A Fast Distributed Algorithm for Uniform Price Auction with Bidding Information Protection

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This manuscript presents a distributed algorithm for solving the winner determination and uniform price problems in multiunit auctions where bidders can bid for multiple identical items with per-unit prices. The proposed approach employs a bisection method to achieve convergence in O(log M) steps while protecting bidders' private information, including both bidding prices and quantities. The algorithm represents a significant improvement over existing RNN-based solutions in terms of convergence speed.

 

Overall, the paper addresses an important problem in auction theory and makes a valuable contribution by developing a fast distributed algorithm with privacy protection features. The theoretical analysis of convergence properties is generally well-presented, and simulation results effectively demonstrate the algorithm's superiority over existing methods. However, there are several areas that require improvement and clarification before the manuscript is suitable for publication.

 

I have the following comments to help improve the manuscript.

 

- The authors only provide a conjecture (Section 6.3) that their algorithm protects bidding information privacy. This is a critical claim that requires formal theoretical proof. The authors should either provide a rigorous privacy analysis using established frameworks or clearly state the limitations of their privacy guarantees. Additional arguments and clarification are necessary.

 

- The current algorithm assumes all per-unit bidding prices are distinct (Assumption 1b), which is unrealistic in practice. While the authors briefly mention using timestamps (Section 6.1), this solution is inadequate. A comprehensive treatment of tie-breaking mechanisms and their impact on privacy and fairness properties is essential. Additional arguments and clarification are necessary.

 

- The paper lacks discussion of practical implementation challenges and validation with real auction data. The authors should include case studies from actual auction markets (e.g., electricity markets mentioned in line 53) or discuss implementation barriers and how to overcome them.

 

- While the proposed approach is interesting, the paper would benefit from a comparative discussion with other modern techniques that pursue similar privacy goals. Including a comparison of computational complexity, communication overhead, and security guarantees would likely make the paper more informative and engaging for readers.

 

- The limitation to uniform price auctions with identical items seems to be restrictive. The authors should discuss the extensibility of their approach to combinatorial auctions, discriminatory price auctions, or auctions with budget constraints, even if only theoretically.

Author Response

This paper has been undergone throughout revision. The major changes are listed below.

1. Many sentences and phrases have been revised so as to better present the ideas.
2. Additional references are added to highlight the differences between our work and the others.
3. Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The article studies uniform price auctions in which multiple bidders request several units of a homogeneous good and proposes a fast distributed algorithm that allocates units and computes the uniform price while preserving bidding privacy. Instead of sending quantities and per unit bids to a central auctioneer as in the standard centralized procedure, bidders repeatedly send only transformed quantities based on a piecewise linear function of the gap between their private bid and a public tentative price, and the auctioneer updates a common price interval using a bisection rule derived from a monotone aggregate function of these transformed quantities; illustrative examples on page 3 and the block diagrams on page 6 show how this contrasts with a centralized and a recurrent neural network based solution. Under mild assumptions on distinct integer bid prices and bounded quantities, the authors prove convergence, derive stopping criteria, and show that the number of iterations grows only logarithmically with the maximum possible bid and is independent of the number of bidders. Simulation results, including the trajectories in Figure 5 and the step comparisons in Figures 6 and 7, confirm that the new method reaches the correct allocation and price in a handful of iterations, while an RNN based scheme needs thousands of steps. A privacy comparison in Table 2 shows that the centralized method reveals all bids, the RNN approach hides only losing bids, and the proposed algorithm allows bidders and the auctioneer to learn only the allocated quantities and a single uniform price, not individual bid values or losing quantities. The paper also discusses communication delays for realistic wireless links, shows how to adapt the scheme when the uniform price is defined as the highest losing bid, notes limitations when multiple bidders submit identical per unit prices, and suggests that energy markets and other multiunit auction settings can benefit from this fast and privacy preserving distributed mechanism.

Intriguing. I find the paper clearly written and technically sound, with a careful comparison to both centralized and RNN based approaches. The assumptions are explicit, the convergence analysis is rigorous, and the simulations convincingly support the theoretical claims about iteration complexity and privacy. Overall, I liked the paper, and in my view its main strength is the elegant use of a simple bisection based scheme to obtain a fast, scalable, and genuinely privacy preserving distributed algorithm for multiunit uniform price auctions, with performance guarantees that are independent of the number of bidders.

However, I have some comments that you might wish to address.

Major comments

1.The analysis relies on discrete, integer bid prices, distinct per unit prices across bidders, and a specific uniform price rule. This narrows the range of auctions where the method can be applied without modification and raises questions about ties and more general bidding formats. So, make the limitations fully explicit and perhaps add a short subsection that outlines how price discretization could approximate continuous bids, describes a simple tie breaking scheme compatible with the algorithm, and sketches how the bisection procedure could be adapted to alternative uniform price definitions (for example, highest losing bid versus lowest winning bid). Even if full proofs are deferred to future work, a clear roadmap would reassure readers that the approach is not overly tailored to a narrow case.

2.The privacy discussion is mostly qualitative, focusing on what the auctioneer and bidders do not see, but it does not clearly specify an adversary model or analyze what can be inferred from the sequence of transformed messages and price updates, especially under collusion. My suggest is to add a concise threat model section that states who may be curious or malicious (auctioneer only, colluding bidders, external eavesdropper), what information they are assumed to have, and what the algorithm guarantees in this setting. Then provide simple information leakage arguments or illustrative examples (for small auctions) showing what can and cannot be inferred. I think this can be done without introducing heavy cryptography.

3.The experiments convincingly illustrate correctness and faster convergence compared to an RNN-based scheme, but they use relatively small instances and synthetic setups. It is not fully clear how the method behaves at larger scales or under more heterogeneous conditions. So, extend the numerical section with scalability experiments that vary the number of bidders, the distribution of bid prices and quantities, and the price grid resolution. Reporting iteration counts, communication rounds, and runtime for hundreds or thousands of bidders would support the claim that complexity is independent of the number of bidders in practice.

4.The algorithm is presented for a synchronous setting where all bidders respond promptly and messages are reliably delivered. The brief discussion of communication delays is based on simple calculations and does not address packet loss, stragglers, or partial participation, which are common in wireless or distributed environments. I suggest this: add perhaps a robustness subsection that discusses how the method could cope with delays and failures using standard mechanisms. For example, describe practical rules such as timeouts and using the last received transformed quantities, or updating the price based on a subset of bidders while bounding the possible error. A small set of simulations with randomly dropped or delayed messages would also help, without requiring a full redesign of the algorithm.

5.The work focuses on computation and privacy but says little about how strategic bidders might behave under the proposed protocol. Although the allocation and price coincide with a standard uniform price auction, we may wonder whether the distributed implementation introduces new opportunities for manipulation based on observing price intervals or iteration counts. My suggestion is to include (if possible) a short economic discussion that connects the protocol to known results on uniform price auctions, emphasizing that the final outcome is unchanged and that incentives are therefore inherited from the standard mechanism. Or, add a simple experiment where a small number of bidders deviate in a naive way (for example, misreport quantities) to illustrate that the protocol itself does not create additional obvious manipulation channels beyond those already present in the underlying auction format.

I really believe that if you could address these points through clarifications, modest theoretical extensions, and a few additional simulations would significantly strengthen the paper without pushing it outside its current scope.

I hope the above comments will be useful.

Author Response

Additional references  have been added. Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

I think you authors satisfactorily addressed my concerns. This is a very nice piece of work. Congrats.