Auctions, Adverse Selection and Internet Display Advertising. By Marissa Beck and Paul Milgrom1. This version: August 8, 2012. Abstract. We study the adverse ...
Auctions, Adverse Selection and Internet Display Advertising By Marissa Beck and Paul Milgrom1 This version: August 8, 2012 Abstract. We study the adverse selection problem that arises in sales of online display advertising to a mix of brand advertisers and betterinformed performance advertisers. We characterize a new class of auction mechanisms that insulate brand advertisers from adverse selection and a subclass of mechanisms that are immune to manipulations by shills. For the case of the power law distribution of match values, this subclass contains a single auction, which awards the impression to a brand advertiser when the ratio of the highest to second-highest bid among performance advertisers is low. This new auction leads to sometimes very large Pareto gains over current practice. Imagine the problem and opportunity facing the owner of a big city newspaper as its readers shift to using its website, rather than its print edition, for news and other content. Traditionally, much of the newspaper’s ad revenue comes from its contracts with brand advertisers, who seek to show their ads to a cross section of the paper’s readers. With an aim to build brand awareness and to associate their brand with a reputable news site, these advertisers typically evaluate opportunities based on reach-and-frequency goals. But there is another, increasingly important class of advertisers in the electronic medium who prefer to place their ads differently. These performance advertisers typically advertise on 1
Both of the Departement of Economics, Stanford University. We thank Xiaoning Liu, Josh Mollner, and Evan Storms for helpful comments.
1
many different websites and target their ads to narrow and highly relevant groups of consumers – for example, ones who have recently searched for flights to San Francisco, or read online articles about home refinancing, or browsed online stores selling high-fashion women’s clothing. Individual performance ads are most valuable when targeted at customers who are already motivated and ready to purchase the right product soon afterwards. Auctions for display advertising have emerged on the Internet as a means to allow these performance advertisers to bid anonymously for their relevant customers.2 If brand advertisers participate in these same auctions without the same selectivity, then adverse selection is a potential problem because many of the customers targeted by performance advertisers have high incomes and high responsiveness to advertising, making them valuable to brand advertisers as well. This correlation among values for brand and performance advertisers challenges the publisher to devise a trading mechanism that both enables effective matching of ads to consumers for performance advertisers and protects brand advertisers from adverse selection. The traditional approach to selling display ads has avoided adverse selection by treating the demand of performance advertisers as a remnant, with auctions only for impressions that have not already been sold to brand advertisers. In our model, this is represented by randomly setting aside a fraction λ of all
2
Performance advertisers and ad auctions are best known in search advertising, which refers to the ads displayed by search engines along with the so-called “organic” search results. Search ad auctions have been modeled by Edelman, Ostrovsky and Schwarz (2007), Varian (2007), and Athey and Ellison (2010). Our focus in this paper, however, is on display advertising, which (unlike search advertising) is widely used by brand advertisers.
2
impressions for the brand advertisers.3 These random set-asides can also be implemented with an auction in which the publisher uses a proxy bidder to bid on behalf of its brand advertisers, randomizing between very high bids (placed with probability λ) and bids of zero (placed with probability 1–λ). The high bids always win, the zero bids always lose, and the resulting assignment gives an adverseselection-free λ-share of the impressions to brand advertisers. Further developing this idea of random set-asides within an overall auction system, Yahoo researchers (Ghosh, McAfee, Papineni and Vassilvitskii (2009), McAfee, Papineni and Vassilvitskii (2010)) have devised a sophisticated method to randomize bids on behalf of brand advertisers in a second-price auction. The method allows a brand advertiser to reduce its average price by including intermediate bids in the support of the randomization. For any average price, the system identifies how to get the most representative selection of impressions subject to spending no more than some target, where “most representative” means that it matches the distribution of highest performance bids in the brand advertiser’s sample as closely as possible to that of the overall sample. Any use of intermediate bids, however, necessarily re-introduces adverse selection. The only adverse-selection-free version of the Yahoo system is the set-aside auction, which assigns impressions to brand advertisers without regard to the peformance advertisers’ bids. That system, however, can miss opportunities to match performance advertisers with their most motivated 3
We assume that the allocation of impressions among the brand advertisers is treated separately and algorithmically, so that there is at most one brand advertiser that has been identified in advance to compete with performance advertisers for each impression.
3
customers. In our model below, the financial loss from those missed opportunities is shared between the advertisers, who earn lower profits on that account, and the publisher, who earns lower revenues. Are such losses avoidable or are they inevitable in any auction mechanism that prevents adverse selection? What is the whole class of adverse-selectionfree, strategy-proof mechanisms? In terms of efficiency, which one is the best? Are any of these adverse-selection-free mechanisms strategy-proof even in the context of Internet display advertising, where anonymous bidders could potentially submit multiple bids? What are the welfare properties of such mechanisms? Compared to set-asides, in what circumstances do they raise or reduce the publisher’s revenue, the performance bidders’ profits, and the welfare of the readers of the publication? Are these effects large? This paper offers a model that allows all of these questions to be analyzed. In our model, a brand advertiser has already contracted with a publisher for some proportion of the available impressions. Performance advertisers, who are better informed than the publisher and brand advertiser, have values for ad opportunities that are determined by mulitiplying two variables. The first, a private match-quality variable, quantifies the intensity of the consumer’s current interest in an advertiser’s products, which may vary over time, and the second is a variable quantifying consumer characteristics that all advertisers value. This second, common value variable may also be correlated with the number of bidders in the auction. In our model, the problem of achieving a good match while avoiding adverse selection can be roughly described as one of distinguishing
4
cases where all performance bids tend to be high (due to a high common value) from ones in which a single performance bid is very high (due to a high individual match value). Our analytical approach is axiomatic. Based on current practice, we identify a set of desirable properties that any acceptable auction should have. Specifically, we require that the auction rules should be deterministic, strategy-proof,4 and anonymous (among performance advertisers),5 and they should take payments only from winning bidders. Qualified mechanisms are ones that satisfy these properties and, in addition, are free of adverse selection. We report several results. First, we describe the set of qualified auction mechanisms. We show that for every qualified mechanism, there is some homogeneous-of-degree-one pricing function such that the highest performance bidder wins and pays the corresponding price when its bid exceeds that price. Next, we introduce the additional requirement that a qualified auction should be false-name proof (Yokoo, Sakurai and Matsubara (2004)), that is, immune to deviations in which a bidder can submit multiple bids. We give a necessary and sufficient condition for qualified auctions to be false-name proof. The bulk of the value of performance ads for any advertiser seems to come from a careful matching of a few ads to a small number of impressions, when particular buyers appear ready to purchase the specific product or service being
4
For advertisers accustomed to bidding in second-price auctions, switching to an unfamiliar auction design with new strategic properties could create large additional participation costs. However, if the new auction were strategy-proof, the additional costs would be minimal.
5
In this paper, anonymous always means as among performance bidders, so that permuting the performance bidder indices in the mechanism permutes the outcome indices in the same way.
5
advertised. This pattern of values suggests modeling match values with a fattailed distribution, of which the proto-typical example is the power law. This logic leads us to feature power law distributions. We show that for these distributions, there is a unique qualified auction that is false-name proof: the modified secondbid auction, in which the highest performance bidder wins if and only if the ratio of its bid to the second-highest performance bid is sufficiently large. If the ratio is low, then the impression is awarded to the brand advertiser. The power law assumption is useful for the derivation of several additional results. The first is that the modified second-bid auction multiplies the expected payoffs of the performance bidders and the publisher/seller by the same amount, so it yields a Pareto improvement over the set-aside auction and shares the gains proportionately. Second, the peformance multiplier can be very large: it goes to infinity as the fraction of performance ads goes to zero. Third, the requirement that the auction be false-name proof is expensive: if one can eliminate manipulations by shills, then the qualified auction that achieves the highest expected total value is the one in which the highest performance bidder wins when the ratio of its bid to the lowest performance bid is sufficiently large. Because the power law assumption is so special, we also establish that there is a larger class of distributions for which modified second-bid auctions perform better than random set-asides. The distributions have a particular statistical property, namely, that a high ratio of the maxium to second-highest order statistic is associated with a high value of the maximum. In such cases, for all λ , the modified second-bid auction has higher total expected value than random set-
6
asides. For the case in which match values are idependent and identically distributed, we characterize this property of order statistics in terms of the log concavity of the distribution. We also give conditions on hazard rates that ensure the modified second-bid auction gives both higher publisher revenues and higher performance bidder profits. The rest of this paper proceeds as follows. Section I introduces our model. Section II characterizes the mechanisms that satisfy our axioms both in the general case and for a power law distribution of match values. Section III relaxes the requirement of false-name proofness and investigates the most efficient mechanisms satisfying the remaining properties. Section IV shows that the modified second-bid auction provides a Pareto improvement over the set-aside auction for the power law case and characterizes a larger class of distributions with this Pareto improvement property. Section V concludes.
I. Model Our model has one brand advertiser, whom we call advertiser 0, and N performance advertisers, N ≥ 2 , whom we label 1,2,…,N .6 The publisher has a contract which, in exchange for a particular payment, requires it to deliver a fraction λ ∈(0,1) of the impressions to the brand advertiser.7 Each impression has a value to advertiser i given by xi = cmi where c is a common random factor 6
Having N ≥ 2 performance advertisers eliminates the need for a reserve price in a second-price auction, which simplifies our analysis. Binding reserve prices in online auctions typically result in the placement of “house ads,” in which the publisher advertises another part of its website to the consumer, trying to attract a visit. The house ad bid might be regarded as the first bid by a performance bidder for each ad impression. The condition N ≥ 2 then means that, in addition, there is at least one “real” performance bidder.
7
Actual advertising contracts typically guarantee delivery of a fixed number of impressions. We simplify the analysis by using this alternative representation of the contract requirement. In practice, λ can be varied adaptively to ensure that the guarantees are satisfied with high probability.
7
that scales the value to every advertiser (including the brand advertiser) and mi is a random matching factor that scales the value of the impression differently for the different advertisers. Each performance advertiser is a risk-neutral, expectedprofit maximizer and knows its own value xi , but does not know the individual components c and mi .8 The brand advertiser has no information about x0 and so does not participate actively in the mechanism. The multiplicative specification plays a critical role in our formal analysis. It is most appropriate for modeling a situation in which the variable c represents a consumer’s propensity to notice and respond to advertising, and it might be stretched to include cases in which c also depends on consumer income. The latter interpretation treats income as a proxy for expected demand: the probability of a purchase times its expected size. This interpretation is stretched because the income elasticities of demand vary among advertised products and can even be negative (think of pay-day loans). Still, the multiplicative specification captures the possibility of adverse selection and lends tractability to our analysis. The number of performance bidders, N, is a random variable that may be correlated with the common value factor c. We assume that there is a random vector m = (m1,...,mN ) of performance matching values with a joint distribution having a continuous density on ℜ N+ . We further assume that c is distributed on ℜ + and normalized so that E[c] = 1 and that, conditional on N, the factors m and
8
This assumption rules out mechanisms in which all bidders report the values separately and the auctioneer can costlessly learn c from the reports.
8
c are statistically independent. We treat m0 as a constant, reflecting the idea that the brand advertiser has no known matching preference except to attract individuals with high values of c. A direct mechanism in this setting is a pair (z,p) in which z specifies the probability that any advertiser i receives the impression and p specifies expected payments by each advertiser. A direct mechanism has the four properties that for
{
}
all N ≥ 2 and all possible reported values xˆ1,…, xˆN , (1) for all i ∈ 0,…,N ,
(
)
{
}
(
)
zi xˆ1,… xˆN ≥ 0 , (2) for all i ∈ 1,…,N , pi xˆ1,… xˆN ∈ℜ , (3)
N
∑ z ( xˆ ,…, xˆ i=0
i
1
N
) = 1, and
(4) for all i=1,…,N, zi ( xˆ1,…, xˆN ) must be left-continuous in xˆi .9,10 We will consider two mechanisms to be equivalent if they lead to the same outcomes with probability one.
II. Axiomatic Characterization Our main objective in this section is to characterize a particular set of “qualified” advertising auctions that have certain useful properties for applications to online advertising. Roughly, the qualified auctions are mechanisms that take reports from performance advertisers, are anonymous, deterministic, strategy-
9
Since we will look for strategy-proof mechanisms, the revelation principle allows us to restrict attention to direct mechanisms. The technical requirement that zi be left-continuous is not standard, but simplifies the statements and proofs of certain necessity results below.
10
Formally, the domain of the mechanism is
∞ N=2
ℜN+ .
9
proof, and free of adverse selection, require no payments to or from losing bidders, and satisfy E ⎡⎣ z0 (x1,...,xN ) ⎤⎦ = λ .11 For advertising auctions, all of the listed properties are desirable ones. Deterministic mechanisms that entail no transfers to or from losing bidders are easier to audit than alternative mechanisms, reducing fraud.12 Strategy-proofness reduces bidders’ costs of participating and bidding. For brand advertisers, freedom from adverse selection offers protection and makes their results more predictable, reducing their costs of performance monitoring and contracting. In many ad markets, performance advertisers use advertising agencies that conceal their identities, which undermines the effectiveness of non-anonymous mechanisms. Finally, E ⎡⎣ z0 (x1,...,xN ) ⎤⎦ = λ ensures the publisher fullfils its contract with the brand advertiser. As a step toward characterizing the mechanisms that comply with the listed properties, we first describe the class of direct mechanisms in this setting that are strategy proof and adverse-selection free.
11
Actual advertising contracts typically provide for delivery of a fixed number of impressions, not a random number with a fixed expectation as formulated here. We adopt our formulation because it leads to a simpler formal analysis.
12
Payments by non-winning bidders could encourage publisher fraud in which the publisher places high bids for imaginary ad opportunities or ones reserved for house ads. Outside advertisers would then still be required to pay, without having access to the impression to verify that it exists and is described correctly. As in many other markets, advertisers typically pay only for what is delivered because that gives them a better opportunity to verify quality. Similarly, transfers to losers encourage fraud by bidders who are not serious advertisers, but nevertheless place bids to collect payments from the seller. The property that losers neither make nor receive payments can be imposed without loss of optimality in devising revenue-maximizing (“optimal”) auctions in some settings, such as that of Myerson (1981).
10
Definition 1. A mechanism (z,p) is strategy proof if
xi ∈argmax xi zi ( xˆi ,x− i ) − pi ( xˆi ,x− i ) xˆi
{
}
∀N ≥ 2, i ∈ 1,…,N , x ∈ℜN+
For a strategy-proof mechanism, no matter what the other advertisers report, each performance advertiser prefers to reveal his true value for the impression. In our model, the brand advertiser makes no report and so we do not apply the truthful reporting constraint to advertiser 0. Definition 2. A strategy-proof mechanism (z,p) is adverse-selection free if, conditional on any N ≥ 2 , the random variables z0 (x1,…,xN ) and c are statistically independent and if E ⎡⎣ z0 (x1,...,xN ) | N ⎤⎦ equals the same fixed constant for all N ≥ 2.
Thus, “adverse-selection free” means that if all advertisers report truthfully, then the probability that the brand advertiser wins the impression does not depend, even statistically, on the common factor. The next theorem shows that adverse-selection free mechanisms take a specific form. Theorem 1. A strategy-proof, direct mechanism (z,p) is adverse-selection free if and only if, there exists γ ∈ ⎡⎣0,1⎤⎦ such that for all N ≥ 2 , z0 (x1,...,xN ) is homogenous of degree zero and E ⎡⎣ z0 (x1,...,xN ) | N ⎤⎦ = γ . Proof: Letting x = (cm1,…,cmN ) for any N ≥ 2 , the random variables z0 (x1,…,xN ) and c are statistically independent if and only if for all c > 0 ,
11
z0 (cm1,…,cmN ) = z0 (m1,…,mN ) . That equation expresses homogeneity of degree
zero. Note that this result does not depend on the distribution of the match values. If a mechanism is anonymous and the probability that the brand advertiser wins is a homogeneous-of-degree-zero function of the performance values, then we can write the allocation and payment rules as a function of ratios of the order statistics of those values. Therefore, it is useful to define the ratios ri =
x( i ) x( N )
for
i = 1,...,N − 1. These ratios will play a key role in the describing the set of qualified advertising auctions. Definition 3. A qualified advertising auction is an anonymous, deterministic mechanism (z,p) that satisfies E ⎡⎣ z0 (x) ⎤⎦ = λ and has three additional properties: the mechanism must be adverse-selection free and strategy proof among performance bidders, and entail no transfers to or from losing bidders. As the next theorem will show, there always exist mechanisms satisfying these properties and they take a particular form. Theorem 2. A mechanism (z,p) is a qualified advertising auction if and only if
{ }
∞
there exists a function g : ∅ ∪ [1,∞)k → ℜ + such that for all N ≥ 2 and all k =1
x ∈ℜ N+ :
12
z0 (x) = 1 x(1) =x( 2)
{
zi (x) = 1⎧
}∪{g(r ,...,r 2
}
N−1 )≥r1
⎫ ⎨ xi >max x j ⎬∩ g(r2 ,...,rN−1 )max x j ⎬ j ≠i ⎩ ⎭
N
N
i=1
⎫ xj ⎬ i=1 ⎨⎩ xi >max j ≠i ⎭
Therefore, z0 (x) = 1− ∑ zi (x) = 1− ∑1⎧
(
(
)
{
}
h x(1) ,…,x(N) for all i ∈ 1,...,N .
(
h x(1) ,…,x(N)
)
)
= 1− 1 x(1) ≠x( 2) h x(1) ,…,x(N) .
{
}
13
Theorem 1 implies that h is homogenous of degree 0 so
(
) (
)
h x(1) ,…,x(N) = h r1,…,rN−1,1 . Strategy-proofness requires that zi (x) be non-
(
)
decreasing in xi ; hence, h r1,…,rN−1,1 must be non-decreasing in r1 . Let
(
{
)
) }
(
g r2 ,…,rN−1 = inf t ∈ℜ + | h t,r2 ,…,rN−1,1 = 1 . Then zi (x) = 1⎧ z0 (x) = 1 x(1) =x( 2)
{
}∪{g(r ,...,r 2
}
N−1 )≥r1
{(
) }
⎫ ⎨ xi >max x j ⎬∩ g r2 ,…,rN−1 x( 2) ∩ g r2 ,…,rN−1
max x j ⎬∩ g r2 ,…,rN−1 max y j ⎬∩ ⎨g ⎜ ( N+m ) ,…, j ≠i ⎩ ⎭ ⎪⎩ ⎝ y y ( 2)
y ( N+m−1) ⎞ y ( N+m )
⎟⎠ < y
{
⎡ x − max y (2) ,y (N+m)g ⎫⎢ i ⎣ ( N+m ) ⎬ y (1)
⎪⎭
(
y ( 2) y
( N+m )
( N+m−1)
,…, yy ( N+m )
)}⎤⎥⎦ .
x) ˆ . We treat two cases: We prove that this condition is equivalent to p(x) ≤ p(x, xi ≤ max x j and xi > max x j . j ≠i
j ≠i
When xi ≤ max x j , the left-hand side of this equation is zero and the rightj ≠i
hand side is non-positive because y (2) ≥ x(2) ≥ xi , so the inequality is satisfied. Next suppose that xi > max x j . Note that the above inequality assumes that j ≠i
xˆi ≥ xˆN+ j for all j = 1,…,m . This is without loss of generality because of anonymity. Also notice that there always exists some xˆi large enough such that the indicator function on the right-hand side equals one (the bid is winning) and
16
any such large xˆi maximizes the right-hand side. Therefore, the above inequality
{
)}
(
is equivalent to: 1 g(r ,…,r ) x(N)g r1,…,rN−1 (2) x
(1)
)} ≤ max {y
(2)
,y (N+m)g
(
y ( 2) y
( N+m )
( N+m−1)
,…, yy ( N+m )
)
{
≤ max y (2) ,y (N+m)g
(
y ( 2) y
( N+m )
( N+m−1)
,…, yy ( N+m )
)} for all x
{
(1)
(
≤ x(N)g r1,…,rN−1
(
Condition (2) holds if and only if max x(2) ,x(N)g r2 ,…,rN−1
{
max y (2) ,y (N+m)g
(
)} for all
y ( 2) y ( N+m )
( N+m−1)
,…, yy ( N+m )
)} for all x
)
)} ≤
(
)
≤ x(N)g r1,…,rN−1 . This is certainly
(1)
{
(
sufficient for (2). It is also necessary because max x(2) ,x(N)g r2 ,…,rN−1
{
(
{
(
max y (2) ,y (N+m)g max y (2) ,y (N+m)g
{
y ( 2) y ( N+m )
y ( 2) y ( N+m )
( N+m−1)
,…, yy ( N+m )
x(1) ∈ ⎡max y (2) ,y (N+m)g ⎢⎣
( N+m−1)
,…, yy ( N+m )
(
y ( 2) y ( N+m )
)} implies x
(N)
(
)} >
)
g r2 ,…,rN−1 >
)} , which means there would exist some ( N+m−1)
,…, yy ( N+m )
)},x
(N)
(
)
g r2 ,…,rN−1 ⎤ , violating (2). ⎥⎦
17
Therefore, a qualified advertising auction is false-name proof if and only if
{
(
p(x) = max x(2) ,x(N)g r2 ,…,rN−1
)} ≤ max {y
(2)
,y (N+m)g
(
y ( 2) y ( N+m )
( N+m−1)
,…, yy ( N+m )
x) ˆ for all )} = p(x,
N ≥ 2 , x ∈ℜ N+ , m ≥ 1 , and ( xˆN+1,..., xˆN+m ) .
Theorem 3 implies an additional virtue for false-name proof mechanisms: the seller will never be motivated to disqualify bidders wrongly because doing so can only reduce prices.13 Since the pricing in the auction and the winner determination both depend on the same function g as described in Theorem 2, it follows from Theorem 3 that the probability of a performance advertiser winning does not increase when losing bids are added. How many mechanisms are both qualified and false-name proof? The answer to that question, unlike any of our previous results, depends on the distribution of match values. Remember that a performance advertiser’s value for an impression is the product of a common component and an individual match value. Placing some distributional assumptions on the match values allows us to specify the exact auction(s) that satisfy the conditions of both Theorems 2 and 3. One case that is technically tractable and incorporates a “fat-tail” hypothesis is when the match values are IID draws from the power law distribution in which, for
{
}
µ ≥ 1 , Pr mi > µ = 1− F( µ ) = µ −a .14 We assume that the expected match value is finite, so a > 1. Given this distribution of match values, there is a single mechanism that is both qualified and false-name proof. 13
This is not always true in auctions. For example, multi-item Vickrey auctions do sometimes provide incentives for the seller to disqualify bidders. See Ausubel and Milgrom (2006). 14 See Wikipedia (2010). This distribution is sometimes called the Pareto distribution or Zipf’s law.
18
Theorem 4. When match values are IID draws from the power law distribution with parameter a > 1, there is a single qualified auction that is false-name proof: for all N ≥ 2 , the highest performance bid wins exactly when x(1) > (1− λ )−1/a x(2) and then pays (1− λ )−1/a x(2) .
(
Proof: Let α = 1− λ
(
)
− a1
> 1 and define g(∅) = α and, for all N > 2 ,
)
g r2 ,…rN−1 = α r2 . Consider the mechanism defined by z0 (x) = 1 (1) ( 2) x =x zi (x) = 1⎧
{
⎫ ⎨ xi >max x j ⎬∩ g(r2 ,...,rN−1 ) 1, this implies:
x( 2) x( N+1)
such that α
(
)
≤ gˆ
x( 2) x( N+1)
(
< gˆ
x( 2) x( N+1)
(
(N )
)
,…, xx( N+1) . Suppose there exists a non-
x( 2) x( N+1)
)
,…, xx( N+1) for every x ∈Y . This (N )
⎡ ⎤ implies that E ⎡⎣ z0 (x) | N + 1⎤⎦ = E ⎢1⎧ x(1) ⎛ x( 2) x( N ) ⎞ ⎫ (1) ( 2) | N + 1⎥ = ⎢⎣ ⎨⎩ x( N+1) ≤gˆ⎜⎝ x( N+1) ,…, x( N+1) ⎟⎠ ⎬⎭∪{x =x } ⎥⎦ ⎡ ⎤ ⎡ ⎤ E ⎢1⎧ x(1) ⎛ x( 2) x( N ) ⎞ ⎫ | N + 1⎥ > E ⎢1 x(1) x( 2) | N + 1⎥ = ⎢⎣ ⎨⎩ x( N+1) ≤gˆ⎜⎝ x( N+1) ,…, x( N+1) ⎟⎠ ⎬⎭ ⎥⎦ ⎣ { x( N+1) ≤α x( N+1) } ⎦ ⎡ ⎤ E ⎢1⎧ x(1) ⎛ x( 2) x( N ) ⎞ ⎫ | N + 1⎥ = λ , a contradiction. Thus gˆ r2 ,…,rN = g r2 ,…,rN ⎢⎣ ⎨⎩ x( N+1) ≤g⎜⎝ x( N+1) ,…, x( N+1) ⎟⎠ ⎬⎭ ⎥⎦
(
)
(
)
except possibly on a set of measure zero. By induction, the qualifed advertising auction described by gˆ is equivalent to that described by g . We call the mechanism characterized by Theorem 4 a “modified second-bid auction” because the performance advertiser wins only if the ratio of the highest to second-highest bids is sufficiently large.15 This mechanism could be implemented as a second-price auction by using a proxy bidder that submits a
15
This mechanism and its extensions are the subject of a recent patent application by Milgrom, Cunninghham and Beck (2011).
21
bid of a constant α = (1− λ )
−1/a
times the second-highest performance bid.
Therefore, a performance advertiser wins the impression when the ratio of the highest to the second-highest performance bids is larger than α . When a performance bidder wins, he pays α > 1 times the second-highest performance bid. As compared to the set-aside auction, which also uses a proxy bidder, the modified second-bid auction uses information about the performance advertiser match values to determine which impressions are assigned to brand advertisers and it results in a higher price for the performance advertiser than the set-aside whenever both would assign it the impression. We compare these auctions more carefully in Section IV.
III. Relaxing False-Name Proofness The requirement that a mechanism be false-name proof in addition to being qualified is a restrictive one. For a power law distribution of match values, the set of admissible mechanisms shrinks from those defined by Theorem 2 to a single mechanism. How much does this restriction cost in terms of the total expected value generated by the mechanism? To evaluate this, we will repeat our analysis with a fixed and known number of bidders N ≥ 2 . This eliminates the need for a mechanism to be false-name proof. We can then recharacterize the set of qualified mechanisms in this setting and determine the mechanism(s) within this class that achieve the highest expected total match value. With a fixed number of bidders, the qualified advertising auctions take a very similar form to those of the previous section. This set is always non-empty when
22
there are at least two performance advertisers and the larger the number of performance advertisers, the larger is the set of qualified advertising auctions. Theorem 5. When N>2 is fixed, a mechanism (z,p) is a qualified advertising auction if and only if there is a function g : ⎡⎣1,∞ z0 (x) = 1 x(1) =x( 2)
{
}∪{g(r ,...,r
zi (x) = 1⎧
2
}
N−1 )≥r1
⎫ ⎨ xi >max x j ⎬∩ g(r2 ,...,rN−1 ) µ |T r1,…,rN−1 = t is a non-decreasing
{(
)
}
function of t. Let α be the solution to Pr T r1,…,rN−1 > α = 1− λ . Then the mechanism with z0 (x) = 1 T r ,…,r
{(
1
N−1
)≤α }∪{x(1) =x( 2) }
maximizes the total expected value
among qualified advertising auctions.
(
)
Proof: Let z0 (x) = 1− R r1,…,rN−1 be the probability that the brand advertiser wins in a qualified auction. Any qualified auction eliminates adverse selection, leading to the same expected value from impressions assigned to brand advertisers. Thus, we will focus on the value generated by performance matches and show
(
)
that E ⎡ x(1)1 T (r ,…,r )>α ⎤ ≥ E ⎡⎣ x(1)R r1,…,rN−1 ⎤⎦ . Using the law of iterated expectations { 1 N−1 } ⎦⎥ ⎣⎢
(
)
and the fact that T r1,…,rN−1 is a sufficient statistic,
(
))
(
(
(
))
E ⎡⎢ x(1) 1 T (r ,…,r )>α − R r1,…,rN−1 ⎤⎥ = E ⎡⎣c ⎤⎦ E ⎡⎢m(1) 1 T (r ,…,r )>α − R r1,…,rN−1 ⎤⎥ = { 1 N−1 } { 1 N−1 } ⎣ ⎦ ⎣ ⎦
(
)
(
))
(
E ⎡⎢E ⎡⎣m(1) | r1,…,rN−1 ⎤⎦ 1 T (r ,…,r )>α − R r1,…,rN−1 ⎤⎥ = { 1 N−1 } ⎣ ⎦
(
)
(
(
)
(
))
(
E ⎡⎢E ⎡⎣m(1) |T r1,…,rN−1 ⎤⎦ 1 T (r ,…,r )>α − R r1,…,rN−1 ⎤⎥ = { 1 N−1 } ⎣ ⎦
(
) (
))
E ⎡⎢E ⎡⎣m(1) |T r1,…,rN−1 ⎤⎦ 1 T (r ,…,r )>α − E ⎡⎣R r1,…,rN−1 |T r1,…,rN−1 ⎤⎦ ⎤⎥ . { 1 N−1 } ⎣ ⎦
24
(
) (
)
Notice that 1 T r ,…,r >α − E ⎡⎣R r1,…,rN−1 |T r1,…,rN−1 ⎤⎦ is non-positive for { ( 1 N−1) }
(
)
(
(
) { }
)
T r1,…,rN−1 ≤ α and non-negative for T r1,…,rN−1 > α because
(
)
(
) (
(
) (
)
R r1,…,rN−1 ∈ 0,1 for all r1,…,rN−1 implies E ⎡⎣R r1,…,rN−1 |T r1,…,rN−1 ⎤⎦ ∈ ⎡⎣0,1⎤⎦ . Also, E ⎡1 T (r ,…,r )>α − E ⎡⎣R r1,…,rN−1 |T r1,…,rN−1 ⎢⎣ { 1 N−1 }
(
) (
E ⎡E ⎡⎣R r1,…,rN−1 |T r1,…,rN−1 ⎣
)⎤⎦ ⎤⎥⎦ = 0 because
)⎤⎦ ⎤⎦ = E ⎡⎣R (r ,…,r )⎤⎦ = 1− λ 1
N−1
{
(
for every qualified
) }
advertising auction. The assumption that Pr m(1) > µ |T r1,…,rN−1 = t is non-
(
)
decreasing in t implies that E ⎡⎣m(1) |T r1,…,rN−1 ⎤⎦ is non-decreasing in
(
)
T r1,…,rN−1 . Therefore, by majorization,
(
)
(
(
) (
))
E ⎡⎢E ⎡⎣m(1) |T r1,…,rN−1 ⎤⎦ 1 T (r ,…,r )>α − E ⎡⎣R r1,…,rN−1 |T r1,…,rN−1 ⎤⎦ ⎤⎥ ≥ 0 .17 { 1 N−1 } ⎣ ⎦ If there exists a function of the ratios of the order statistics that is a sufficient statistic for the largest match value and such that higher values of this function are associated with higher realizations of the largest match value, then the “optimal” qualified advertising auction – the one that maximizes the expected total match value – assigns the impression to the highest performance bidder when that sufficient statistic is high and assigns the impression to the brand advertiser when that sufficient statistic is low. The existence and form of such a sufficient statistic will depend on the distribution of the match values. For the
The relevant majorization inequality is this one: if (1) f : ℜ → ℜ is a function that is positive for arguments x > α and negative for arguments x ≤ α and such that E ⎡⎣f (Z) ⎤⎦ = 0 , (2) g : ℜ → ℜ is a non-decreasing function, and (3) Z is a real-valued random variable, then E ⎡⎣ f (Z)g(Z) ⎤⎦ ≥ 0 .
17
25
special case of the power law distribution, the ratio of the highest to the lowest order statistics is a sufficient statistic for m(1) . In that case, when the number of bidders is fixed, it is optimal to allocate the impression to performance bidders as a function of the ratio of the highest bid to the lowest bid. Corollary 1. If the match values are IID draws from a power law distribution with paramenter a , then there exists some β > 1 such that the mechanism that maximizes total expected value among qualified advertising auctions has the assignment rule z0 (x1,…,xN ) = 1 x(1) =x( 2)
{
}∪{x
(1)
≤ β x( N )
}
.
Proof: By assumption, each match value mi has pdf f (mi ) = ami−a−1 . Thus, the
(
)
(
)
joint pdf of m(1) ,r1,…,rN−1 is f m(1) ,r1,…,rN−1 =
(m ) N!f m(1) f ( ) r (1)
N−1
N 1
( )
⎡ N (1) ⎢⎣N!a m
−Na−1
( ) ( )f ( )1{ m(1)r2 r1
m(1)rN−1 r1
f
m(1) r1
} . This expression simplifies to
r1≥…≥rN−1≥1
(
)(
⎤ −a−1 1 m(1) ≥r ⎥ ⎡⎢r1(N−1)a−1r2−a−1rN−1 1{r ≥≥r ≥1} ⎤⎥ = g m(1) ,r1 h r1,…,rN−1 . 1 N−1 ⎦ { 1} ⎦ ⎣
)
Therefore, by the Fisher-Neyman Factorization Theorem,18 r1 is a sufficient
{
}
statistic for m(1) . Also, Pr m(1) > µ | r1 is a non-decreasing function of r1 . So, by Theorem 6, the qualified auction with assignment rule z0 (x1,…,xN ) = 1 r ≤β
{ 1 }∪{x(1) =x( 2) }
{
= 1 x(1) ≤β x( N )
{
}∪{x
(1)
=x( 2)
}
where β is defined by
}
Pr x(1) ≤ β x(N) = λ maximizes by the total expected value among qualified advertising auctions. 18
See Halmos and Savage (1949).
26
As the statistical analysis of Theorem 6 highlights, because r1 =
x(1) is a x(N)
sufficient statistic for m(1) for any fixed N, the “optimal” qualified auction uses it as the basis for allocating impressions between the brand and performance advertisers. This is in sharp contrast to Theorem 4, in which the unique qualified advertising auction that is false-name proof bases its allocation rule on the value of
x(1) x(1) . One can further show that given and any of the other ratio of order x(2) x(2)
statistics, say
x(1) for some k > 2 , the latter is a sufficient statistic for m(1) , so (k ) x
x(1) x(1) x(1) x(1) , ,…, is in that sense the least informative statistic among . In x(2) x(2) x(3) x(N) view of Theorem 4, this suggests that the need to protect the auction against false-name bids may sharply limit the gains to adopting an adverse-selection-free mechanism. Yet, as shown below, if the baseline for avoiding adverse selection is the set-aside auction, then the gains to switching to even a modified secondbid auction can be very large.
IV. Comparison to Set-Asides In this section, we study just two auctions – the set-aside auction and the modified second-bid auction. First, we compare them under the class of power law distributions of match values to show how a Pareto improvement is possible when switching from the set-aside auction to the modified second-bid auction and to give an idea of the magnitude of the possible improvement. Then, to highlight the assumptions about the match value distribution on which these
27
conclusions rest, we look at general distributions and give conditions under which the modified second-bid auction will outperform the set-aside auction. Since both of the mechanisms studied completely eliminate adverse selection, the expected value of impressions assigned to brand advertisers is the same. Our analysis, therefore, focuses on the values associated with the impressions assigned to performance advertisers.
{
}
For the power law distribution defined by Pr m > µ = 1− F( µ ) = µ −a for µ ≥ 1 , it is straightforward to show that, conditional on any realization of m(2) , the distribution of r = m(1) / m(2) is the same as that of the individual match values:
{
}
Pr r > ρ = ρ −a for ρ ≥ 1 . Since this does not depend on N or c, the random variables x(2) and r are statistically independent. Also, for any α ≥ 1 and ρ ≥ 1 ,
{
}
{
}
Pr r > αρ | r > α = Pr r > ρ . It follows that E ⎡⎣r | r > α ⎤⎦ = α E ⎡⎣r ⎤⎦ . We utilize these properties of the power law distribution in calculating three welfare indexes: expected total value from performance matches, expected revenue from performance matches, and performance bidders’ expected profits. Our first welfare index is the expected total value of the performance matches for the modified second-bid auction. This index is α times higher than for the set-
{
}
aside auction because E ⎡⎣ x(1)1{r >α } ⎤⎦ = E ⎡⎣ x(2)r1{r >α } ⎤⎦ = E ⎡⎣ x(2) ⎤⎦ α E ⎡⎣r ⎤⎦ Pr r > α =
α (1− λ )E ⎡⎣rx(2) ⎤⎦ = α (1− λ )E ⎡⎣ x(1) ⎤⎦ . The first term is the expected value of performance matches in the modified second-bid auction and the last term is α times the value in the set-aside auction.
28
Second, the expected revenue from performance matches in the modified second-bid auction is also α times higher than in the set-aside auction because, using independence, E ⎡⎣α x(2)1{r >α } ⎤⎦ = α (1− λ )E ⎡⎣ x(2) ⎤⎦ . The performance bidder’s total expected profit, which is the excess of the expected total value over expected revenue from performance matches, is therefore also multiplied by α. How large might α be? There is no upper bound on α, which increases in λ and tends to infinity as λ → 1 . To illustrate what may be plausible magnitudes, we use the special case of the power law known as the “80-20 rule,” according to which 80% of the expected value comes from the top 20% most valuable impressions.19 This ratio corresponds to a power law distribution with parameter a = 1.16. For that case, if the fraction of impressions set aside for brand advertisers is λ = 0.8 or λ = 0.5 , then the corresponding values for the multiplier are α = 4.0 or α = 1.8, respectively. These represent increases in match value, performance advertiser profit, and publisher revenues of 300% or 80% compared to the set-aside auction. The modified second-bid auction provides a Pareto improvement over the setaside auction for the class of power law distributions, for which it is the unique mechanism that is both qualified and false-name proof. To compare these auctions for more general distributions, we next develop a series of sufficient conditions on the distribution of match values to imply the conclusions that the 19
The 80-20 rule, which is sometimes called the “Pareto principle” after Pareto’s observation that 80% of Italy’s land was owned by 20% of its population, has been claimed to apply to a wide range of business statistics, especially ones related to sales. See Wikipedia (2012). It is typically modeled using power law distributions.
29
modified second-bid auction has a higher expected total match value, higher expected seller revenues, and higher expected bidder profits, respectively, than the set-aside auction. These conditions employ versions of a statistical selection condition, according to which the conditional distribution of m(1) , given the ratio
r=
m(1) m( 2)
, is stochastically higher when r is higher. The first of these results,
Theorem 7, affirms the intuitive proposition that when this selection condition holds, the modified second-bid auction, which picks performance bidders as winners when r is high, results in higher total expected value (from matches to performance bidders) than the set-aside auction. For our analysis, it will be useful to recall that the modified second-bid auction is a qualified advertising auction in which z0 (x) = 1 α (N)x( 2) ≥x(1) and α (N) is defined
{
}
⎡ ⎤ by the equation E ⎢1 α (N)x( 2) ≥x(1) | N ⎥ = λ . Unlike with the class of power law } ⎦ ⎣{ distributions of match values, the appropriate α for a general distribution may need to vary with N to guarantee that the auction is adverse-selection free. In contrast, the set-aside auction is not deterministic: it assigns the impression to ∞
the brand advertiser with probability z0 (x) = λ for all x ∈ ℜ k+ . Thus, the k =2
expected value generated by performance matches is E ⎡⎣ x(1) (1− λ ) ⎤⎦ for the setaside auction and E ⎡⎣ x(1)1{r >α (N)} ⎤⎦ for the modified second-bid auction.
30
{
}
Theorem 7. Suppose that for all µ, the conditional probability Pr m(1) > µ | r,N is
{
}
a non-decreasing function of r. Let α (N) be such that Pr r > α (N) = 1− λ . Then, E ⎡⎣ x(1) (1− λ ) ⎤⎦ ≤ E ⎡⎣ x(1)1{r >α (N)} ⎤⎦ . Proof: Notice that 1{r >α (N)} − (1− λ ) is non-positive for r ≤ α (N) and non-negative
{
}
for r > α (N) . The stochastic dominance assumption that Pr m(1) > µ | r,N is nondecreasing in r implies that E ⎡⎣m(1) | r,N ⎤⎦ is also non-decreasing in r. By the same majorization inequality used in Theorem 6,
(
(
)
)
E ⎡m(1) 1{r >α (N)} − (1− λ ) | N ⎤ = E ⎡E ⎡m(1) 1{r >α (N)} − (1− λ ) r ,N ⎤ | N ⎤ = ⎦ ⎦ ⎣ ⎦ ⎣ ⎣
(
)
E ⎡E ⎡⎣m(1) r,N ⎤⎦ 1{r >α (N)} − (1− λ ) | N ⎤ ≥ 0 This implies that ⎣ ⎦
(
)
(
)
E ⎡ x(1) 1{r >α (N)} − (1− λ ) ⎤ = E ⎡cm(1) 1{r >α (N)} − (1− λ ) ⎤ = ⎣ ⎦ ⎣ ⎦
(
)
E ⎡E ⎡⎣c | N ⎤⎦ E ⎡m(1) 1{r >α (N)} − (1− λ ) | N ⎤ ⎤ ≥ 0 because both terms inside the ⎣ ⎦⎦ ⎣ expectation are non-negative. Theorem 7 works directly with the selection condition on m(1) and r. Later theorems will rely instead on an IID model for match values, which allows us to use standard methods to analyze how performance advertisers and the publisher share the value generated by performance matches. The following theorem connects the IID model to the general condition, providing a sufficient condition in the IID model for the selection property to hold.
31
Theorem 8. Suppose that the logarithms of match values ln(mi ) are independent draws from the cdf Fˆ with log-concave density fˆ . Then, r and m(1) are
{
}
affiliated.20 In particular, Pr m(1) > µ | r,N is nondecreasing in r. Proof: We compute the joint distribution of m(1) and r conditional on N as follows:
{
} ∫ (F
Pr m(1) ≤ µ,r ≤ ρ | N =
µ
0
)
(s) − F N−1(s / ρ ) Nf (s)ds . The joint density of r and
N−1
m(1) conditional on N is ∂2 Pr m(1) ≤ µ,r ≤ ρ | N = N(N − 1)µρ −2f ( µ )f ( µ / ρ )F N−2 ( µ / ρ ) . Define σ = ln µ ∂µ ∂ ρ
{
}
and τ = ln ρ . Then the joint density of σ and τ conditional on N is N(N − 1)e 2σ −τ f (eσ )f (eσ −τ )F N−2 (eσ −τ ) = N(N − 1)fˆ(σ )fˆ(σ − τ )Fˆ N−2 (σ − τ ) . Since
affiliation is invariant under order-preserving transformations of the variables, m(1)
( )
and r are affiliated exactly if ln(r ) and ln m(1) are affiliated. Affiliation holds if the log of the joint density is supermodular, that is if
∂2 ln ⎡⎣N(N − 1)fˆ(σ )fˆ(σ − τ )Fˆ N−2 (σ − τ ) ⎤⎦ ≥ 0 , which follows since ln(fˆ) is concave ∂σ ∂τ ˆ is concave, see An (1995)). (and hence ln(F)
Log-concavity of the density of the logarithm of the match values guarantees a higher expected total match value under the modified second-bid auction than under the set-aside auction. To give conditions such that both the publisher and
20
See Milgrom and Weber (1982).
32
performance advertisers share these gains, we must first calculate expressions for their profits when the match values are IID. Theorem 9. Suppose a mechanism is such that the highest performance bidder is assigned the impression as a function only of the ratio of the two highest performance bids and the number of bidders: zi (x) = 1⎧
⎫ ⎨ xi >max x j ⎬ j ≠i ⎩ ⎭
R(r,N) . If the match
quality variables mi are independent and identically distributed with cdf F and corresponding density f, then, conditional on N, the expected total value of the allocation to performance bidders, the performance bidder profits, and the seller’s expected revenues from performance bidders are as follows:
Expected Performance Value = E[c | N]E ⎡⎣m(1)R(r,N) N ⎤⎦
( ) ( )
⎤ ⎡ 1− F m(1) ⎢ R(r,N) N ⎥ Expected Performance Bidder Profits = E[c | N]E ⎥ ⎢ f m(1) ⎦ ⎣ ⎡ ⎛ 1− F m(1) ⎞ ⎤ (1) ⎢ Expected Performance Revenues = E[c | N]E R(r,N) ⎜ m − ⎟ N⎥ (1) ⎢ ⎜⎝ ⎟⎠ ⎥ f m ⎣ ⎦
( ) ( )
Proof: The first equation holds because, conditional on N, expected value of the allocation to performance bidders is ⎤ ⎡N ⎡N ⎤ R(r,N) | N E ⎢ ∑ xi zi (x) | N ⎥ = E ⎢ ∑ xi 1⎧ ⎥ = E ⎡⎣ x(1)R(r,N) | N ⎤⎦ = ⎫ xj ⎬ ⎥⎦ ⎢⎣ i=1 ⎨⎩ xi >max ⎣ i=1 ⎦ j ≠i ⎭
E ⎡⎣cm(1)R(r,N) | N ⎤⎦ = E ⎡⎣c | N ⎤⎦ E ⎡⎣m(1)R(r,N) | N ⎤⎦ . The last equation combines the first and second equations, so it remains only to prove the second equation.
33
For any realizations of c and of a performance bidder’s match value t, if that match value is highest and the second highest is s, then the bidder’s probability of winning is R(t / s,N) . So, conditional on its own match value t and the number of bidders N, the bidder’s probability of winning the impression is t
∫ R(t / s,N)dF 0
t
(s) = (N − 1) ∫ R(t / s,N)F N−2 (s)f (s)ds . Hence, by the envelope
N−1
0
theorem, when the common factor is c, the expected profit of a bidder with match value m is (N − 1)c ∫
m
0
t
∫ R(t / s,N)F 0
N−2
(s)f (s)ds dt . Thus, conditional on N, the total
expected profit of any single performance bidder is:
E ⎡(N − 1)c ∫ 0 ⎣⎢
m
=
∫
∞
0
t
∫ R(t / s,N)F
N−2
0
(N − 1)E[c | N] ∫
m
0
t
(s)f (s)ds dt | N ⎤ ⎦⎥
∫ R(t / s,N)F 0
N−2
(s)f (s)ds dt f (m)dm
∞ ∞ t = E[c | N] ∫ (N − 1) ⎛ ∫ f (m)dm⎞ ⎛ ∫ R(t / s,N)F N−2 (s)f (s)ds ⎞ dt ⎝ t ⎠⎝ 0 ⎠ 0
= E[c | N]
1 ∞ t 1− F(t) R(t / s,N)N(N − 1)F N−2 (s)f (s)f (t)ds dt N ∫0 ∫0 f (t)
( ) ( ( )
(1) ⎤ 1 ⎡⎢ 1− F m (1) (2) ⎥. = E[c | N] E R m / m ,N N N ⎢ f m(1) ⎥ ⎦ ⎣
)
So, the total expected profit of all N performance bidders is the expression in the theorem. Given these expressions, we can now state a theorem about the advantage of the modified second-bid auction over the set-aside auction for the IID model, using the same criteria we discussed for the power law distribution.
34
{
}
Theorem 10. Let α (N) be such that Pr r > α (N) | N = 1− λ for all N ≥ 2 . Suppose that the match quality variables are independent and identically distributed with cdf F and corresponding density f. Assume that the density fˆ of the logarithm of match values ln(mi ) is log-concave. 1. The modified second-bid auction generates higher total expected value
(
)
than the set-aside auction: E ⎡⎣ x(1) 1− λ ⎤⎦ ≤ E ⎡⎣ x(1)1{r >α (N)} ⎤⎦ .
2. If
f (m) is a non-increasing function, then the modified second-bid 1− F(m)
auction generates higher expected performance bidder profits than the set-aside auction: ⎡ ⎡ ⎡ 1− F(m(1) ) ⎡ 1− F(m(1) ) ⎤⎤ ⎤⎤ 1− λ N 1 N E ⎢E[c | N]E ⎢ E E[c | N]E ≤ ⎥ ⎢ ⎢ ⎥ ⎥⎥ . {r >α (N)} (1) (1) ⎢⎣ f (m ) ⎢⎣ f (m ) ⎥⎦ ⎥⎦ ⎢⎣ ⎢⎣ ⎦⎥ ⎥⎦
(
3. If m −
)
1− F(m) is a non-decreasing function, then the modified second-bid f (m)
auction generates higher expected seller revenues from performance bidders than the set-aside auction:
⎡ ⎡ ⎛ (1) 1− F(m(1) ) ⎞ ⎤ ⎤ N ⎥⎥ ≤ E ⎢E[c | N]E ⎢ 1− λ ⎜ m − f (m(1) ) ⎟⎠ ⎥⎦ ⎥⎦ ⎝ ⎢⎣ ⎢⎣
(
)
⎡ ⎡ ⎛ 1− F(m(1) ) ⎞ ⎤ ⎤ N ⎥⎥ . E ⎢E[c | N]E ⎢1{r >α (N)} ⎜ m(1) − f (m(1) ) ⎟⎠ ⎥⎦ ⎥⎦ ⎝ ⎢⎣ ⎢⎣
35
Proof: Note that both the modified second-bid auction and the set-aside auction satisfy the conditions of Theorem 9 with R(r,N) = 1{r >α (N)} and R(r,N) = 1− λ , respectively. Therefore, the proof uses Theorems 8 and 9 and then follows the same lines as the proof of Theorem 7, with each conclusion using a different non-decreasing function of m. The power law case studied earlier meets all of the conditions of Theorem 10.
{
}
Each private match value m is distributed so that 1− F( µ ) = Pr m ≥ µ = µ −a and
{
}
ˆ = Pr ln(m) ≥ s = e −as . The hence ln(m) is exponentially distributed: 1− F(s) corresponding density is fˆ(s) = ae −as , which is log-linear. For a finite mean, the power law requires that a > 1, so the two hazard rate conditions are also satisfied. The requirement of Theorem 10 that fˆ is log-concave is a strong one that helps to characterize some of the cases in which the modified second-bid auction is a Pareto improvement over the set-aside auction. The fact that fˆ is log-linear in the power law case, so that the log-concavity condition is barely met even though the welfare gains compared to the set-aside auction are large, suggests that superior results may be found for a considerably broader range of probability distributions than just those for which fˆ is log-concave.
V. Discussion We have introduced a new auction mechanism for online display advertising: the modified second-bid auction. Within our model, it eliminates adverse
36
selection against brand advertisers and, if the match value distribution has a fat tail, it enables performance bidders to receive a large proportion of the impressions for which they have especially high match values. For the case of power law distributions, the percentage increase in expected match surplus for performance advertisers grows without bound as the fraction of impressions reserved for brand advertisers goes to one. We have adopted an axiomatic approach to our problem in the sense that our various requirements, including our novel requirement that the auction should be adverse-selection free, are not derived from any optimal auctions analysis. These requirements are interesting because of their mutual consistency and tractability and because of their appeal to practitioners, as described earlier. We have introduced a tractable model that permits us to study the restrictions imposed on ad-matching by the need to avoid adverse selection. Yet tractability comes at a cost: not only do we rely throughout on the assumption that the common value element enters valuations in a multiplicative way, but some of our results also depend on the assumption that match values are drawn IID from a power law distribution. In contrast, the set-aside auction procedure is robustly adverse-selection free, requiring none of these assumptions. For our new design is to be appealing in practice, despite the robustness advantage of the set-aside auction, it needs to offer the possibility of substantial performance improvements. Our analysis indicates that, for the IID power law cases, the modified second-bid auction can lead to quite large improvements, particularly when the fraction of impressions allocated to brand advertisers is large.
37
An empirical assessment of the performance advantages of the modified second-bid auction could use simulations guided by data from existing set-aside auctions. Since the second-price and modified second-bid mechanisms are both strategy-proof, the bids collected in the former can be assumed to express bidder values. Consequently, it is reasonable to assume that the bids in the two kinds of auctions would be the same. With those assumptions and data from existing auctions, one can estimate performance bidder profits and seller revenues under the alternative auction rules. Testing whether adverse selection is actually eliminated would be harder, since the brand advertiser’s value is not observed in the auction data.
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References An, Mark Yuying (1995). "Log-concave Probability Distributions: Theory and Statistical Testing." Duke University Department of Economics Working Paper: No. 95-03. Athey, Susan and Glenn Ellison (2010). "Position auction with consumer search." Quarterly Journal of Economics Forthcoming. Ausubel, Lawrence and Paul Milgrom (2006). The Lovely but Lonely Vickrey Auction. Combinatorial Auctions. P. Cramton, Y. Shoham and R. Steinberg. Cambridge, MA, MIT Press. Edelman, Benjamin, Michael Ostrovsky and Michael Schwarz (2007). "Internet Advertising and the Generalized Second Price Auction: Selling Billions of Dollars Worth of Keywords." American Economic Review 97(1): 252-259. Ghosh, Arpita, Preston McAfee, Kishore Papineni and Sergei Vassilvitskii (2009). "Bidding for Representative Allocations for Display Advertising." Proceedings of the 5th international Workshop on Internet and Network Economics (WINE). Halmos, Paul R. and L. J. Savage (1949). "Application of the Radon-Nikodym Theorem to the Theory of Sufficient Statistics." The Annals of Mathematical Statistics 20(2): 225-241. McAfee, Preston, Kishore Papineni and Sergei Vassilvitskii (2010). Maximally Representative Allocations for Guaranteed Delivery Advertising Campaigns. Yahoo Working Paper. Milgrom, Paul, Stephan Cunninghham and Marissa Beck (2011). Impression Allocation System and Method Using an Auction That Considers Losing Bids. Patent Pending, OpenX Corp. Milgrom, Paul and Ilya Segal (2002). "Envelope Theorems for Arbitrary Choice Sets." Econometrica 70(2): 583-601. Milgrom, Paul and Robert J. Weber (1982). "A Theory of Auctions and Competitive Bidding." Econometrica 50: 463-483. Myerson, Roger B. (1981). "Optimal Auction Design." Mathematics of Operations Research 6(1): 58-73. Varian, Hal R (2007). "Position Auctions." International Journal of Industrial Organization 25(6): 1163-1178. Wikipedia (2010) "Power Law." --- (2012) "Pareto Principle." Yokoo, Makoto, Yuko Sakurai and Shigeo Matsubara (2004). "The effect of falsename bids in combinatorial auctions: new fraud in internet auctions." Games and Economic Behavior 46(1): 174-188.
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