Optimal Auction Design in the Joint Advertising
Authors: Yang Li
Yuchao Ma
Qi Qi
Venue: ICML 2025 poster (2025)
Keywords: Joint Advertisement
Auction Design
BundleNet
Primary area: applications->everything_else

Abstract: Online advertising is a vital revenue source for major internet platforms. Recently, joint advertising, which assigns a bundle of two advertisers in an ad slot instead of allocating a single advertiser, has emerged as an effective method for enhancing allocation efficiency and revenue. However, existing mechanisms for joint advertising fail to realize the optimality, as they tend to focus on individual advertisers and overlook bundle structures. This paper identifies an optimal mechanism for joint advertising in a single-slot setting. For multi-slot joint advertising, we propose **BundleNet**, a novel bundle-based neural network approach specifically designed for joint advertising. Our extensive experiments demonstrate that the mechanisms generated by **BundleNet** approximate the theoretical analysis results in the single-slot setting and achieve state-of-the-art performance in the multi-slot setting. This significantly increases platform revenue while ensuring approximate dominant strategy incentive compatibility and individual rationality.

Optimal Auction Design in the Joint Advertising
Yang Li Yuchao Ma
Qi Qi 123
Abstract
fic thus becomes fundamental research direction for these companies.
Online advertising is a vital revenue source for major internet platforms. Recently, joint adver- tising, which assigns a bundle of two advertis- ers in an ad slot instead of allocating a single advertiser, has emerged as an effective method for enhancing allocation efficiency and revenue. However, existing mechanisms for joint advertis- ing fail to realize the optimality, as they tend to focus on individual advertisers and overlook bun- dle structures. This paper identifies an optimal mechanism for joint advertising in a single-slot setting. For multi-slot joint advertising, we pro- pose Nund a novel bundle-base neural net- work approach specifically designed for joint ad- vertising. Our extensive experiments demonstrate that the mechanisms generated by BundleNet ap- proximate the theoretical analysis results in the single-slot setting and achieve state-of-thh-ar art per- formance in the multi-slot setting. This signifi- cantly increases platform revenue while ensuring approximate dominant strategy incentive compat- ibility and individual rationality.
In recent years, a novel auction scenario known as "joint advertisement" emerge on online advertising platforms such as Facebook (Facebook, 2024). This joint advertisement scenario (Ma et al., 2024), where both stores and brands contribute bundle for advertising, allows for a more integrated approach to advertisement placement. As illus- trated in Figure 1, in traditional advertising pages, each product is bid on solely by the retailer as single bidder. However, in joint advertisement, the retailer and the supplier jointly bid on an ad. The ranking of the advertisement is thus influenced by the combined bid rather than by a single party's 's bid, and the platform can charge both participants accordingly. By accommodating the interests of multiple stakeholders including platforms, retailers, and brand sup- pliers- pliers-joint advertisements create a mutually beneficial ecosystem, fostering win-win situation for all parties in- volved.
Existing approaches for improving revenue in joint adver- tisements can be categorized into two types: the first is the use of BeckreyCllrk-GGovvee (VCG) mechanism (Vickrey, 1961; Clarke, 1971; Groves, 1973; Ma et al., 2024) and VCG-like mechanisms such as JAMA (Ma et al., 2024), and the second is the application of automated mechanism design methods to joint advertisements, known as Jeeg (Zhang et al., 2024). Compared to VCG, JAMA struggles to adapt to the flexible and complex bipartite relationships between retailers and suppliers that often arise in joint ad- vertisement scenarios. JRegNet, on the other hand, suffers from poor model generalization and robustness, and in many cases, its performance is inferior to VCG, leading to reduced rather than increased revenue.
1. Introduction
Online advertising emerges as the principal revenue source for internet platforms like Google, Amazon, and Facebook. With online advertising revenue reaching approximately $225 billion in 2023, plays a critical role sustaining the growth and development of these platforms. A preva- lent method for allocating advertisement slots is sponsored search auctions, wherein advertisers submit bids to the plat- form. The platform subsequently employs predetined auc- tion mechanisms to determine ad placements and pricing. Maximizing the monetization efficiency of advertising traf-
Nevertheless, these methods have the following issues:
These methods are similar to traditional methods in that they only establish incentive compatibility (IC) in the brand (or store) dimension. However, in the scenario of joint advertising, the final ad slot will be allocated to a bundle consisting of brand and a store. Previous work did not fully characterize the bundle, resulting in space for optimization in the actual ad slot allocation process.
Gaollng School of Artificial Intelligence, Remmin University of China, Beijing, China BBiijing Key Laboratory of Research on Large Models and Intelligent Governance 'Enginerrig Research Center next-Generation Intelligent Search and Recommenda- tion, MOE. Correspondence to: Qi Qi <qi.qi@ruc.edee edu..ed...n.....
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
In addition, since these methods cannot ensure the perfect

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Optimal Auction Design in the Joint Advertising
Platform
SuapteenPattiiippnnss the ad anction)
Bundle Bid
Sa
Bid - Sb
Dog Flen Tick Control
Search
Q
Dog Flen Tick Control
Search
Pluifom
Supplies Plus
Activate II
Hundle Bid Sa -Sb
(etailerrPar inciontstt the auciion) BidrSa
SUPPLIES
Accepes B:B
Htess
Retailen Particintat the ad suction) Bid-Sa
Ascipt ERT
S26.97
stock
Supplier[Non- parricipunts in ad tnctions)
Seresto
Joint
Supplic Plax
Ancipe EST
$53.98
Smart
store
PETSMART
Vien all Heess
Seresto
Scresta
FRONTUNE
Plus
Frentline
Adrandi
Pridway
Pet Smaa AccIpeCBIT
SAL.
$34.96
2
(a) Traditional advertising
(b) Joint advertising
Figure 1. A Comparison Between the Traditional Advertising Model and the Joint Advertising Model. The image on the left illustrates traditional advertising, where retailer submits a for the ad, while the supplier (brand owner) does not participate in auction. this case, the platform only receives the bid from the retailer. In contrast, shown in the right image, for joint advertisement, both the retailer the supplier participate the auction. They submit bids simultaneously, and the platform receives bids from both parties.
fulfillment of IC when maximizing the platform's revenue, the regret value is used to measure the degree of IC viola- tion. Although the lower the regret value, the closer the mechanism is the optimal, may not be close to the optimal in the parameter space, there is the possibility of locally optimal solution.
(Edelman et al., 2007). Variations like adding reserved
prices (Thompson & BeytonBBrown,, 2013; Roberts et al., 2016) and "squashing" (Lahaie Pennock, 2007: Charles et al., 2016) have been introduced boost revenue. But GSP lacks the IC property.
In auction mechanism design, Myerson (Myerson, 1981) characterized the revenue maximizing single- parameter auc- tion mechanism. However, the multi-parameter optimal auction design problem remains unsolved even after 40 years. Automated mechanism design (Conitzer Sandholm, 2002; 2004: Sandholm & Likhodedov, 2015) addresses the multi-item, multi-bidde auction design challenge by find- ing approximate optimal solutions. There are three main ap- proaches in this field: the RegretNet-likk approach (Dütting et al., 2024; Curry et al., 2020; Peri et al., 2021; Rahme et al., 2021; Duan et al., 2022; Ivanov et al., 2022) constructing IC constraints; the affine maximizer auctions (AMA) (Roberts, 1979) and related methods (Likhodedov & Sandholm, 2004; Guo et al., 2017; Curry et al., 2022; Duan al., 2023) mod- ifying allocation with weights for higher revenue; and the approach characterizing utility functions for IC and strategy- proofness (Dütting et al., 2024; Shen et al., 2019; Wang et al., 2024). Automated mechanism design is increasingly used in online advertising auctions (Zhang et al., 2021; Liu et al., 2021; Liao et al., 2022).
To address these challenges, propose novel and ef- ficient automated mechanism learning approach, termed BundleNet, specifically tailored for joint advertisements.
Concretely, the contributions of our paper can be summa- rized as follows:
First, identify the optimal mechanism for single-slot joint advertisement. Additionally, propose novel neural net- work architecture and introduce a new IC constraint method for multi-slot joint advertisements. Our approach not only improves platform revenue but also ensures approximate IC and individual rationality (IR). Extensive experiments demonstrate that our method achieves state-of-the-ar per- formance.
2. Related Work
In traditional advertising auctions, the generalized second price (GSP) auction, following early work (Aggarwal et al., 2006; Varian, 2007), has been popular in the past two decades for its simplicity, feasibility, and good revenue
(Ma al.. 2024) proposed the joint advertisement setting, Revised VCG mechanism for revenue improvement, and
2

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Optimal Auction Design in the Joint Advertising
JAMA (a menu-based AMA approach (Curry et al., 2022)) to further increase revenue. Our approach is closer to JReg- Net (Zhang et al., 2024), RegretNet-like method for joint ads. JRegNet encodes bipartite relationships in neural net- works and transforms allocation to payment functions, out- performing VCG in small-bidder bipartite settings, But both JAMA and Jet have limitations. JAMA can't adapt to complex bipartite relationships in joint ads, and JRegNet has poor generalization in large, complex scenarios. (Aggarwal et al., 2024) proposed learning algorithms for repeated joint ads to minimize regret and maximize revenue in different environments, not focusing on single-round ad revenue.
senting bundles that exclude specific retailer r or supplier 8, respectively.
Definition 3.1 (Joint Auction Mechanism). The Joint Auc- tion Mechanism is represented by a pair of rules M = (x, p). The allocation rule of bundle e is denoted by xf V + 2M, and the payment rule of bidder i for bundle e is denoted by p V + Rgo. The joint auction mechanism M - (x, p) is then defined with the following components:
1. Allocation and Payment Rules:
For each bundle e (r) E E, the allocation and payment are shared be- tween the participating bidders r E R and E S as follows:
3. Preliminaries
=e= (1, E E, pf (v) - (v) + pZ(0), Ve= (r, E E. The total allocation and payment for a participant R RUS S are:
(v) - =
x,) x1]),,
In this section, we mainly introduce the basic setting of joint advertising system.
In the context of joint advertisements, the sets of re- tailers R and suppliers S are mutually exclusive, with R EnS S = 0. When a user submits a query, the advertis- ing system retrieves n advertisements represented by the set fei....,.n. ), where each advertisement links a retailer r E R with supplier & S. Simultaneously, there m slots .M- = {I..... m available for displaying ad- vertisements. The tlick-through rate (CTR) for the k-th slot is denoted as As. where & E M, and we assume that the CTRs are ordered such that 0 S Am .... S A I S A1 S 1. We represent these CTRs using a vector, (AI...., Am.. relationship between and S is modeled as bipartite graph G SR E), with edges R x S representing advertisements.
(,)
Σ (V), EE
Pi(O) Σ pf(v). (I)
EEF.
Since each slot can only be allocated to a single bundle, the allocation must satisfy the constraint:
Σ EEE
x (v) S 1m
2. Expected Quasilinear Utility: The expected utility for a participant i E R U S is given by:
U,(v;, u)) = Ey EVLINv (rai(1(,,-.))) - p;(0,, v-:)
E [
The joint valuation distribution domain is defined as V = V x vs, where V represents the domain of all pos-
3. Incentive Compatibility: A mechanism satisfies IC if:
sible retailer valuation distributions, with RR = (u, rren. and vs is the domain of all possible supplier valuation distributions, with v (va).es. Retailer valuations e, and supplier valuations u, are independently drawn from their respective cumulative distribution functions F, and F., with probability density functions (v,) and [x())). The domain excluding retailer r is V_, = VR x V$, where VR. = II (r) VRR and the domain excluding supplier is V_, = R x VSS.. where v.S. = [I> V.S.
Ug(V;, u,) 2 Us(2s, uf), RUS, U, 2/. V.
4. Individual Rationality: A mechanism satisfies IR if:
U.(vi, u/) 2 0, E Us E V..
5. Expected Revenue: The expected total revenue of the mechanism is:
1001-1 1- (v) XT >
Ev(v X
Un EVNV
V0
The auctioneer knows the distribution F= (Fi)eeuus but not the realized valuation profile v (us).ERUS. Bidders
EEE
BE
4. Optimal Joint Auction Design with Single Slot
report their valuations bids b (b.) where b, E V, for retailer and b, E V, for supplier s. The auctioneer's personal value for each slot, if unallocated, is denoted by Ug.
We say that mechanism M is feasible if and only if the mechanism M satisfies the conditions of IR and IC. For the single-item environment, the most well-known theory for this case is Myerson's 's Lemma (Myerson, 1981), which pro- vides a necessary and sufficient condition for the feasibility of a mechanism.
Joint ad advertisements involving retailers or suppliers are de- fined as E, = ((r",s) E E r" = r} for retailer r and E. = ((r,5) El $f for supplier s. Excluded joint advertisements are denoted as E-, and E-,, repre-
3

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Optimal Auction Design in the Joint Advertising
Lemma 4.1 (Myerson's Lemma). In the single-item setting, mechanism (x. p) is feasible if and only if the following conditions hold:
(i) Step Function:
{
1
if > 6,(w=))
x7 (ot, =
otherwise
(a) An allocation rule x is monotonically non-decreasing.
(b) If is value-monotonie then there is unique payment rule p for which ().P) is IR and IC and the winner pays her critical bid and the losers pay zero.
(i) Critical Value:
(or, u-1) 0:(v<() if > 0,(v-))
otherwise
Extending this framework single-slot joint advertise- ment scenario, the feasibility conditions remain consistent with traditional auction mechanisms. The primary differ- ence lies in the allocation and payment rules, which are adapted to account for joint advertisement characteristics. Myerson-like mechanisms can thus be applied to the joint advertisement setting through appropriate modifications.
where the critical value 0z(U=i) is defined as follows:
For E R, the critical value E, (v-_r) is:
(v-r) (b, 1c (br, Uapt) 2 vn
In this paper, we consider distributions that satisfy the regu- lar condition, which is crucial in Myerson's 's auction theory: Definition 4.2. A distribution V is termed regular if its associated virtual value function
e (6r, Uppt) 2 è Uz), VE E E-r).
For E S, the critical value (v(v is defined similarly:
1 - F(v)
6. (v-e) = (bg c (ury, ba) 2 voA
<(v) = 0
f(v)
(r,r, (vp, VE E_a).
monotonically increasing function of v in the support of V.
See Appendix B for detailed proofs.
To formulate these modifications, we first introduce the concept of bundles and neighbors, which are critical for defining the allocation and payment rules in the optimal joint auction mechanism. The neighbors of node are defined as (r)) = (s S (r,8) E E} for retailer r and (s) = (r E R (r, E E} for supplier s. These structures allow us to define the virtual value func- tions for retailers and suppliers as cv,)) u, 1-F, (n,)
5. Optimal Joint Auction Design as Learning Problem
In multi-slot joint advertisement, bids for bundles are jointly determined by two bidders, making them interdependent and challenging to handle. To address this, we propose BundleNet, a novel neural network architecture with cor- responding learning methodology for optimal mechanism design. Section 5.1 introduces bundle-based IC constraint, Section 5.2 details BundleNet's architecture, and Section 5.3 elaborates on its loss function and training process.
f.(o,)
and (v,) = U - 1=F,(e,),
f.(v)) respectively. Using these virtual values, we identify the maximum adjacent nodes, where ar = arg Nax) (v,) for retailer and r = arg (()) e (u,) for supplier s. The corresponding bundles are e,t (r, ) and e. (r-A, 1), and the vir- tual value of bundle
5.1. Differentiable Approximation Approach for Joint Auction Design
(r, is defined the sum of the virtual values of its nodes: vu,, c,(u,, + c, (v.). These bundle and neighbor definitions serve as the foun- dation for the allocation and payment rules in the optimal mechanism.
In the domain of automated mechanism design, particularly within the class of RegretNet- based approaches (Dütting et al., 2024; Duan et al., 2022; Ivanov et al., 2022; Zhang et al., 2024), the optimization problem typically involves balancing two conflicting objectives: maximizing revenue and minimizing regret. The trade-off between these objec- tives is controlled by hyperparameters, such the initial values and schedules of the Lagrangian multipliers.
Building on these structures, we establish the necessary and sufficient conditions for optimal joint auction mechanism in the single-slot scenario. The optimal mechanism aims to achieve both feasibility (satisfying IR and IC) and revenue maximization.
Theorem 4.3. For the single-slot joint advertisement with regular bidders, deterministic joint auction mechanism M is optimal if and only if for all E RUS,
In the context of joint advertisements, (Zhang et al., 2024) extended this framework by introducing IC constraints for all participants, including both retailers and suppliers. Their

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Optimal Auction Design in the Joint Advertising
model is formalized as:
misreporting exceeds that of truthful reporting. When the utility from misreporting is less than the utility from truth- ful bidding, the IC constraint becomes inactive. Therefore, rgt((w) 2 O. When the sum of penalties imposed by the bundle IC constraints approaches zero, the IC penalties for all bidders also approach zero. With Lemma 5.1 guarantee- ing the desired properties, we reformulate the optimal joint advertisement design as the following constrained optimiza- tion problem:
min WER
Σ pi(u;
LiERUS
S.L. rgt((w)=0) =0 O, RUS,
where w represents the parameters of the neural network used for the automated mechanism. The regret function rgt((w) is defined as:
rgt;(w) = Evov max uu(e;; (-)))=)); w) (u( (vey w)));
min WER"
- EEOV
Σ (v; LEEF
(3)
s.t. rgt (w) =0, O, Ve E.
In joint advertisement scenarios, the bidding strategies of bidders have different impacts on the mechanism, as their de- grees in the bipartite graph are not identical. However, from the perspective bundles in joint advertisements, we ob- serve that the properties of each bundle are generally similar. Each bundle contains bids from both retailers and suppli- ers. Therefore, we propose redefining the IC constraints by constructing them for bundles instead of individual bidders. Although it may seem that bundle, consisting of both re- tailers and suppliers, could be modeled as a wwo-parameter auction problem, the bundle itself does not have a strategy: depends on the bidding strategies of the connected retailer and supplier. As a result, directly defining an IC constraint for bundles is highly challenging. To address this, pro- pose method to expand the feasible domain, which derives an IC constraint from the bundle's perspective. The key feature of this new constraint is that it encloses the feasible region defined by the original IC constraints. By forcing the new constraint to approach zero, the original IC constraints also approach zero, ultimately ensuring the incentive com- patibility of the entire mechanism. We define the ex-post regret for bundles follows:
To solve this, we aim to use sampling-hases methods to learn the mechanism M = (a,, p). Given a sample S of L valuation profiles drawn from a distribution F, we estimate the ex ex-post regret for each bundle as:
rgt (w)=
+ Σ
,EV, { (ut (0/, (() - (t(( ,000) (w)
max
max (uā(vb) (1b,,(1):: - (v( 10 w) VVVV
Finally, we reformulate the original optimization problem as minimizing the empirical loss of negated revenue, subject to the constraint that the empirical ex-post regret for all bundles is zero:
L
min WER
pe (u(0)) m)
(4)
I
s.t. rgt (w) =0 Ve E E.
rgt (w)
Additionally, we utilize neural network architecture to ensure that the mechanism satisfies the conditions of IR in See 5.2.
E [OEV,
max (ur (1/, w-)) uf(er, (Ur, w))
max [ul( (1', [-)): (L (vo, eeV,
(t,, w))
5.2. Neural Network Architecture
This section introduces our neural network-based approach for mechanism design in joint advertisements. Our archi- tecture comprises two primary components: Allocation Network and Payment Network. As illustrated in Figure 2, we employ a graph-based approach to model the inter- actions between retailers and suppliers effectively. This process, termed Graph Feature Fusion, involves aggregat- ing node features from a bipartite graph into edge features that capture the combined characteristics of retailer-supplier pairs (bundles).
Lemma 5.1 demonstrates the relationship between the origi- nal constraints and the newly introduced constraints:
Lemma 5.1. In a joint advertisement, the sum of the IC constraints for bundles is always greater than or equal to the sum of the IC constraints for individual bidders, as follows:
Σ rgla(w) S >Ergh (rg((w)) S
EEE
The detailed proof is provided in Appendix C. IC constraints are designed to penalize situations where the utility from
Each node in the bipartite graph G = (R, S, E) is associated with a feature vector representing the bidder's cost per click
5

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Optimal Auction Design in the Joint Advertising
Graph Feature Fuslon
Allocation Network
MLP
Row Softmak
Retailer Bids
Stacked Bids for Bundles
Alocation of Bundies
2M
Mr1 Column Softmax
Diviced Bids for
Payment
Bipartte Graph
Retailer Payments
MLP
M Supplier
Normalized Divided Payments for Bundles
Divided Paymentss for Bundles
Supplier Payments
Figure 2. The neural network architecture BundleNet. Details are shown in Sec 5.2.
The aggregated edge features SB serve the input to (CPC), defined as X, b, R. for retailer X, b,A E 20 for supplier
multi-layer perceptron (MLP), producing an intermedi- ate output denoted as Y Rdy, where dy represents the dimensionality of the output feature space:
To capture the combined influence of retailers and suppliers on each bundle, we aggregate their node features into edge features, which we refer to as Divided Bids for the Bundle. This process mathematically formalized as follows:
Y= MLP(SBB) E RR..
The output Y contains values that reflect the potential allocation results for each bundle across available slots. To normalize these values, we apply two distinct trans- formations using matrices Wrow E x (m+1)mm+1) and W E R. (x+)x(mm))mm=m)))
DB [X,, X.]
2m R
Ve (r, E E,
20>
where (r, s) represents edge between retailer and supplier We denote DBE as the features of all edges in the bipartite graph.
The first transformation involves multiplying Y by Wow- resulting matrix that captures the allocation potential for each slot:
By summing up the aggregated edge features, we derive the Stacked Bids for the Bundle, which is defined as
SB = x X, Rm
Ve (r,s) E E.
MR W.... RR. (m+++1mm+1)
Similarly, we use F of edges in the bipartite graph.
F. to represent the aggregated features
Next, we apply the row-wise softmax function to R. The softmax function transforms the raw scores into probability distribution across the bundles for each slot.
In the Allocation Network, similar RegretNet, we uti- lize doubly stochastic matrix approach ensure that each bundle is allocated to exactly one slot and each slot is as- signed to only one bundle, reflecting the joint advertisement scenario. A key property of doubly stochastic matrices is encapsulated in the following lemma:
exp(M
DRa = Ex=T exp(MR,,)
Vi (1...,,,24
n n 1}.
Similarly, we perform second transformation by multiply- ing Y by West
Lemma 5.2 (DDütting et al., 2024)). The matrix o DS (x, x²) doubly stochastic Vx, E Rmm, For any doubly stochas- tic matrix [0, 10.1 there exist x, x! E Ram for which DS
MC = YWo. E g(++1)mm+1)
( (x,x²).
We then apply the column- softmax function to C. For each column j in C, the softmax is defined as:
DS (I,x²) min ai by
e
E
exp(MC,,)
a1l kml
ea
mm+
6
DC Σ9 kk=1 exp()
Vj E (l...., m + 1}.
6

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Optimal Auction Design in the Joint Advertising
Alg.
Setting
U2
U3
U1
Us
E2
E3
E₁
Eg
Ours
BundleNet
0.5286
0.6681
0.7805
0.8802
0.4248
0.5460
0.6354
0.7215
IC Violation
0.0006
< 0.001
0.001
0.001
< 0.001
0.001
< 0.001
0.001
IC Baselines
RVCG
0.3811
0.6003
0.7455
0.8607
0.2820
0,4649
0.5927
0.7041
Optimal
0.5247
0.6705
0.7826
0.8819
0,4249
0,5479
0,6470
0.7376
Baselines with IC Violation
JRegNet
0.5622
0.7287
0.7791
0.7882
0.4727
0,5892
0.6306
0.6943
IC Violation
0.00054
< 0.001
<
0.001
< 0.0001
< 0.001
0.001
< 0.001
0.001
Table 1. The experimental results compare BundleNet, JetgNet, Revised VCG, and the Optimal Mechanism the number of bundles increases under different settings the single-slot scenario with CTR À = (1). The notations U2, Us, U4, Us represent cases where the number of bundles is 2, 3, and 5, respectively, under the uniform distribution (O, 1). Similarly, E2, Ea, Ea, Es correspond to scenarios with 2, 3, and bundles under the truncated exponential distribution E2). In this table, we use bold to indicate the method among BundleNet, JegNett, and RVCG that is closest the optimal mechanism, rather than the one with the highest revenue.
Finally, to ensure consistency in the allocation, we take the element-wise minimum of the two resulting matrices to obtain the final allocation matrix A, where:
where represents the neural network parameters, A represents the Lagrangian multipliers associated with the constraints, while is nyper-parameter controlling the weight of the quadratic penalty term. During optimiza- tion, we utilize the Adam optimizer to update our parame- ters as well the misreports of(0) and v.(() in turn, i.e.,
Ag min(DRa) DCij), Vi E n..j......... E (1, m)
In the Payment Network, to ensure that the auction satis- fies ex-post IR, generate normalized payments for each bundle corresponding to the retailer and supplier, denoted as Dr and p [0, 1]. We employ an MLP with Sigmoid activation function in the final layer to represent this map- ping:
update wew arg min, Cp (w pald) and update pnew pold + rggt (w"e"), Ve E E. The detailed algorithmic specifications can be found in Algorithm 1.
6. Experiments
p = Sigmoid( R
In this chapter, we present empirical experiments to demonstrate the effectiveness of BundleNet. All experi- ments are conducted on Linux machine equipped with NVIDIA Graphics Processing Unit cores.
Subsequently, we derive the corresponding payments follows:
p
X,
Baseline methods: We compare BundleNet with the fol- lowing baselines:
m Ag X.I.
Optimal Joint Auction Mechanism, Myerson-lin like method for optimal mechanism design in joint advertising auctions with single slot (See Sec. 4).
Finally, determine the payment for each participating bidder in the advertising auction using the equations defined in Equation (1).
VCG (Vickrey. 1961; Clarke, 1971; Groves, 1973), classic mechanism satisfying DSIC and IR. In our experi- ments, we apply the RVCG mechanism (Ma et al., 2024) to the joint advertising.
5.3. Optimization and Training
We optimize the constrained objective (4) by introducing the augmented Lagrangian method. Our loss function is formulated as follows:
JRegNet (Zhang al., 2024), a neural network architec- ture for near DSIC mechanism design in the joint adver- tising auction settings which can achieve the near-optimal revenue.
Lp(w;
1 EE (v(0)
(w) Σ
rgt c (w)
Evaluation: We generate training and test data from dif- ferent distributions. The training set consists of 204,800
ElEEE

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Optimal Auction Design in the Joint Advertising
Alg.
Setting
Usxs
U6x5
U7x5
Ugxs
U9x5
U10x5
U11x5
U12x5
Ours
BundleNet
1.4982
1.7162
1.9233
2.0890
2.2210
2.4047
2.5649
2.6495
IC Violation
0.001
< 0.001
<< 0.001
< 0.001
< 0.001
0.001
< 0.001
< 0.001
IC Baseline
RVCG
0.8420
1.2854
1.6142
1.8831
2.0991
2.2800
2.4564
2.5868
Baselines with IC Violation
JRegNet
1.4972
1.6849
1.8244
1.9350
1.9804
1.9622
1.9763
1.9973
IC Violation
< 0.001
< 0.001
<
0.001
< 0.001
< 0.001
<
0.001
<
0.001
< 0.001
Table The experimental results of BundleNet, JRegNet, RVCG the number of bundles increases under different settings in the multi-slots scenario. Similar to those of Table I, the notation Usxs. ... U12xx represent the settings where the number of bundles varies from to 12, while letting the CTRs of these 5 slots (1, 0.8, 0.6, 0.4, 0.2).
samples, while the test set contains 20,480 samples. To assess the performance of each method, we utilize the average empirical ex-post regret of the mechanism: rgt- = In LiE RUS rgt,, Since in real-world scenarios, Uo = 0, we also consider empirical revenue: rev := I E ( In all synthetic data experiments, the joint relationship matrix between stores and brands is randomly generated for each search request sample. We evaluate the revenue performance of BundleNet, JRegNet, and RVCG across different distributions.
6.2. Experimental Results for the Multi-Slot
We have also conducted extensive experiments to evaluate the performance of BundleNet under the multi-slot scenario from experiments real-world dataset (Zhang al., 2024).
Concretely, we explore different variations of this setting, where 10 bundles compete for 5 slots. We fix the number of slots and vary the number of bundles to evaluate the performance of BundleNet in comparison with baseline mechanisms. The bidders' value profiles are sampled from three different probability distributions: Setting U, where values follow the uniform distribution U(0, 1); Setting LN, where values follow the truncated lognormal distribution (O 1.44) over the interval (0, 1); Setting N, where values follow the truncated normal distribution (0.55, 0.1) over the interval (0, 1). For each distribution, we assess different mechanisms by varying the number of bundles from 5 12, while keeping the CTRs of the 5 slots fixed (1, 0.8, 0.6, 0.4, 0.2).
6.1. Experimental Results for the Single Slot
The experimental setup considers the optimal joint auction mechanism in single-slot setting, evaluated under three different probability distributions commonly studied in the literature: Setting U: The uniform distribution U(, 1). Set- ting E: The truncated exponential distribution Exp(22 over the interval (0, 1). Setting N: The truncated normal distribu- tion N (0.5, 0.1) over the interval (o, 1). For each distribu- tion, we examine auction scenarios with varying number of
Regarding the experimental results for Setting U, as shown in Table 2, we conclude that BundleNet achieves higher rev- enue compared to other baseline mechanisms. BundleNet consistently outperforms VCG across all scenarios, whereas JRegNet surpasses VCG only in some cases. The experi- mental results for Setting LN and Setting N. provided in Appendix D.2, lead to similar conclusions.
bundles, specifically ranging from 2 to 5, with a single-slot CTR of 1.
The experimental results, reported in Table 1 and Table 3 illustrate the performance of different auction mechanisms under these settings. The experimental results demonstrate that BundleNet consistently approximates the optimal mech- anism across various settings, whereas JRegNet does not always exhibit such proximity. This indicates that, although both methods are based on RegretNet, BundleNet improves upon it by modifying the neural network architecture and optimization approach, enabling it to better learn the optimal mechanism. We provide a visual analysis in the Appendix E. The allocation results of BundleNet are much closer to the optimal mechanism, while the allocation results of JRegNet significantly different from the optimal mechanism.
7. Conclusion
We propose two solutions for the optimal mechanism design in joint advertisement. The first solution in the single-slot scenario, Myerson- -ased approach for the single-slot scenario and automated RegretNet-inspired method for the multi-slot case. Empirical results show that BundleNet learns a Myerson-like mechanism in the single-slot setting and finds near-optimal solution outperforming baselines in the multi-slot setting.
8

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Optimal Auction Design in the Joint Advertising
Acknowledgments
neural network for auction design. In International Con- ference Machine Learning, pp. 5609-5626. PMLR, 2022.
This work was supported by National Natural Science Foun- dation of China No. 62472428, Public Computing Cloud Renmin University of China and fund for building world- class universities (disciplines) of Renmin University of China.
Duan, Z., Sun, H.,, Chen, Y., and Deng, X. A scalable neural network for dsic affine maximizer auction design. Advances in Neural Information Processing Systems, 36: 56169-56185, 2023.
Impact Statement
Dütting, P., Feng, Z., Narasimhan, H., Parkes, D. C., and Ravindranath, S. S. Optimal auctions through deep learn- ing: Advances differentiable economics. Journal of the ACM, 71(1):1-53, 2024.
This paper presents work whose goal advance the field of Machine Learning. There are many potential societal consequences of our work, none which feel must be specifically highlighted here.
Edelman, B., Ostrovsky, M., and Schwarz, M. Internet ad- vertising and the generalized second- pricd--rieeeee auction: Sell- ing billions of dollars worth of keywords. American economic review, 97(1))242-255, 2007.
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Optimal Auction Design in the Joint Advertising
A. Optimization and Training Procedures
Algorithm 1 BundleNet Training
Input: Minibatches , of size C
Parameters: Vt, Ps 0, 0, D, KEN
Initialize: w0 R. _0 Rm
= 0 do Receive 8, = (GO) )
Initialize v, (9) < Vr, (0) & V., vl e
for 0 to do
for
do
VA E [C], RUS:
- 1 + +
hu (v (0%, e
end for end for
Compute Lagrangian gradient and update w'
w + wt V (w", p²)
if is a multiple of then
net1 t- ple + pergt ), Ve E
else
X +
end if end for
B. Proof of Theorem 4.3
The joint auction with single slot remains a single-parameter auction, with the primary distinction from traditional auctions lying in the allocation and payment rules. As a result, the necessary and sufficient conditions for feasible auction, as established in Myerson's work (Myerson, 1981), still hold in this setting.Before Before presenting the proof, we first introduce an assumption: suppose a; and b; are the lower and upper bounds the domain the probability density function f; for bidder That is, 0 [a,, by]. Given a joint auction mechanism, we define (v;); as:
(v(+:) Ev_,,F_,I, Vi RUS,
where represents the expected probability that bidder i wins, given their value estimate U/-- Lemma B.1 ((Myerson, 1981)). In the single-slot setting, a joint auction mechanism (x, p) is feasible if and only if the following conditions hold:
(a) of S vs, then Qi(1f) S Qi(v)), Vi RUS, V. (b) U,(v,,v)) V,(a,,,,) + f Q(z)dz, RUS.
(c) U,(Vi, U1) 2 0, Viei RUS.
(d) E.EF x" (v) 11 1
xz (v) 0
M = (x, p) an optimal joint auction mechanism and only if maximizes Un while ensuring the feasibility of the auction. We provide a simpler condition to characterize optimality.
Lemma B.2. Suppose that the function x V 2n maximizes the following objective:
/
Σ c=(r,)EEB
tu 1 --Fr(rr) F-FaUus) no ()))((u)d
max M
fr(1,)
fo(Ue)
por (/ (2(2 VIERUS
s.t.
pi(u) = (v
(5)
of < u. then (1((()) S Q,())), Vi RUS, of V.
Σ x (v) 1, (v) 20 0
E

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Optimal Auction Design in the Joint Advertising
Then, the pair M (x, represents an optimal joint auction.
Proof. To prove that the pair M = (x. p) represents optimal joint auction, we analyze the expected revenue function Un and show that it maximizes the given objective function while satisfying feasibility and incentive compatibility constraints. The expected revenue Ug is given by:
1-Ex(v)
+ Σv(M) cEE
Un = ErAF
10
cEE
= 1 vo (v)(u
E1/ (V,(1)
(oo (2(0) (w))((
(6)
TT
cE
Σ (v)(v, u. (o)
EEE
Using the conclusion of Lemma B.1, we can simplify the following equation.
Σ / /((0)) (x(())
(o) =
Σ
(v) (o))(
cEE
rEn
Σ
(U,(U, (,)
TER
(r) 1" (.(e)d=lff(l..dd.
Σ
rER Or
Σ (U-(ar, -.e
br ( ( (r(=)f
TER
> [Ur(ar, ar) +
(1=
Jar
VER
br
Σ [Ur(ar; ar) +
(1- ((-F
rER
Σ (ar ar) + 1/mm bh)
eEE
Jrr
Substituting Equation 7 into Equation 6 gives us:
Un = / (v)(v + Σ
/ (0(((), +
1- Fr
1- 1- Fs(vs) f((rr) fz(v))
vo). (u)de
c=(ra)EB
(8)
Σ U,(a,, ap) EU((a,)))) rEn
SES
The first term in Equation& a constant, while the third and fourth terms are always non-negative due to the IR property of the auction. When these terms are equal to zero, the objective function reaches its maximum value without affecting the value of the first and second terms. Thus, by setting the third and fourth terms zero, obtain:
pr(v) (.
Fe (2, v-r)dz
(9)
Pv(u) v,
Is (s(2,
12

---

Optimal Auction Design in the Joint Advertising
Next, we proceed with the proof of Theorem 4.3:
Proof of Thheorem 4.3. In this paper, we consider distributions that are regular. Since distributions in the regular class ensure the monotonicity required by Lemma B.2, they guarantee the IC property of the joint auction. As a result, we can simplify the objective function follows:
Σ (T,))E
(u,, Us) - (0))z()))
This implicitly indicates that the slot will be allocated to the bundle with the highest virtual value. For single-slot joint auction, each bidder's bidding strategy depends solely on the neighbor node with the highest virtual value. This is because any bundle formed with other neighbors has no chance of winning the auction. Therefore, for each bidder, their critical value is given by:
For r R, the critical value or (v-r) is:
(v-r) = inf {or ct (br, Uage) 2 No, and c²* (br, Uapt) 2 (1p, = (f,8) E E-r
For s E S, the critical value 0, (v))) )) is similarly:
(v = inf (b, (u,2e, b.) 2 vo, and c= (r,ge, ba) 2 (1p, v4 (f,8) E E..)
We take node R as example. The allocation of all bundles connected to r is given by:
M ah (br, v-r) = (1 1 if 2 (,(u-r) if br < (y(U-r)
x (br, Ve" E, \
(cpd) M
Thus, we can derive the allocation result for r as follows:
if 2 (r(r) = 10 if (v
Ir
(br, = x²
v-r)
v-r+
Σ
x (br,
E*EE-/ieM1
We can derive the final payment result by solving the payment formula based on the Equation 9, We integrate the allocation rule to determine the payment function. Thus, the final payment for bidder r is given by:
2 (c-r)
if (x(())) if br (v(r)
Pr (br, U-r)
For node S, the conclusion follows similarly. The allocation and payment rules apply symmetrically due to the structure of the joint auction mechanism, where each participant's 's strategy depends on their highest virtual value neighbor. Thus, the proof is complete.
13

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Optimal Auction Design in the Joint Advertising
C. Proof of Lemma 5.1
Proof.
Σ
rgt.
Σ EN-F
max u, (15; (1/, v-1); w) (145 (1%, 2-1); veE.
AERUS
VERUS
Σ B R
max ur (Des (1/s v-r): (or: o (Urr U )=)):2) LeEV,
max ue (a (M (v,, m-)); us (ms: (Me: u-0): "'EV,
Σ B
max Σ EV,, eEEr
(v., v-)): Σ
(1r) (Er, v-r): m)
rell
eEEr
(10)
Σ E
max E eVV eEE,
Σ
(vsi (Va, U-
B
max uf (uys (1/ v-r): w) - (ers (er, )ー ) C,E,V
cEE
BeWF
max ul(us: (wh, v=a); w) us (us: (2'; Uーo (==);;0]] EVV
cCE
Σ E
pgt((w)
D. Additional Experiments
D.1. Experimental Results Under Single-Slot Setting N
Alg.
Setting
N₂
Ng
N4
N5
Ours
BundleNet
0.7750
0.8692
0.9134
0.9561
IC Violation
< 0.001
0.001
0.001
0.001
IC Baselines
RVCG
0.5492
0.8114
0.8956
0.9444
Optimal
0.7789
0.8656
0.9188
0.9582
Baselines with IC Violation
JRegNet
0.8183
0,9091
0.8747
0.9117
IC Violation
< 0.001
0.001
< 0.001
< 0.001
Table The experimental results compare BundleNet, JRegNet, Revised VCG, and the Optimal Mechanism as the number bundles increases under different settings in the single-slot scenario with CTR X = (1). The notations N₂, Ns. .V. Nss represent cases where the number bundles is 2, 3, and 5, respectively, under the normal distribution N (0.5, 0.1). this table, we use bold to indicate the method among BundleNet, JRegNet, and RVCG that is closest to the optimal mechanism, rather than the one with the highest revenue.
14

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Optimal Auction Design in the Joint Advertising
D.2. Experimental Results Under Multi-Slot Setting
Alg.
Setting
LNsxs
LN6x3
LN7xs
LN8xs
LNgx5
LN10x5
LN11xs
LN12x5
Ours
BundleNet
2.6560
3.0093
3.4346
3.7031
3,9952
4.2366
4.4618
4.6038
IC Violation
0.001
< 0.001
0.001
0.001
0.001
0.001
0.001
0.001
IC Baseline
RVCG
1.4365
2.1764
2.7130
3.1579
3.5104
3,8332
4.1019
4.3449
Baselines with IC Violation
JRegNet
2.6020
2.9237
3.1845
3,4139
3,441
3,2535
3,2747
3.3848
IC Violation
< 0.001
< 0.001
<
0.001
<
0.001
<
0.001
<
0.001
< 0.001
< 0.001
Table The experimental results of BundleNet, JRegNet, VCG the number of bundles increases under different settings in the multi-slots scemario. Similar to those of 1, the notation I.N5x5.-.. I.N12x5 represent the settings where the number of bundles varies from to 12, respectively, under the truncated lognormal distribution LN (0.1, 1.44) over the interval (0, 1), while letting the CTRs these 5 slots as (1, 0.8, 0.6, 0.4, 0.2). In most of the scenarios, BundleNet report the better performance of JRegNet.
Alg.
Setting
Naxs
N6x5
N7x5
N8x5
N9x5
N10x5
N11x5
N12x5
Ours
BundleNet IC Violation
2.1393
2.3690
2.4914
2.6287
2.7020
2.7916
2.8428
2.8841
0.001
< 0.001
< 0.001
< 0.001
< 0.001
0.001
< 0.001
< 0.001
IC Baseline
RVCG
1.3110
2.0132
2.3565
2.5343
2.6395
2.7228
2.7840
2.8383
Baselines with IC Violation
JRegNet
2.2071
2.3537
2.4814
2.4740
2.5185
2.4711
2.4082
2.3272
IC Violation
< 0.001
< 0.001
<
0.001
<
0.001
< 0.001
< 0.001
< 0.001
< 0.001
Table The experimental results of BundleNet, JRegNet, VCG as the number of bundles increases under different settings in the multi-slots scenario. Similar those of the notation ".x .N12xt represent the settings where the number of bundles varies from
5 to 12, respectively, under the truncated normal distribution N (0.5, 0.1) over the interval (0, I), while letting the CTRs of these 5 slots as (1, 0.8, 0.6, 0.4, 0.2). In most of the scenarios, BundleNet report the better performance of JRegNet.
E. Comparison of Visualized Allocation Results Under the Setting U2
To further validate our hypothesis, we visualize the allocation rules of BundleNet and JRegNet to examine their differences. Since the bipartite graph structures in joint advertisement scenarios can be highly complex, we select the Setting U2, which consists of only two types of heterogeneous bipartite graphs. Based on these structures, we design two different sets of experiments.
In the first experiment, we consider bipartite graph where Set R contains two nodes (2), Set S contains single node ($1), and there are two bundles, e = (r1, and e2 = $1). . To observe the impact of bidding on the allocation outcome of é1, we fix the bid for 81 at 0, 0.25, 0.5, and 0.75, while varying the bids of the two nodes rı, 72.
In the second experiment, we extend the complexity by introducing bipartite graph where set R contains two nodes (r1, r2). Set contains two nodes 82), and there are two bundles, e1 (r1, $1) and e2 (r2, $2). To analyze how the bids in ej affect its allocation outcome, we fix the bid for e2 at (0,0). (0.25,0.25), (0.5,0.5), and (0.75,0.75) and observe the impact of the bids from r1 and 8 in e1-
As shown in Figure 3, we present the allocation results of BundleNet and JRegNet in two experimental settings. We observe that BundleNet exhibits clearer boundary awareness than JRegNet, accurately identifying scenarios where Bundle should not be allocated.
To better demonstrate BundleNet's capability in learning the optimal mechanism, we compare its deterministic allocation
15

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Optimal Auction Design in the Joint Advertising
a. The allocation results BundleNet in the first experiment.
b. The allocation results of JRegNet in the first experiment.
The allocation results of BundleNet in second experiment.
d. The allocation results of JRegNet the second experiment.
Figure The figure presents the allocation rules learned by BundleNet and JRegNet under the Setting Uz. Subfigures (a) and (c) show the allocation results of BundleNet in two experiments, while (b) and (d) display the results of JRegNet in the same experiments. The solid regions in all subfigures depict the probability of the single-slot being allocated bundle ex, with the white dashed line representing the boundary of the allocation results derived from Myerson-like mechanism.
results with those RegretNet-like methods and the theoretical optimum in Figure 4. Unlike stochastic allocation, deterministic allocation provide clearer visual differentiation between these mechanisms. This visualization confirms BundleNet's superior approximation of the optimal mechanism, while JRegNet exhibits inconsistent performance. The deterministic view climinates random noise, making the comparative advantages more apparent.
16

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Optimal Auction Design in the Joint Advertising
Optimal
BundleNet
JRe
Optimal
BundleNet
JRegNet
(1)
(2)
Figure 4. The figure presents the allocation rules learned by BundleNet and JRegNet under the Setting U2. Subtigure (1) show the allocation results of optimal mechanism, BundleNet and JRegNet in the first experiment, while subfigure display the results of three mechanism the second experiment.
17


Reviews:
[{"review_id": "pQgcIH4PU0", "reviewer": "Reviewer_BtzC", "summary": "This paper proposes an optimal mechanism for joint advertising in a single-slot setting and introduces BundleNet, a bundle-based neural network for multi-slot joint advertising. Through extensive experimentation, the paper demonstrates that BundleNet effectively approximates theoretical results in single-slot settings and outperforms existing methods in multi-slot scenarios.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We would like to thank you for the constructive comments and suggestions! Below are our responses to your questions.\n\n**Q1.** We conducted experiments using real-world data from a leading Internet company. In practice, advertisements will go through the stages of recall, rough sorting, and fine sorting. A maximum of 10 advertisements and 5 advertising spaces will be reserved after rough sorting and will go into the fine sorting stage. JregNet and BundleNet are both used in the fine sorting phase. The dataset consists of 102,679 samples, with 80% used for training and 20% for validation. We compared the revenue differences among RVCG, JRegNet, and BundleNet. During preprocessing, we normalized the bidding data to facilitate neural network learning. After obtaining the final results, we scaled the values back to the original bid price range. We evaluated the revenue of these three methods, and for methods with IC Violation, $\\widehat{rgt}$ during testing on normalized data was less than 0.0005. The results are presented below：\n\n|         | RVCG    | BundleNet | JRegNet |\n| ------- | ------- | --------- | ------- |\n| Revenue | 26.7173 | 38.1939   | 36.2950 |\n\n**Note:** In the real world data bidding data, in the fine sorting stage, there are actually less than 50% of the cases that involve more than 5 bundles. Instead of treating them separately, we use a single unified neural network to process all bundle sizes, which contributes to JRegNet’s superior performance over VCG.\n\n**Q2.** In our experiments, the hyperparameter settings are consistent with the open-source implementations of JRegNet and RegretNet.\n\nFor both JRegNet and BundleNet, we use a neural network with 6 hidden layers, each containing 100 neurons, for training and testing. We initialize our Lagrange parameters as $\\forall e \\in E, \\lambda_e = 2.0$. In each gradient ascent step, the update parameter for the Lagrange multiplier and the quadratic penalty term is set to $\\rho = 2.0$. The learning rate for searching the optimal misreport value is $\\gamma = 0.1$, and the neural network learning rate is $\\eta = 0.001$. We use the Adam optimizer for all updates. The primary modification we made was adjusting the batch size for training both networks to 1024. We will release our code upon the acceptance of this paper to facilitate reproduction and further research.\n\n**Q3.** Based on the aforementioned hyperparameters, we present a sensitivity analysis of the batch size and the initial value of the Lagrange multiplier in the $U_2$ scenario：\n\n   - Sensitivity Analysis of Batch Size：\n\n     |           | 128    | 256    | 512    | 1024   | 2048   | 4096   |\n     | --------- | ------ | ------ | ------ | ------ | ------ | ------ |\n     | BundleNet | 0.4993 | 0.4971 | 0.5174 | 0.5262 | 0.5286 | 0.5295 |\n     | JRegNet   | 0.5535 | 0.5594 | 0.5641 | 0.5638 | 0.5685 | 0.5658 |\n\n   - Sensitivity Analysis of Initial Lagrange Multiplier\n\n     |           | 1.0    | 2.0    | 3.0    | 4.0    | 5.0    | 6.0    |\n     | --------- | ------ | ------ | ------ | ------ | ------ | ------ |\n     | BundleNet | 0.1480 | 0.5330 | 0.5373 | 0.5267 | 0.5229 | 0.5240 |\n     | JRegNet   | 0.5669 | 0.5659 | 0.5653 | 0.3837 | 0.1386 | 0.1165 |\n\n**Q4.** Thank you for your feedback. We will provide a more detailed description of the joint advertisement in the label for Figure 1.  The image on the left illustrates traditional advertising, where the **retailer** submits a bid for the ad, while the **supplier** (brand owner) does not participate in the auction. In this case, the platform only receives the bid from the **retailer**.  In contrast, as shown in the right image, for a joint advertisement, both the **retailer** and the **supplier** participate in the auction. They submit bids simultaneously, and the platform receives bids from both parties for the same ad.\n\nThank you for recognizing our work. Here we re-emphasize the key contributions and findings of our work:\n\n1. **Novel solution**: We propose a novel solution for automated mechanism design in joint advertising markets, and empirically demonstrate the alignment between neural network outputs and theoretical predictions.\n2. **Better performance**: For automated mechanism design, we introduce a fundamentally different approach from conventional neural network adaptations. Our core methodology works by **reformulating the problem structure** and **redesigning the optimization process**, thereby overcoming neural networks' inherent limitations in processing sparse graph data while achieving performance improvements.\n3. **Interesting Findings** : We theoretically and empirically show that directly applying RegretNet-like methods to new problem domains may fail to discover optimal mechanisms, as significant gaps can persist between learned mechanisms and theoretically optimal solutions. Our analysis demonstrates that successful application requires problem-specific modifications with rigorous theoretical validation.", "claims_and_evidence": "yes", "methods_and_evaluation_criteria": "The optimal mechanism should be helpful to the bidding advertisement area.", "theoretical_claims": "Yes", "experimental_designs_or_analyses": "The paper lacks experiments on real-world datasets.", "relation_to_broader_scientific_literature": "NA", "essential_references_not_discussed": "The essential references are discussed in this paper.", "other_strengths_and_weaknesses": "Strengths：\n1. The paper is well-written and is easy to follow.\n2. The claims of this paper is validated via experiments\n3. A detailed theoretical proof is provided, which adds rigor to the proposed approach and validates the claims made in the paper.\n\nWeaknesses:\n1. The study is validated only on a simulated dataset, which raises concerns about its practical applicability. Evaluating the method on real-world data would strengthen its reliability and impact.\n2. The experimental settings are not clearly described, particularly regarding model parameters and related configurations. Providing more details would improve the reproducibility and clarity of the study.\n3. The experiments appear relatively simple and may not be comprehensive enough. Including additional tests, such as parameter sensitivity analysis, would provide a more thorough evaluation of the proposed method.", "other_comments_or_suggestions": "1.  The abbreviation \"IC\" is introduced before the full term \"incentive compatibility\" is defined. It would be clearer to first provide the full term before using the abbreviation.\n2. It would be helpful to provide more explanation about joint advertisement in the Figure 1 label, as the relevant background has not been clearly introduced.\n3. the paper presents interesting findings, but the contribution is not sufficiently emphasized. It would be helpful to clarify how this work advances the field.", "questions_for_authors": "1. The study is validated only on a simulated dataset, which raises concerns about its practical applicability. Evaluating the method on real-world data would strengthen its reliability and impact.\n2. The experimental settings are not clearly described, particularly regarding model parameters and related configurations. Providing more details would improve the reproducibility and clarity of the study.\n3. The experiments appear relatively simple and may not be comprehensive enough. Including additional tests, such as parameter sensitivity analysis, would provide a more thorough evaluation of the proposed method.\n4.  It would be helpful to provide more explanation about joint advertisement in the Figure 1 label, as the relevant background has not been clearly introduced.\n5. the paper presents interesting findings, but the contribution is not sufficiently emphasized. It would be helpful to clarify how this work advances the field.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "vIdXgjVqgy", "reviewer": "Reviewer_DGzw", "summary": "The paper addresses auction design in joint advertising settings, where both retailers and suppliers jointly bid on ad slots. It extends classical auction theory by identifying an optimal mechanism for single-slot joint advertisements (via a Myerson-inspired framework) and introduces BundleNet—a novel neural network architecture that leverages a bundle-centric, graph-based approach—for multi-slot scenarios. This method also incorporates a new bundle-level incentive compatibility (IC) constraint, enabling the model to capture the interdependencies between the two parties. Experimental results show that BundleNet not only approximates the theoretical optimum in single-slot settings but also outperforms existing methods in multi-slot environments while maintaining approximate IC and individual rationality.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Q1.** This is a very good question. I think this method may limit the space of discoverable mechanisms, but the difference is not significant.\n\n1. Our core idea is to improve performance by limiting the space of discoverable mechanisms, avoiding the neural network's processing of sparse graph data. In the RegretFormer paper, their improvement likely relies more on expanding the discoverable mechanism space to find a better mechanism. However, for the more challenging problem of joint auctions, our experiments show that as the number of bundles increases, the performance of JRegNet gradually worsens. The main reason for this could be the sparsity of the graph. As the number of bundles increases, the sparsity of the graph significantly strengthens. Encoding this into the neural network inevitably degrades performance.\n 2. We argue that the inequality in **Lemma 5.1** becomes an equality when the following condition holds: For every agent $i \\in R \\cup S$, the utility functions $u_i^e$ across all edges $e$ can be simultaneously maximized by the same strategy $v'_i$.\n3. Our method does not significantly limit the space of discoverable mechanisms. In our training process, we observed that increasing the number of misreport points to 10 does not lead to a significant difference in the sum of individual regrets versus the sum of bundle regrets. Additionally, the result of BundleNet converges to approximately the theoretical optimal.\n4. The inequality in **Lemma 5.1** is difficult to extend into an equality.\n\n**Q2.** Regarding the single-slot scenario, we have added a new visualization to comprehensively demonstrate BundleNet's capabilities. Based on the original experimental setup, we took two well-trained models (with individual average regret values below 0.001, where the current JRegNet even shows higher revenue than BundleNet) and made their allocation results deterministic rather than randomized. The results show that BundleNet's allocations align more closely with the optimal solution. Here is the figure:https://anonymous.4open.science/r/BundleNet-6E3E/visualpicture.jpg\n\n**Q3.** We conducted experiments using real-world data, including 10 bundles and 5 slots. The dataset consists of 102,679 samples, with 80% used for training and 20% for validation. JRegNet and BundleNet are both used in the fine sorting phase. During preprocessing, we normalized the bidding data to facilitate neural network learning. After obtaining the final results, we scaled the values back to the original bid price range. We evaluated the revenue of the three methods, and for methods with IC Violation, $\\widehat{rgt}$  during testing on normalized data was less than 0.0005. The results are presented below:\n\n|      | RVCG    | BundleNet | JRegNet |\n| ---- | ------- | --------- | ------- |\n| Rev  | 26.7173 | 38.1939   | 36.2950 |\n\n   Note: In the real world data bidding data, there are actually fewer than 50% of the cases that involve more than 5 bundles. Instead of treating them separately, we use a single unified neural network to process all bundle sizes, which contributes to JRegNet’s superior performance over VCG.\n\n**Q4.** BundleNet's approach is very transferable:\n\n1. While the idea of joint auctions originated in online advertising, it can be applied to more general scenarios, such as government procurement projects where a single project requires collaboration between companies from different domains. Extending this to allow more participants to form coalitions for joint bidding presents an interesting theoretical problem. Our modeling approach could be similarly adapted for such cases.\n2. As demonstrated in an AAAI 2025 paper (building upon the new scenario introduced in KDD 2024's \"Ad vs Organic: Revisiting Incentive Compatible Mechanism Design in E-commerce Platforms\"), JRegNet has been successfully applied to merged ranking mechanism design, achieving excellent results.\n3. We particularly emphasize our core research insight: When adapting RegretNet to broader mechanism design problems where neural networks struggle with certain features, our method of constraining the discoverable mechanism space can effectively improve performance.\n\n**Q5.** I consider this an open problem in automated mechanism design. Existing methods for multi-agent, multi-item auctions face various limitations:\n\n   - RegretNet can only guarantee approximate IC\n   - GemNet may suffer from over-allocation issues\n   - AMenuNet might not sufficiently approximate theoretical solutions\n\nFrom my perspective, the key may lie in characterizing feasible auctions through utility function properties (analogous to Rochet’s theorem for single-item cases). Similar to RochetNet’s approach, we could:\n\n   - Formalize such properties for multi-bidder settings\n   - Integrate them into neural architecture design\n   - Develop a new automated mechanism design framework\n\nWe are currently investigating concrete implementations, though technical details remain under study.", "claims_and_evidence": "The major claims of the paper are well-supported by theoretical analysis and experimental results:\n- The claim of identifying an optimal mechanism for single-slot joint advertising is supported by a theoretical derivation extending Myerson's framework to the joint advertising scenario (Theorem 4.3).\n- The claim that BundleNet approximates the theoretical optimal mechanism in single-slot settings is shown numerically in Table 1 and Figure 3. \n- The claim of superior performance in multi-slot settings is supported by comparative experiments against RVCG and JRegNet across different distribution settings (Tables 2 and 3).\n- The theoretical foundation for the bundle-centric approach relies on Lemma 5.1, which shows that bundle IC constraints provide an upper bound on individual IC constraints.\n\nQuestions and potential issues:\n- For the last point, it remains unclear to me when reading Lemma 5.1 whether this approach might restrict the space of discoverable mechanisms, can the authors elaborate on this?\n- The proposed framework convincingly outperforms the major baseline of JRegNet, especially in larger settings. However, Figure 3, where the authors visualize the allocation rules learned by BundleNet and JRegNet in a particular setting, show the two methods learning comparable allocation rules with only minor differences. This makes me wonder whether the improved performance could be due to optimization or hyperparameter issues. Can the authors provide any examples where learned allocation structure exhibits a noticeable difference over the baselines, or further justify the superiority of their method, ruling out optimization or hyperparameter tuning as confounding factors?", "methods_and_evaluation_criteria": "The evaluation methods align well with current standards in automated mechanism design; however, the synthetic nature of the dataset may oversimplify the dynamics of real advertising environments, particularly in capturing complex bidder dependencies and realistic click-through rate behaviors. I don't view it as a major problem, but I feel this is worth mentioning. The work would benefit from validating the approach on more complex or real-world datasets to assess its robustness and practical applicability.", "theoretical_claims": "- Theorem 4.3 (optimal mechanism for single-slot joint advertisement) extends Myerson's framework to the joint advertising setting.\n- Lemma 5.1 (relationship between bundle IC constraints and individual IC constraints) demonstrates that the sum of bundle IC constraints upper-bounds the sum of individual IC constraints. \n\nI read through, but didn't check correctness in detail for the two proofs.", "experimental_designs_or_analyses": "The experimental design is robust, with comparisons against strong baselines such as VCG and JRegNet. However, as mentioned above, while the synthetic datasets serve well for controlled comparisons, for the specific problem of ad auctions, the realism of these experiments could be enhanced by incorporating more complex simulations or real-world data to better capture practical ad auction scenarios.", "relation_to_broader_scientific_literature": "The paper effectively builds upon and advances several research streams, though with some limitations in transferability:\n- It builds upon automated mechanism design approaches (Duetting et al., 2019) via neural network\n- It advances work on joint advertising mechanisms (Ma et al., 2024; Zhang et al., 2024), specifically addressing limitations in existing approaches like JAMA and JRegNet.\n- The approach shares RegretNet's limitation of only approximating IC rather than guaranteeing it, which remains an open problem in neural mechanism design.\n- The transferability of the bundle-centric approach and graph-based architecture to other mechanism design problems remains unclear, as the approach is heavily tailored to the specific bipartite structure of joint advertising.", "essential_references_not_discussed": "None major.", "other_strengths_and_weaknesses": "See above sections.", "other_comments_or_suggestions": "The symbols are dense at times, readability could be improved.", "questions_for_authors": "Some questions above are restated here for readability:\n- Is the focus on bundle IC constraints without loss of generality, or might it restrict the space of discoverable mechanisms? Does the upper bound relationship in Lemma 5.1 extend to equivalence under certain conditions?\n- Although BundleNet shows significant numerical performance improvements, the visual differences in allocation boundaries depicted in Figure 3 appear subtle. Could you provide specific examples or additional evidence demonstrating that structural characteristics of BundleNet's learned allocation rules (e.g., sharper or more distinct boundaries) directly contribute to its performance gains? Alternatively, if these structural differences are minimal, can you elaborate on how you rule out potential confounding factors—such as optimization issues or hyperparameter effects—as the primary drivers behind the observed improvements?\n- How transferable is the bundle-centric approach and graph-based architecture to other mechanism design problems beyond joint advertising? \n- Like RegretNet, BundleNet only approximates IC rather than guaranteeing it. Have you explored or have ideas for mechanisms to provide stronger theoretical guarantees on the degree of IC violation?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "nfMUc9EhzR", "reviewer": "Reviewer_EM98", "summary": "The paper considers the sponsored search auctions with joint advertising. In this setting, advertisers are partitioned into suppliers and retailers, with a bipartite graph representing whether it is possible for the pair of advertisers to form a joint advertisement. Each advertisement is then assigned to a retailer and a supplier. The goal is to maximize the revenue of the platform, under incentive compatibility and individual rationality constraints. The authors first study the single-item case. Assuming monotone hazard rate value distributions of the advertisers, they propose an optimal mechanism for this case. The construction of the optimal mechanism is an adaptation of Myerson's result for single-item auctions without joint advertising. For the general case, the authors propose a neural network-based method to design the mechanism. The authors conduct experiments on synthetic data to show the effectiveness of the proposed method.\n\n## Update after rebuttal\n\nI stand by my original positive evaluation.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Q1.** To comprehensively validate our approach, we have conducted additional experiments across diverse settings:\n\n| Single-slot ｜LN(0.1,1.44) | $LN_2$     | $LN_3$     | $LN_4$     | $LN_5$     |\n| -------------------------- | ---------- | ---------- | ---------- | ---------- |\n| JRegNet                    | 0.9666     | 1.2519     | 1.2044     | 1.2415     |\n| BundleNet                  | 1.0132     | 1.2634     | **1.4042** | 1.4339     |\n| RVCG                       | 0.6337     | 0.9982     | 1.2388     | 1.4191     |\n| Optimal                    | **1.0227** | **1.2786** | 1.3916     | **1.4358** |\n\n| Single-slot ｜N(0.5,0.1) | $N_2$      | $N_3$      | $N_4$      | $N_5$      |\n| ------------------------ | ---------- | ---------- | ---------- | ---------- |\n| JRegNet                  | **0.8183** | **0.9091** | 0.8747     | 0.9117     |\n| BundleNet                | 0.7750     | 0.8692     | 0.9134     | 0.9561     |\n| RVCG                     | 0.5492     | 0.8114     | 0.8956     | 0.9444     |\n| Optimal                  | 0.7789     | 0.8656     | **0.9188** | **0.9582** |\n\n| Multi-Slots ｜N(0.5,0.1) | JRegNet    | BundleNet  | RVCG   |\n| ------------------------ | ---------- | ---------- | ------ |\n| $N_{5\\times5}$           | **2.2071** | 2.1393     | 1.3110 |\n| $N_{6\\times5}$           | 2.3537     | **2.3690** | 2.0132 |\n| $N_{7\\times5}$           | 2.4814     | **2.4914** | 2.3565 |\n| $N_{8\\times5}$           | 2.4740     | **2.6287** | 2.5343 |\n| $N_{9\\times5}$           | 2.5185     | **2.7020** | 2.6395 |\n| $N_{10\\times5}$          | 2.4711     | **2.7916** | 2.7228 |\n| $N_{11\\times5}$          | 2.4082     | **2.8428** | 2.7840 |\n| $N_{12\\times5}$          | 2.3272     | **2.8841** | 2.8383 |\n\n| Multi-Slots ｜E(2) | JRegNet | BundleNet  | RVCG   |\n| ------------------ | ------- | ---------- | ------ |\n| $E_{5\\times5}$     | 1.1590  | **1.1884** | 0.6379 |\n| $E_{6\\times5}$     | 1.2435  | **1.3718** | 0.9726 |\n| $E_{7\\times5}$     | 1.2054  | **1.5497** | 1.2436 |\n| $E_{8\\times5}$     | 1.2620  | **1.7162** | 1.4775 |\n| $E_{9\\times5}$     | 1.3269  | **1.7937** | 1.6675 |\n| $E_{10\\times5}$    | 1.4370  | **1.9463** | 1.8347 |\n| $E_{11\\times5}$    | 1.5389  | **2.1050** | 2.0016 |\n| $E_{12\\times5}$    | 1.6002  | **2.1869** | 2.1189 |\n\n**Q2.** We agree this is an excellent suggestion. In fact, we noticed other reviewers raised similar questions. Here we clarify our key findings:\n\n1. **Single-slot scenario**:  \n         BundleNet achieves allocations **closer to the optimal solution** compared to baseline methods (as demonstrated in **Fig. 4**).\n\n2. **Multi-slot scenario**:  \n         BundleNet consistently yields **the highest revenue** among all compared approaches (shown in **Table 2**).\n\nFor the camera-ready version, we will follow conventional practice by:  \n\n   - **Bolding** the highest-revenue method in all tables  \n   - Adding explicit textual markers (e.g., \"★best\") where appropriate  \n   - Including a footnote to explain these annotations  \n\n   This will eliminate potential ambiguity while maintaining rigorous comparison standards.\n\n**Q3.** We sincerely appreciate the reviewer's careful reading regarding the writing details. We agree that the social welfare (SW) formulation could be more precise. In the revised version, we will correct it to:\n$$\n \\text{SW} =E_{\\mathbf{v} \\sim F}\\left[v_0\\left(\\mathbf{1}  -\\sum_{e \\in E}x^e(\\mathbf{v})\\right) \\boldsymbol{\\lambda}^T + \\sum_{e=(r,s) \\in E}x^e(\\mathbf{v}) (v_r + v_s)\\boldsymbol{\\lambda}^T \\right]\n$$", "claims_and_evidence": "I think the claims made in the submission are supported by clear and convincing evidence.", "methods_and_evaluation_criteria": "The proposed methods and evaluation criteria make sense.", "theoretical_claims": "The paper does not claim strong theoretical results. The optimality in the single-item case is an adaptation of Myerson's result, whose correctness is very intuitive, and the neural network-based method in the general case is a heuristic.", "experimental_designs_or_analyses": "In general, the experimental design is solid and provides a good explanation for the effectiveness of the proposed method by comparing it with both previous method (JRegNet) and an fully IC mechanism. (RVCG). However, I have the following comments for the presentation of the experimental results: (1). Normally, when people make data points bold in tables, it means that the data point is the best among the methods. However, as far as I can see, in this paper, the bold data points indicate the method proposed in this paper. To see this, one could refer to Table 1. For $U_2, U_3, E_2, E_3$, the bold data points are not the best among the methods. This might give the reader a wrong impression. (2). The value distributions of the advertisers in the experiments of Sections 6.1 and 6.2 are different. I could not find the reason for this difference.", "relation_to_broader_scientific_literature": "This paper is a nice contribution to the automated mechanism design literature.", "essential_references_not_discussed": "I am not aware of any essential references that are missing.", "other_strengths_and_weaknesses": "- This paper is overall solid and well-written.\n- Like other papers in the automated mechanism design literature, the method's scalability beyond small instances is not good.", "other_comments_or_suggestions": "- Line 123 (right): The notation $E_i$ is used without definition. \n- Line 153 (right): Why the expression of $\\mathrm{SW}$ does not contain the term $v_0$?\n- Line 364 (right): Missing line break before \"Baseline\".\n- Line 403 (left): \"::\" should be \":\".", "questions_for_authors": "- Do you have a reason for the difference in the value distributions of the advertisers in the experiments of Sections 6.1 and 6.2?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "XTQxHB48tf", "reviewer": "Reviewer_X7Et", "summary": "This paper studies the auction design problem in the joint advertising scenario. For the single-slot setting, the optimal mechanism is derived. For the multi-slot setting, the authors propose BundleNet which achieves good performance.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Q1.** We agree this is an excellent suggestion. In fact, we noticed other reviewers raised similar questions. Here we clarify our key findings:\n\n1. **Single-slot scenario**:  \n         BundleNet achieves allocations **closer to the optimal solution** compared to baseline methods (as demonstrated in **Table 1**). Regarding the single-slot scenario, we have added a new visualization to comprehensively demonstrate BundleNet's capabilities. Based on the original experimental setup, we took two well-trained models (with average individual  regret values below 0.001, where the current JRegNet even shows higher revenue than BundleNet) and made their allocation results deterministic rather than randomized. The results show that BundleNet's allocations align more closely with the optimal solution.Here is the figure:https://anonymous.4open.science/r/BundleNet-6E3E/visualpicture.jpg\n\n2. **Multi-slot scenario**:  \n         BundleNet consistently yields **the highest revenue** among all compared approaches (shown in **Table 2**).\n\nFor the camera-ready version, we will follow conventional practice by:  \n\n   - **Bolding** the highest-revenue method in all tables  \n   - Adding explicit textual markers (e.g., \"★best\") where appropriate  \n   - Including a footnote to explain these annotations  \n\n   This will eliminate potential ambiguity while maintaining rigorous comparison standards.\n\n**Q2.** I think this is a good question that is innovative enough for the article. Let me provide a more systematic response:\n\n1. We propose **two novel solutions** for optimal mechanism design in joint advertising, and **empirically verify** the consistency between neural network results and theoretical outcomes.\n\n2. For automated mechanism design, we present **a new approach** that differs from traditional neural network modifications. Our core methodology is as follows: By **reformulating the problem structure** and **optimization process**, we circumvent neural networks' limitations in handling sparse graph data, thereby improving performance.", "claims_and_evidence": "I think so.", "methods_and_evaluation_criteria": "Generally yes. However, it seems to me that for the multi-slot case, the major improvement of this work is to model the IC constraint from a bundle's perspective rather than an agent's perspective. As for me, it seems a bit unclear why this works. However, it is sufficient for an applied paper. T", "theoretical_claims": "I didn't check the proof. Yet in Theorem 4.3, why do the authors write the theorem from the agent's perspective, rather than the bundle's perspective? The current writing makes understanding a bit hard, since the final allocation function is defined on the bundle, right? Meanwhile, $v_0$ appears in page 3, right, line 145, but not introduced before that. The same circumstance also appears for $U_i(v_i, v'_i)$.", "experimental_designs_or_analyses": "It seems a bit strange to me that the authors bold all results of BundleNet in Table 1, even though they are not optimal in all cases.", "relation_to_broader_scientific_literature": "The key contributions are related to automated mechanism design.", "essential_references_not_discussed": "I am not really sure.", "other_strengths_and_weaknesses": "No.", "other_comments_or_suggestions": "No.", "questions_for_authors": "Please see the above.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The joint advertising problem arises in sponsored search auctions where two parties, typically a supplier and a retailer, jointly benefit from an ad impression and co-bid for the same ad slot. The paper proposes an optimal mechanism for single-slot joint advertising, and introduces BundleNet, a graph-based neural architecture for multi-slot settings that incorporates bundle-level IC constraints. Theoretical contributions are well-motivated and extend classical auction design. BundleNet outperforms baselines on synthetic data and approximates theoretical optima.

On the negative side, all experiments are on simulated data, limiting practical applicability. The bundle IC constraint’s impact is not fully clarified, and experimental details are sometimes unclear. Scalability and generalizability remain open questions. Despite these issues, the paper is well-written, and is a solid contribution to the sponsored search auction literature, so it can be accepted if there is room.
Decision: Accept (poster)