BundleFlow: Deep Menus for Combinatorial Auctions by Diffusion-Based Optimization
Authors: Tonghan Wang
Yanchen Jiang
David C. Parkes
Venue: NeurIPS 2025 poster ()
Keywords: Combinatorial Auctions
Continuous Flow Matching
Optimization
Dominant-Strategy Incentive Compatible
Revenue-Maximizing
Primary area: deep_learning
TL;DR: We propose the first deep menu-based method for single-bidder combinatorial auctions that is dominant-strategy incentive compatible and revenue-optimizing, by adapting continuous flow matching for optimization problems.

Abstract: Differentiable economics—the use of deep learning for auction design—has driven progress in multi-item auction design with additive and unit-demand valuations. However, there has been little progress for combinatorial auctions (CAs), even in the simplest and yet important single bidder case, due to exponential growth of the bundle space with the number of items. We address this challenge by introducing a deep network architecture for a menu-based CA, which supports the first dominant-strategy incentive compatible (DSIC), revenue-optimizing single-bidder CA. Our idea is to generate a bundle distribution through an ordinary differential equation (ODE) applied to a tractable initial distribution. The BundleFlow method learns suitable ODE-based transforms, one for each menu element, to optimize expected revenue. BundleFlow achieves up to 2.23$\times$ higher revenue than baselines on standard CA testbeds and scales up to 500 items. Compared with other menu-learning baselines, BundleFlow also reduces training iterations by 3.6-9.5$\times$ and cuts training time by about 80% in settings with 50 and 100 items.

BUNDLEFLOW: Deep Menus for Combinatorial Auctions by Diffusion-Based Optimization
Tonghan Wang
Yanchen Jiang
Harvard University 0g harvard. edu
Harvard University
yanchen_jiangee harvard. edu
David C. Parkes
Harvard University
parkes@eecs. harvard. edu
Abstract
Differentiable economics- tte use of deep learning for auction design- has driven progress in multi-item auction design with additive and unit-demand valuations. However, there has been little progress for combinatorial auctions (CAs), even in the simplest and yet important single bidder case, due to exponential growth of the bundle space with the number of items. We address this challenge by introducing a deep network architecture for menu-based CA, which supports the first dominant- strategy incentive compatible (DSIC), revenue-optimizing single-bidder CA. Our idea is to generate a bundle distribution through an ordinary differential equation (ODE) applied to tractable initial distribution. Our method, BUNDLEFLOW. learns suitable ODE- transforms, one for each menu element, to optimize expected revenue. BUNDLEFLOW achieves up to 2.23 higher revenue than baselines on standard CA testbeds and scales up to 500 items. Compared with other menu-learning baselines, BUNDLEFLOW also reduces training iterations by x.5 and cuts training time by about 80% in settings with 50 and 100 items.
Introduction
When selling multiple items simultaneously. bidders often have complex valuations that exhibit synergies among items. For instance, some items may act as complements, with their collective value to bidder exceed the sum of their individual values. Combinatorial auctions (CAs) support such
valuations by allowing bids on bundles of items. Recognized as early as 1922 [54] and formally
defined in 1982 43 CAs have found lot of application, from allocating airport runways to 41, efforts whose far-reaching impact underpinned the 2020 Nobel
auctioning spectrum licenses
Prize in economics [46).
While multi-bidder CAs represent the most general framework, many real-world settings are single- bidder CAs: streaming platform curating a collection of movies for a viewer (c.g., Prime Video UK publicly listed 65+ channels on March 2. 2022 M.. a cloud provider packaging compute features (hundreds of AWS EC2 instance types with CPU/GPU, memory, bandwidth, and storage options (2). an integrated facilities vendor offering configurable service lines (custodial, HVAC, security, pest control, energy. etc.) across multiple sites []. This problem has also been studied as the so-called "FedEx problem" [5]. The single-bidder CA model is also applicable in settings with multiple bidders long as supply constraints apply independently to each bidders; e.g., with information goods, or services that can be replicated and sold any number of times.
Despite their prominence, designing revenue optimal CAs remains fundamentally challenging. As with auctions for additive or unit-demand valuations, it is typical to seek mechanisms that are (1)
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Samples from the minal distribation
Samples from the fisal dusimbationn (y(s))
Vector field that defines ODC Possible Bundles
Stage Flow Inirialization
Stage
Optimization Menu Klement
Menu Element
Menu Element
Menu Element
Heraatio 210, Rey. 36 30
Prioz
Price
Price
Derration 1:00 Rev, de. 31
Price
92.01
Figure 1: Demonstration of the BUNDLEFILOW method. The X- and y-axes represent the bundle variables for two specific items: I = 1 and y = correspond to item and B, respectively, being in the bundle. An ODE ds, = ptt, 8t )tt (blue curves) generates bundle distributions aT((s) (blue dots) from simple initial distributions (5)) (green dots). The opacity represents probability density. Left. Stage I: Fix (50) and train the vector field, so the final distribution has all feasible bundles as its support. Right. Stage 2: Fix the vector field and update initial distributions of menu elements to manipulate bundle distributions. We present snapshots of four menu elements (organized in columns) at different training iterations (organized in rows), showing the bundle distribution and price for each element, along with the test-time auctioneer revenue from the entire menu the corresponding iteration (see Sec.
dominant-strooe syrrinant-stratgyy incentive compatible (DSIC, also tirategy-proots ensuring that bidders benefit most by reporting their true values, and (2) revenue-maxin mexinue-axiniiiing from the auctioneer's 's perspective. However, even the seemingly simpler single-bidder combinatorial setting-a foundational building block for multi-bidder scenarios-lack a comprehensive theoretical characterization.
In recent years, researchers have explored the use of deep learning techniques for optimal auction and mechanism design, commonly referred to as differentiable economics [3]. In particular, deep learning together with menu-based methods show promise. This learns menu of options for bidder, guaranteeing strategy- strategy-proofness provided the menu remains self-bid independent and agent- optimizing (20]. For a CA, each option in a menu corresponds to a bundle of items (or a distribution on bundles) and a price. Following RocherNer [13), various methods have been developed for demonstrating the ability to discover
the single-bidder, non-combinatorial setting 110 auctions that are provably optimal.
However, differentiable economies has made little headway on the problem of optimal CA design due to the fundamental challenge of handling a number of bundles exponential in the number of items.
Dütting al. (13) deal with two items and do not guarantee exact DSIC. Duan et [[2] restrict their attention to the class of the virtual valuation combinatorial auction (VVCA), which is not fully general, and only scale to 10 items. Ravindranath al.
44 consider a sequential CA setting, which is generally suboptimal when items can be simultaneously auctioned. No previous work provides a path towards DSIC and flexible, i.e., fully general, mechanisms for tens or hundreds of items, and novel methodology is needed to make progress.
To understand the challenge of optimal CA design we compare menus for simpler valuations with menus for combinatorial valuations. For a bidder whose value additive on items, it suffices to specify in a menu element the item-wise allocation probabilities, such as [0.3, 0.6] for two items, along with price. However, an item-wise allocation is ambiguous for a bidder with a more general valuation function. For example, [0.3, 0.6] could be 0.3[L. 0] +0.600, 1] or 0.3 1]+0.3 1]. with the ttem-wise allocation decomposed as different bundle mundle-wise allocations, leading to distinct valuations. One could predefine how to interpret marginal adopting product distribution semantics. But such interpretations sacrifice flexibility- wise allocations cannot be represented; e.g., 0.5(1, 0] + 0.5[0, 0, 1] cannot be represented as a product distribution. One way to address this is for menu element to directly specify an allocation probability for each possible bundle. However, the number of bundles grows exponentially with the number of items, making an explicit representation intractable.
allocation avoid ambiguity, for example
bundle-

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We solve this problem, developing menu based, deep learning method for DSIC and fully general, single-bidder CAs that scales to as many as 500 items. The core idea is to avoid the need directly specify probability for every possible bundle. Instead, we represent a distribution on bundles for a menu element by a tractable and oow-dimensional initial distribution, Oup (o((8)) on initial bundle variables, 80 E Rm, and an ordinary differential equation (ODE): ds, (tt, 8t )dt. The ODE operates on bundle variables, 8, E IK"", through the vector field (tt, ). Here m is the number of items and E [0, T] is the ODE time. A feasible bundle corresponds to bundle variable where the entries are all Os or 1s.
su Formally, 8; is generated from sample by applying the ODE: (s() = so + (TT dy) We omit the dependence on So and write 8, for simplicity. In this way, the ODE transforms the initial distribution Ogg (5n) to a final distribution (s((s)) at time T. By designing the functional form of the vector field and using the Liouville equation [3]. we get a tractable closed-form expression of aor (8T) that is fully differentiable and supports gradient-based training. Based on this, our method proceeds in two steps: (1) train the vector field so that the support of the final distribution or (8T) corresponds to feasible bundles; (2) fix this trained vector field (p(t, 1), and backpropagate the
loss through the flow to update the initial distribution (80) (which updates the final distribution over bundles) and the price for each menu element. A demonstration of these two steps is shown in Fig. I
This method draws inspiration from diffusion models 22 32 50
51 60 used in generative Al and 37 38. The technical novelty to extend generative
especially continuous normalizing flow 87 models to solve an optimization problem, seeking bundle distribution (and price) for each menu element that optimizes expected revenue. particular, there is no known target distribution in our work. By contrast, in classic generative Al tasks, the target distribution is known and observed as the data distribution in large-scale pre-training datasets of natural images [14. 45), language (39), and videos [53]. We call our method BUNDLEFLOW to emphasize the core idea of using continuous normalizing flow to optimize bundle distributions.
We evaluate BUNDLEFLOW using single-bidder instantiations of CATS 35 widely adopted CA testbed. Experimental results demonstrate that our method consistently and significantly outperforms all baselines across all benchmark settings and scales to auctions involving up to 500 items. For auctions with 50 to 150 items, BUNDLEFILOW achieves 1.11- 22.23 x higher revenue. Moreover, this substantially improved revenue does not compromise training efficiency. On the contrary, compared to the baseline RochetNet that also learns allocations in menu elements, our method typically requires 3.6 fewer training iterations and reduces training time by about 80% in settings with 50 or 100 items.
We also make a critical observation when each menu element is forced to deterministically assign a single bundle. The revenue in this case drops sharply-by as much as down to baseline levels. This underscores the necessity of supporting randomized bundle distributions in differentiable economics for CAs, a feature that is unique our method. We discuss direction to extend this work to multi-bidder CA design in Appx. B
performance
2 Preliminaries
Combinatorial Auction. We consider CAs with a single bidder and m items, M = (1I...,m.
The bidder has valuation function, v 2M + Rzo- v is drawn independently from a distribution F defined on the space of possible valuation functions V, determining how valuable each bundle S S M is for the bidder. We consider bounded valuation functions: v(S) E [0, Umax], S c 2 with Umax > O, normalized so that v(Ø)= 0. The auctioneer knows distribution F' but not the valuation v. The bidder reports their valuation function, perhaps untruthfully, as their bid (function), E V.
We seek an auction (g. p) that maximizes expected revenue. Here, V + X' is the allocation rule, where X' is the space of feasible allocations (i.e., no item allocated more than once), so that g(b) C M denotes the set of items (perhaps empty) allocated to the bidder at bid b. Also, p V + Rzo is the payment rule, specifying the price associated with allocation gbb). This is challenging design problem that has not been solved computationally or analytically in prior work. The utility to the bidder with valuation function v lunsi-tin so that values are in effect quantified in monetary units, say dollars. In full generality,
bid & u(v; b) = v(g(b)) p(b), which is the standard model of
3

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the allocation and payment rules may be randomized, with the bidder assumed to be risk neutral and seeking to maximize their expected utility.
In a dominant-strane utility is maximized by bidding their true valuation v, whatever this valuation; i.e., u(v; v) 2 u(u; b). for Ve E V, v6 E V. An auction is individually rational (IR) if the bidder receives a non-negative utility when participating and truthfully reporting: u(v; v) 2 O, for Vv V. Following the revelation principle, it is without loss of generality to focus on SP and IR auctions, as any auction that achieves a particular expected revenue in a dominant-stratess IR auction with the same expected revenue. Optimal auction design therefore seeks to identify an SP and IR auction that maximizes expected revenue, i.e., Ey [p(()]]
incentive compatible (DSIC) or strategy-proof (SP) auction, the bidder's
equilibrium can be transformed into an SP and
Based CAs. In a menu-based auction, allocation and payment rules are represented by a menu, B. consisting of K Z 1 menu elements. We write B = (B(1), (Be...... B(K)), , menu element, B(k), specifies allocation probabilities on bundles, a 2 - [0, 1], and a price, g(8) E RR. We allow randomization, where a (+) E [0, 1] is the probability that bundle S E 2M is assigned in element k. The menu B corresponds to a menu-based representation of an auction. A bidder with bid is assigned the element from menu B that maximizes their utility according to the reported valuation: k- E arg max, Ese2" (k) (S)b(S) - g( We denote this optimal element by (a (b)). The use of menu-based representations for auction design is DSIC and without loss of generality (20). The optimal auction design problem is to find a menu-based representation Menu-based architectures
and the kth
that maximizes expected revenue, i.c.,
E F
F
8
(v)
[33 48 in the
differentiable economics literature [IO. 16
63 learn to generate such menus by neural networks but have not been applied to anything other than very small CA problems.
24
29
30
42
56
61
Diffusion Models and Continuous Normalizing Flow. Diffusion models perform a forward noising process in which noise is incrementally added training data over multiple steps, gradually corrupting the original sample. A reverse diffusion process is then learned to iteratively remove noise, thereby reconstructing data from near-random initial states. A deterministic reverse process modeled by an ordinary differential equation (ODE) preserves the same marginal probability densities as a reverse process modeled by a stochastic differential equation (SDE) [51]. A continuous normalizing flow also learns to transform and manipulate distributions by an ODE, with a continuous normalizing flow transporting an input 2g E Re to o; = o(t, 2o) at timestep t E [0. 7). Here, <tt,-) RR R the flow, and is governed by the ODE,
d
dt x. 4 (t,
where the vector field o
[0, x RC Re specifies the rate of change of the state æ. The flow & transforms an initial distribution (= to a final distribution (x)) at time T. Continuous normalizing flow methods 8 suggest to represent the vector field o with a neural network.
Rectified Flow. A bottleneck that restricts the use of continuous normalizing flow in large-scale problems is that the ODE (Eq. [ is hard to solve when the vector field y is complex. The rectified flow 38 addresses this by encouraging the flow to follow the linear path:
min Exo-pol)) [II pr - σo) - (tt, =z]] dt, I = txg + -T-
(2)
Here, the target distribution pr(z) and the initial distribution (o(2)) are known. x; for t [0, T| is the interpolated point between By and Zu, and the rectified flow encourages the vector field to align as closely as possible with the straight line æT - Wo.
3 The Flow-Based Combinatorial Auction Menu Network
The major challenge in learning menus for CAs is to provide a suitable representation of bundle distri- butions. This representation should be efficient to avoid bottlenecks associated with the exponential number of bundles, while remaining differentiable to support gradient- based training.
3.1 Menu representation
Our key idea, inspired by score-based diffusion models and continuous normalizing flow, to construct concise and differentiable representation of a bundle distribution by modeling this through

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the solution of an ODE. Specifically, the kth menu element generates its bundle distribution by the ODE,
(k) dsp = = 86 dtt,
for a suitable vector field yo. The bundle variable time 8i (k) IR" is generated from 80 (R)
(k)
by 8, (8) (s0 90 + Jo G(T, st )dT. For clarity, we omit the superscript (k) and the explicit dependence on 8o and write 8, when it is clear from the context.
We train the vector field ip so that, at time T, a bundle variable sT has all its entries being Os or 1s and thus represents valid bundle. By the Liouville equation [36], the allocation probability of this bundle, ar derived from Eq. satisfies:
log OCT (8T) = log 0rg (8o) f0 V pl(, 8,)d1,
(4)
where (s))s) is the initial probability and V (t(t, 8t) is the divergence of yp.
Training scheme. Both the vector field o and the initial distribution agg can influence the final distribution OT- Our method proceeds in two stages, involving the training of each of these two components in turn: (l) Flow Initialization. We fix the initial distribution og and train the vector field (t,. -) so that the final distribution, or,, provides reasonable coverage over bundles. (2) Menu Optimization. We fix the vector field from Stage 1, and backpropagate the revenue-maximinn loss through the flow update the initial distribution ai for each menu element
and (8o) play a crucial role in ensuring a concise and easily differentiable representation for efficient training. We next propose specific functional forms that meet these criteria.
Vector field. We adopt the following functional form for the vector field,
(t,, S E, 0) (( 8)ss:
(5)
where Q RTM + mm written function of so, and the scalar factor 17 R - R. written as a function of the ODE time t E [0, 7". are neural networks with learnable parameters 0 and &. respectively. We omit dependence on @ and & when the context is clear. This formulation's 's advantage becomes apparent when we consider yps divergence:
Dec I8t.
sp(t, 8x) Σ
Σ
(())( Σ n(t)Qi((0)) ry(t) [r()(s))]
(6)
Here, ip and 8t,i are the ith element of & and 8t, respectively, Qi the ith row Q, and Qu the ith diagonal element of Q. Thus, the probability density at T (Eq. becomes
log ar (ST) log (ogannss)) Tr(Q(sa)] n())di.
The integral in Eq. is tractable as only involves scalar function, instead of bundle variables. We can efficiently estimate this integral by time discretization.
Initial distribution. Stage 1, we use a mixture-of-Gaausian distribution model ao(s)), with 80 ~ Eb waN(Hu, Im). where, for components, μ4d E IR"", aa E R> Im is the m m identity matrix, and w. > 0, Eb= Wa 1. In Stage 2, as discussed later, we ensure DSIC by adopting a mixture-of of-Dirac distribution for (so).. which is operationalized by setting a very small variance aa in the mixture-of- Gaussian distribution.
3.2 Stage 1: Flow initialization
Stage ensures that the flow maps any initial bundle variable su to feasible bundle S E 2M Denote the indicator vector of S by (Ifi E S}). practice, numerical issues make challenging to obtain exactly 8. To account for this, we allow small region around to be approximated as 8 by modeling the bundle as a Gaussian variable, S., = (ss o We thereby define the target distribution oz (ST) = 2TT Eseg S., as uniform mixture-of of-GGsssiannn model, with components centered around each feasible bundle. This
if item is in S and 0 otherwise. In
5

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target distribution only applies in Stage I, where it serves to encourage balanced coverage of the final distribution over feasible bundles. In Stage have an optimization problem, and there is no longer fixed target distribution.
For Stagel, fix the initial distribution (0 to mixture-of-GGassaan model (x Ed (wa, Im) with D components. The flow training loss
LrlOw (0.g) g) = (a. at). []](ST -80) ((t,, 8 0, E)
(8)
8t =t ST +TT so:
ST
-0; (pt,, 8z; 0,)) r7(:: {s) 0) 8)
This loss is used update the neural networks and 7) encourage the vector field at interpolated points 8; to point from &p to 87- The expectation in the flow training loss taken over (oop oz ), but directly sampling from oT is intractable it involves 2n bundles. Using flow-based representation provides a workaround. We first draw 8n (pp, which is straightforward given that o comprises manageable number of components (D). We then round 80 the nearest feasible bundle, 8
(( 2 0.5) E (0, 1) and sample 8T ~ (8, 2 Im)
3.3 Stage 2: Menu optimization
In Stage 2, we train the menu to optimize the expected revenue. Trainable parameters are prices (36),, and (A) ft that define initial distributions 0g The vector field yp is fixed in Stage 2 and shared among all menu elements. We always maintain null menu element (zero allocation, zero price), which ensures individual rationality (IR), so that the bidder non-negative expected utility.
(k)
Computing the expected utility corresponding to menu element with bundle distribution (*) is central to evaluating the revenue objective but intractable when done with a direct calculation, because (*) (v) = Σs Esga (k) (S){(S) requires enumerating 2m bundles for general valuation function. By contrast, with flow, operation to both sides of Eq. 7 and writing
get the bundle allocation probabilities by applying the exponential
(so) exp (-
1."'' n(t)dt) o ([ST(80)))
u (k) (v) = E
Sy aag (2)
(k) ag
( [-I[(Q(s))
(9)
Here, [sr()))] = I(O(T, sp) 2 0.5) is the rounded final bundle, and ss)) is the solution of the ODE, and solved by forward Euler,
8T(8)) ΦTT, so) Q(00) ( ((())sdd.
(10)
Due to its simple form, modern ODE solver efficiently solve the ODE (Eq. 10 in just few steps. In this way, and since also make the initial distribution simple, the calculation of u (v)
(v)
becomes tractable.
To ensure DSIC, we need to accurately calculate the expectation in Eq. to get the exact utility to the bidder. We accomplish this by employing a mixture-of-Dirac distribution as the initial distribution, which has finite support. To implement this in practice, we set, for Stage 2 only, very small variance the Gaussian components, with aa = 1e-20 for every component d. In this way, the utility can be obtained by enumerating over the finite support of the initial distribution:
D
exp (-INQQUU(T) (1) S. n(t)dt) v u((oT(u())))) (1)
u (k)
Σ
an (pa
(1))
(k)) )
(1I)
where (1+ (k)
T( (4) (k) 2 0.5). That is, the support of a (k) consists, in effect, of the set worth noting that in Eq. I does not need to be the same
of means, one for each component. D in Stage 1, and it could even vary across menu elements.
With (*) in Eq.
given a set of bidder valuations V, the revenue-maxxiiiaate loss is defined as
(1818)) (w (1) de(D)
LREV
(k) {ur
deE
M
Σ
Σ KE[K]
2k)) (1)3(())
(12)
kE(K] ke [)
BEY

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Table 1: Revenue comparison on CATS, When m = 10, menus of baselines are large enough to accommodate all possible bundles. VCG has zero revenue for single-bidee bidder CAs.
CATS-Regions Uniform Valuations
CATS Regions Normal Valuations
Methods
m=10
m=
m==75
m== 100
m==10
T/ 10
m=
m=75
m=100
m= 150
BUNDLEFL.OW
196.19
555.05
454.96
417.83
385.68
173.93
603.70
448.35
389.51
394.82
VCG
0
Rochetliet
189.17
288.41
290.14
292.11
312.65
167.16
270.23
300.55
270.23
291.83
Grand Bundle
162.57
316.27
321.10
317.00
314.93
142.54
319.06
328.89
305.06
309.73
Nenu+
202.26
399.85
354.48
322.68
329.17
181.93
459.97
405.36
306.60
312.21
Nenu-
202.16
322.85
318.33
326.76
334.20
181.75
342.31
339.82
303.31
315.02
CATS-Arbitrary Uniform Valuations
CATS-Arbitrry Normal Valuations
Methods
m=10
m=
v/ 100
mr 150
17um 10
m==00
m=
m=100
m/ 150
BUNDLEFLOW
211.35
560.71
467.79
434.77
420.75
235.80
646.37
490.03
428.54
394.37
VCG
Rochetliet
205.02
316.79
312.41
313.76
334.51
221.49
348.41
312.79
304.00
316.81
Grand Bundle
175.09
329.62
340.40
343.66
345,98
186.59
354.86
345.88
329.99
336.45
Nenut 233.26
396.78
351.70
357.44
351.07
248.16
478.67
349.34
335.75
339.10
Kenu-
222.74
353.20
355.73
364.33
360.41
248.13
376.31
352.80
343.58
348.39
where (xx) (e)= = SoftMax: (AsorTMAX
ul (v)) is a scaled SoftMax over bidder utilities and AsorrMax is a scaling factor. In Stage 2, we fix the vector field y (Q and n networks) in Eq. [1 and update trainable parameters associated with the price and initial distribution (k)
0o i.e., 3(0,, {w))
D J==1- and
( Jd=1-
(k) (
We formally prove that the learned mechanism always DSIC.
Theorem 1. [Exact DSIC] The BUNDLEFI.OW fraDDEeeoowrammewoo
framework ensures the learned auction mechanisms
are DSIC.
The detailed proof can be found in Appx. A
4 Visualization on a Didactic Example
We present an example to visualize the BUNDLEFLOW training process, adopting the Regions environment from CATS (35] and considering uniform value distribution and 5 items. For the flow network architecture, the Q and a networks in this example each have three and two 128-dimensional, tanh-activated, fully-connected hidden layers, respectively. We use the Adam optimizer with learning rate 5e-3 to train these networks, with 20K samples and 60K iterations. Fig. left shows the effect of Stage 1, where we fix the initial distribution and update the vector field. The and y-axes are the bundle variables for the 3rd and 4th items, respectively (æ: = 1 and y = 1 mean that the 3rd and 4th items are in the bundle). The vector field gradually learns to cover all possible bundles in this projected, two-item subspace, as indicated by points (0, 0), (0. 1), (1,0). and (1,1) in the plots. Fig. r-ihht illustrates the dynamics of Stage 2. The support size of the initial distributions is set to 512, the menu size to 8, and ASOFIMAX to 1. The subplot is organized into four columns, each displaying changes in the bundle allocation (depicted by blue dots) and price of distinct menu element. Changes in the bundle distribution for a menu element result from updates to the initial distribution for this element. We observe that each menu element learns to allocate different bundles. For example, Menu Element 3 manages bundles (0. 1) and (1, 0), and sets a price of 88.245 after Train Iteration 1020, at which point the revenue reaches 64.31.
Experiments
Benchmark. We evaluate our method CATS 35 standard benchmark in CA research | Consistent with previous works [5] 6 17 28
47 49
59 62). we focus our experiments on
Arbitrary and Regions environments, which represent the most challenging problem instances (34). Valuations are expressed in the CATS XOR bidding language as sets of bundles paired with their corresponding values (sets of atoms). We test different numbers of items: 10, 50, 75, 100, and 150
'We used the latest CATS v2.2 as is distributed under the "CATS License Agreement" (non nommccomeercial research use): see hattss://wwe ubc. wac..ccaaa... ht//kkkkvlbbAAs..
7

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Menu-
Menu+
Blod
RochetNet
Grand Bundle
50 Items
(b) 100 Items
Test Reventi
600.0-
400.0-
300.0
400.0
200.0
200.0
100.0-
0.0
0.0
100k
200k 0
400
800
10M
20M
100k
200k 0
400
800
0
10M
20M
# Iteration
Wall time (s)
Train Samples
Iteration
Wall time (s)
Train Samples
(c) 150 Items
(d) Items, XOR atoms
Test Rovenue
300.0
400.0
200.0-
200.0-
100.0
0.0
0.0
200k
20M
200k
1000
20M 40M
# Iteration
Wall time (s)
Train Samples
# Iteration
Wall time (s)
# Train Samples
Figure 2: Learning curves on CATS Arbitrary with normal valuations. (a-c) Results for 50, 100, and 150 items, respectively. (d) Results for 50 items, increasing XOR atoms per value function from 5 to 50. Columns show the changes in test revenue as a function of the number of training iterations, wall time in seconds, and the number of training samples, respectively.
Table 2: Effect of the support size of the initial distribution (D) on BUNDLEFLOW in different CATS environments. The revenue drops dramatically when changing from D to D = 1.
CATS-Rcgion Nornal Valuations
CATS Aabs-Arbytrayy Uniform Valuations
# Trems (m)
D=1
D=2
D=4
D=8
D=16
D=1
D=2
D=4
D=8
D=16
50
278.82
589.50
596.67
603.70
595.14
287.98
557.06
563.70
560.71
561.51
100
258. 18
364.29
372.49
389.51
388.06
279.75
425.94
426.80
434.77
428.53
150
291.69
338.52
368.00
394.28
383.24
318.97
383.44
396.71
420.75
411.86
across all environments. On the Regions environment with normal value distributions, we further test 200 and 500 items. When varying the number of items, we set the maximum XOR atoms per bid to (the default value). We also experiment with increasing the maximum number of atoms to 50. Appx. D. discusses how to set up single-bidder auctions in CATS. All experiments are conducted on a single NVIDIA A 1100 GPU. Our code is available online
(VCG) mechanism remains truthful in this single bidder CA context but charges the bidder her externality, which zero when there only one participant. Myerson's virtual-value approach (40) is designed for single-item auctions and is not well defined when items exhibit complementarities.
Baselines. The classic Vickrey-Clarke-Gnen
Learning-ban based methods that are DSIC fall in two broad categories. First, there are menu- based methods with ttem-wise allocations such as RochetNet (13). These methods have limited power for CAs because they do not express general bundle distributions. We use RochetNet as the first baseline:
Interpret the item-wise allocation as a product distribution. Calculating bidder values remains intractable due to the need to enumerate all bundles. We address this by employ- ing the Gumbel-Softtaa technique [1I, which enables sampling from product distributions and backpropagation through the samples to update item-wise allocations.
(1) RochetNet 13
Second, there are methods such as the affine maximizer auction (AMA) and its deterministic variant the VVCA [2). These explore a restricted family of affine mechanisms, with the result that they exhibit restricted expressiveness and achieve suboptimal revenue. We include baselines that are simplified versions of the AMA and VVCA methods, where we necessarily fix the menu to contain a subset of all possible bundles in order to scale to instances with large number of items:
(2) Menu+: The menu contains as elements the grand bundle (the bundle of all items), well the largest bundles immediately smaller in size. When the menu size cannot accommodate all bundles of a certain size, the selection of bundles of this size is random.
(3) Menu-: Similar to Menut, but we begin by including elements that consist of single-item bundles, expanding as the menu size permits, and always including the grand bundle.
8

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Table 3: Effect of increasing D, the support size of initial distributions, under varying menu sizes K, the number of elements in menu.
Menu Size
K/4
K/2
K
K #2
Environments
D=1
D=2
D=1
D=2
D=1
D=2
D=1
D=2
CATS-Regions
50
228.34
575.52
259.95
568.94 94
278.82
589.50
315.88
602.98
Normal
100
213.26
324.04
226.11
345.51
258.18
364.29
270.65
419.12
Valuations
150
245.22
312.82
268.78
318.21
291.69
338.52
302.04
385.08
CAT'SArbitrayy
50
238.96
546.50
261.61
551.78
287.98
557.06
303.53
561.28
Uniform
100
230.84
357.99
254.20
408.45
279.75
425.94
305.11
438.27
Valuations
150
297.22
355.07
303.08
361.01
318.97
383.44
343.19
404.30
(4) Grand Bundle: This menu has the grand bundle as the only menu element.
For (2) and (3), we learn the prices corresponding each element using gradient-based optimization. For (4), the price on the single element, the grand bundle, is determined through grid search to maximize training revenue. Performance, as with all methods, is reported on the test set.
Hyperparameters. Detailed hyperparameter settings of BUNDLEFLOW and baselines can be found in Appx. D For BUNDLEFLOW, menu size K to 5000 when m I 100, and 20000 otherwise. We did not extensively fine-tune the remaining typerparameters, such as neural network sizes, the initial trial guided by prior practice already yielded satisfactory results.
set the support size of the initial distribution D to 8, and the
5.1 Experimental Results
Performance. Table T shows the revenue of our method and baselines. BUNDLEFLOW performs
consistently and significantly better when the number of items is large (10 < < 200), achieving significant revenue gains of 1.11- over baselines. remains superior at scale: for 200 S m < 500, BUNDLEFLLOW continues to outperform all the baselines as shown in Table 4 The learning curves in Fig. 2 reveal another advantage of BUNDLEFLOW. Compared to RochetNet, which is the baseline that also learns allocations in a menu, our method achieves 3.6-9.5x improvement in the number of training iterations in CATS Arbitrary with normal value distributions (for 10 < m < 200). Our method can also reduce training time in some settings. For example, when m = 50, BUNDLEFLOW reduces the training duration from about 700 seconds to 140 seconds, achieving 5x improvement in training speed. These results highlight that BUNDLEFLOW, while allowing improved representational capacity over baselines, formulates an optimization problem that tractable, enabling improved revenue as well training efficiency. We discuss inference time in Appendix D.4
When the number of items is small (m=10), baselines (Menut, Menu-, RochetNet) with menu size of 5K can already cover all possible bundles (210- = 1024). Although these settings are not our primary focus, BUNDLEFLOW achieves comparable revenue these baselines.
The support size of bundle distributions. Table provides insights into why BUNDLEFLOW has superior revenue. When we reduce the support size of initial distributions (D) to 1, each menu element deterministically assigns single bundle and no longer represents bundle distribution. Correspondingly. we observe dramatic drop in revenue when D is decreased from D 2 to D = 1. For example, in the CATS Regions with normal value distributions and 50 items, the revenue declines sharply from 589.50 to 278.82. This trend remains when we vary the menu size, as shown in Table This pronounced performance gap between D = and D = highlights the importance of maintaining randomized distribution over bundles in the application of differentiable economics to CAs-a capability uniquely enabled by our method.
In Table we vary the menu size in smaller range (10, 20, 100, 200) and present the test revenue of our method with D 1, D 2, and D 16. When the menu size small, increasing beyond can triple the test revenue. With increased menu sizes, the gain for D larger than 2 diminishes but remains positive. We interpret this observation as follows. The representation capacity depends on both the menu size and the support size of each menu element. For small menu size, increasing D beyond continues to help. For a large menu size, the main gain comes from allowing some randomization (increasing D from to 2).

---

Table 4: Test revenue (large number of items).
Table 5: A support size of D > 2 contributes more significantly when the menu size small.
Methods
m 200
500
Menu Size
D
D 2
D 16
BUNDLEFLOW
375.90
397.33
Menu+
298.0
312.9
Menu Size 10
43.86
106.00
303.51
Menu- 302.8
315.0
Menu Size 20
67.84
272.12
337.82
RochetNet
365.0
375.0
Menu Size 100
130.46
402.85
449.27
Grand Bundle
296.3
308.5
Menu Size 200
148.47
439.61
449.18
Menu size and valuation function size. TableBpresent the influence of menu sizes (K), specifically when they are halved, quartered, or doubled. A randomized distribution over bundles is crucial in each of these settings, with revenue falling sharply when D decreases to 1. Furthermore, increasing the menu size tends to improve performance, although the gains are modest for small number of items. For example, in CATS Regions with normal value distributions and menu size from K/4 to K x 2 leads to 29.34% revenue boost, in contrast to a merely 4.77% boost when 50 Increasing the maximum number of XOR atoms per valuation from 10 50 has
100, increasing the
minimal effects on our method and baselines, as evidenced in Table 6 and Fig.
Closing Remarks
The key contribution in this paper is ODE-based framework that enables concise yet flexible representation and efficient optimization for bundle distributions in CA deep learning. By keeping these distributions tractable- avoiding an exponential representation size wwe dramatically increase revenue over baselines. Appx. [ also suggests an interesting direction in extending this approach to multi-bidder CA design, although its current complexity is prohibitive. Reducing this complexity is key for future work.
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Appendix
A Proof
Theorem 1. [Exact DSIC] The BUNDLEFLOW framework ensures the learned auction mechanisms are DSIC.
establishes necessary and sufficient conditions for strategyproof menu-based auction: (I) Self-bid independent: the menu is independent of the bidder's bid; and (2) Agent-optimizing; the bidder is assigned the menu element that maximizes their utility. As we analyze here, our method satisfies these two properties.
Proof. The seminal work by Hammond [20)
In BUNDLEFLOW, all element prices, well bundle allocations, which depend initial distribu- tions and the vector field, are trained on values sampled from the distribution F, without using any information about the bidder's specific valuation. Therefore, menus learned by BUNDLEFLOW self-bid independent.
As discussed in Sec. 3.3 we require the initial distribution for each menu element to have finite support, which means that the bundle distribution for each menu element can be reconstructed without any approximation error. This guarantees exact utility calculation for every menu element. Moreover, unlike the SoftMax during training, we use hard argmax at test time, thereby selecting the menu element with the highest utility the bidder. In this way, BUNDLEFLOW is agent-optimizing. O
B Discussion
Representation flexibility. In Stage I, we initialize the vector field y. The final distribution can in principle cover all 2n bundles and is trained to seek to achieve this. In Stage 2, since the initial distribution for a menu element has finite support of size D, the bundle distribution for a menu element is also limited to finite support of size D. What is crucial, though, that we can learn which (up to) D bundles are represented in the distribution that corresponds to menu element. In practice, we find that a bounded D that is much smaller than 2n still gives very high expected revenue.
Extension to multi-bidder settings. By providing a tractable and yet fully flexible representation of single-bidder menus for the CA setting, BUNDLEFLOWW opens up the possibility of developing general DSIC multi-bidder CA mechanism. A principled approach is to adapt the idea of GemNet (57). First, we can learn separate BUNDLEFLOW menu for each bidder. The modification in the network architecture is that these menus should now also depend on other bidders' bids b.;. To achieve this, we can condition the vector field, specifically the Q and a networks, on b-; by concatenating them to the inputs. For the price of each menu element, we can model them as the output of a neural network whose input b.; During training, we can also introduce compatibility loss the same way as that used in GemNet. This loss penalizes any over-allocation of items in the selected agent-optimizing elements from individual menus.
The major challenge in adapting GemNet to the multi-bidder CA setting will arise during the post- training stage of GemNet, which adjusts prices of menu elements so that there is provably never any over-allocation of items. For this, GemNet constructs a grid over the space of bidder values. On each grid point, GemNet formulates a mixed-integer linear program (MILP) adjust prices to ensure that, the utility of the best element that is compatible with the choices of others in the sense of not allocatin items is larger than that of all other elements by safety margin. These safety margins prevent an incompatible menu element from being selected in the regions between grid points. Although the concise BUNDLEFLOW menu representation makes possible for this MILP to be directly adapted to the CA setting and used to adjust BUNDLEFLOW menus to obtain DSIC multi-bidder CA, the main issue is that the space of bidder values exhibits exponential dimensionality in the CA setting, resulting in an excessively large grid. Reducing this complexity represents the crucial remaining step in future work to enable the use of differentiable economics for learning a general, DSIC, and multi-bidder CA mechanism.
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C Additional Background
Bidding language. In CAs, suitable bidding language is critical to allow a bidder to report their bid without needing to enumerate a value for every possible bundle. There are many ways to do this (including text-based preference elicitation via an LLM [25 52), but common approach is to use the XOR bidding language, which allows bidders to submit bid prices for each of multiple bundles under an exclusive-or condition; in effect, only one bid price on a bundle can be accepted. Popular CA testbeds such as CATS [35] and SATS [58] employ this bidding language extensively The semantics of the XOR bidding language is that the value on a bundle S is the maximum bid price on any bundle S" submitted as part of the XOR bid, and for which S C S. XOR bids succinct for valuation functions in which the bidder is only interested in a bounded number of possible bundles.
Diffusion models. Diffusion models have emerged as a powerful class of generative AI methods, spurring notable advances in a wide range of tasks such as image generation [4. 45), video gen- eration 7 21 23], molecular design [8], text generation [39]. and multi-agent learning (55). At their core, these models perform a forward noising process in which noise is incrementally added to training data over multiple steps, gradually corrupting the original sample. A reverse diffusion process is then learned to iteratively remove noise, thereby reconstructing data from near-random initial states. In our setting, instead of reconstructing data, extend the diffusion process to develop a tractable and differentiable method that optimizes a high- nigh-dimensional distribution.
In particular, score-based diffusion models enjoy strong mathematical and physical underpinnings. The forward noising process is an Itô stochastic differential equation (SDE).
dx fxx + h(t)dw,
(13)
where xtt E IR' is the state at time t, for some E E Z f(..t)::: R' IR' is the drift coefficient, h(-): : R + E is the diffusion coefficient. and is the standard Wiener process (Brownian motion). Different forward processes are designed by specifying functional forms for f(,,t) and h(.). The generation of data then based on the reverse process, which is a diffusion process given by the reverse-time SDE 3
dx= = [f(,,t) - h(t)²V log (x))ddt + h(t)du.
(14)
where dt is an infinitesimal negative timestep, v is the standard Brownian motion with reversed time flow, and qe (x) is the distribution of state xtt) ar time The principal task diffusion models is to learn the score function, Va log Ze (z), which has been effectively achieved using neural networks in recent work. This enables solving the reverse-time SDE and generating new data samples. Notably, in the diffusion model (and more broadly, generative AI) literature, qo (z) is typically known target distribution over data samples from pre-training dateset.
D Experiments
D.1 Benchmark
In CATS, the valuation functions of bidders are recorded as bundle-bid pairs in an output file, with bundles from the same bidder identified by appending a dummy item tagged with a unique identifier. In effect, the bundle-bid pairs in an output file involving the same dummy item form an XOR representation of the bidder's valuation function. To obtain single-bidder valuations, we generate 100,000 such files and extract valuation functions identified by consistent dummy item. Of these, 95% are used for training, with the remaining 5% reserved for testing.
D.2 Hyperparameters
We do not extensively fine-tune hyperparameters. This suggests that our formulation of the ODE, including its functional form (Eq. 55) and initial conditions, is well-suited to the needs of the CA setting, making the optimization of the flow model relatively straightforward. Specifically, the Q
'When representing the values of multiple bidders these testbeds often also introduce so so-called dummy items for distinguishing the bids of different bidders. Still, the semantics for single bidder is, in effect, that of the XOR language.
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Table 6: Revenue comparison against baselines across different CATS environments and value distributions. The number of items fixed at m = 50, and we increase the valuation function size: 10, 20, 30, 40, and 50 as maximum XOR atoms per valuation, corresponding to the maxbid parameter in CATS.
Environment
Baseline
10
20
30
40
50
Grand Bundle
314.46
309.11
316.04
320.61
314.35
Menu+
374.28
369.29
357.91
372.39
363.13
CATS-Reeion
Menu-
328.95
328.92
323.44
333.00
326.58
Uniform Private Valuations
RochetNet
308.81
313.24
317.14
318.15
310.60
BUNDLEFLOW
533.42
528.46
528.26
539.92
537.36
Grand Bundle
322.74
301.54
310.38
304.38
334.95
Menu+
434.32
375.51
404.74
381.05
430.24
CATS- CATSSRegions
Menu-
326.98
307.81
332.39
313.76
342.23
Normal Private Valuations
RochetNet
301.47
297.34
317.10
302.04
327.32
BUNDLEFLOW
564.13
524.24
535.35
525.68
566.51
Grand Bundle
331.21
334.43
330.33
351.01
336.68
Menu+
355.92
351.02
341.76
347.26
348.92
CATS-Arbitray Uniform Private Valuations
Menu-
352.34
354.58
346.36
354.87
353.58
RochetNet
329.61
345.06
338.44
344.76
350.75
BUNDLEFLOW
565.54
583.60
568.01
579.01
581.39
Grand Bundle
381.51
335.44
323.52
320.28
358.83
Menu+
470.63
349.70
346.39
340.59
381.13
CATS-Arbitrary Normal Private Valuations
Menu-
402.23
341.79
337.44
332.90
368.26
RochetNet
367.44
329.28
333.88
327.57
365.40
BUNDLEFLOW
664.94
564.90
552.77
548.22
615.93
network comprises three 128-dimensional tanh-activated fully connected layers. When > 100, we increase the width the last layer to 256. The a network simpler and has two 128 -i8dimmnsional tanh-activated fully connected layers.
Two important hyperparameters are D. the support size the initial distribution, and K, the menu size. By default, D is set to 8, relatively small number. K is 5000 when m S 100 and is 20000 otherwise. The same menu is used for our method, Menu-, Menut, and RochetNet. Notably, the menu size K is adequate encompass all possible bundles for smaller numbers of items, such as m or 10. We show the impact of different values of and in ablation studies.
Menu optimization for BUNDLEFLOW is conducted using the Adam optimizer with learning rate of 0.3. ASoFTMAx is increased from 0.001 0.2 over the course of training. For comprehensive details on our hyperparameter settings, please refer to the codebase. For the baselines, we fine-tuned their hyperparameters
that they perform significantly better than the default RochetNet setting. The modifications are achieved by performing grid search to obtain the optimum combination of Asoftmax and learning rate that yields the best revenue and also guarantees convergence. Both Menu- and Menu+ use learning rate of 0.3 and AsofTmax of 2, while RochetNet uses a learning rate 0.05 and ASoFTMAx of 20.
All experiments are conducted on a single NVIDIA A100 GPU.
D.3 Baselines.
We give example of allocations in the menu of Menut, With 3 items and menu size of 3, the menu would include [1, 1] (the grand bundle) and a random subset of bundles containing items, such [1, 1, 0] and [0, 1].
D.4 Inference speed.
Inference for our method fast absolute time: for m 100 with batch size of 200 samples, BUNDLEFLOW runs in 4.74 74 x 10-3 + 2.27 x 10-1s s, compared with RochetNet at 6.39 10 +
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Small-Bundle Bundle
Big-Bundle
BundleFlow
Bundle-RochetNet
Grand-Bundle
500.0
Ted Revenue
400.0
300.0
200.0
50k
100k
150k
200k
200
400
600
800
10M
20M
# Iteration
Wall time (seconds)
# Training Samples
(a) a-10
500.0
Test Revenue
400.0-
300.0
200.0
100k
200%
300%
400k
1000
1500
20M
40M
Iteratioo
Wall time (seconds)
# Training Samples
(b)
Figure 3: Learning curves of BUNDIEFFLOW and baselines CATS Regions with uniform valuation distributions with different valuation function sizes. The two rows are results for a = 10 and 50 XOR atoms per valuation, respectively. The three columns show the changes of test revenue as a function of the number of training iterations, wall time in seconds, and the number of training samples, respectively.
3.01 x 10 s, Menut at 3.70 x 1 4.70 X 10 6 and Menu- at 3.60 X 10 2.30 x 10-6g under the same conditions. While method has relatively higher inference time, it requires significantly fewer training iterations to converge, maintaining clear advantage in terms of overall training time. After training, method requires deploying menu, where the allocation and price of each menu element can be calculated before deployment. Users only need to check which bundles (whose number is upper bounded constant) in menu element, calculate its utility, and assign the best menu element.
24


Reviews:
[{"review_id": "BP9b0zI0MK", "reviewer": "Reviewer_B4jQ", "summary": "This paper presents BUNDLEFLOW, the first DSIC and revenue-optimizing deep learning method for single-bidder combinatorial auctions (CAs). It overcomes the exponential bundle space challenge by generating bundle distributions using ODE-based transforms, one per menu element. The method significantly outperforms baselines in revenue (up to 2.23×), reduces training iterations by up to 9.5×, and cuts training time by ~80% for large item sets (e.g., 50–150 items).", "questions": "NA", "limitations": "yes", "rating": 5, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strength:\n1. Scalable and efficient: Achieves superior performance and faster training in large-scale CA settings.\n\n2. Novel technique: Introduces an ODE-based randomized menu approach that balances flexibility and tractability.\n\nWeakness:\n1. The usage of ODE training and hyperparameter tuning may pose a barrier for practical deployment.", "was_revised": true, "final_justification": "I maintain my score for Accept. \n\nThe authors are able to provide sound details regarding technical considerations and a commitment to including the recommended parameters in the source code, which will support the technical progress in the area.", "rebuttal": "Thank you for your valuable comments.\n\n> Question: `The usage of ODE training and hyperparameter tuning may pose a barrier for practical deployment.`\n> \n\nThanks for the question. We provide a detailed analysis of which parts might be barriers to practical deployment and make recommendations about how to mitigate barriers.\n\n**1. The impact of ODE training on practical deployment**\n\n- __Compact closed form of the ODE__. We understand that an ODE might raise the concern because solving it or calculating the final probability distribution involves an integral. To deal with this issue, we propose a special functional form for the ODE (Eq. 5), which simplifies the integral so that it involves only a scalar function. Such a formulation enables fast inference.\n- __Training and deployment in practice__. The lighter ODE computation contributes to the training speed. Compared to RochetNet, which is a baseline that also trains both allocation and prices in a menu of the same size, our method reduces the training duration from about 700 seconds to 140 seconds on the same GPU. After training, our method requires deploying a menu, where the allocation and price of each menu element can be calculated before deployment. Users only need to check which bundles (whose number is upper bounded by a constant $D$) are in a menu element, calculate its utility, and assign the best menu element.\n\n**2. The impact of hyperparameter tuning on practical deployment**\n\n- __Robustness to some hyperparameters__. We find that our method is robust to some hyperparameters, e.g., the neural network architecture, the batch size for menu optimization, and those regarding the initial distributions. For these design choices, the default hyperparameter settings in our codebase are a sensible choice for accelerating the development process.\n- __Recommendations for the others__. There are other hyperparameters that need fine-tuning. During our experiments, we find that the learning performance is sensitive to softmax temperature $\\lambda_{\\rm SoftMax}$ and the learning rate for menu optimization. As detailed in Appendix D.2, we provide generally effective configurations. The published code will include recommended settings for these hyperparameters tailored to CA problems of different scales, enabling users to select appropriate values based on their specific applications."}, {"review_id": "hEwgNs5oxR", "reviewer": "Reviewer_iwrK", "summary": "This paper studies menu-based auctions for single-bidder combinatorial auctions. The authors proposed a diffusion-based computational techniques for generating menus. Experimental results show that the technique scales really well. It can handle 150 items with ease (less than 1 hour). The achieved revenue is significantly better a few baselines.", "questions": "Can you give some practical scenarios where we need to solve single-bidder CA auction for 50-150 items? I am happy to change my rating if I see a good practical scenario. As I mentioned earlier, I think the techniques used in this paper are of high quality and original.", "limitations": "This paper studies a theoretical auction model. The data are synthetic. I don't see any potential negative societal impact.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 2, "originality": 4, "strengths_and_weaknesses": "Since the authors focused on single-bidder menu-based auctions. Strategy-proofness and individual rationality are essentially non issues. As long as the menus are presented to the bidder (with the option of not paying and get nothing), and the bidder picks the best menu option, then SP and IR are achieved. Even when restricted to a single bidder, the setting is well motivated and has significance in practise.\n\nThe authors presented the main computational challenge via an example. What does [0.3, 0.6] mean for two items? As these can be achieved via different ways to group the deterministic bundles. The main compuational challenge is there are exponential number of ways to achieve a given item vector. To address this, the authors' main technique is to use an ODE to describe how the bundle breakdown is generated, and this ODE is characterized via a vector field. The first step of the proposed computational technique is to learn this vector field so that it generates feasible determinsitic bundle breakdowns.  The second step is to maximize revenue via menu optimization. Again this step is technically sophisticated, relying on ODE manuveour for the actual training.\n\nThe main downside of the proposed approach is its practical significance. Technically, the proposed technique makes a lot of sense. It is sophisticated and it does what is designed to do well, i.e., proposing good solutions for *very large* number of items. When we have very large number of items, we cannot use AMA/VVCA (and other heuristic methods for revenue maximization), the authors used baselines like menu+, menu-, grand, rochetnet, which are designed for scalability. The proposed method convincingly beat all existing baselines for large number of items. However, when the number of items is small, say, 10 items, the proposed method does not perform better than all baselines, as shown in Table 1. I totally understand that for this diffusion-based method, its advantage is to handle large number of items as it handles scalability really well. My main concern is whether single-bidder 50 item CA is an important or even a practically realistic setting. We are talking about 2^50 bundles. What is a realistic situation for this kind of scale? The main contribution of this paper is a technique to push up scalability. My question is whether scalability is actually an issue for the studied model.", "was_revised": true, "final_justification": "The authors' rebuttal contains very convincing examples on the practicality of the studied model.", "rebuttal": "Thank you for the valuable comments.\n\n> Main question:  `The main downside of the proposed approach is its practical significance. Can you give some practical scenarios where we need to solve single-bidder CA auction for 50-150 items?`\n\n1. Many examples of single-bidder CA problems involve a large number of items. Here we give three real-world examples with references.\n\n    - ___(Scenario 1) Selling on-demand cloud compute___. AWS [1] is a typical example. In this setting, an item is characterized by the number of GPUs, CPUs, Instance Memory (GiB), GPU Memory (GiB), Network Bandwidth, PUDirect RDMA, Instance Storage (TB), EBS Bandwidth (Gbps), etc. This yields hundreds of items. AWS already lists 850+ instances (”pre-defined bundles”) for users to choose from.  Single-bidder CAs can be used to design and price the instances.\n    - ___(Scenario 2) Selling a bundle of information goods___, e.g., selected from artists or TV shows or podcasts. As an example, on March 02, 2022, Prime Video UK [2] offered 65+ channels (such as Discovery, ITV Hub+, Eurosport Player, hayu, STARZPLAY and more). Users can pay for the ones they want. Optimally pricing and configuring each of the different subsets is an example CA of the size analyzed in this paper.\n    - ___(Scenario 3) Selling configurable managed services___. This is quite common in daily life, for example, a building operations service, where choices involve weekly frequency of trash pickup for different types of trash, different types and sizes of security patrols, method and frequency of cleaning of floors, different types of printers, and different types of A/V IT support. As an instance, Aramark [3] provides at least 9 service lines (ops & maintenance, custodial, engineering, HVAC, electrical, pest control, building intelligence, staffing, energy). Selecting a service level (e.g., daily, weekly, monthly) for each line across 15 buildings easily yields hundreds of items.\n\n2. Consider the next step of research, where the goal will be to extend our method to multi-bidder cases. Wireless spectrum rights auctions, run by governments around the world, handle 3,000-12,000 licenses. \n\n3. We have also completed additional experiments to evaluate the usefulness of our method on smaller problems. Although our method does not surpass baselines in 10-item CAs, these results show that it starts to outperform all baselines when the number of items increases to 12.\n\n[1] https://aws.amazon.com/ec2/instance-types\n\n[2] https://press.aboutamazon.com/uk/2022/3/prime-video-pledges-10m-to-support-training-and-development-in-the-uk-tv-and-film-industry\n\n[3] https://www.aramark.com/our-services/facilities-management-services\n\n> Comment 1: `Strategy-proofness and individual rationality are essentially non issues.  As long as the menus are presented to the bidder (with the option of not paying and get nothing), and the bidder picks the best menu option.`\n\nThe technical conditions required for strategy-proofness and IR of single-bidder auctions are standard. What is challenging in our setting is that it must be tractable to compute the best menu option (at inference time). A menu element can, in theory, involve a randomized allocation over an exponential number of bundles. This is also one motivation for us to introduce ODEs to represent the bundle distribution.\n\n> Comment 2: `What does [0.3, 0.6] mean for two items? `\n\nThis is an item-wise allocation, meaning that the bidder has a probability of 0.3 and 0.6 of getting item 1 and item 2, respectively. As the reviewer pointed out, this item-wise allocation can be achieved in different ways of grouping deterministic bundles."}, {"review_id": "4tD4ErAqGA", "reviewer": "Reviewer_TcJ8", "summary": "The paper deals with single-bidder combinatorial auctions. The paper proposes a new method inspired by diffusion which consists of two stages. In the first stage a diffusion model is trained so that the entire space of bundles is in its support. The second stage uses this distribution for optimization by manipulating the initial distributions in order to maximize the revenue.", "questions": "-", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "### Strengths:\n- Well written and clearly motivates the single bidder setting of combinatorial auctions whereas most methods I am familiar with only discuss the multi-bidder case. Figure 1 is very didactic and explains the method beautifully.\n- Strong empirical results, clearly demonstrating the method's superior performance compared to baselines.\n\n### Weaknesses:\n- Can you clarify why your method only scales to 150 items? Do you see a degradation in performance or does it become too computationally expensive? \n- Going from D=1 to D=2 seems lead to most of the performance improvement which I didn't fully understand. Can you clarify why increasing D further only provides marginal benefits?", "was_revised": true, "final_justification": "The authors have answered my questions and I am maintaining my score.", "rebuttal": "Thank you for your valuable comments.\n\n> Question 1: `Can you clarify why your method only scales to 150 items? Do you see a degradation in performance or does it become too computationally expensive?`\n\nAlthough the main paper reports experiments up to 150 items, our method scales further. In the tables below, we show the test revenue on the CATS regions environment with normal value distributions, for 200 and 500 items, respectively.\n\n**Table 1: With 200 items, compare the test revenue of our method against baselines.**\n\n| Menu+ | Menu- | RochetNet | Grand Bundle | BundleFlow (Ours) |\n| --- | --- | --- | --- | --- |\n| 298.0 | 302.8 | 365.0 | 296.3 | **375.90** |\n\n**Table 2: With 500 items, compare the test revenue of our method against baselines.**\n\n| Menu+ | Menu- | RochetNet | Grand Bundle | BundleFlow (Ours) |\n| --- | --- | --- | --- | --- |\n| 312.9 | 315.0 | 375.0 | 308.5 | **397.33** |\n\nOur method continues to outperform all baselines. Despite this success, a challenge that arises for larger numbers such as 500 items is with the first training stage, where we seek to initialize the flow so that the induced distribution covers all possible bundles. Empirically, we find through sampling that the induced distribution becomes less uniform and more skewed towards larger bundles. We conjecture that this phenomenon is attributable to the representational capacity of the flow network (and can be addressed in future work through increased capacity).  \n\n> Question 2: `Going from D=1 to D=2 seems lead to most of the performance improvement which I didn't fully understand. Can you clarify why increasing D further only provides marginal benefits?`\n\n- **Why $D=2$ makes a big difference compared to $D=1$**. When $D=1$, each menu\nelement must deterministically assign a single bundle. $D=2$ is the smallest possible value that allows a randomized bundle distribution. This is a discontinuity, and a qualitative change.\n- **Behavior for $D>2$ depends on the menu size**. In Table 3, we show the results of a new experiment, where we vary the menu size in a lower range (10, 20, 100, 200) and present the test revenue of our method with $D=1$,  $D=2$, and  $D=16$.\n\n**Table 3: A support size of $D>2$ contributes more significantly with a small menu size.**\n\n| Menu Size | D=1 | D=2 | D=16 |\n| --- | --- | --- | --- |\n| Menu Size 10 | 43.86 | 106.00 | 303.51 |\n| Menu Size 20 | 67.84 | 272.12 | 337.82 |\n| Menu Size 100 | 130.46 | 402.85 | 449.27 |\n| Menu Size 200 | 148.47 | 439.61 | 449.18 |\n\n*Observations:* When the menu size is small, increasing $D$ beyond 2 can *triple* the test revenue. With increased menu sizes, the gain from $D>2$ diminishes but remains positive. \n\n*Analysis:* We interpret this new observation as follows. The representation capacity, in regard to menus, depends on both the menu size and the support size of each menu option. For a small menu size, increasing $D$ beyond 2 continues to help. For a large menu size, the main gain comes from allowing some randomization (increasing  $D$ from 1 to 2)."}, {"review_id": "vDsPFx6Kq3", "reviewer": "Reviewer_kwyj", "summary": "This paper attempts to solve the combinatorial auction problem and proposes bundflow, a menu-based mechanism that represents the bundle distribution of each menu option via a continuous normalizing flow. This method avoids the exponential blow-up in explicit bundle representations and preserves exact dominant-strategy incentive compatible. Experiments and theoretical analysis are conducted.", "questions": "Please see the weaknesses section.\n\nBased on the current presentation and considering the novelty, I tend to recommend acceptance. However, I'm not an expert in this domain and am open to revising my decision according to future discussions.", "limitations": "N/A", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strengths**\n\n1. Using the normalizing flow to solve the combinatorial auction problem seems new in the literature.\n\n2. The sample codes are provided to help the reviewer verify independently. \n\n\n**Weaknesses**\n\n1. The presentation of this paper could be improved. For example, the proof of Theorem 1 is not self-contained. Based on the current version, Theorem 1 seems to be a simple extension/corollary of results in [17].\n\n2. It would be better to include the wall-clock training and inference times (and GPU type) for all baselines to support the 80 % saving claim clearly.", "was_revised": true, "final_justification": "During the rebuttal stage, my concerns regarding presentation improvements and ablation studies have been largely addressed. While I maintain some reservations about the presentation improvements, since the authors have only committed to enhancing them rather than demonstrating concrete changes. After thoroughly reviewing all reviewer comments and author responses, I believe this is not a significant concern to change my rating. Therefore, I choose to maintain my original rating and vote for acceptance.", "rebuttal": "Thank you for your valuable comments.\n\n> Question 1: `The presentation of this paper could be improved. For example, the proof of Theorem 1 is not self-contained. Based on the current version, Theorem 1 seems to be a simple extension/corollary of results in [17].`\n\nWe will reorganize the proof argument of Theorem 1 in the main body of the paper. The crucial part is the discussion presented between Lines 226 and 239, which builds upon Eqs. 11 and 12. The analysis surrounding Eqs. 11 and 12 is used to prove that our method can guarantee the bidder-optimizing condition, thereby establishing strategy-proofness. \n\nThe technical conditions required for strategy-proofness and IR of single-bidder auctions are standard. What is challenging in our setting is that it must be tractable to compute the best menu option (at inference time) to fulfill the bidder-optimizing condition. A menu element can, in theory, involve a randomized allocation over an exponential number of bundles. This is one motivation for us to introduce ODEs to represent the bundle distribution.\n\n> Question 2: `It would be better to include the wall-clock training and inference times (and GPU type) for all baselines to support the 80 % saving claim clearly.`\n\nThe only baseline for which we do not report wall-clock training times in Figures 2 and 3 is *Grand Bundle*. The reason is that *Grand Bundle*  is not a deep learning method. It performs a grid search over a single price for a fixed bundle that contains all items. When claiming the speed advantage of our method in the main paper, we made it clear that we are comparing with other *menu-learning* approaches, and the reported 80% reduction refers specifically to training time.\n\nIn Appendix D.2 we reported that all experiments, including our method and the baselines, are conducted on a single NVIDIA A100 GPU. We will move this detail to the main text to improve clarity.\n\nAs for the inference time, for 100 items and a batch of 200 samples, our method takes $0.00474 \\pm 0.000227$ seconds per batch on an A100 GPU. For comparison, the inference time of *RochetNet*, *Menu+*, and *Menu-* under the same conditions is $0.000639  \\pm 3.01e-05$, $3.7e-05\\pm 4.7e-06$, $3.6e-05 \\pm 2.3e-06$ seconds, respectively. We will add this inference time information to the paper. While our method has a higher inference time, it requires significantly fewer training iterations to converge, maintaining a clear advantage in terms of overall training time. After training, our method requires deploying a menu, where the allocation and price of each menu element can be calculated before deployment. Users only need to check which bundles (whose number is upper bounded by a constant) are in a menu element, calculate its utility, and assign the best menu element."}]

Author rebuttal:

Meta-review: This paper introduces a DSIC and revenue-optimizing deep learning approach for single-bidder combinatorial auctions (CAs). The core challenge addressed is the exponential bundle space, which has traditionally made scalable mechanism design in this setting intractable. It leverages an ODE-based continuous transform to generate bundle distributions for each menu element, enabling both tractability and expressivity.

The paper makes a novel and technically rigorous contribution to the design of revenue-optimal, DSIC mechanisms in combinatorial auctions. Despite some concerns about practical applicability and presentation (which are partially resolved during rebuttals), the scalability results, theoretical grounding, and methodological innovation clearly outweigh the weaknesses.
Decision: Accept (poster)