Coupon Design in Advertising Systems - Weiran Shen,1 Pingzhong Tang, 2 Xun Wang, 2 Yadong Xu, 2 Xiwang Yang 3 - PRELIMINARY ...
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PRELIMINARY VERSION: DO NOT CITE
The AAAI Digital Library will contain the published
version some time after the conference
Coupon Design in Advertising Systems
Weiran Shen,1 Pingzhong Tang, 2 Xun Wang, 2 Yadong Xu, 2 Xiwang Yang 3
1
Renmin University of China
2
Tsinghua University
3
ByteDance
shenweiran@ruc.edu.cn, kenshinping@gmail.com, wxhelloworld@outlook.com,
xuyd17@mails.tsinghua.edu.cn,yangxiwang@bytedance.com
Abstract auction (Vickrey 1961; Clarke 1971; Groves et al. 1973)
which is used by Facebook and ByteDance. The difference
Online platforms sell advertisements via auctions (e.g., VCG between them is that the VCG auction is truthful, i.e., ad-
and GSP auction) and revenue maximization is one of the
vertisers submit bids that truthfully reveal their valuations,
most important tasks for them. Many revenue increment
methods are proposed, like reserve pricing, boosting, coupons while the GSP auction is not. However, it has been proved
and so on. The novelty of coupons rests on the fact that that the GSP auction has a Nash equilibrium whose outcome
coupons are optional for advertisers while the others are com- is equivalent to that of the VCG auction. Therefore, in this
pulsory. Recent studies on coupons have limited applications paper, we study how to maximize the revenue in the VCG
in advertising systems because they only focus on second auction.
price auctions and do not consider the combination with other In marketing, a coupon is a ticket or document that can
methods. In this work, we study the coupon design problem be redeemed for a financial discount or rebate when pur-
for revenue maximization in the widely used VCG auction.
Firstly, we examine the bidder strategies in the VCG auction
chasing a product. It is effective in promoting sales and
with coupons. Secondly, we cast the coupon design problem attracting new customers in numerous realistic scenarios,
into a learning framework and propose corresponding algo- and consumers can find coupons everywhere, ranging from
rithms using the properties of VCG auction. Then we further the wrappings of food, to webpage banners. Coupons have
study how to combine coupons with reserve pricing in our been widely studied in economics (Bester and Petrakis 1996;
framework. Finally, extensive experiments are conducted to Moraga-González and Petrakis 1999; Kang et al. 2006;
demonstrate the effectiveness of our algorithms based on both Board and Skrzypacz 2016). The most common arguments
synthetic data and industrial data. developed by economists to explain the use of coupons
are twofold: 1) Coupons allow for price discrimination. 2)
Coupons allow for peak-load pricing (Mckenzie 2008). In
Introduction this work, coupons play the role of price discrimination.
Online advertising has been one of the most important in- The intuition of revenue increment is that low-valued ad-
dustry on the Internet. In the United States, its revenue has vertisers would spontaneously bid higher if they are pro-
grown by 15.9 percent in 2019 compared to 2018, surpassing vided with coupons, which in turn forces the high-valued
124$ billion. 1 Advertising has become one of the key rev- advertisers to pay more when they win based on the pay-
enue sources for many Internet companies, such as Google, ment rule of the VCG auction. In fact, many methods can
Facebook, ByteDance. For example, more than 83.8 percent be used to increase revenue, such as reserve price (Hartline
of Google’s revenue comes from online advertising. And the and Roughgarden 2009), boosting (Golrezaei et al. 2017),
market still shows no signs of slowing down. There are var- squashing (Lahaie and Pennock 2007a), anchoring (Lahaie
ious types of online advertising, for example, social media and Pennock 2007b). Coupons are significantly different
advertising (e.g., Instagram and Facebook), paid search ad- from them. As for coupons, advertisers hold the right to de-
vertising (e.g., Google and Baidu), and native advertising cide whether to use them. However, they cannot refuse to use
(e.g., BuzzFeed and Tiktok), and so on. These advertise- the other methods, e.g., advertisers always want to remove
ments are usually sold via auction mechanisms. The mecha- the reserve price but they can not. In fact, sometimes reserve
nisms determine which advertisements will be presented and prices may have a negative effect on revenue (Ostrovsky and
how much the corresponding advertisers need to pay. There Schwarz 2011). Higher reserve prices make the auction less
are two widely used auctions, i.e., the GSP auction (Edel- attractive which results in fewer participated advertisers.
man, Ostrovsky, and Schwarz 2007) (generalized second In this work, we generalize (Shen et al. 2020b) to the VCG
price) which is used by Google and Baidu, and the VCG auction, including the combination with reserve pricing. The
Copyright c 2021, Association for the Advancement of Artificial intuition is that the VCG auction (with heterogeneous slots)
Intelligence (www.aaai.org). All rights reserved. can be decomposed into some tractable sub-VCG auctions
1 (with homogeneous slots). Our contributions can be sum-
https://www.statista.com/statistics/183816/us-online-
advertising-revenue-since-2000/ marized as follows.• We generalize the application of coupons for revenue Preliminaries
maximization in advertising systems to more realistic sce- In this section, the setting of VCG auction with coupons in
narios (i.e., VCG auctions). advertising systems will be introduced first. Then the formal
• We propose algorithms for coupon optimization using the problem of learning to design coupons will be defined.
properties of the VCG auction, along with the combina-
tion with reserve pricing. VCG Auction with Coupons in Advertising
• Based on both synthetic data and industrial data, exten- Systems
sive experiments are conducted to demonstrate the effec- There are n bidders [n] = {1, 2, . . . , n} and m slots (usually,
tiveness of our algorithms. n m). We will refer to ‘ad i’ as the advertisement submit-
ted by bidder i. Each bidder can occupy at most 1 slot and
Related Works each slot can be allocated to at most 1 bidder. Let vi be the
Our work is related to the revenue-maximizing mechanism private value which quantifies the value of a click for bidder
design. Myerson (1981) studies the optimal auction where i. And vi is drawn from a publicly known distribution Fi .
advertisers should be ranked in descending order of their For convenience, let v = (v1 , . . . , vn ) denote the value pro-
virtual values. If nobody has positive virtual values, the item file, and let v−i = (v1 , . . . , vi−1 , vi+1 , . . . , vn ) denote the
would not be sold to anyone. However, Roughgarden and value profile of all bidders except bidder i. Similarly, we can
Schrijvers (2016) argues that this setting relies heavily on define the bid profile b and b−i . We only use αj ∈ [0, 1] to
the exact estimation of the advertisers’ value distribution and denote the click-through-rate of ad i when it is allocated to
can be sensitive to estimation errors. Thus, simpler mecha- slot j as (Edelman, Ostrovsky, and Schwarz 2007) do, which
nisms are applied in industry, e.g., VCG and GSP auction. implies that the number of times a particular slot is clicked
Many methods have been proposed to increase revenue in does not depend on the ad in this slot. The reason is that the
these auctions. Reserve pricing is one of the most widely click-through-rate can be decomposed into position effect
studied methods: Hartline and Roughgarden (2009) presents and ad effect while the ad effect can be taken into account in
how to use a reserve-price-based VCG auction to approxi- the private valuation. Besides, αj ≥ αj 0 holds when j < j 0 .
mate the optimal one. Jin et al. (2019) proves a tight approx- In response to the bid profile, VCG auction determines how
imation ratio for anonymous reserve prices. In (Zeithammer to allocate each bidder and how much each bidder has to
2019), there is a bid level below which a winning bidder pay according to some allocation rule and payment rule. To
has to pay according to the first-price auction instead of the be specific, the allocation rule is a function π that maps the
second-price auction with reserve. Another method is boost- bid profile b to an n-dimensional vector indicating the quan-
ing (Golrezaei et al. 2017) where the seller assigns a boost tity of clicks allocated to each bidder, i.e., π : Rn 7→ [0, 1]n ;
value to each advertiser which can transform his regular bid The payment rule is a function p : Rn 7→ Rn+ that maps the
into a boosted one. Besides, squashing (Lahaie and Pennock bid profile b to an n-dimensional non-negative vector spec-
2007a) and anchoring (Lahaie and Pennock 2007b) are also ifying the payment for each bidder. For simplicity, we use
widely used methods in industry (Chawla, Fu, and Karlin π(b) and π, p(b) and p interchangeably. If bidder i is allo-
2014; Huang, Mansour, and Roughgarden 2015; Tang and cated to slot j, then πi = αj , and if bidder i does not win
Wang 2016; Tang and Sandholm 2011). the auction, then πi = 0. These rules are specified as:
Pn
Our work is also related to automated mechanism de- • Allocation rule: π(b) ∈ arg maxπ0 i=1 πi0 bi .
sign. Conitzer and Sandholm (2002) introduces the auto-
• Payment rule: pi (b) = maxπ0 i0 6=i πi00 bi0 − i0 6=i πi0 bi0 .
P P
mated mechanism design approach, where the mechanism
is computationally created for the specific problem instance Coupons work as follows. Before submitting bids, each
at hand. Learning theory tools such as pseudo-dimension bidder i will be informed that he will be provided with a
have been used to prove strong guarantees in auction set- constant coupon ci ≥ 0. This coupon can get ci off the
tings (Morgenstern and Roughgarden 2015, 2016). In a sim- payment when he wins. Hence bidders would change their
ilar direction, bounds on the sample complexity have also regular bids depending on their values and coupons, i.e.,
been developed (Hsu et al. 2016). Mohri and Medina (2014) bi = bi (vi ; ci ). Therefore, the allocation rule and payment
uses DC programming to learn anonymous reserve price rule of VCG auction with coupons would be:
for the second-price auction. Derakhshan, Golrezaei, and n
Leme (2019); Derakhshan, Pennock, and Slivkins (2020) X
π(b) ∈ arg max πi0 bi ,
use linear programming to compute personalized reserve 0 π
i=1 (1)
prices in auction mechanism. Duetting et al. (2019) uses X X
deep learning to design revenue-maximizing mechanisms pi (b) = max
0
πi00 bi0 − πi0 bi0 − πi ci .
π
and deals with the incentive constraint via regret network. i0 6=i i0 6=i
Shen, Tang, and Zuo (2019) also handles the same problem
but can always return exactly incentive compatible mecha- The utility of bidder i is ui (b) = πi vi − pi , and the revenue
nisms. Recently, Tang (2017) uses reinforcement learning to of platform would be:
design and optimize mechanisms in dynamic industrial en- n
vironments and has achieved success in industry (Cai et al.
X
Rev(v, c) = pi . (2)
2018a,b; Shen et al. 2020a). i=1In what follows, let (i) denote the order of bidders satisfying Definition 1 (No-feature case). In this case, features are
if i < i0 , then [b(i) > b(i0 ) ] ∨ [(b(i) = b(i0 ) ) ∧ (c(i) ≤ c(i0 ) )], not considered, and the coupon optimization problem is to
that is, (i) denotes the bidder with the i-th highest bid (if minimize the following empirical loss:
there are more than one, then choose the bidder with the T
least coupon). 1X
LS (c) = − Rev(v t ; c), (5)
T t=1
Learning Formulation where c is an n−dimensional vector specifying coupons
There are two sets of data – the training data and the testing which need to be optimized.
data, containing features and bids. Both sets of data are gen- Proposition 2. In the no-feature case, the optimization of
erated by VCG auction without coupons. And our goal is to coupons in VCG auction (i.e., Equation (5)) is NP-hard.
train a model which can provide coupons for bidders in each
Proposition 2 points out coupon optimization is hard,
auction and maximize the revenue for platform. Since VCG
so we use the coordination descent method to optimize
auction is a truthful mechanism, in these sets of data, we can
coupons. In this method, instead of optimizing coupons si-
regard bids as bidders’ values.
multaneously, we optimize one of them while fixing the oth-
Let us define the problem formally. We consider a generic ers at each time step.
feature space X with the label space B = Rn+ consisting
of the value profile v. We regard (xt , v t ) as a data instance Comparison with Related Work
where xt , v t denote the feature vector and value profile in As proposition 1 demonstrated, although coupons are op-
auction t, respectively. The hypothesis function h : X → tional for advertisers, they still accept and use all coupons
Rn+ is used to set coupons ct = h(xt ). Thus, the objective is in equilibrium. We will compare the difference between
to select a hypothesis function h out of some hypothesis set coupons and other involuntary discount mechanism, like
H to minimize the corresponding empirical loss: boosting (Golrezaei et al. 2017) and bidding credits in
T squashing (Lahaie and Pennock 2007a). Boosting and bid-
1X ding credits both multiply the submitted bids with a factor,
LS (h) = − Rev(v t ; h(xt )). (3)
T t=1 i.e., ai ∗ bi . As for boosting, there is no restriction on ai .
While for bidding credits, ai is between 0 and 1. Then in the
where S = ((x1 , v 1 ), . . . , (xT , v T )). Let αjt , pti and πit de- VCG auction, there are two differences.
note the corresponding notations in auction t. The goal is to • When 0 ≤ ai < 1, advertiser i prefer not to accept boost-
learn the predictor which can work for any advertiser if cor- ing or bidding credits. However, as for coupons, advertis-
responding features can be provided. For the sake of descrip- ers’ dominant strategy would always be accepting them.
tion, we use a new hypothesis function hi to set the coupon
• Coupons can increase the revenue in some cases where
for bidder i as cti = hi (xti ), where xti denotes the feature for
boosting can not. For example, consider one slot and two
bidder i in auction t. For convenience, in what follows we
bidders whose values are 0 and 1. Then, boosting can not
will omit the superscript t when there is no confusion.
increase the revenue since ai multiplies 0 still equals 0.
It is worth noting that all proofs can be found in supple- While for coupons, this would not be a problem.
mentary materials due to limited space.
Comparing with the second-price setting, several non-
trivial technical issues are raised in the VCG setting. For
Problem Analysis example, Shen et al. (2020b) relies heavily on the fact that
In this section, we will introduce the property of coupons given value profile v and coupons profile c−i , Rev(v; c) (as
and show the difficulty of coupon optimization first. Then, a function of ci ) has at most one discontinuous point. Hence
the comparison with related work would be presented. the surrogate loss function can be decomposed as a differ-
ence of two convex functions and DC programming can be
Proposition 1. In VCG auction with coupons, the dominant applied. However, in the VCG setting, the revenue function
strategy of bidder i is using coupon ci and bidding vi + ci . is more complicated as Proposition 3 denotes.
Proposition 1 denotes the dominant strategy of bidders in Proposition 3. In VCG auction with coupons, given value
VCG auction with coupons. Therefore, we can use vi + ci to profile v and the coupon profile c−i , Rev(v; c) (as a func-
denote the bid of bidder i when he is provided with coupon tion of ci ) is not continuous, and is neither convex nor con-
ci in experiments. Thus we can define the revenue of VCG cave. Actually, there can be m discontinuous points.
auction with coupons as Equation (4). We use Example 1 to demonstrate Proposition 3. As illus-
m X
X m m
X trated in Figure 1, Rev(v; c) (as a function of ci ) is a piece-
Rev(v; c) = αi b(i+1) − αi b(i) − αj c(j) wise function. There can be multiple discontinuous points.
j=1 i=j i=j+1 They divide the function into multiple line segments, which
m h i can have different slopes.
X
= j · (αj − αj+1 )(v(j+1) + c(j+1) ) − αj c(j) Example 1. Let n = 6, m = 4, v = (5, 4, 3, 2, 1, 0) and
j=1 α = (1.0, 0.4, 0.3, 0.1). When c−6 = 0, the value of revenue
(4) with respect to c6 is demonstrated in Figure 1.have many local minima (Mohri and Medina 2014). We ex-
5 tend the technique in (Mohri and Medina 2014; Shen et al.
2020b) to tackle the special case:
The value of revenue
4
1. We smooth Equation (6) to derive a surrogate loss func-
3 tion Lγ (y; v, c−i ), which is training-friendly and has a
2 good approximation of Equation (6) theoretically.
1 2. We decompose Lγ as the difference of two convex func-
tions, i.e., Lγ = g1 −g2 . Then we apply DC programming
0 1 2 3 4 5 6 and coordination descent method to optimize the coupons.
The value of c6
Due to limited space, detailed formulas of Lγ and algorithms
Figure 1: The value of revenue w.r.t. c6 are presented in the supplementary materials.
Solution for the General Case
Furthermore, when reserve price is taken into considera- In this section, we will discuss how to extend the afore-
tion, the revenue function can be much more complicated, mentioned special case into the general case, i.e., the click-
making the coupon optimization harder. through-rates of different slots are also different. We decom-
pose each VCG auction into m sub-VCG auctions which
Warm up belong to the special case, and complete the coupon opti-
In this section, we will introduce a special case of VCG auc- mization based on the decomposition. Then we study how
tion with coupons. As Proposition 4 denotes, we can extend to combine coupons with reserve prices in our framework.
the technique in (Mohri and Medina 2014; Shen et al. 2020b)
to tackle the special case. Constructing m sub-VCG Auctions
Definition 2 (Special case). In this case of VCG auction
with coupons, the slots are homogeneous, i.e., α1 = α2 = Algorithm 1 Constructing m sub-VCG auctions
· · · = αm = α. 1: Initialize Q = ∅.
Let subscript (j) only represent the order of bids. We 2: for j = 1 to m do
use b−i,(j) = v−i,(j) + c−i,(j) to denote the j-th high- 3: Let dj = αj − αj+1 , and construct a sub-VCG auc-
est bid in bid profile b−i , that is, if j < l, then the order tion Aj as: there are n bidders, j slots, and each slot
[b−i,(j) > b−i,(l) ] ∨ [(b−i,(j) = b−i,(l) ) ∧ (c−i,(j) ≤ c−i,(l) )] has the same click-through-rates dj .
is ensured. Let L(y; v, c−i ) = −Rev(v; (y, c−i )) be the 4: Q = Q ∪ {Aj }.
negative number of the revenue as a function of ci = y, 5: end for
which can be rewritten as:
L(y; v, c−i ) The decomposition process is demonstrated in Algo-
Pm rithm 1. There are m steps. At the j-th step, we construct
−mαb−i,(m+1) + α
m
j=1 c−i,(j) , y ≤ b−i,(m+1) − vi the j-th sub-VCG auction, where there are j slots and n
bidders, and all slots share the same click-through-rates dj .
P
[−mαb−i,(m) + α j=1 c−i,(j)
= −α(c−i,(m) − y)+ ], y = b−i,(m) − vi Thus, the j-th sub-VCG auction belongs to the special case.
Pm−1 Furthermore, Theorem 1 denotes that given value profile v
αy − mαb−i,(m) + α j=1 c−i,(j) , y > b−i,(m) − vi
and coupon profile c, the results of these m sub-VCG auc-
Pm
−mαy − mαvi + α j=1 c−i,(j) , otherwise
tions are equivalent to those results of the original VCG auc-
(6) tion for both platform (in terms of revenue) and each bidder
(in terms of allocation and payment).
Here x+ means max {x, 0}. As Proposition 4 states,
Theorem 1. Given value profile v and coupon profile c, the
L(y; v, c−i ) has at most one discontinuous point, which is
original VCG auction and these sub-VCG auctions would
a reduction of Proposition 3.
receive the same bid profile b. Besides, for bidder i, his total
Proposition 4. In the special case, given the value pro- payment and allocation (the sum of obtained click-through-
file v and the coupon profile c−i , L(y; v, c−i ) has at most rates) within these sub-VCG auctions are equivalent to those
one discontinuous point at y = b−i,(m) − vi . When y < in the original VCG auction.
b−i,(m) − vi , L(y; v, c−i ) is non-increasing and concave. As a result, we present Theorem 2 to denote that by com-
When y > b−i,(m) − vi , L(y; v, c−i ) is an increasing lin- puting the optimal coupon for bidder i in sub-VCG auctions,
ear function. What’s more, for all the cases, L(y; v, c−i ) the optimal coupon for bidder i in the original VCG auction
achieves its minimum at y = (b−i,(m) − vi )+ . can also be obtained.
However, L(y; v, c−i ) is still not a good choice for opti- Theorem 2. Given v and c−i , the optimal coupon c∗i for
mization since L(y; v, c−i ) is neither convex nor concave, bidder i in the original VCG auction belongs to the follow-
but quasi-convex in fact. And a sum of quasi-convex func- ing sets
tions does not maintain the quasi-convex property and can {0} ∪ {(b−i,(j) − vi )+ }j∈[m] ,where (b−i,(j) − vi )+ is the optimal coupon for bidder i in Algorithm 3 Algorithm for the general case
the j-th sub-VCG auction Aj . 1: Run Algorithm 1 to construct mT sub-VCG auctions.
Combine with the coordinate descent method, we propose 2: Initialize ω ← 0 and kout = 0, and generate a random
Algorithm 2 to optimize coupon for the no-feature case. permutation of 1 to n as N̄ .
Here λ in line 7 denotes the learning rate, and Kout in 3: repeat
4: Set ωp ← ω and kout ← kout + 1.
5: for i = N̄1 to N̄n do
Algorithm 2 Algorithm for the no-feature case
6: Use ω to initialize ωi0 , and calculate coupons for
1: Run Algorithm 1 to construct mT sub-VCG auctions. bidders except i via ctj = ω · xtj + c0 .
2: Initialize c ← 0 and kout = 0, and generate a random 7: for k = 1 to K do
permutation of 1 to n as N̄ . 8: ωik ← DCA(ωik−1 ).
3: repeat 9: end for
4: Set cp ← c and kout ← kout + 1. 10: Use λωiK + (1 − λ)ω to update ω.
5: for i = N̄1 to N̄n do 11: end for
6: Fix c−i , calculate the optimal coupon via evaluat- 12: Set Ψ = {0.1 · (10∆/kωk)0.1j |j = 0, 1, . . . , 10},
ing Equation (5) at the following O(mT ) points compute η ∗ ∈ arg minη∈Ψ LS (η · ω) and use η ∗ · ω
and returning the best y ∗ . to update ω.
t∈[T ]
13: Generate a new random permutation N̄ if LS (ω) ≥
{0} ∪ {(bt−i,(j) − vit )+ }j∈[m] LS (ωp ).
14: until kω − ωp k ≤ or kout = Kout
7: Use λy ∗ + (1 − λ)ci to update ci .
8: end for
9: Generate a new random permutation N̄ if LS (c) ≥ following optimization problem (Equation (8)),
LS (cp ).
T X
m
10: until kc − cp k ≤ or kout = Kout X
min stj − δG2 (ωik−1 ) · ωi s.t. ∀j ∈ [m]
||ωi ||≤Λ,s
t=1 j=1
line 10 are used to determine when the algorithm terminates. t
ci = ωi · xti + c0 , t ∈ [T ]
t t t t
Pj−1 t j j j
s /d ≥ c − jb + c , t ∈ C ∪ C 3 ∪ C4
Algorithm Design for the General Case
j j i −i,(j) l=1 −i,(l) 1
t,j
t ∈ C2j
t t t t
sj /dj ≥ −jci − jvi + fi,1 ,
The hypothesis set H consists of linear functions whose un- et,j
biased term is bounded, i.e., H = {h : xi 7→ ω · xi + stj /dtj ≥ (1 + γdit,j )(cti − dt,j t t,j
i,1 ) − jb−i,(j) + fi,1 , t ∈ C2
j
i,1
c0 |kωk ≤ ∆} and c0 is a positive constant. Besides, ∆
et,j
stj /dtj ≥ ( it,j − j)(cti − dt,j t t,j
i,1 ) − jb−i,(j) + fi,2 , t ∈ C3
j
is chosen to satisfy ∆ ≤ maxckx0
ik
, which guarantees that
γdi,1
φt,j
sj /dj ≥ αit (cti − dt,j t,j
t ∈ C4j
t t t
coupons are non-negative because i,1 ) − jb−i,(j) + fi,2 ,
(8)
ω · xi + c0 ≥ c0 − |ω · xi | ≥ c0 − kωkkxi k
where G2 (ωik−1 ) = k−1
· xti + c0 ; v t , ct−i , Atj )
P
≥ c0 − ∆ · max kxi k ≥ 0. t,j g2 (ωi
k−1
and δG2 (ωi ) denotes an arbitrary element of the sub-
It is worth noting that max kxi k can be bounded once we
gradient ∂G2 (ωik−1 ). Besides, Ckj , dt,j t,j t,j t,j
i,1 , di,2 , ei , fi,1 , fi,2
t,j
normalize the feature space.
Note that Theorem 1 states that the results of the m sub- and φt,ji are similar to the definitions in the special case with
VCG auctions are equivalent to those of the original VCG just a substitution from m to j as Equation (9).
auction, which holds over the T VCG auctions, thus the em- dt,j t t t,j t t t,j t t
i,1 = b−i,(j) − vi , di,2 = b−i,(j+1) − vi , ei = v−i,(j) − vi ,
pirical surrogate loss can be written as Xj Xj−1
t,j t,j
fi,1 = ct−i,(l) , fi,2 = ct−i,(l) + dt,j
i,1 ,
T X
X m l=1 l=1
Lγ (ωi · xti + c0 ; v t , ct−i , Atj ), (7) φt,j t t,j t t t,j
i = dj /γdi,1 · (j(b−i,(j+1) − b−i,(j) ) + ei ),
t=1 j=1
C1j = {t|vit ≥ bt−i,(j) }, C2j = {t|vit < bt−i,(j) , vit ≤ v−i,(j)
t
},
where Atj denotes the j-th constructed sub-VCG auction of C3j = {t|vit < bt−i,(j) , vit > v−i,(j)
t
, (1 − γ)dt,j t,j
i,1 ≥ di,2 },
the t-th VCG auction by executing Algorithm 1, and the cor-
responding click-through-rate is dtj = αjt − αj+1
t
. It is worth C4j = {t|vit < bt−i,(j) , vit > v−i,(j)
t
, (1 − γ)dt,j t,j
i,1 < di,2 }.
t
noting that Aj belongs to the special case. Follow the work (9)
in (Mohri and Medina 2014; Shen et al. 2020b), we also use
the technique DC programming for optimization. Therefore, Combination with Reserve Price
we complete the algorithm design for the general case as We combine reserve prices with coupons to further improve
Algorithm 3. Here DCA(ωik−1 ) (line 8) means to solve the the revenue by eliminating the negative payments caused bylarge coupons. Let ri + ci denote the eager reserve price for of features for each bidder in each auction is 178. Besides,
bidder i where ri is independent of ci , the eager reserve price all the bids are transformed into the interval (0, 10), and
works as below. features are normalized to satisfy kxi k ≤ 10. We choose
1. The auction first eliminates the bidder who does not clear c0 = 10 so that ∆ = 1.
his reserve price, then the remaining bidders form the can-
Surrogate Loss vs. Real Loss
didate set S = {i|bi ≥ ri + ci }.
We use synthetic data to show the difference between Lγ and
2. Bidders in the candidate set S are allocated according to L. For a target bidder, we randomly select related T = 50
the allocation rule of VCG auction. value profiles and generate others’ coupons. Without loss of
3. The payment rule can be modified as follows. If bidder i generality, the click-through-rates are all 1. As Figure 2(a)
gets the j-th slot, then he need to pay show, LγS is a lower bound for LS , and the difference be-
m
tween two functions is relatively small. Besides, Figure 2(b)
X denotes that we can approach the real loss function as we
(αk − αk+1 ) max{b(k+1) , ri + ci } − αj ci , (10)
select sufficiently small γ. Similar results are also shown
k=j
on the industrial data (see in supplementary materials). Al-
where b(k+1) = 0 holds for k = |S|, . . . , m if |S| ≤ m. though small γ is necessary to guarantee the precision accu-
racy, too small γ could make the surrogate loss function tend
It is worth noting that the payment (Equation (10)) must to be discontinuous. γ is chosen to be 0.1 in the experiments.
be non-negative. Furthermore, we use Proposition 5 to
show the dominant strategy of bidders in VCG auction with
coupons when we combine reserve prices. Thus we can still
use vi + ci to denote the bid of bidder i when he is provided
with coupon ci in experiments.
Proposition 5. In VCG auction with coupons {ci } and re-
serve prices {ri + ci }, the dominant strategy of bidder i is
using coupon ci and bidding vi + ci .
Note that ri is independent of ci , we can naturally inte- (a) (b)
grate the reserve prices into the framework of VCG auction
with coupons once we optimize the coupon for each bidder. Figure 2: Plot of the difference between LS and LγS on the
For example, if we simply select ri = 0 for bidder i, then synthetic data. (a) LS and LγS as the function of coupon y
the reserve price for bidder i is ci . Bidder i will always clear when γ = 0.09, where x-axis denotes the coupon y while
his reserve price, but his payment can never be negative. y-axis denotes the value of loss. (b) average difference be-
tween LS and LγS as the function of γ, where x-axis denotes
Experiment the choice of γ while y-axis denotes the difference.
In this section, we first verify the validity of the design of
surrogate loss function by comparing it with the real loss Demonstration of Training Phase
function. Then we examine the impact of hyper-parameters We implement the algorithms for coupon optimiza-
of our algorithms and verify the properties of coupons. Fi- tion, where Algorithm 3 is implemented using Gurobi
nally, we conduct some experiments to show the effective- 9.0 (Gurobi Optimization 2020). We first examine the im-
ness of our algorithms against state-of-the-art algorithms. pact of learning rate λ in terms of revenue. The results
are shown as Figure 3(a), where Alg-2 and Alg-3 repre-
Data Description sent Algorithm 2 and 3 respectively and λ is selected from
We use both synthetic data and industrial data to demon- {0.01, 0.05, 0.1, 0.5}. We can see that Alg-3 always yields
strate the results of our experiments.
As for synthetic data, we choose three different types of
distribution to sample the value data, i.e., the uniform dis-
tribution, the Pareto distribution and the lognormal distri-
bution. To be specific, we sample T numbers from a stan-
dard distribution with fixed parameters for each bidder (i.e.,
X ∼ U (0, 1) or ln X ∼ N (0, 1)), and then each bidder
is associated with a random scale size to multiply these T (a) (b)
numbers as his values in T auctions.
As for industrial data, it comes from one of the biggest Figure 3: Plot of training process. (a) revenue curve, where
short-form mobile video community in the world. We ran- x-axis denotes the iteration while y-axis denotes the corre-
domly extract 177 million auctions from the log. Each ad- sponding revenue gain. (b) ratio curve, where x-axis denotes
vertiser instance contains many features, including labels, the iteration while y-axis denotes the corresponding ratio of
industry category, pricing type (i.e., cost per click, cost per negative payments.
mille or cost per action), budget and so on, and the number better performance than Alg-2 since it can utilize features.Algorithms may not converge within 20 iterations when λ Table 1: Performance (i.e. ρa ) on synthetic data when m =
is small, i.e., λ ≤ 0.1 in Alg-2 and λ = 0.01 in Alg-3. 4. Numbers in the brackets denote standard deviations.
As for Alg-2, larger λ can achieve better performance, thus
we choose λ = 0.5 in Alg-2 to guarantee convergence and Method Uniform Pareto Lognormal
achieve better performance. While for Alg-3, although larger ARP 2.58(1.05) 8.88(2.38) 2.92(3.24)
λ can have higher revenue in some iterations, it is unstable SRP 12.16(5.98) 9.80(3.36) 21.86(9.44)
during the training phase. Hence λ is set to 0.05 in Alg-3 BVCG 7.51(3.32) 1.83(1.27) 6.94(2.80)
to maintain robustness and obtain comparable performance. Alg-2 13.91(5.77) 3.18(2.57) 10.36(4.77)
Besides, in the remaining experiments, we use = 0.001 Alg-2a 13.94(5.75) 3.28(2.57) 10.41(4.75)
and Kout = 20 in both algorithms. Alg-2b 15.75(6.07) 9.94(3.36) 22.54(9.15)
We further verify the property that the payment can be
negative for some bidder when the coupon provided for him
is too large. We calculate the ratio of negative payments after erson 1981) via producing similar allocation (see in sup-
#{(i,t)|pti ≤0}
each iterations, i.e., mT . As shown in Figure 3(b), plementary materials). For other distributions, Alg-2 always
the payment of some bidder can be negative, but the ratio has better performance than BVCG. Thirdly, results show
of negative payments is small (< 0.5%). Besides, the ratio the effectiveness of combining reserve prices with coupons.
increases to improve the revenue in the beginning, but then Alg-2(a) outperforms Alg-2 by eliminating the negative pay-
tends to decrease. What’s more, the final ratio of negative ments and Alg-2(b) performs the best among above methods
payments in Alg-3 is smaller than that in Alg-2. in all datasets by setting bidder-specific reserve prices. Sim-
ilar conclusions can also be seen in the results when m 6= 4,
Revenue Comparison we demonstrate them in the supplementary materials.
We conduct some experiments to show the effectiveness As for industrial data, We first divide 177 million into 59
of coupons and our algorithms. The performance metric is groups. Then in each group, auctions are divided into train-
a −Rev0
ρa = RevRev × 100%, where Reva represents the rev- ing data and testing data. After training, we summarize the
0
average results on these 59 groups as Table 2. On the indus-
enue achieved through the corresponding algorithm a and
Rev0 denotes the revenue obtained without these methods
in VCG auction. The following methods are compared: Table 2: Performance (i.e. ρa ) on the industrial data.
• (ARP) This method sets anonymous reserve price in VCG Method training data testing data
auction (Mohri and Medina 2014). ARP 0.00(0.00) 0.00(0.00)
• (SRP) This method sets bidder-specific reserve price in SRP −1.05(1.13) −2.86(4.01)
VCG auction via maximizing vi (1−Fi (vi )) (Hartline and BVCG 1.19(0.36) 0.13(0.75)
Roughgarden 2009). Alg-2 1.33(0.57) 0.32(0.29)
• (BVCG) This method assigns a boost value for each bid- Alg-2a 0.36(0.28)
der in VCG auction (Golrezaei et al. 2017). Alg-3 4.74(3.95) 4.86(3.81)
Alg-3a 4.87(3.81)
• (Alg-2a/Alg-3a) This method combines Algorithm 2 or 3
with reserve prices ri + ci where ri = 0.
trial data, Alg-3(a) outperforms others by a large margin,
• (Alg-2b/Alg-3b) This method combines Algorithm 2 demonstrating the effectiveness of our mechanisms. More
or 3 with reserve prices ri + ci where ri belongs to details can be found in supplementary materials.
arg maxvi vi (1 − Fi (vi )).
Note that (Mohri and Medina 2014; Golrezaei et al. 2017) Conclusion
only study the second price setting, we extend their methods
In this paper, we study how to design coupons in VCG auc-
to VCG auction as ARP and BVCG (details are provided in
tion via a learning framework. Firstly, we derive the dom-
supplementary materials).
inant strategy of each bidder and characterize the coupons.
As for synthetic data, given the type of distribution and Then we start from a special case where slots are homoge-
the number of slots m, we randomly sample the value data neous and extend this special case to the general one via auc-
and divide the whole dataset into training data and testing tion decomposition and complete the algorithm design. We
data, with training data makes up about 70%. Then we run also combine coupons with reserve prices in our framework.
different algorithms on the training data and calculate ρa on Finally, extensive experiments are conducted to demonstrate
the testing data. This procedure is repeated for 20 times and the effectiveness of the algorithms.
the results when m = 4 are summarized as Table 1 shows.
We can draw the following conclusions from Table 1.
Firstly, coupons work well on these synthetic datasets as Acknowledgments
Alg-2 can increase the revenue by a large margin. Sec- This work is is supported by Beijing Academy of Artifi-
ondly, Alg-2 always outperforms ARP, SRP and BVCG on cial Intelligence (BAAI), Science and Technology Innova-
the datasets sampled from uniform distribution. This is be- tion 2030—“New Generation Artificial Intelligence” Major
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