bv′2(b) + v1(b) + v2(b)︸��︷︷��︸[b(v1(b)+v2(b))]′

+ A, where A � M(1 − α)α

.

(A.15)

Integrating both sides yields the following relationship betweenthe two functions, up to a constant B:

v2(b)v1(b) � b[v1(b) + v2(b)] + bA + B. (A.16)

Because v1(h) � h, the constant B must satisfy B � −h(h + A).Plugging this expression for B into Equation (A.16) and rear-ranging yields v2(b) in terms of v1(b):

v2(b) � b[v1(b) + A] − h(h + A)v1(b) − b

. (A.17)

Finally, plug the v2(b) in Equation (A.17) into the first-ordercondition of the high bidder (the second equation in 21), andobtain a differential equation that includes only the v1(b):

v′1(b)(b − h)(b + h + A) � v1(b)[v1(b) − b]. (A.18)

To solve Equation (A.18), divide both sides by v21(b)(b−h)1+θ(b + h + A)2−θ and rearrange to obtain

v′1(b)v21(b)(b − h)θ(b + h + A)1−θ +

bv1(b)(b − h)1+θ(b + h + A)2−θ

� 1(b − h)1+θ(b + h + A)2−θ.

(A.19)

Observe that when we set θ � hA+2h, the LHS of Equation

(A.19) can be expressed as a derivative:

v′1(b)v21(b)(b − h)θ(b + h + A)1−θ +

b + [Aθ − h(1 − 2θ)]︷��︸︸��︷0

v1(b)(b − h)1+θ(b + h + A)2−θ�( −1v1(b)(b − h)θ(b + h + A)1−θ

).

(A.20)

The RHS of Equation (A.19) integrates, up to a constant, as∫1

(b − h)1+θ(b + h + A)2−θ db

� −1(b − h)θ(b + h + A)1−θ

(A + b

h(A + h))+ const. (A.21)

Combining Equations (A.20) and (A.21) thus yields the fol-lowing solution of the differential Equation (A.18) up to someconstant C:

v1(b) �[ (A + b)h(A + h) + C(b − h)θ(b + h + A)1−θ

]−1. (A.22)

The fact that v1(b) � 1 fixes the constant to C �1− (A+b)

h(A+h)(b−h)θ(b+h+A)1−θ , so the solution becomes

v1(b) � h(A + h)A + b + [h(A + h) − (A + b)]

(b − h

b − h

)θ(b + h + A

b + h + A

)1−θ .

(A.23)

Because v1(b) � 1 and v2(b) � M, Equation (A.17) fixes the

upper support of bidding to b � M+h(h+A)1+M+A � hM+α(h2+M−hM)

α+M .Observe that limα→0 b � h, so limα→0 E(πI) � h(1 − h). Q.E.D.Proposition 5.

Proof of Lemma 2. When h < 1, the distribution G of sellerrevenue πI is

G(x) ≡ Pr(πI < x) � Pr(β1(v1)< x)[1 − α + αPr(β2(v2)< x)]� v1(x)[1 − α + αv2(x)].

(A.24)

The expected revenue then follows from G by a single in-tegration over possible revenue levels:

E(πI |h< 1) �∫ b

h

xdG(x) � −∫ b

hx(1 − G(x))′dx

� −b(1 − G(b) )︸��︷︷��︸ 0+ h (1 − G(h))︸��︷︷��︸

Pr(vi<hor high out)+∫ b

h1 − G(x)dx

� h[1 − h(1 − α + αh)] +∫ b

h1 − v1(x)[1 − α + αv2(x)]dx.

(A.25)Q.E.D. Lemma 2.

Endnotes1The use of soft floors is widespread. At least AdX, OpenX, andAppNexus exchanges have used them in the U.S. market. Whenused, they tend to affect many transactions: Yuan et al (2013) analyzea bidding platform on which more than half of the transactions involvea price equal to the winner’s bid, that is, an active soft floor.2Note that randomly present high-valuation bidders are just onepossible asymmetry. This paper does not attempt to characterize theimpact of soft floors in all possible asymmetric markets, but insteadfocuses on a particular asymmetry used in the industry to justify thesoft-floor practice.

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