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BundleFlow: Deep Menus for Combinatorial Auctions by Diffusion-Based Optimization
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Proposes BundleFlow, a menu-based deep mechanism for single-bidder combinatorial auctions that represents bundle distributions via a continuous normalizing flow (ODE) and optimizes revenue, scaling to 150 items on CATS.
Problem
Designing DSIC, expressive revenue-maximizing combinatorial auctions with many items is hard because the bundle space grows exponentially with the number of items, so explicit bundle-wise allocation distributions are intractable to learn.
Approach
Represent each menu element's randomized bundle distribution as a mixture-of-Dirac initial distribution pushed through a shared learned ODE (continuous normalizing flow with divergence-trick vector field); stage 1 trains the flow to cover feasible bundles via rectified flow, stage 2 optimizes only initial-distribution supports, weights, and prices for revenue.
Key results
- On CATS with 50-150 items, BundleFlow achieves 1.11-2.23x higher revenue than the best baseline (e.g., 555.05 vs 399.85 for Regions uniform at m=50).
- Compared with Bundle-RochetNet, BundleFlow needs 3.6-9.5x fewer training iterations and reduces training time by about 80% at 50 or 100 items.
- Reducing the initial distribution support size D from 2 to 1 drops revenue sharply (e.g., 589.50 to 278.82 for Regions normal, m=50), showing the value of randomized bundle distributions.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use BundleFlow instead of the strongest reported baseline | single-bidder CATS combinatorial auctions · CATS Regions and Arbitrary; Uniform and Normal; m=50, 75, 100, 150 | revenue | Helped |
| Set initial-distribution support size D to 1 instead of D=2 (ablation) | BundleFlow · CATS-Regions Normal, m=50 | revenue: 589.50 -> 278.82 | Hurt |
| Set initial-distribution support size D to 1 instead of D=2 (ablation) | BundleFlow · CATS-Regions Normal, m=100 | revenue: 364.29 -> 258.18 | Hurt |
| Set initial-distribution support size D to 1 instead of D=2 (ablation) | BundleFlow · CATS-Regions Normal, m=150 | revenue: 338.52 -> 291.69 | Hurt |
| Increase menu size from K/4 to K*2 (ablation) | BundleFlow with D=2 · CATS-Regions Normal, m=100 | revenue: 324.04 -> 419.12 | Helped |
| Increase menu size from K/4 to K*2 (ablation) | BundleFlow with D=2 · CATS-Regions Normal, m=50 | revenue: 575.52 -> 602.98 | Helped |
| Increase maximum XOR atoms per valuation from 10 to 50 (ablation) | BundleFlow and baselines · CATS, m=50; tested environments and value distributions | revenue | No clear effect |
| Reduce initial-distribution support size from D=2 to D=1 (ablation) | BundleFlow · CATS-Arbitrary Uniform, m=50, 100, 150 | revenue | Hurt |
Surprising findings
- Fixed-allocation baselines (bundle, Big-Bundle) outperform the fully learned Bundle-RochetNet, attributed to optimization difficulty with large menus.
- At m=10 the baselines with menu size 5000 cover all bundles and BundleFlow offers no clear advantage.
Limitations
- Single-bidder only; extension to multi-bidder DSIC requires solving a grid-based MILP whose size grows exponentially with items.
- Performance drops when the initial distribution support is reduced to one bundle per menu element (D=1).
- At small item counts (m=10) no advantage over baselines whose menus can cover all bundles.
Experimental setup
Datasets: CATS Regions (uniform and normal) CATS Arbitrary (uniform and normal)
Benchmarks: CATS Regions CATS Arbitrary
Metrics: test revenue training iterations wall-clock training time
Baselines: Bundle-RochetNet Big-Bundle bundle Grand-Bundle
Builds on: RochetNet GemNet rectified flow continuous normalizing flow CATS
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