Computing Bayes-Nash Equilibria in Combinatorial Auctions with Continuous Value and Action Spaces
Bibliographic record
Abstract
Combinatorial auctions (CAs) are widely used in practice, which is why understanding their incentive properties is an important problem. However, finding Bayes-Nash equilibria (BNEs) of CAs analytically is tedious, and prior algorithmic work has only considered limited solution concepts (e.g. restricted action spaces). In this paper, we present a fast, general algorithm for computing symmetric pure ε-BNEs in CAs with continuous values and actions. In contrast to prior work, we separate the search phase (for finding the BNE) from the verification step (for estimating the ε), and always consider the full (continuous) action space in the best response computation. We evaluate our method in the well-studied LLG domain, against a benchmark of 16 CAs for which analytical BNEs are known. In all cases, our algorithm converges quickly, matching the known results with high precision. Furthermore, for CAs with quasi-linear utility functions and independently distributed valuations, we derive a theoretical bound on ε. Finally, we introduce the new Multi-Minded LLLLGG domain with eight goods and six bidders, and apply our algorithm to finding an equilibrium in this domain. Our algorithm is the first to find an accurate BNE in a CA of this size.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.005 | 0.029 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".