BundleFlow: Deep Menus for Combinatorial Auctions by Diffusion-Based Optimization

TL;DR

BundleFlow addresses the challenge of designing DSIC, expressive combinatorial auctions with exponentially many bundles in the single-bidder setting by representing bundle distributions through an ODE-driven flow. A flow-based approach uses the Liouville equation to connect an easily specified initial distribution to a tractable final bundle distribution, enabling efficient, differentiable optimization of per-menu-element allocations and prices. Training occurs in two stages: first shaping the flow to cover feasible bundles, then fixing the flow and optimizing initial distributions and prices to maximize revenue; DSIC is enforced via finite-support initial distributions and hard selection at test time. Empirically, BundleFlow achieves $1.11$–$2.23\times$ revenue gains over baselines on the CATS benchmark up to $m=150$ items and substantially reduces training iterations, demonstrating scalable, expressive, and DSIC-compliant CA design with potential extensions to multi-bidder settings.

Abstract

Differentiable economics -- the use of deep learning for auction design -- has driven progress in the automated design of multi-item auctions with additive or unit-demand valuations. However, little progress has been made for optimal combinatorial auctions (CAs), even for the single bidder case, because we need to overcome the challenge of the bundle space growing exponentially with the number of items. For example, when learning a menu of allocation-price choices for a bidder in a CA, each menu element needs to efficiently and flexibly specify a probability distribution on bundles. In this paper, we solve this problem in the single-bidder CA setting by generating a bundle distribution through an ordinary differential equation (ODE) applied to a tractable initial distribution, drawing inspiration from generative models, especially score-based diffusion models and continuous normalizing flow. Our method, BundleFlow, uses deep learning to find suitable ODE-based transforms of initial distributions, one transform for each menu element, so that the overall menu achieves high expected revenue. Our method achieves 1.11$-$2.23$\times$ higher revenue compared with automated mechanism design baselines on the single-bidder version of CATS, a standard CA testbed, and scales to problems with up to 150 items. Relative to a baseline that also learns allocations in menu elements, our method reduces the training iterations by 3.6$-$9.5$\times$ and cuts training time by about 80% in settings with 50 and 100 items.

Figures (5)

• Figure 1: (a) A menu for additive or unit-demand valuations only needs to specify allocation probabilities for each item. However, item-wise allocation probabilities are too inflexible for CAs, as bidder values are specified for bundles. (b) The space complexity for representing an explicit distribution on bundles (a bundle-wise allocation) grows exponentially with the number of items. (c) We represent a bundle distribution through a tractable initial distribution and an ordinary differential equation (ODE).
• Figure 2: Evolution of the vector field $\varphi$ (represented by blue curves) during the first stage of menu training: Flow Initialization. The x- and y-axes represent the bundle variables for two of items. $x=1$ means item A is in the bundle, and $y=1$ means item B is in the bundle. We employ an ODE $d{\bm{s}}_t = \varphi(t, {\bm{s}}_t) dt$ to generate the final distribution $\alpha_T({\bm{s}}_T)$ (a distribution over bundles), represented by blue dots, from a simple initial distribution $\alpha_0({\bm{s}}_0)$, represented by green dots. During the first stage, $\alpha_0({\bm{s}}_0)$ is fixed as a mixture-of-Gaussian distribution. Dot opacity represents probability density. The aim of this first stage is to train the vector field so that the final distribution has all feasible bundles as its support (see Sec. \ref{['sec:viz']}).
• Figure 3: Visualization of the second stage of training: Menu Optimization. The figure presents 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 at the corresponding iteration. The x- and y-axes represent the bundle variables for two of items. $x=1$ means item $A$ is in the bundle, and $y=1$ means item $B$ is in the bundle. We fix the vector field (blue curves) and update initial distributions of elements to manipulate distributions over bundles (refer to Sec. \ref{['sec:viz']}).
• Figure 4: Learning curves of BundleFlow and baselines on CATS Arbitrary with normal valuation distributions with different numbers of items. The three rows are results for 50, 100, and 150 items, 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.
• Figure 5: Learning curves of BundleFlow and baselines on 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.