Truthful Auctions for Arbitrary Sized Ads
Kate Larson, Tyler Lu
David R. Cheriton School of Computer Science, University of Waterloo
Abstract. We consider designing truthful auctions allowing bidders to
place advertisements of any desired size on sponsored search results. This
may be a remedy for large companies wanting more visibility and consequently higher click through rates. It might also be a way to prevent a
company’s competitors from bidding on the company’s name. Our model
assumes a more general notion of single-minded bidders where bidders
are only interested in displaying ads of a certain size. We examine three
cases: (1) ad sizes are publicly known and the position of ad placement
matters and (2) ad sizes are private but position of ad placement doesn’t
matter and (3) ad sizes are private and ad position matters. We design a
truthful auction for (1) via an application of Aggarwal et al.’s [1] auction.
And a truthful auction for (2) by extending the framework of Lehmann
et al. [4]. This auction also has applications to knapsack auction problems with unknown sizes, and therefore differs from the work in [2].
Designing a truthful auction for (3) implies truthful auctions for (1) and
(2), however, we pose this as an open problem and give issues that needs
to be overcome in its solution.
1
Introduction
We consider designing truthful auctions allowing bidders to place advertisements
of any desired size on sponsored search results. This can be useful in situations
where a company wishes to prevent competitors from bidding on their keywords.
For example, automobile maker Ford may not want Dodge bidding on the keyword Ford. It may also be applicable when companies would like more visibility
on the results page causing higher click through rates.
Our model is based on a slightly different single-minded setting where bidders
are only interested in displaying ads of a certain size. This notion of singlemindedness differs from the conventional definition in combinatorial auctions
where a bidder is interested in a set of goods or in this context, slots, B =
{a1 , a2 , . . . , am } which may correspond to non-contiguous slots in the sponsored
search space – that is, B 6= {s ∈ Z : min(B) ≤ s ≤ max(B)}.
We design truthful auctions for two important cases: (1) ad sizes are publicly
known and the position of ad placement matters and (2) ad sizes are private but
position of ad doesn’t matter.
The auction for (1) is an extension of the auction from Aggarwal et al. [1], and
for (2) is an extension of the framework of Lehmann et al. [4]. The second auction
we design can be used in more general settings such as auctioning off a large
chunk of rectangular land where bidders are interested in smaller rectangular
pieces, regardless of position. Note that this differs from the work in Aggarwal
et al. [2] where they assume sizes are public information.
Finally we pose as an open problem the third case where ad sizes are private
and ad position matters. We will give an example of why the two previous
auctions fail and discuss issues that needs to be addressed in the solution. A
summary of our contributions is given in Table 1.
Ad size is
public knowledge
Ad size is
private knowledge
Ad position doesn’t matter
Can use truthful
auction of § 4 or § 3
Extension of
Lehmann et al. [4] see § 3
Ad position matters
Extension of
Aggarwal et al. [1] see § 4
Open problem, see
discussion in § 5
Table 1. A summary of our contributions.
2
Model of Single-Minded Bidders
We assume our bidders (advertisers) are single-minded in the sense that they
are interested in exactly k contiguous slots regardless of the position where their
advertisement gets displayed. This is slightly different from the conventional notion of single-mindedness in generalized VCG where a bidder would discriminate
between different sets of goods, which in our context, means different positions
in sponsored search results.
Let N = {1, . . . , n} be the bidders and K be the number of available slots.
Each bidder i submits αi = (si , bi ) where si is the number of slots needed for her
advertisement and bi is her bid. Also, let vi be bidder i’s true private valuation
for her true goal of receiving ti slots. We examine the special cases when ti is
either public or private knowledge.
3
Unknown Single-Mindedness and Advertiser
Indifference
In this model we are only concerned with advertisers who only care whether
their ad gets displayed and not on its relative position in the K slots. We call
such advertisers indifferent. Because of this, we do not take into consideration
the various ranking procedures that rely on click-through rates (or relevance).
Also, we assume free disposal so that getting more than si slots results in the
same valuation as getting exactly that many slots. This would also mean that
the click through rates of ads larger than what is desired remains the same. This
assumption is indeed strong, as it also implies that the click through rates won’t
change for larger ads.
Suppose that our intention is to maximize social welfare. It is not hard to see
that this optimization problem is the 0-1 Knapsack problem: we have capacity K,
and the si and bi correspond to the sizes and valuations of each object associated
with bidder i (here we assume bi is the valuation, i.e. auction is truthful).
Remark 1. It is NP-hard to maximize social welfare in a setting where there are
K identical goods with single-minded bidders.
The Knapsack Auction model in [2] is not adequate for our situation because
it assumes that every bidder’s si is publicly known (i.e. Known Single-Minded
model) such an assumption is unreasonable here. We use the framework of [4]
to design truthful auctions for the Unknown Single-Minded knapsack auctions.
It is well known that the generalized VCG auction is social welfare maximizing and is NP-hard to compute this efficient allocation. However, a greedy
approximate truthful mechanism of [4] can be adapted to this situation. Note
that our notion of “single-mindedness” differs from that of [4] for the simple reason that each bidder is not interested in a specific position within the K slots,
but rather is interested in getting his ad of size si displayed.
Our greedy allocation algorithm is very simple: first sort the bidders decreasingly in terms of the average bid per slot ai = bi /si . WLOG, assume that bidder
i has the i-th highest average bid per slot. Then for each i = 1, 2, . . . bidder i will
be allocated his bid if there is enough room left for si to fit into the remaining
slots. This greedy algorithm has approximation ratio 1/K.
It turns out that by changing the VCG allocation scheme to the greedy one
above, the Clarke pricing no longer makes the mechanism truthful, and in fact
it is no longer individually rational (i.e. bidders can obtain negative utility by
bidding their true valuations) [4]. Thus, a new pricing is required for the greedy
allocation.
Here, we will modify their pricing scheme and show that our “greedy auction”
is indeed truthful by providing four criteria (in the spirit of [4]) and showing
that they are sufficient for an auction to be truthful.
Let n(i) ∈ N denote the losing bidder (advertiser whose ad is not shown)
with the highest bid per slot such that if i was to exit the auction, n(i) would
get his slots. Of course, n(i) need not exist. Note that n(i) is not necessarily the
losing bidder with the highest bid per slot.
Definition 1. (Greedy Knapsack Auction) The allocation is determined as follows:
– Sort the bidders decreasingly by ai = bi /si
– We service the bidders in the order that they were sorted. If a bidder’s desired
ad size can fit into the remaining ad space then grant the bid, otherwise reject
the bid.
The price, pi that bidder i pays is as follows:
– If i does not get his bid, or if he does but n(i) does not exist, then pi = 0.
b
– Otherwise i gets his bid, and he must pay pi = si an(i) = si sn(i)
n(i)
Before proving the truthfulness of the above payment scheme in conjunction
with the greedy allocation, let us reformulate the criteria from [4] adapted to
our model.
Definition 2. (Exactness) A bidder is either allocated exactly his si goods (and
no more) or none at all.
Note that this definition differs a little from the one in [4] in that the bidder can
be allocated any of the contiguous si slots available.
Definition 3. (Monotonicity) Suppose that i’s bid is successful (ad is shown)
when he submits (si , vi ), then for any bid i makes of the form β = (s0i , vi0 ) where
s0i ≤ si and vi0 ≥ vi , we have that bid β is also successful.
Again, this definition has been reworked to suit our situation.
The following lemma comes from [4] and also applies to our model.
Lemma 1. If Exactness and Monotonicity is satisfied, then for each bidder i,
there exists a vi∗ such that if v < vi∗ then i does not receive his slots, and if
v > vi∗ he does.
Proof. Nearly identical to the original one in [4].
Definition 4. (Critical) A successful bidder i must pay vi∗ .
Definition 5. (Participation) An unsuccessful bidder pays nothing.
Before proving that an auction satisfying the above is truthful, we need several lemmas as in [4].
Lemma 2. In a mechanism that satisfies Exactness and Participation, a bidder
whose bid is denied has utility zero.
Proof. The same as in [4]. By Exactness, the bidder gets nothing and his valuation is zero. By participation, his payment is zero.
Lemma 3. In a mechanism that satisfies Exactness, Monotonicity, Participation and Critical a truthful bidder’s utility is non-negative.
Proof. The same as in [4]. Assume i’s bid is granted (otherwise Lemma 2 does
it). Since he is truthful, his declaration is βi = (si , vi ). Because his bid is granted,
Lemma 1 tells us vi ≥ vi∗ . By Critical, i pays vi∗ and so his utility is vi − vi∗ ≥ 0.
Lemma 4. In a mechanism that satisfies Exactness, Monotonicity, Participation and Critical, a bidder i that submits a truthful bid (si , vi ) is never better off
by submitting a lying bid (si , vi0 ).
Proof. If i’s lying bid is rejected then by Lemma 2 his (lying) utility is zero and
by Lemma 3 his truthful bid gives him non-negative utility.
Consider now when i’s lying bid is granted. If the truthful bid is also granted,
then in both bids the same valuation is obtained (by bidder indifference, i.e. any
set of si goods give same valuation) and the same price of vi∗ is paid which
gives the same utility in both bids. If the truthful bid is denied, but the lying
bid granted, then Lemma 1 says vi ≤ vi∗ ≤ vi0 . This means the lying utility is
vi − vi∗ ≤ 0.
Lemma 5. In a mechanism that satisfies Exactness, Monotonicity and Critical,
a bidder declaring β = (s, v) whose bid is granted pays a price p that is at least
the price p0 that he would have paid had he declared β 0 = (s0 , v) for s0 ≤ s.
Proof. The proof is similar as that in [4]. By Monotonicity, β 0 is granted. Suppose
by contradiction that p0 > p. Then by bidding (s, x) where p < x < p0 the bid is
granted by Critical, but the bid (s0 , x) is not granted by Critical, and this would
contradict Monotonicity.
This leads us to prove the that:
Theorem 1. If an auction satisfies Exactness, Monotonicity, Participation, and
Critical, then the auction is truthful.
Proof. The proof is similar as that in [4]. Compare the utilities of the truthful
bid (si , vi ) of i with a lying bid (s0i , vi0 ). We consider case when the lying bid is
granted otherwise the lying utility is zero, and the truthful utility is non-negative
by Lemma 3.
If s0i < si , then i’s lying valuation is zero and he pays non-negative amount
by Critical, so his utility is non-positive. Suppose s0i ≥ si , by Monotonicity the
bid (si , vi0 ) would also have been granted and in both bids i obtains the same
valuation. But Lemma 5 the lying bid pays at least as much as the truthful bid.
Finally, Lemma 4 says bidding (si , vi0 ) is not better than bidding (si , vi ).
The above Theorem 1 has applications for more general situations such as
“2-dimensional” knapsack, and geometrical goods as in [3].
Theorem 2. The Greedy Knapsack Auction in Definition 1 is truthful.
Proof. It is enough for us to check that the auction satisfies the conditions given
in Theorem 1.
(Exactness) The greedy allocation as defined either grants exactly the goods
desired by the bidder or none at all.
(Monotonicity) If a successful bid (si , vi ) by i is granted, then by bidding
s0i ≤ si and vi0 ≥ vi we get that vi0 /s0i ≥ vi /si . This means the new bid will be
served no later than the old bid in the greedy allocation, and since s0i ≤ si then
there’s enough remaining space to grant the new bid.
(Critical) Consider a successful bidder i. If n(i) does not exist, then we claim
the critical value vi∗ = 0. Suppose for a contradiction that i bids vi0 > 0 but is
denied. This means a bidder j who previously was served later than i by the
greedy allocation is now served earlier than i and furthermore is granted the bid.
But this would mean j’s bid conflicts with i’s bid since i’s bid would have been
granted before. This contradicts non-existence of n(i).
Say n(i) does exist. We claim that the critical value vi∗ = si an(i) . Let r(j)
denote the rank (ordered by decreasing average bids) of bidder j when each
bidder l ∈ N bids (sl , vl ). Now suppose i bids vi0 > si an(i) . The new rank of i is
still higher than that of n(i). If i’s new bid is denied, then ∃j ∈ N : r(i) < r(j) <
r(n(i)) and j’s bid is now granted but was previously denied. But this means in
the absence of i, j’s bid would have been granted, so it conflicts with j’s bid.
Also we have r(j) < r(n(i)) but this contradicts definition of n(i). Suppose that
i bids vi0 < si an(i) , then i would be served later than n(i). By definition n(i)’s
bid will be granted, but then i’s bid would have to be denied otherwise n(i)
would never have conflicted with i.
(Participation) Lemma 2 tells us that a denied bidder has zero utility.
Also, Theorem 1 has an application to sponsored search auctions where bidders are indifferent to their allocated positions in the ad space. Suppose that
there are K slots and bidders are only interested in contiguous slots where their
ad can be displayed. That is, an advertiser i is interested in placing an ad of size
si and we assume that this information is private.
Let CTRi,j denote the click through rate of advertiser i whose ad is positioned starting at slot j. We give the following natural definition of advertiser
indifference.
Definition 6. An advertiser i is indifferent if his click through rates CTRi,j =
CTRi,1 for all j.
It is easy to check that indifferent advertisers fit into the model in this section.
The allocation algorithm will be changed to take into account an advertiser’s
relevance. In this model the expected utility per impression of an advertiser i is
ui = CTRi,j (vi − pi ) = CTRi,1 vi − CTRi,1 pi
(1)
where vi and pi are the valuation and price charged for i when his ad is clicked
on.
Thus, the expected valuation is CTRi,1 vi and expected payment is CTRi,1 pi
per impression. The greedy allocation would rank the bidders by CTRi,1 vi /si ,
and the rest of the allocation is the same as the first come first serve from above.
The critical expected valuation occurs when
CTRn(i),1 vn(i)
CTRi,1 vi∗
=
si
sn(i)
so the price charged to i per click is
pi = vi∗ =
∗
CTRn(i),1 vn(i)
si
CTRi,1 sn(i)
It is easy to check that this auction is indeed truthful as in Theorem 2.
4
Known Single-Mindedness, Advertiser
Non-Indifference, and Rank by Relevance
In this model, we assume that the size of ads si of which bidder i desires is
publicly known, but that the valuation vi per click is private. This also means
that click through rates only depend on the bidder and the position allocated
since ad size is fixed. Therefore we aim to develop truthful auction that reveals
the private valuation. In this case we also assume that the click through rates for
different slots is not necessarily the same for each bidder (i.e. the bidders are not
indifferent). Therefore, there is a question of how to rank the advertisers whose
ads are shown. In this section we consider a modification to Rank by Relevance,
an approach taken by Google and more recently Yahoo!. Our auction that we
design uses this ranking function, but can be easily extended to a more general
weighted ranking found in [1].
The expected utility per impression of each bidder i when his ad is allocated
a slot beginning at position r(i) is
ui = CTRi,r(i) vi − CTRi,r(i) pi
(2)
and let yi = CTRi,r(i) vi and qi = CTRi,r(i) pi be the expected valuation per
impression, and the expected price per impression respectively. Note that if an
advertiser’s ad is not displayed, then his valuation is 0.
Suppose that we want to preserve the Google advertisement ranking function
while keeping the auction truthful (note that a similar result can be given for any
weighted ranking function as in [1]). Since the Google ranking function does not
completely apply to our context, we must consider an extension. A natural and
reasonable extension is to rank by CTRi,1 vi /si . We can use a different “fudge
factor” than simply CTRi,1
We can exploit a weighted VCG where the auctioneer can assign weights to
each bid in considering the maximization of social welfare.
First we’ll need the separability assumption before using weighted VCG.
Definition 7. The click through rate, CTRi,r(i) is separable if, there are values
γ1 , γ2 , . . . , γn and δ1 ≥ δ2 ≥ · · · ≥ δK such that
CTRi,r(i) = γi δr(i)
(3)
Let wi be the weight the auctioneer assigns to bidder i’s bid. Therefore the
auctioneer wants to maximize the social welfare, which in this case is
X
F (N ) =
wi yi
(4)
i∈N
if we choose wi = 1/si we get that
F (N ) =
X 1
X
X δr(i)
X δr(i)
vi
vi
vi
CTRi,r(i) vi =
γi δr(i) =
γi δ1 =
CTRi,1
si
si
δ1
si
δ1
si
i∈N
i∈N
i∈N
i∈N
assume for the moment that under Google ranking, bidder i is the i-th ranked
K
. Now one can
advertiser. That means CTR1,1 vs11 ≥ CTR2,1 vs22 ≥ · · · ≥ CTRK,1 vsK
check that F (N ) is maximized exactly when r(i) = i due to the non-increasing
δj ’s. But that means weighted VCG will produce the exact same allocation as
the Google ranking.
Theorem 3. The weighted VCG mechanism with weights wi = 1/si produces
the same advertisement ranking as rank by relevance (Google)
Using weighted VCG, we will now calculate the payments which will make
this auction truthful. Recall that in VCG a bidder pays the damage in social
welfare he does by being in the auction. We use this payment scheme to calculate
an advertiser’s payment when his ad gets clicked on. If an advertiser i’s ad is
not shown (i.e. r(i) + si − 1 > K) then he pays nothing. Consider an advertiser
i whose advertisement gets shown, and let
Pi = {j ∈ N : r(i) < r(j) ≤ K − sj + 1}
(5)
be the set of advertisers whose ads are shown below that of i’s. Also, let
Qi = {j ∈ N : K < r(j) + sj − 1 ≤ K + si }
(6)
be the set of advertisers whose ads are not shown, but gets shown when reranking is done in the absence of i.
Informally, i pays (per impression) for the increase in social welfare (per
impression) obtained when every advertiser ranked below is moved up si slots.
Formally,


X
X
1 
vj
vj
qi = CTRi,r(i) pi =
(δr(j)−si − δr(j) )CTRj,1 +
δr(j) CTRj,1 
δ1
sj
sj
j∈Pi
j∈Qi
(7)
and the pay per click cost can be obtained by dividing through by CTRi,r(i) .
Note that because we are using weighted VCG, valuations correspond to each
term in the sum in (4).
5
Open Problem: Unknown Single-Mindedness,
Advertiser Non-Indifference
In this section, we do not assume that si are publicly known nor do we assume that bidders are indifferent about slot locations. Therefore, the auctions
developed in Sections 3 and 4 does not apply.
Remark 2. The auction in Section 4 is not truthful under the assumption that
si are private.
Bidders
1
2
3
si
1
1
2
vi
2
3
1
vi
CTRi,1
si
1
1.5
0.1
CTRi,1 CTRi,2 CTRi,3
0.5
0.25
0.1
0.5
0.5
0.5
0.2
0.1
0.1
Proof. Consider the following example with K = 3: When everyone bids truthfully, 1 and 2 are allocated first and second slots respectively, and 3 is denied
(third slot empty). In the absence of 2, the allocation would be to give 1 the first
slot and 3 the second and third slots. Also, by removing 2 from the first slot we
get that 1 is the only ad shown in the second slot. This means
v1
v3
v1
CTR2,1 p2 = (CTR1,1 + CTR3,2 ) − (CTR1,2 )
s1
s3
s1
= (0.5 · 2 + 0.1 · 0.5) − (0.25 · 2) = 0.55
u2 = v2 CTR2,1 − p2 CTR2,1 = 3 · 0.5 − 0.55 = 0.95
Suppose bidder 2 now lies and changes his desired ad size to s = 2. The new
allocation will have 1 at the first slot and 2 at the second slot (since CTR2,1 vs2 =
0.75) and 3 is denied. Again, in the absence of 2, the allocation would be to give
1 the first slot and 3 the second and third slots. Also, by removing 2 from the
second slot we get that 1 is the only ad shown in the first slot. We get
v1
v3
v1
CTR2,2 p02 = (CTR1,1 + CTR3,2 ) − (CTR1,1 )
s1
s3
s1
= (0.5 · 2 + 0.1 · 0.5) − (0.5 · 2) = 0.05
u02 = v2 CTR2,2 − p02 CTR2,2 = 3 · 0.5 − 0.05 = 1.45
which is a better utility 2 gets by lying about his size.
Potential issues include:
– This tells us that the price should somehow incorporate the size of i’s bid.
– The price should also incorporate the damage done to the advertisers ranked
below i and not just the advertisers who have their bids denied.
– The click through rates will likely change depending on the size of the ad
(i.e. CTR is a function of advertiser, rank, and size of ad). The case of known
ad sizes we did not need to worry about CTR changing as ad size changed.
However, realistically CTR will increase as ad sizes increase. Therefore in
the unknown ad sizes case we should ideally incorporate this issue into the
model. But of course, doing so would further change our meaning of “singlemindedness”.
6
Open Problem: An Interesting Mixed Type
Single-Minded Auction Design Problem
The work in [4] gives us a truthful auction for single-minded bidders for a finite set of goods. The work in [3] uses the framework of [4] to design mech-
anisms for single-minded bidders for geometrical goods (i.e. goods is R2 and
each bidder wants a convex object at a specific location). What if we want to
design mechanisms for goods that have are both geometrical and discrete? E.g.
Goods = {[0, 1], a, b, c, d} bidder 1 is interested in {[0, 0.3], b, d}.
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