A Scalable Neural Network for DSIC Affine Maximizer Auction Design
Authors: Zhijian Duan
Haoran Sun
Yurong Chen
Xiaotie Deng
Venue: NeurIPS 2023 spotlight (2023)
Keywords: Automated Mechanism Design
Auction Design
Affine Maximizer Auctions
Deep Learning
Game Theory
TL;DR: We propose AMenuNet, a scalable neural network for DSIC affine maximizer auction design.

Abstract: Automated auction design aims to find empirically high-revenue mechanisms through machine learning. Existing works on multi item auction scenarios can be roughly divided into RegretNet-like and affine maximizer auctions (AMAs) approaches. However, the former cannot strictly ensure dominant strategy incentive compatibility (DSIC), while the latter faces scalability issue due to the large number of allocation candidates. To address these limitations, we propose AMenuNet, a scalable neural network that constructs the AMA parameters (even including the allocation menu) from bidder and item representations. AMenuNet is always DSIC and individually rational (IR) due to the properties of AMAs, and it enhances scalability by generating candidate allocations through a neural network. Additionally, AMenuNet is permutation equivariant, and its number of parameters is independent of auction scale. We conduct extensive experiments to demonstrate that AMenuNet outperforms strong baselines in both contextual and non-contextual multi-item auctions, scales well to larger auctions, generalizes well to different settings, and identifies useful deterministic allocations. Overall, our proposed approach offers an effective solution to automated DSIC auction design, with improved scalability and strong revenue performance in various settings.

A Scalable Neural Network for DSIC Affine Maximizer Auction Design
Zhijian Duan
Haoran Sun
CFCS, School of Computer Science Peking University
Peking University sunhaoran0301 10tttu pku..du edu. en
zj jduan@pku.edi edu.
Yurong Chen
Xiaotie Deng
CFCS, School of Computer Science Peking University chenyurong@pku. edu. cn
CFCS, School Computer Science & CMAR, Institute for AI Peking University xiaotie@pku.c edu.
Abstract
Automated auction design aims to find empirically high-revenue mechanisms through machine learning. Existing works on multi item auction scenarios can be roughly divided into RegretNet-like and affine maximizer auctions (AMAs) approaches. However, the former cannot strictly ensure dominant strategy incen- tive compatibility (DSIC), while the latter faces scalability issue due to the large number of allocation candidates. nuNet, scalable neural network that constructs the AMA parameters (even in- cluding the allocation menu) from bidder and item representations. AMenuNet is always DSIC and individually rational (IR) due to the properties of AMAs, and it enhances scalability by generating candidate allocations through a neural network. Additionally, AMenuNet is permutation equivariant, and its number of parameters is independent of auction scale. We conduct extensive experiments to demonstrate that AMenuNet outperforms strong baselines in both contextual and non-contextual multi-item auctions, scales well to larger auctions, generalizes well different settings, and identifies useful deterministic allocations. Overall, our proposed approach offers an effective solution to automated DSIC auction design, with improved scalability and strong revenue performance in various settings.
address these limitations, we propose AMe-
Introduction
One of the central topics in auction design is to construct mechanism that is both dominant strategy incentive compatible (DSIC) and individually rational (IR) while bringing high expected revenue to the auctioneer. The seminal work by Myerson [ЛРЛ]] characterizes the eevenue-mxiimiiig mech- anism for single-parameter auctions. However, after four decades, the optimal auction design prob- lem in multi-parameter scenarios remains incompletely understood,
in simple settings such as two bidders and two items [Dutting et all 2019) To solve the problem, recently, there has been significant progress automated auction design [Sandholm and Likhodedos, 2015 Dütting et all 2019). Such a paradigm formulates auction design as an optimization problem subject to DSIC and IR constraints and then finds optimal or near-optimal solutions using machine learning.
The works of automated auction design can be roughly divided into two categories. The first cat- egory is the RegretNet-like approach [Curry et ali, 2020, |eer et all 2021, Rahme all 202lb Duan et all 2022, Vanoy et all 2022], pioneered by RegretNet [Dütting et al, 2019] These works
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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represent the auction mechanism as neural networks and then find near- optimal and approximate DSIC solutions using adversarial training. The second category is based on affine maximizer auc- tions (AMAs) [Roberts, 1979, Likhodedov and Sandholm. 2004, Likhodedov etl 2005, Sandholm and Likhodedov, 2015a, Guo et all 2017 Curry et all 2023). These methods restrict the auction mechanism to AMAs, which are inherently DSIC and IR. Afterward, they optimize AMA parame- ters using machine learning to achieve high revenue.
Generally, RegretNet-like approaches can achieve higher revenue than AMA-based approaches. However, these works are not DSIC. They can only ensure approximate DSIC by adding a regret term in the loss function as a penalty for violating the DSIC constraints. There are no theoretical results on the regret upper bound, and the impact of such regret on the behaviors of strategic bidders. Even worse, computing the regret term is time-consuming [Rahme et all 20216]. AMA-based ap- proaches, on the other hand, offer the advantage of guaranteeing DSIC and due to the properties of AMAs. However, many of these approaches face scalability issues because they consider all de- terministic allocations as the allocation menu. The size of the menu grows exponentially, reaching (n+1) for bidders and m items, making it difficult for these approaches to handle larger auctions. Even auctions with 3 bidders and 10 items can pose challenges to the AMA-based methods.
To overcome the limitations above, we propose a scalable neural network for the DSIC affine max- imizer auction design. We refer to our approach as AMenuNet: Affine maximizer auctions with Menu Network. AMenuNet constructs the AMA parameters, including the allocation menu, bidder weights, and boost variables, from the bidder and item representations. After getting the parameters, we compute the allocation and payment results according to AMA. By setting the representations as the corresponding contexts or IDs, AMenuNet can handle both contextual [Duan et all, 2022] and non-contextual classic auctions. As AMenuNet only relies on the representations, the resulting mechanism is guaranteed to be DSIC and IR due to the properties of AMAs.
Specifically, we employ two techniques to address the scalability issue of AMA-based approaches: (I) Firstly, we parameterize the allocation menu. We predefine the size of the allocation menu and train the neural network to compute the allocation candidates within the menu, along with the bidder weights and boost variables. This allows for more efficient handling of large-scale auctions. (2) Secondly, we utilize transformer-basee permutation-equun equivarraan architecture. Notably, this architecture's 's parameters remain independent of the number of bidders or items. This enhances the scalability of AMenuNet, enabling it to handle auctions of larger scales than those in training.
We conduct extensive experiments to demonstrate the effectiveness of AMenuNet. First, our per- formance experiments show that in both contextual and classic multi-item auctions, AMenuNet can achieve higher revenue than strong DSIC and IR baselines. AMenuNet can also achieve comparable revenue to RegretNet-like approaches that can only ensure approximate DSIC. Next, our ablation study shows that the learnable allocation menu provides significant benefits to AMenuNet, from both revenue and scalability perspectives. Thirdly, we find that AMenuNet can also generalize well to auctions with a different number of bidders or items than those in the training data. And finally, the case study of winning allocations illustrates that AMenuNet can discover useful deterministic allocations and set the reserve price for the auctioneer.
2 Related Work
Automated mechanism design[[ [Conitzer and Sandholm, 2002 2004, Sandholm and Likhodedov, 20156 has been proposed to find approximate optimal auctions with multiple items and bidders [Ba can et all 2008, Lahaie, 2011, Dütting et all, 2015). Meanwhile, several works have analyzed the
sample complexity of optimal auction design problems [Cole and Roughgarden, 2014), Devanun alll 2016. Balcan aaall 2016, Guo 2019, Gonczarowski and Weinberg, 2021]. Recently, pioncered by RegretNet [Dütting ef al,, 2019) there is rapid progress on finding (approximate) optimal auction through deep learning [[urry et all 2020, Per el all 202T Rahme all 20216a. Duan et all 2022,
Ivanoy et all, 2022)). However, as we mentioned before, those RegretNet-like approaches can only ensure approximate DSIC by adding a hard-to-computt regret term.
Our paper follows the affine maximizer auction (AMA) [Roberts, 1979 based approaches. AMA is weighted version of VCG [Vickrey, [16]]], which assigns weights to each bidder and assigns boosts to each feasible allocation. Tuning the weights and boosts enables AMAs to achieve higher revenue than VCG while maintaining DSIC and IR. Different subsets of AMA have been studied

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in various works, such as VVCA [Likhodedov and Sandholm. 2014 Likhodedov all 2015, SandE
holm and Likhodedov, MUSSE, X-auction [eehell et all 2007 mixed bundling auction [[ang and Sandholmi 2012), and bundling boosted auction [Balcan et all 2021]. However, these works set all the deterministic allocations as candidates, the size of which grows exponentially with respect to the auction scale. To overcome such issue, we construct a neural network that automatically computes the allocation menu from the representations of bidders and items. Curry et all [2023] is the closest work to our approach that also parameterizes the allocation menu. The key difference is that they optimize the AMA parameters explicitly, while we derive the AMA parameters by a neural network and optimize the network weights instead. By utilizing a neural network, auctions by incorporating representations as inputs. Additionally, the trained model can generalize to auctions of different scales than those encountered during training.
handle contextual
Contextual auctions are more general and realistic auction format assuming that every bidder and item has some public information. They have been widely used in industry IZhang et al]. 2021 all 2021I. In the academic community, previous works on contextual auctions mostly focus on the online setting of some well-known contextual repeated auctions, such as posted-price auc- tions [Amin etlll 20 Mao ef all 2018, Druisal 2020. Zhryanov and Drutsa, 2020]. where the seller prices the item to sell to a strategic buyer, repeated second-price auctions [Golrezaer et all, 2021I. In contrast, our paper focuses on the offline setting of contextual sealed-bid auctions, similar to Duan al [2022]] RegretNet-like approach.
3 Preliminary
Sealed-Bid Auction. We consider a sealed-bid auction with v bidders and m items. Denote [n] = (1. n3. Each bidder i E [n] is represented by a dy-dimensional vector Φ; E Rd,, which can encode her unique ID) or her context (public feature) [Duan et all, 2022]. Similarly, each item j E [m] is represented by a dyydimeesiona vector Yy E TRdv, which can also encode its unique ID or context. By using such representations, we unify the contextual auction and the classical Bayesian auction. We denote by X = [x,, .,,,....................... xx," E Raddd and Y = [y1.3/2.... 3/2..... ymm' E gmxdy the matrices of bidder and item representations, respectively. These matrices follow underlying joint probability distribution Fx.y. In an additive valuation setting, each bidder values each bundle of items S C [m] with valuation UgS = Ejes Vij- The bidder has to submit her bids for each item j [m] as b; = (61, (61,62...., bm)) The valuation profile V (1i)))e[ll,JE[m) E =x is generated from conditional distribution FvIx,y Y that depends on the representations of bidders and items. The auctioneer does not know the true valuation profile V but can observe the public bidder
representations X, item representations Y, and the bidding profile B = (bez) (m)))em E R R
Auction Mechanism. An auction mechanism (g.P) consists of an allocation rule g R X
Raxd2 x Rm + [0, 1| and a payment rule p R=xm x Raxd2 x Rmxdd + R 20. Given the bids B, bidder representations X, and item representations Y, 2y [B X.Y) E J0, 1] computes the probability that item j is allocated to bidder i. We require that Σ Li=1 Diy [B, X, Y.Y) < 1 for any item j to guarantee that no item is allocated more than once. The payment PB X,B, the price that bidder i needs to pay. Bidders aim maximize their own utilities. In the additive valuation setting, bidder i's utility is (u( X, ==
Y) X.Y) 2 O computes
T Σラ 9is (B, X, Y)uūj p,,BB X, Y), given v, B, X, Y. Bidders may misreport their valuations to benefit themselves. Such strategic behavior among bidders could make the auction result hard to predict. Therefore,
require the auction mechanism to be dominant strategy incentive compatible (DSIC), which means that for each bidder E [n]. reporting her true valuation is her optimal strategy regardless of how others report. Formally, let B_; = (b,....,,,... b.41,.... bx) be the bids except for bidder i. A DSIC mechanism satisfies
u,(u,, (v:, B- X))) 2 u, (b,, X,Y)), Vi, U,,, X,Y.b.
(DSIC)
Furthermore, the auction mechanism needs to be individually rational (IR), which ensures that truth- ful bidding results in a non-negative utility for each bidder. Formally,
ue(ws, (w,, B B-)); 2 O, Vi, v. B_, X,Y.
(IR)
Affine Maximizer Auction (AMA). AMA RRbeers 1979 a generalized version of Vickr- cyClarkeGroves (VCG) auction [iikkryy 196] that is inherently DISC and IR. An AMA consists
3

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of positive weights w, R+ for each bidder and boost variables X(A) E R for each allocation A E where A the allocation menu, i.., the set of all the feasible (possibly random) allocations. Given B, X, Y, AMA chooses the allocation that maximizes the affine welfare:
g(B,X,Y) = A arg max AEA
w,b((A) *(A),
(Allocation)
where b₂(A) 25=1 bij Aij in additive valuation setting. The payment for bidder & is
Ph (B,
Σ i
wabb(AAk) +
Σ w.b.(A-) + (wdd(4+)+A44))) +A(+)) ig
(Payment)
We
wk
Here we denote by A = arg ACX Eisk (A) - X(A) the allocation that maximizes the affine welfare, with bidder k's utility excluded. Our definition of AMA differs slightly from previous literature [Roberts [979], whose allocation candidates are all the (n + 1)th deterministic solutions. Instead, we explicitly define the allocation menu so that our definition is more general. As we will show in Appendix Al such AMAs are still DSIC and IR.
Methodology
In this section, introduce the optimization problem for automated mechanism design and proposed approach, AMenuNet, along with its training procedure.
4.1 Auction Design as an Optimization Problem
We begin by parameterizing our auction mechanism (g".p"), where 0 represents the neural net- work weights to be optimized. We denote the parameter space as θ 0 and the class of all pa- rameterized mechanisms M The optimal auction design seeks find the avrenue-mximmizinn mechanism that satisfies both DSIC and [[]:
Revlg'pp') B(v.XYY)
Σ Exl(v.x,))
max DE0
(OPT)
s.t. ( is and
where Rev(g",p) p²) is the expected revenue of mechanism (g" "pp)). However, there is no known characterization of a DSIC general multi-item auction [Dütting et al. 2019]. Thus, we restrict the search space of auctions to affine maximizer auctions (AMAs) [Rbbett 1979]. AMAs are inherently DSIC and IR, and can cover a broad class of mechanisms: Lavi el all [200]]] has shown that every DSIC multi-item auction (where each bidder only cares about what they get and pay) is almost an AMA (with some qualifications).
The AMA parameters consist of positive weights w E I2T for all bidders, an allocation menu A, and boost variables À IRA-A for each allocation in A. While the most straightforward way to define A is to include all deterministic allocations, as done in prior work [Lakhodedovv and Sandhom AMAL Lkhhodedo all 2005, Sandholm and Likhodedox, 20156), this approach suffers from scalability issues as the number of allocations can be as large as (n + 1)m". To address this challenge, we propose AMenuNer, which predefines the size of A and constructs it along with the other AMA parameters using a coimutation-equivrriaa neural network.
4.2 AMenuNet Architecture
Denote by |A| the predefined size of the allocation menu. AMenuNet takes as input the repre- sentations of bidders X and items Y, and constructs all the allocations, the bidder weights w, and boost variables X E IR. through permutation-equivarine neural network architecture as illus- trated in Figure II. The architecture consists of an encode layer, a menu layer, weight layer, and boost layer. The final allocation and payment results can be computed by combining the submitted bids B according to AMA (see Equation (Allocatio)) and Equation (Paymen())

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Represoniation itams (m
I ((N x m.x.o
((w xcm 1-1
Altocatin Karts A
(r.
gf X
Spii
Bate B (n X mN)
Layer
Eirbed&
(Mddlipp))
Assuntii
cone
Interaction Medidee
Weight
Avecage
(+)
Allocation (ts x m)
Paument (+)
Layer
Figure schematic view of AMenuNet, which takes the bidder representations X (including the dummy bidder) and item representations Y inputs. These representations are assembled into a tensor E E R 1x-++)xxx((xd (4,, Two 1 convolution layers are then applied to obtain the tensor Following L, multiple transformer- based interaction modules are used model the mutual interactions among all bidders and items. The output tensor after these modules is denoted as J. J is further split into three parts: A E [2(+1)xxxxx _ E R [n++]Xxm and J E R (=++1)xm These parts correspond the allocation menu A, bidder weights w, and boosts A, respectively. Finally, based on the induced AMA parameters and the submitted bids, the allocation and payment results are computed according the AMA mechanism.
We first introduce a dummy bidder n + handle cases where no items are allocated any of the bidders. We set such dummy bidder's representation In+1 as de dy-dimensional vector with all elements
Encode Layer. The encode layer transforms the initial representations of all bidders (including the dummy bidder) and items into joint representation that captures their mutual interactions. In contextual auctions, their initial representations are the contexts. In non-contextual auctions, similar word embedding [Mikolov all 2OIK), embed the unique ID of each non-dummy bidder or item into a continuous vector space, and use the embeddings as their initial representations. Based on that, we construct the initial encoded representations for all pairs of + 1 bidders and m items E E
+d,) where
Fay = [a,;; yy] E RR
is the initial encoded representation of bidder i and item
We further model the mutual interactions of bidders and items by three steps. Firstly, we capture the inner influence of each bidder-item pair by two 1 x convolutions with a ReLU activation. By doing so, we get
Conv2 ReLU Conv,
[((+1)xmxdd
where d the dimension the new latent representation for each bidder-item pair, both Conv, and Conv2 are convolutions, and ReLUU()) max(x; 0).
Secondly, we model the mutual interactions between all the bidders and items by using the trans- former [Vaswanr et all 2012 based interaction module, similar to Duan et [2022]. Specifically, for each bidder i, we model her interactions with all the m items through transformer on the i-th row of L, and for each item transformer on the j-th column of
model its interactions with all the bidders through another
7.ow transformer((:: E IR. mxdd
reolumn transformer((...)) E IR. (n+1)xx +J
where d, is the size of the hidden nodes of transformer; For all the bidders and items, their global representation obtained by the average of all the representations
x+1
@llobal
Lig-
(n + 1 m
1=1
Thirdly, we get the unified interacted representation
[1 1 coolumn,
global
2d,
5

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by combining all the three representations for bidder and item j. Two x 1 convolutions with ReLU activation are applied on to encode I into the joint representation
your = Conv4 o ReLU Conv3 T E [R(++))xxxx
By stacking multiple interaction modules, we can model higher-order interactions among all bidders and items.
For the final interaction module, we set dout 2s + 1 and we denote by E ( its output. We then partition into three tensors: JA E (g(=+))xmx. xx J E R ( and
J [(++]xxmxx, and pass them to the following layers.
Menu Layer. To construct the allocation menu, we first normalize each column of J through softmax function. Specifically, Vj E [m], E [s], we denote by J t E R=+ the j-th column of allocation option k, and obtain its normalized probability J. J.J. by
J=h Softmax(T J_ Jey.s)
where Softmax(2)) eth /< k= e**) (0, 1) for all R=+1 the softmax function, and >> O is the temperature parameter. Next. we exclude the probabilities ussociated with the n - 1-th dummy bidder obtain the allocation menu:
A= J E 1
Thus, for each allocation A E A we satisfy A:j E [0, 1] and E Aij S for any bidder i and any item j.
Weight Layer: In this work, we impose the constraint that each bidder weight lies in (0, 1]. Given Jw E R [x++))xm, for each bidder i S n we get her weight by
m
w. = Sigmoid(
J
m
j=1
where Sigmoid(s) = 1/(1 + =) E (0, 1) for all
the sigmoid function.
Boost layer. In boost layer, first average J across all bidders and items. We then multi- layer perceptron (MLP), which is fully connected neural network, to get the boost variables A. Specifically, we compute by
n++ 1 m
A MLP
ER.
After get the output AMA parameters A, À from AMenuNet, result according to Equation (Allccaioo)) and the payment result according to Equation (Payment). The computation of A, w, X only involves the public representations of bidders and items, without access to the submitted bids. Therefore, the mechanism DSIC and IR (see Appendix [ for proof):
compute the allocation
Theorem 4.1. The mechanism induced by AMenuNet satisfies both DSCC and T
Moreover, As AMenuNet is built using equivariant operators such transformers and 1 con- volutions, any permutation of the inputs to AMenuNet, including the submitted bids B, bidder rep- resentations X, and item representations Y. results in the permutation of the allocation and payment outcomes. This property is known as permutation equivariance Bahmm al 2021E Dnen
all 2023]:
Definition 4.2 (Permutation Equivariance). We say (g.p) is permutation equivariant, if for any B, X, Y and for any two permutation matrices HI {0, 1] jxxn and IT,, E (0. mmm have I(T,, BI, X, IT = TT.g( B, X, Y) and p (II,, BTT,,, II, II = TT, (,,
Permutation equivariant architectures are widely used in automated auction design [Rahme et all 202 Duan et all, 2022, [vanoy et all, 2022]. Qin et all [2022] have shown that this property can Iead to better generalization ability of the mechanism model.

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4.3
Optimization and Training
Since the induced AMA is DSIC and IR, we only need to maximize the expected revenue Rev(g" p² To achieve this, following the standard machine learning paradigm [Shalev-Shwariz and David. 2014), we minimize the negative empirical revenue by set the loss function as
|
T
S) =
Pi (V ( v(h)).
where S the training data set, and 9 contains all the neural network weights in the encode layer and the boost layer. However, the computation of E 5) involves finding the affine welfare-manne maximizing allocation scheme A (and x)) which is non-differentiable. To address this challenge, softmax function as an approximation. During training, we compute an approximate A by
the
A
Σ AEA
exp
TA
Σ w.b + (.) +
A,
where Z == A''e exp (TA (Ei=1 (ab.(A') X(A')) is the normalizer, and тд is the softmax temperature. We can control the approximation level of A by tuning TAΣ when TA + co, A recover the true A", and when - 0, A tends to uniform combination of all the allocations in A. The approximation A of A is similar. By approximating A" and A through the differentiable A and A R. we make it feasible to optimize ((0, through gradient descent. Notice that in testing, we still follow the standard computation in Equation (Ellocation) and Equation (Payment).
5 Experiments
In this section, we present empirical experiments that evaluate the effectiveness of AMenuNet All experiments are run on a Linux machine with NVIDIA Graphics Processing Unit (GPU) cores. Each result is obtained by averaging across 5 different runs. In all experiments, the standard deviation of AMenuNet across different runs less than 0.01.
Baseline Methods. We compare AMenuNet against the following baselines:
1. VCG [Vickrey, [961]], which is the most classical special case of AMA;
2. Item- Item-Myyrson, strong baseline used in Dütting al. [209), which independently applies
Myerson auction with respect to each item;
3. Lottery AMA [Curry et all, 2023], AMA-based approach that directly sets the allocation menu, bidder weights, and boost variables as all the learnable weights.
4. RegretNet [Dutting et al, 201%), the pioneer work of applying deep learning in auction design, which adopts fully-connected neural networks to compute auction mechanisms.
5. ClTransNet [Duan et all, 2022). RegretNet-like transformer-based approach that supports contextual auctions.
Note that both RegretNet and ClTransNet can only achieve approximate DSIC by adding a regret term in the loss function. We train both models with small regret, less than 0.005.
Hyperparameters. We train the models for a maximum of 8000 iterations. with 32768 gener- ated samples per iteration. The batch size is 2048, and we evaluate all models on 100000 samples. We set the softmax temperature as 500 and the learning rate in f32, 64, 128, 256, 512, 1024) For the boost layer, we use a two-layer fully connected neural network with ReLU activation. Given the induced AMA parameters, our implementation of the remaining AMA mechanism is built upon the implementation of Curry et all [203]] Further imple- mentation details can be found in Appendix B
10-4 We tune the menu size
I Our implementation is available kttps://githin cow.HHaarrar0000//Mellle...o
We explore the impact of different menu sizes in Appendix Q
7

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Table 1:
The experiment results of average revenue. For each case, we use the notation n x m to represent the number of bidders n and the number of items m. The regret of both CiTransNet and RegretNet less than 0.005. The best revenue highlighted in bold, and the best revenue among all DSIC methods is underlined.
(a) Contextual auctions (Setting A and (B).
We omit Lottery AMA and RegretNet since they are designed for classic auctions. "Randomized Allocations" shows the ratio of randomized allocations output by AMenuNet.
Method
DSIC?
2x2
2x5
2x10
3x2
3x5
3x10
4x2
5x2
6x2
7x2
(A)
(A)
(A)
(A)
(A)
(B)
(B)
(B)
CTTransNet
No
0.4461
1,1770
2.4218
0.5576
1.4666
2,9180
0.7227
0.7806
0.8396
0.8997
VCG
Yes
0,2882
0.7221
1,4423
0.4582
1.1457
2.2967
0.5751
0.6638
0.7332
0,7904
Wem-Myeeson
Yes
0.4265
1.0700
2.1398
0.5590
1,3964
2.7946
0.6584
0.7367
0.8004
0.8535
AMenuNet
Yes
0.4398
1.1599
2,3935
0.5601
L4458
2,9006
0,7009
0,7548
0.8127
0,8560
Randomized Allocations
1.42%
7.37%
18.01%
3.52%
22.25%
54.38%
12.41%
14.81%
10.76%
17.02%
(b) Classic auctions (Setting (CHF)
Method
DSIC?
2x5(C)
3x10(C) 5x5(C)
3x1(D)
1x2(E)
1x2(F)
Optimal
2.7490
9.7810
0.1706
CTTransNet
No
2.3788
5,9191
3,4759
2.7541
9.7551
0.1691
RegretNet
No
2.3390
5.5410
3.4260
2.7264
9.7340
0.1732
VCG
Yes
1.6645
5.0002
3.3319
2.4954
Item-Myerson
Yes
2.0755
5.3141
3.3566
2.7490
8.8359
0.1482
Lottery AMA
Yes
2.2354
5.3450
2.8527
2.7238
9.3139
0.1544
AMenuNet
2.2768
5.5896
3.3916
2.7382
9.6219
0.1701
Auction Settings. AMenuNet can deal with both contextual auctions and classic Bayesian auc- tions. We construct the following multi-item contextual auctions:
(A) We generate each bidder representations 2; E RR10 and item representations y, E R 10 inedpen-
dently from uniform distríbution in [-1, (i.e., U[-1, 1). The valuation Bay is sampled from U0, (igmoid(<= y )] This contextual setting is also used in Duan et all [2022]. We choose the number of bidders n E {2, 3} and the number of items m E {2, 103.
(B) We set 2 items, and the representations are generated from the same way in Setting A For valuations, we first generate an auxiliary variable uy from UO, 1] for each bidder i, and then we set 0i = v/-- Sigmoid (a:f y1) and 0:2 = (1 vt) Sigmoid (xr y2). We do this to make the valuations of the 2 items highly correlated. We choose the number of bidders
{4, 5, 6, 7).
For classic auctions, we can assign each bidder and item a unique ID and embed them into multi- dimensional continuous representation. We construct the following classic auction settings:
(C) For all bidders and items, Uay U [0. 1]. Such setting is widely evaluated in RegretNet-like approaches [Dütting et all 201%]. We select n x m (the number of bidders and items) in
(2 x 5,3 10, 5 x 5).
(D) 3 bidders and 1 item, with Un ~ Exp(3) (i.e. the density function is f(x) = 1/3e-1/3a for all E {1, ())) The optimal solution given by Myerson auction.
(E) bidder and 2 items, with U11 U[4, 7] and 212 U4, 16). The optimal auction is given by Daskalakis al [2015].
(F) bidder and 2 items, where Un has the density function f(x) = 5/(1 + z²). and V12 has the density function f(z) = 66((1 + x²?. The optimal solution is given by Daskalakis all [2015]
Revenue Experiments. The results of revenue experiments are presented in Table II We can see that in settings with unknown optimal solutions, AMenuNet achieves the highest revenue among all the DSIC approaches. The comparison between AMenuNet and VCG reveals that incorporating affine parameters into VCG leads to higher revenue. Notably, AMeunNer even surpasses the strong baseline Item- which relies on prior distribution knowledge, an over strong assumption that may not hold especially in contextual auctions. In contrast, AMenuNet constructs the mechanism solely from sampled data, highlighting its data efficiency. Furthermore, AMenuNet demonstrates its
8

---

Table The revenue results of ablation study. For each case, we use the notation X m to represent the number of bidders n and the number of items m. Some results of Adtm are intractable because the menu size is too large.
2x2(A)
2x10(A)
3x2(B)
5x2(B)
2x3(C)
3x
AMenuNet
0.4293
2.3815
0.6197
0.7548
1.3107
5,5896
With w =
0.4209
2.3848
0.6141
0.7010
1.2784
5,5873
With A 0 0.3701
2.2229
0.3892
0.3363
1.2140
5,4145
With Admm
0.3633
0.5945
0.7465
1,2758
With FCN
0.4124
2.3416
0.5720
0.6963
1.2524
5.0262
2.0
0.9
Revenue
1.5
0.8
Revenue
0.7
1.0
0.6
Trained on 2x2
Trainad on 5x2
Trained on 3x10
0.5
Trained on 2x5
0.5
Trained on 8x2
Trained on 5x10
Itamn-yyersan
Itoo-Myyrsoo
hom-Myyrson
10
Number of Items
Number of Bidders
Number Items
(a) Setting [ with fixed n = (b) Setting [B] with fixed m = 2. (c) Setting [Ci with fixed n = 3.
Figure 2: Out-of-setting generalization results. We use the notation n X m to represent the number of bidders and the number of items m. We train AMenuNet and evaluate it on the same auction setting, excepts for the number of bidders or items. For detailed numerical results, please refer to Appendix C
ability to approach near-optimal solutions in known settings (Setting DHF. even when those so- lutions are not AMAS themselves. This underscores the representativeness of the AMAS induced by AMenuNet. Remarkably, the comparison between AMenuNet and Lottery AMA in classic auctions showcases the effectiveness of training a neural network to compute AMA parameters. This sug- gests that AMenuNet captures the underlying mutual relations among the allocation menu, bidder weights, and boosts, resulting in superior mechanism after training. Lastly, while CITransNet and RegretNett may achieve higher revenue in many cases, it is important to note that they are not DSIC. Such results indicate that AMenuNet's zero regret comes at the expense of revenue. Compared to RegretNet-based approaches, AMenuNet's distinctive strength lies in its inherent capacity to ensure DSIC by design.
Ablation Study. We present an ablation study that compares the full AMenuNet model with the following three ablated versions: AMenuNet with fixed w = 1, AMenuNet with fixed À = 0, AMe- nuNet with Adtm (all the (n + 1)m deterministic allocations). and AMenuNet with the underlying architecture to be 4-layers fully connected neural networks (FCN). We set the number of layers in FCN to be 4, each with 128 hidden nodes. The revenue results are presented Table M For large values of and the experiment result of AMenuNet with Autm can be intractable due to the large size the allocation menu (59049 for 2 10 and 1048576 for 3 10). Such phenomenon indicates that will face scalability issue if we consider all deterministic allocations. In contrast, the full model achieves the highest revenue in all cases except for 2 x 10(A), where it also performs compa- 1 is coincidently the optimal solution due
rably to the best result. Notice that, in Setting [C], to the symmetry the setting. The ablation results underscore the benefits making the allocation menu, weights, and boost variables learnable, both for effectiveness and scalability. Moreover, with significantly fewer learnable parameters, MeemuNetconsistently outperforms AMenuNetwith FCN, further highlighting the revenue advantages of transformer-based architecture.
Out-of-Setiiee -Sttinn Generalization. The architecture of AMenuNet is designed to be independent of the number of bidders and items, allowing be applied to auctions with varying sizes. To evaluate such out-of-settigg generalizability [Rahme all 202 Rl, conduct experiments whose results are

---

allocations
allocations
0.9 0. 0/D 8. 91
0.0
04 0.6
6.3
0.0 0.0 00
0.0
9.
00 0.000
0.0
0.0
0.0
0.0 0.
0.0
0.0
0.0
0.
0.0
0.
06
0.0
0.
12.
0.1 1..9
12.11
12.31
(bidder, item) pairs
(bidder, item) pairs
(a) 2 with 128.
(b) x
C
with 1024.
Figure 3: The top-10 10 allocations (with respect to winning rate) and the corresponding boosts among 100, 000 test samples.
shown in Figure Q The experimental results demonstrate that the generalized AMenuNet achieves revenue comparable that of Item-Myerson, particularly in cases with varying items where AMe- nuNet often outperforms Item-Myerson. This highlights the strong out-of-setting generalizability of AMenuNet. Furthermore, The fact that AMenuNet can generalize larger settings enhances its scalability.
Winning Allocations. We conducted experiments to analyze the winning allocations generated by AMenuNet. In order to assess the proportion of randomized allocations, recorded the ratio of such allocations in the last row of Table IE Here, we define an allocation as randomized it contains element in the range of [0.01, 0.99]. The results indicate that randomized allocations account for a significant proportion, especially in larger auction scales. For instance, in settings such 2 x 10(A), 3 x 10(A), and 7 2(B), the proportion of randomized allocations exceeds 17%. Combining these findings with the results presented in Table D. we observe that the introduction of randomized allocations leads to improvement in the revenue generated by AMenuNet.
Additionally, we present the top-10 winning allocations based on their winning rates (i.e., allocations that are either A' A for some & E [n]) in Figure E. specifically for the 2 x 5 and 5 x 5[C] settings. Notably, the top-10 winning allocations tend to be deterministic, suggesting that AMenuNet is capable of identifying useful deterministic allocations within the entire allocation space. Furthermore, in the x C setting, we observed instances where the winning allocation was the empty allocation with substantial boost. This can be seen reserve price, where allocating nothing becomes the optimal choice when all submitted bids are too small.
6 Conclusion and Future Work
In this paper. we introduce AMenuNet, a scalable neural network for the DSIC affine maximizer auction (AMA) design. AMenuNet constructs AMA parameters, including the allocation menu, bidder weights and boosts, from the public bidder and item representations. This construction en- sures that the resulting mechanism is both dominant strategy compatible (DSIC) and individually rational (IR). By leveraging the neural network to compute the allocation menu, A.MenuNet offers improved scalability compared to previous AMA approaches. The architecture of AMenuNet is per- mutation equivariant, allowing it to handle auctions of varying sizes and generalize to larger-scale auctions. Such out-of-setting generalizability further enhances the scalability of AMenuNet. Our various experiments demonstrate the effectiveness of AMenuNet, including its revenue, scalability, and out-of-setting generalizability.
As for future work, since we train MenuNet using an offline learning approach, a potential direction is to explore online learning methods for training AMenuNet. Additionally, considering that the allocation menu size still needs to be predefined, would interesting to investigate the feasibility of making the allocation menu size learnable as well.
10

---

Acknowledgments and Disclosure of Funding
This work supported by the National Key R&D Program of China (2022ZD011400), We thank all anonymous reviewers for their helpful feedback.
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A Proof of Theorem
Theorem 4.1. The mechanism induced by AMenuNet satisfies both ISSI and VR
Proof. In AMenuNet, the computation of allocation menus A. bidder weights and allocation boosts are based on public bidder representation X and item representation Y, without the chance to access the bids B. So we only have prove that when A, w and À are fixed, the mechanism is both dominant strategy incentive compatibility (DSIC) and individually rational (IR).
The proof of DSIC. Denote by us (kk, (0k, B-)); X,); X)) the utility bidder i can get she truthfully bids and un(ue, (br, B B-x); X, Y) the utility bidder k can get if she bids by. In fact, un (Rk, (Uk; B-x); ]i Y) and ue(uk. (bn, B_R); X, Y) can also be written as un (Ug, (va, B_ A, w. X) and (u (b,, B A); w. A), respectively, since the parameters are determined by X and
Let A = ((v,, B-x); A, w. A) and A", = ((v,. B-x): A, w, A) be the affine-welfare- maximizing allocation and the dlfine-welfare-maxiniee allocation excluding bidder ks when bid- der k truthfully bids. Formally,
A" arg max AEA
Σ w,b.(A) + (,b.(A) +(A)) AA' arg max AEA
Σ w.bb((A) + X(A) isf
Similarly, define A bidder i and her bids by. Besides, let p& pu((Uk, B-x);; A, w. X) be bidder k's payment defined by AMA. By definition,
A BB));
w, 2) and A
ALx ((bs, B=k): A) for
Pa (VV,, A,
Σ w.b.(A w.d.(AA))) + Ky
Σ Ew:b((A+++AA)) x
Wk
First notice that A and A is the same since bidder k's bid has no influence others affine welfare. Hence,
un (Uk, (Dk, B B_ X,) un (UL, (bs, B x); Y)
=uu (Uks (tk, B x)j A, X) un. (bi, B A, w A)
=ox(A") - pk (1 B A. w. A) (ux(A') (At) - Pk (u((bk,
A, w,
(A²( 1
Σ
w.b (A + A(Aix)
Σ (A') + +1(A
+ Wi
Wkk
iz
-Ax(A")
1
Σ
(ibi(A + A(AE
Σ w.bb((A)) + +
Wk
Wk
(A") 1
Σ
(abb(A*) + *(AA)
)
(A+)
1
Σ webb + 2(F7)
wk
Wkk
w.bb( + A(A7))
1
Σ
w. + A(A-)
Wkk
wh
0,
where (1) holds by A and A is the same, and (2) holds by the definition A*.
The proof of IR. With the same notations above, we aim to prove that for any given representa- tions X, Y and others' bids B_g, we have us (vx, (vx, B :) X, Y) > O. We use the same notation
14

---

Table 3: The detailed parameters for all the settings (Setting A (F).
Parameters
2x2(A)
2x5(A)
2x10(A)
3x2(A)
3x5(A)
3x 3x10(A)
Menu Size s
32
64
64
256
256
256
Temperature
5
5
5
5
1
1 8000
Iterations
3000
3000
3000
3000
8000
Parameters
3 3x2(B
4x2(B)
5x2(B)
6x2(BB
7x2(B)
8x2(B)
Menu Size s
64
64
64
64
128
256
Temperature
3
3
3
3
1
Iterations
3000
3000
3000
8000
8000
8000
Parameters
2x5(C)
3x 10(C)
5x5(C)
1(D
1x2(E)
1x2(F)
Menu Size s
128
512
1024
16
128
40
Temperature Iterations
10
10
50
10
10
10
3000
2000
2000
2000
2000
2000
as in the proof of DSIC.
un. (uk. (Uk, B_x); X, X,
(0k (Uk, B B_b); A, w. A)
(Ax(A") p).((u Ba)); A, w. A)
=Ax(A")
Σ
wobe (x A(Ain)
+
1
Σ w,bb((A))) -2(49) d
wkke
wkk
1
wabi(A") + +
1
Σ (, +((Aa)
Wk
Wk
iyk
(1)
1
Σ (At) + +A(AA))
1
Σ wyby wrb:(AIx) A(A2s)
16k
uik
0,
where (1) holds by We 2 and (-) 2 0, and (2) holds by the definition of A".
B Further Implementation Details
AMenuNet. For all settings, the learning rate is 3 10-4, We also apply a linear warm-up the learning rate from x 10-8 to 3 x 10-4 during the first 100 iterations. Totally in AMenuNet there two softmax temperatures. Denote by тл the lerfare-maximzzingg allocation and r the one used in Menu Layer to generate allocations. We set Ta as 500 for all settings. We train the models for maximum of 8000 iterations with 2 32784 new samples each iteration. For those we train more than 3000 iterations, we manually reduce the learning rate to 5 x 10-5 after 3000 iterations. We test the model on a fixed test with 100000 samples. The menu size and varies in different settings, and we present these numbers in EL For classic auctions, including Setting [CHF], we embed each bidder and item ID into 16-dimension space. For encode layer, the output channel of the first 1 x 1 convolution in both the input layer and interaction modules are set to 64. The number of interaction modules is 3 or 5, and in each interaction module we adopt transformer with 4 heads and 64 hidden nodes. The boost layer is two-layer fully-connected network with ReL. activation. To enable the out-of off-o--setting generalization ability, we set both its channel and the hidden size as menu size.
in the softmax function for deciding social-
15

---

2.455
2x30/4)
3a54C2
1x30/0
53
2.75
5.8
Revenue
2.355
2.70
5.5
2,350
2.325
533
2225
2.55
5.2
2250
2.50
Meru
Menu
Menu
-tyr-Mn There are no parameters in Myerson auction and VCG auction. We just implement the classic form of these two mechanisms. For Item-Myerson, we first compute the virtual value (v;) according to (v;) = U, f((ve) distribution. Then allocate the item to the bidder with the highest virtual value if it is greater than O. Figure Revenue results of AMenuNetunder different menu sizes.
Table The experiment results average revenue out-of-setting generalization experiments.
Method
2x2(A)
2x3(A)
2x4(A)
2x5(A)
2x6(A)
2x7(A)
2x8(A)
2x9(A)
2x 10(A)
Myy-snn
0,4265
0.6405
0.8559
1,0700
1.2861
1.4993
1.7111
1,9227
2.1398
Trained 2x2
0.4380
0.6371
0.8401
1.0503
1,2606
1.4706
1.6801
1,8900
2,0935
Trained on 2x5
0.1212
0.4593
0.8286
1,1013
1.3193
1.5357
1.7579
1.9850
2.2034
Method
2x2(B)
3x2(B)
4x2(B)
5x2(B)
6x2(B)
7x2(BB
8x2(B)
9x2(BB
10x2(B)
Item-Myerson
0.4274
0.5580
0.6584
0.7366
0.8004
0.8535
0.8990
0,9386
0.9696
Trained on
0.4800
0.6120
0.6946
0.7547
0.8005
0.8346
0.8603
0,8815
0,8975
Trained on 8x2
0.4656
0.5972
0.6868
0.7524
0.8076
0.8519
0.8902
0.9247
0,9542
Method
3x2(C)
3x3(C)
3x4(C)
3x5(C)
3x6(C)
3x7(C)
3x8(C)
3x9(C)
3x 10(C)
Mem-MMyrson
1.0628
1.5941
2.1256
2.6570
3.1883
3.7197
4,2512
4.7825
5.3140
Trained on x 10
1.0221
1,5686
2.1433
2.7365
3.3267
3,9116
4.4776
5.0440
5.5799
Trained on 5x10
1.0366
1.5947
2.1766
2.7630
3.3398
3,9034
4,4598
5.0098
5,5498
F((e:) where F; and fi are the edf and pdf function i's value
The payment by the winning bidder equal the minimal bid that she would have had to make in order to win.
VCG. VCG auction is special case of AMA. It is equivalent to AMA by setting A as all deter- ministic allocations, up as and À
Lottery AMA. We use the implementation of Curry et all 2023
The hyperparameters and the
results mainly consistent with that reported in Curry et [02 We train all models for 10000 steps with 215 newly generated samples. The learning rate is 0.001 and the softmax temperature is 100. We set menu size as (256, 4096, 2048, 16, 40, 40) for the six experiments in Setting (C) respectively.
CTTransNet and RegretNet. ting all [201%)
We follow the same hyperparameters in Duan et all [222 and DMi
C Further Experiments Results
Impact of Menu Size. In Figure El we present the experimental results of AMennNNtunderr various menu sizes. We conduct such experiment in setting x 10[A], 3 x 5[C and x 10[C] Generally, a larger menu, while entailing higher computational costs, can lead increased revenue.
Out-of-Setting Generalization. In Table H. we present the detailed numerical results of our out- of-setting generalization experiments.
16

---

allocations
0.0
0.0
0.0
0.
0.
9,9
0.0
00
04
6.0
0.0 0.0 00
6
60
00
0.0
0.0
00
0.0
10
0.0 0.0 0.0 0.0 0.0 0.0 6.0 0.0
0. 0.
0.0 0.0
00
60
60
00
0.0
0.0
00
0.
0.0
6.0
5.0
0.0
0.0
0.0 0.0 0.
0. 0.0
8.0 0.0 0. B.
0.
68
6.0
0.0
0. 0.
0.0
18
19 10 14
1.0
6.0
6.
5.0
0.0
00
0.9
0.0 0.
0.0
6.0
0.0 0. 0.
00
00
6.0
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Figure 5: The top-10 allocations (with respect to winning rate) and the corresponding boosts among 100, 000 test samples in 10TC] with menu size
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Winning Allocations. We further present the top-10 winning allocations of 3 x 10 in Figure S. The experimental results maintain the same phenomenon: the top-10 10 winning allocations are all deterministic.
17


Reviews:
[{"review_id": "RGXg87ZoQ9", "reviewer": "Reviewer_Zst9", "summary": "Revenue maximizing strategyproof auction design with multidimensional types has proven to be extremely challenging. The lack of theoretical progress even in simple problem instances has motivated the use of machine-learning-based techniques to approximately learn high-performing auctions. One approach, typified by “RegretNet” and its followup works, involves the use of neural networks as function approximators to directly represent auction mechanisms — the training process optimizes revenue and strategyproofness. However, although the mechanisms learned by these neural network based approaches are qualitatively good and likely near to the true optimal mechanism, the enforcement of the strategyproofness constraints is not perfect — a significant limitation. Another line of work searches within some restricted class of mechanisms, all of which are guaranteed to be strategyproof, for one that performs well. \n\nThe current work builds on this latter approach. Like some previous work, they focus on the class of affine maximizer auctions, which are guaranteed to be strategyproof. However, previous work directly optimized the parameters and possible outcomes of the auction. They instead treat the mechanism itself as the outcome of a neural network. Crucially, this neural network does not see the bids as input — at most, it sees some informative “context” features — so strategyproofness is preserved. Also, even for auctions without context, training using this architecture seems to perform better than optimizing the parameters directly. The neural network uses a transformer architecture, which comes with some advantages — it allows for permutation-equivariance, a useful property which is satisfied by optimal auctions when the bidders are anonymous. \n\nIn experiments, the authors find that their method performs very well -- getting higher revenue than previous strategyproof approaches and even performing comparably to the unconstrained neural network approaches in some cases.\n", "questions": "Where do you get the reported revenues for the other methods? Are these experiments you ran yourself or taken from another paper? If reproduced yourself, how do they compare to reported results in other papers?\n\nYour table 1 has some methods in bold even when RegretNet/CITransNet outperform them. I admit it does say in the caption that bold means “best among DSIC methods”, and the DSIC methods are separated by bars, but when reading the paper, the first thing I did was immediately look at that table, so I still found it confusing. I think it would be good to find a way to make this even more obvious.", "limitations": "The authors adequately discuss the limitations of their approach (the main one being that it deals with a restricted class of auctions, unlike the RegretNet approaches).", "rating": 8, "confidence": 5, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "The experiments are done quite well and follow standard methodology, the baselines are well-chosen, the method is explained clearly, the results show a clear improvement on existing work, and the technique opens up capabilities in new settings.", "weaknesses": "There have since been many improvements on RegretNet in addition to CITransNet, with and without contexts. It could be interesting to compare to some of these as well.\n\nThe section describing the auction architecture is extremely dense. It's possible this had to be compressed a lot for the NeurIPS page limit, but I found it hard to follow. I ended up looking at the diagram+code and found this much easier to understand than the written part.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your positive feedback! We value your affirmation and we will address the questions and concerns you have listed.\n\n**Q1: About the comparison of RegretNet-based approaches.**\n\n> There have since been many improvements on RegretNet in addition to CITransNet, with and without contexts. It could be interesting to compare to some of these as well.\n\nWhile introducing more RegretNet-based methods into our experiments could be interesting, our primary focus is comparing our method with other DSIC (Dominant-Strategy Incentive-Compatible) approaches. \nWe use the comparison with RegretNet and CITransNet to give extra information, not to make them the main focus.\nWe suggest reading the detailed discussion in Q1 of our general response.\n\n**Q2: About the results of baselines.**\n\n> Where do you get the reported revenues for the other methods? Are these experiments you ran yourself or taken from another paper? If reproduced yourself, how do they compare to reported results in other papers?\n\nFor the reported revenues of other methods, we used their previously reported results for settings that appeared in other papers. For settings not found in prior work, we conducted our own experiments. You can find detailed descriptions of hyperparameter settings in Appendix B.\n\n**Q3: About the presentation of Table 1.**\n\n> Your table 1 has some methods in bold even when RegretNet/CITransNet outperform them. I admit it does say in the caption that bold means “best among DSIC methods”, and the DSIC methods are separated by bars, but when reading the paper, the first thing I did was immediately look at that table, so I still found it confusing. I think it would be good to find a way to make this even more obvious.\n\nThank you for your feedback! We will make the presentation of Table 1 more clear in the revision."}, {"review_id": "QXfnhhqzZP", "reviewer": "Reviewer_Dfp6", "summary": "This paper proposes a new architecture for learning auctions that are DSIC (but not necessarily revenue optimal) auctions. The authors' main contribution lies in a transformer-based permutation equivariant architecture designed to calculate the allocations, weights, and boosts variables utilized by AMA-based approaches. ", "questions": "In the context of classical auctions, is there any intuition for why optimizing the neural network that outputs an allocation works better than directly optimizing the allocation variables?", "limitations": "N/A", "rating": 5, "confidence": 5, "soundness": 4, "presentation": 4, "contribution": 2, "strengths": "The primary strength of this novel architecture resides in its capacity to effectively handle the following aspects:\n\n1. **Contextual and non-contextual auctions:** The architecture demonstrates the ability to accommodate both contextual and non-contextual auction settings, allowing for a broader range of applications.  \n2. **Equivariance for varying auction sizes:** The architecture exhibits permutation equivariance, enabling it to handle auctions of different sizes without requiring significant modifications.\n3. **Better scalability:** In comparison to existing approaches, the proposed architecture exhibits improved scalability, providing a more efficient and scalable solution for auction learning tasks.", "weaknesses": "This paper shares similarities with the work of Curry et al. [2022] on Differential Economics for Randomized AMA auctions. However, a key distinction is that the allocation menus in this paper are parameterized by a neural network, in contrast to directly being parameterized through autograd variables. Additionally, the proposal of contextual auction design through the transformer architecture has been previously suggested by Dual et al. [2022]. Consequently, the contributions of this paper may not be considered highly novel. Nevertheless, the authors demonstrate improved revenue performance over existing approaches, as evidenced in Table 2.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your constructive review! We will address the questions you have listed.\n\n**Q1: About the contribution**\n\n> This paper shares similarities with the work of Curry et al. [2022] on Differential Economics for Randomized AMA auctions. However, a key distinction is that the allocation menus in this paper are parameterized by a neural network, in contrast to directly being parameterized through autograd variables. Additionally, the proposal of contextual auction design through the transformer architecture has been previously suggested by Dual et al. [2022]. Consequently, the contributions of this paper may not be considered highly novel.\n\nIn contrast to the work of Curry et al. [2022] (Lottery AMA), our contribution lies in introducing a novel deep neural network-based AMA method. The underlying neural network architecture empowers our method to handle classical and contextual settings, offering flexibility and adaptability.\nFurthermore, our method enjoys several advantages beyond revenue maximization. Its inherent permutation equivariance ensures consistent outcomes regardless of the order of bidders or items, enhancing its practicality. Additionally, the scalability of our approach is noteworthy, as the neural network can effectively process input data of various sizes, making it applicable to a wide range of auction scenarios.\n\n\nCompared to the work of Duan et al. [2022], our AMA-based method can ensure Dominant-Strategy Incentive Compatibility (DSIC) by construction.\nThis property enhances the reliability and trustworthiness of the proposed mechanism, providing bidders with strong incentives to reveal their true valuations.\n\n\n**Q2: About the benefits of AMenuNet with respect to lottery AMA.**\n\n> In the context of classical auctions, is there any intuition for why optimizing the neural network that outputs an allocation works better than directly optimizing the allocation variables?\n\nWe have extensively discussed the reasons for AMenuNet's superior performance over Lottery AMA in our general response (Q2). In summary, the key factors contributing to AMenuNet's advantage are:\n1. AMenuNet adds more inductive bias by capturing the correlation of AMA parameters through the neural network.\n2. AMenuNet's over-parameterization from the neural network offers a better optimization landscape."}, {"review_id": "fJ9fgmwM7O", "reviewer": "Reviewer_MRew", "summary": "This paper introduces AmenuNet, a scalable NN for the AMA design, which ensures DSIC and IR. And the experiments demonstrate the effectiveness of AMenuNet, including its revenue, scalability, and out-of-setting generalizability.", "questions": "1.Lack of modeling of full contextual externalities.\n2.Lack of comparison with WVCG, NMA.", "limitations": "No.", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "Originality:\n1.Proposes a new automated auction design method: The paper proposes a new automated auction design method that uses a neural network to construct AMA parameters, thereby improving scalability and revenue performance.\n2.Combines deep learning and game theory: The paper combines deep learning and game theory to propose a new method for solving the DSIC problem in multi-item auctions, which is a novel approach.\n\nQuality:\n1.Experimental results demonstrate the effectiveness of the proposed method: The paper demonstrates the effectiveness of the proposed method in extensive experiments, including performance in different environments and scalability in large auctions.\n2.Theoretical proof ensures the DSIC property of the auction: The paper ensures the DSIC property of the auction through theoretical proof, thereby improving the quality of the auction.\n\nClarity:\n1.Clear paper structure: The paper has a clear structure, strong logic, and is easy to understand.\n2.Detailed experimental section: The experimental section of the paper is detailed, including experimental settings, results, and analysis, making it easy for readers to understand and reproduce.\n\nImportance:\n1.Contribution to the field of automated auction design: The proposed method in the paper makes an important contribution to the field of automated auction design, helping people design high-revenue auction mechanisms.\n2.Inspirational significance for the combination of deep learning and game theory: The paper explores the combination of deep learning and game theory, providing inspiration for related research in the field.", "weaknesses": "1.Lack of comparison with existing methods: The paper does not compare the proposed method with some existing industrial methods, such as DNA, NMA, which makes it difficult to evaluate the novelty and effectiveness of the proposed method in industry.\n2.Limited applicability: The proposed method is only verified on small-scale data sets, lacking demonstrations on large-scale industrial scenarios.\n3.Expression problem: some typos need to be optimized.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your constructive comments! We appreciate your positive feedback and we will address the questions you listed.\n\n**Q1: About the comparison with industrial methods.**\n\n> The paper does not compare the proposed method with some existing industrial methods, such as DNA, NMA, which makes it difficult to evaluate the novelty and effectiveness of the proposed method in industry.\n\n> Lack of comparison with WVCG.\n\nIn case of misunderstanding, we refer DNA as the method proposed in the paper \"Neural auction: End-to-end learning of auction mechanisms for e-commerce advertising\", NMA as the one proposed in the paper \"NMA: Neural Multi-slot Auctions with Externalities for Online Advertising\" and WVCG as the one presented in the paper \"Truthful learning mechanisms for multi-slot sponsored search auctions with externalities.\"\n\nIt's worth noting that all of these methods focus on a different type of auction problem, namely, advertising auction.\nAdvertising auctions are single-parameter auctions, which involve bidders placing a single parameter as their valuation.\nThe DSIC characterization of such auctions is based on Myerson Lemma [Myerson, 1981].\n\nIn contrast, our AMenuNet is specifically designed for more complex multi-parameter auctions, where bidders bid multiple parameters as their valuations. \nSuch auctions introduce greater complexity due to the lack of DSIC characterizations, making them distinct from single-parameter auction settings. \nTherefore, while DNA, NMA and WVCG are relevant for advertising auctions, the nature of the auction setting addressed by AMenuNet is inherently different.\n\n**Q2: About the scalability.**\n\n> The proposed method is only verified on small-scale data sets, lacking demonstrations on large-scale industrial scenarios.\n\nAs highlighted earlier in Q1, our primary focus is on multi-parameter auctions, a more intricate setting than industrial advertising auctions. The complexity inherent to multi-parameter auctions poses challenges for scaling to large industrial scenarios. \nNotably, the auction scales we've considered in this paper already exceed those of prior AMA-based methods, such as Sandholm and Likhodedov [2015] and Curry et al. [2022], and are comparable to previous RegretNet-based approaches.\n\n**References**  \n[1] Tuomas Sandholm and Anton Likhodedov. Automated design of revenue-maximizing combinatorial auctions. Operations Research, 63(5):1000–1025, 2015  \n[2] Michael Curry, Tuomas Sandholm, and John Dickerson. Differentiable economics for randomized affine maximizer auctions. IJCAI 2023.  "}, {"review_id": "Sw6A1l4Axc", "reviewer": "Reviewer_1hHm", "summary": "AMAs is a parametric family of auctions that generalizes VCG and that is exactly DSIC (unlike mechanisms found with alternative regret-based approaches) and IR. This paper proposes a deep variant of AMAs. In particular, AMAs parameters are learned as outputs of a permutation-equivariant attention-based network. Near-optimal mechanisms within AMA family can be found by optimizing this network. it can be applied to both contextual and classic auctions. The theory is provided to show that the learned auctions are indeed DSIC and IR. Extensive experiments and ablations support the method’s usefulness.\n\nI will begin by saying that a deep variant of AMAs is a glaring gap in the literature, considering the popularity of deep regret-based approaches. Someone has to do it, and if done correctly, a paper like this absolutely should be published. Unfortunately, in my opinion, this particular version is not ready to be published yet.", "questions": "- How are hyperparameters selected for RegretNet in novel settings (that did not appear in its original paper)?\n- Why do the authors think the learned menu is so diverse? If all allocations are trained on the same loss function, why is there no mode collapse (i.e., I could expect one “best” allocation to be repeated in all menu positions)?\n- 176-177: “In non-contextual auctions, similar 177 to word embedding [Mikolov et al., 2013], we embed the unique ID of each bidder or item…”. This looks like positional encoding. Doesn’t it make learned mechanisms not permutation equivariant? The network could discriminate items and participants. Shouldn’t the same dummy ID be provided for all participants and items for true permutation equivariance?\n- Why does AMenuNet outperform Lottery AMA, given that the parameters of AMAs are learned in both cases? We are used to attention outperforming everything else, but there is nothing to attend to in non-contextual auctions (as I understand, the input is always the same combination of IDs), so this is surprising to me.\n", "limitations": "For reasons mentioned in the weaknesses section, I think the method is a bit oversold.", "rating": 7, "confidence": 5, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "- The paper is clear and well-written. It cites and discusses the relevant literature and places itself within the context of the literature.\n- The goal of the paper, namely, a deep variant of AMAs, is worth pursuing. The paper is relevant to the conference.\n- The experiments and ablations that are provided are extensive and look sound. I especially like the case study in Fig 3. I was surprised to see such diverse allocations, and the allocation acting as a reserve price is especially cool.\n- The theory about the auctions being DSIC and IR is an essential part of the paper.\n", "weaknesses": "- My biggest worry is that experimental comparison is unfair. Authors claim in Table 1b that they “omit CITransNet since it is designed for contextual auctions” and proceed to compare their approach to RegretNet. RegretNet is based on fully-connected layers, whereas AMenuNet uses attention layers. Ivanov et al. (2022) show that attention layers improve revenue for the same regret levels in auctions with larger input sizes. So CITransNet (and, ideally, RegretFormer) should be included for comparison in auctions without context, as they are expected to achieve higher revenue than RegretNet and AMenuNet. For example, in the 3x10 auction, the paper reports 5.59 revenue of AMenuNet vs 5.54 revenue of RegretNet. Judging from these results, it might seem that AmenuNet is strictly better than RegretNet (same revenue but always DSIC). However, RegretFormer in its paper reports more than 6.1 revenue in 3x10 for the regret of <= 0.005 (as indicated in Table 1, “The regret of both CITransNet and RegretNet is less than 0.005”). By the way, this only takes 2 attention-based blocks, as opposed to 3-5 used by AMenuNet. If this is included in the comparison with AMenuNet, the conclusion would be that zero regret actually comes at a cost of revenue. This would not diminish the authors' contributions or the usefulness of their method, but would simply be a more sound experimental design.\n- As Rahme et al. (2021) show, RegretNet is hyperparameter-sensitive. Choosing the wrong hyperparameters may decrease performance. The paper includes some settings that were not explored in the original RegretNet paper, like 3x1 and 5x5. However, in the appendix, it is only stated: “CITransNet and RegretNet. We follow the same hyperparameters in Duan et al. [2022] and Dütting et al. [2019].”. So for 3x1 and 5x5, it is not reported how the hyperparameters are selected, potentially resulting in unfair comparison. I would appreciate it if the authors could clarify this for me. As a suggestion, both Rahme et al. (2021) and Ivanov et al. (2022) propose alternative loss functions that are less sensitive to hyperparameters.\n- A potential issue with AMenuNet is that it requires a hand-selected menu size. I wonder if larger settings would require larger menu sizes and if this introduces scaling issues. An ablation would be welcome where AMenuNet is given different menu sizes in some small and large settings and we examine revenue as a function of the menu size. E.g., we could see that the approach performs well in settings both large and small given the same small menu sizes, in which case there are no scaling issues.\n- Compared to CITransNet or RegretFormer, there is nothing novel about the architecture, besides the softmax trick with annealed temperature to work with deterministic allocations. This is not a weakness per se, just a neutral fact.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your review! Your comments raise very good questions, and we will address them and the other concerns that you have raised.\n\n**Q1: About the experiments on RegretNet-based methods.**\n\nWe understand the suggestion to include CITransNet and RegretFormer in the comparison for auctions without context, as they are expected to achieve higher revenue than RegretNet. \nHowever, it's important to note that our paper focus on comparing our methods against other DSIC approaches instead of the non-DSIC RegretNet-based algorithms.\nThe comparison with RegretNet and CITransNet serves as supplementary insights rather than central facets of our study. \nWe invite you to read the detailed discussion in Q1 of our general response for a more comprehensive discussion.\n\n> If this is included in the comparison with AMenuNet, the conclusion would be that zero regret actually comes at a cost of revenue. \n\nWhile incorporating CITransNet and RegretFormer in the comparison would indeed support the conclusion that \"zero regret comes at a cost of revenue,\" our experimental results already demonstrate this trend:\n\n1. In the contextual auction experiments, CITransNet outperformed AMenuNet in 9 of 10 instances.\n2. In the classical auction experiments, RegretNet outperformed AMenuNet in 4 of 6 instances. \n\n**Q2: About the hyperparameters selection of RegretNet in novel settings**\n\nFor 5x5(C), we set the hyperparameters of RegretNet as reported in the original paper for  '5x10 uniform' setting. Such choice was made due to the close relationship between these two cases. As indicated in Table 1(b), RegretNet with these hyperparameters performed well in 5x5(C), outperforming all other methods in terms of revenue.\n\nSimilarly, for 3x1(D), we set the same configuration used for '1x2 uniform' in the original paper. Table 1(b) shows that RegretNet achieved revenue close to the optimal solution under these hyperparameters.\n\n**Q3: About the ablation study with respect to menu size.**\n\nWe appreciate the suggestion for conducting an ablation study to analyze the impact of different menu sizes on AMenuNet's performance. We conduct experiments on AMenuNet using various menu sizes in two uniform (setting C) scenarios:\n\n| Menu Size    | 32     | 64     | 128    | 256    | 512    | 1024   | 2048    |\n| ------------ | ------ | ------ | ------ | ------ | ------ | ------ | --- |\n| 3x5 uniform  | 2.6336 | 2.7125 | 2.7470 | 2.7848 | 2.8051 | 2.8099 |  2.8110   |\n| 3x10 uniform | 5.1853 | 5.2348 | 5.4042 | 5.4932 | 5.5941 | 5.6714 |  5.7349   |\n\nWe can see that AMenuNet's revenue performance improves as the menu size increases in both the 3x5 uniform and 3x10 uniform settings.\n\n\n**Q4: About why the learned menu is diverse.**\n\nThe learned menu exhibits diversity for several reasons:\n\n1. The menu includes allocations that are not only candidates for the winning allocation $A^*$ but also candidates for the allocation $A^*_{-k}$ that maximizes the affine social welfare for each bidder $k \\in [n]$. This requirement for non-zero revenue leads to at least two different allocations within the menu, resulting in no unique allocation.\n2. Although all allocations are trained using the same loss function, gradients for each allocation within a training batch differ. The use of the softmax function to approximate the winning allocation $\\hat{A}^*$ during training means that only the best allocation (i.e., the one with the highest score) receives a significant gradient during backpropagation.\n3. The diversity is further enhanced by sampling valuation profiles from a prior distribution during training. Different valuation profiles within the training batch can lead to various best allocations.\n\n**Q5: About the embeddings of the IDs**\n\nTo ensure the mechanism can handle diverse and heterogeneous participants and items effectively, the use of unique ID embeddings is essential.\nFor instance, in Myerson auctions [Myerson, 1981], the auctioneer relies on prior knowledge of each bidder's valuation profile, making it essential to correctly identify each bidder for accurate prior distribution estimation.\n\nUsing the same dummy ID for all participants and items would hinder the mechanism's ability to differentiate between different bidders and items, especially in cases of heterogeneity. Consequently, the induced auction mechanism would be limited to symmetric settings, as seen in Rahme et al. 2021. \n\n\n**Q6: About the comparison between AMenuNet and Lottery AMA.**\n\nWe discussed the potential reasons for AMenuNet's superior performance compared to Lottery AMA in Q2 of our general response. \nIn short, AMenuNet's neural network effectively captures the correlation of AMA parameters, and its over-parameterization provides advantages for optimization, leading to improved results."}]

Author rebuttal:
We thank all reviewers for their careful comments and constructive suggestions. Here are our responses to some common questions in the reviews.

**Q1: About the comparative experiments on AMenuNet and RegretNet-based methods.**

Our primary focus in the experiments is to compare AMenuNet with other DSIC (Dominant-Strategy Incentive-Compatible) approaches, such as Lottery AMA and Item-Myerson.

While RegretNet-based methods like CITransNet and RegretFormer cover a broader range of mechanisms, they do not guarantee DSIC. These methods might achieve higher revenue levels but are at the cost of sacrificing the DSIC property.

In contrast, our DSIC approach strictly ensures that the mechanisms are DSIC, resulting in a more limited range than Regret-based methods.
Despite this narrower range, the primary advantage of AMenuNet over RegretNet-based approaches lies in its ability to ensure DSIC, prioritizing truthful bidding over revenue maximization.

Nevertheless, we still conducted the comparison experiments involving the non-DSIC RegretNet. 
These experiments serve as supplementary insights rather than central facets of our study. 
We do so to highlight AMenuNet's capability to achieve substantial revenue while maintaining DSIC.

**Q2: About why AMenuNet performs better than Lottery AMA.**

AMenuNet holds several advantages over Lottery AMA:
1. Handles contextual settings, adapting to diverse auction environments where bidder valuations may depend on contextual information.
2. Permutation equivariant, ensuring consistent outcomes regardless of the order of bidders or items.
3. Demonstrates generalization capabilities, performing well in auctions with varying scales and sizes.

Although AMenuNet and Lottery AMA possess the same capabilities in classical auctions given identical menu sizes, our empirical experiments demonstrate that AMenuNet excels over Lottery AMA. 
Several factors potentially contribute to this superiority:
1. **Benefits of inductive bias**: While the Lottery Allocation Mechanism (Lottery AMA) calculates the allocation menu, bidder weights, and boosts as separate entities, AMenuNet takes a different approach. AMenuNet leverages an underlying neural network to compute these parameters, thereby capturing the intricate interdependencies among them. This characteristic of AMenuNet is an additional inductive bias. Adding more inductive biases usually improves the generalization ability of machine learning models [1,2].
2. **Benefits of over-parameterization**: The parameters of Lottery AMA are just the allocation menu, bidder weights, and boosts. In contrast, AMenuNet is over-parameterized with the underlying transformer-based neural network. Extensive research in deep learning has consistently demonstrated that over-parameterization offers several benefits, including improved optimization landscapes [3], better generalization [4], and less sensitivity to parameter initialization [5]. As a result, the over-parameterization of AMenuNet makes it easier to optimize than Lottery AMA, leading to better revenue performance in practice.


**References**  
[1] Shalev-Shwartz, Shai, and Shai Ben-David. Understanding machine learning: From theory to algorithms. Cambridge university press, 2014.  
[2] Mitchell, Tom M. "The need for biases in learning generalizations." (1980).  
[3] Buhai, Rares-Darius, et al. "Empirical study of the benefits of overparameterization in learning latent variable models." International Conference on Machine Learning. PMLR, 2020.  
[4] Allen-Zhu, Zeyuan, Yuanzhi Li, and Yingyu Liang. "Learning and generalization in overparameterized neural networks, going beyond two layers." Advances in neural information processing systems 32 (2019).  
[5] Frankle, Jonathan, and Michael Carbin. "The lottery ticket hypothesis: Finding sparse, trainable neural networks." ICLR 2019.

Meta-review: This paper looks at automated mechanism design through the lens of differentiable economics, roughly speaking the use of differentiable programming in economic design.  The paper looks at affine maximizer auctions (AMAs), a class of mechanism known to be representable by neural networks in a way that maintains exact DSIC and IR, but may not contain the revenue-maximizing auctions.  This is an interesting focus as most work in this space has focused on approximate DSIC with the possibility to represent the revenue-maximizing auction.  Reviewers unilaterally appreciated the experimental results showing strong revenue performance as well as the architecture that scales less dramatically with auction scale.
Decision: Accept (spotlight)