Bicriteria Multidimensional Mechanism Design with Side Information

Summary

This paper presents a novel mechanism design based on the WVCG and proved that the mechanism could achieves both welfare and revenue guarantees parameterized by errors in the side information. Compared with previous work, it shows that the proposed the mechanism could degrade gracefully as the prediction errors increase. And finally, it also gives the application of the theory in the situation where the agent's type belongs to some low-dimensional subspace.

Strengths

1. The paper presents a novel approach to mechanism design that takes into account both welfare and revenue. This bicriteria approach is innovative and could potentially lead to more effective mechanisms in various contexts. 2. The authors present a method for determining the quality of the side information used by the mechanism, which they refer to as "predictions". This focus on the quality of predictions is a significant strength, as it could improve the performance of the mechanism and lead to more accurate outcomes. 3. The authors extend their techniques to handle more expressive forms of side information that allow for varying degrees of uncertainty. This allows for finer-grained beliefs and can express quantiles of certainty, precise distributional beliefs, and arbitrary mixtures of these. This extension to more expressive side information further enhances the versatility and practicality of their proposed mechanism.

Weaknesses

1. The paper primarily focuses on theoretical aspects and lacks experimental validation. 2. The allocation space in some previous seminal work is in the form of the probabilities assignments instead of allocating some items to some agents surely, such as the ‘p’ in Myerson's paper, so that the allocation space could be quite complex. In that case the results of the paper may be hard to use.

Questions

1. In Figure 1, the dashed blue line appears to be outside the polytope $\widetilde{\Theta}$, whereas according to the definition, it should be inside? 2. The definitions of (a,b)-consistency for welfare and revenue rely solely on OPT, while (c,d)-robustness incorporates both OPT and VCG. Could you provide some illustrations to clarify this?

Rating

5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.

Confidence

2: You are willing to defend your assessment, but it is quite likely that you did not understand the central parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

3 good

Presentation

3 good

Contribution

2 fair

Limitations

see weakness and questions.

Summary

The authors studied the problem of mechanism design that leverages side information on agent types to obtain both welfare maximization and revenue maximization. The paper generalized the previous results on the generalized VCG mechanism to obtain guarantees on both welfare and revenue depending on the quality of side information. Finally, the authors provided results in a setting where the principal knows the common constant-dimensional subspace of agent types.

Strengths

- The studied problem of mechanism design with side information is interesting. - The paper is well-written. The example applications in Section 2 are useful to help the reader understand the problem. - The theoretical results on welfare and revenue guarantees are strong contributions. The guarantees' dependence on the quality of the prediction is clear and is a useful observation that can be used for future research.

Weaknesses

- There is a lack of empirical experiments that can help support the theoretical claims. - There is a lack of discussion around the computation complexity of constructing the weakest-competitor hull.

Questions

N/A.

Rating

7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.

Confidence

4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.

Soundness

4 excellent

Presentation

3 good

Contribution

4 excellent

Limitations

The authors have addressed the limitations of their work.

Summary

The paper is concerned with mechanism design to maximize two objectives (welfare and revenue) when side information is available. The paper introduces a mechanism that is both incentive compatible and individually rational with lower bound guarantees for welfare and revenue even if the side information is incorrect.

Strengths

-The topic is interesting and impactful. I think the contribution is important given that side information is well motivated and welfare and revenue are perhaps the two most prominent objectives. -The fact there is a lower bound on welfare even if side information is incorrect is significant. -The model captures many scenarios as indicated in page 4.

Weaknesses

-what is the run-time of the algorithm in general? Can it be exponential? I see that theorem 3.6 gives a characterization for polytopes. -It would be interesting to see some empirical tests of the theory even for simulated examples. Minor Point: -line 280: typo for E[revenue] \ge b VCG not OPT

Questions

-Some as first point in the weaknesses (what is the run-time?).

Rating

7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.

Confidence

2: You are willing to defend your assessment, but it is quite likely that you did not understand the central parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

3 good

Presentation

3 good

Contribution

4 excellent

Limitations

Yes

Summary

This paper studies the problem of mechanism design with side information. The authors develop a meta mechanism based on the classic VCG mechanism. The authors show that by incorporating a proper randomization scheme, the meta mechanism can achieve strong welfare and revenue guarantees parameterized by the errors in the side information. The authors further apply the meta mechanism to a setting where each agent's type is determined by a constant number of parameters.

Strengths

1. This paper is generally well-written, clear, and easy to follow. The problem studied in this paper is interesting and relevant to the algorithmic game theory community. 2. It is nice to have a mechanism with performance guarantee decaying gracefully as the quality of the side information drops.

Weaknesses

1. The idea of using randomization seems to be a bit straightforward. The paper would be stronger if the authors can provide matching lower bounds and/or provide improved bounds (maybe via assuming more structural properties of the side information). 2. The notion of "weakest competitor" is a bit confusing.

Questions

1. The notion of "weakest competitor" is a bit confusing; in particular, what does "competitor" mean here? It is unclear to the reviewer why the payment is interpreted in this way. If the reviewer understands the payment rule correctly, in the special case of second price auctions, it is equivalent to charging the winner the maximum between the second highest bid and the winner's minimum possible value. It is true that agent i's value is "replaced" by another value in the payment calculation. However, the value is not from a competitor; instead, it is a possible value from bidder i (based on the side information) that leads to the minimum social welfare. 2. Is it possible to show some matching lower bounds?

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.

Soundness

3 good

Presentation

3 good

Contribution

3 good

Limitations

The authors adequately addressed the limitations.

Summary

The article proposes a new method for designing mechanisms that can achieve high welfare and revenue by using side information about the agents' types. Side information can be any kind of data or prediction that is available to the mechanism designer, such as historical data, expert advice, or machine learning models. The authors do not assume any prior distribution on the agents' types or the side information. They introduce a meta-mechanism that combines the VCG mechanism with a novel notion of a weakest competitor, which is an agent that has the least impact on welfare. They show that their meta-mechanism can achieve strong bicriteria (i.e. social welfare and revenue) guarantees that depend on the quality of the side information, and that it can handle settings where the agents' types are determined by a constant number of parameters that lie on known subspaces. They provide examples and simulations to illustrate their results.

Strengths

The article addresses an important and timely problem of multidimensional mechanism design with side information, and provides a novel and versatile framework for achieving bicriteria objectives. It is well-written and organized, and the main results are clearly stated and proven. The authors also provide intuitive explanations and examples to motivate their approach and illustrate their findings. It makes several original contributions, such as introducing the concept of a weakest competitor, designing a meta-mechanism that integrates side information with VCG, and obtaining the first welfare and revenue guarantees for subspace-type settings.

Weaknesses

As the authors mentioned in the last section, the computational complexity of the proposed mechanism has not been sufficiently discussed. The article could also benefit from more empirical evaluation of their mechanisms, such as testing them on real-world data sets or benchmark problems, and analyzing their robustness and scalability.

Questions

In the meta-mechanism, it says that, if agent i\notin \mathcal{I}, then i is excluded and receives zero utility (zero value and zero payment). Can you explain more about this step? What if the allocation \alpha^* with max social welfare would allocate some items to agent i? Then where these items will go? So it will not have the max social welfare right? What is the complexity for computing \hat{\Theta_i} from \tilde\{\Theta_i}?

Rating

7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.

Confidence

2: You are willing to defend your assessment, but it is quite likely that you did not understand the central parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

3 good

Presentation

3 good

Contribution

3 good

Limitations

N/A

Summary

This work focuses on studying social welfare and revenue maximization in multi-dimensional auctions under predictions. Each agent is associated with a private type $\theta_i$ drawn from a distribution $\Theta_i$​, and the mechanism designer utilizes predictions $\tilde{\theta}_i \in \Theta_i$​ as additional side information for the agent types. These predictions offer insights, such as constraining the sum of values for multi-item auctions to be within a constant limit. The authors' objective is to design incentive-compatible mechanisms that ensure both consistency, the worst-case multiplicative error when predictions are accurate, and robustness, the worst-case multiplicative error regardless of quality of the side informations. The primary focus of the authors centers around maximizing social welfare and obtaining maximum revenue through efficient mechanisms. On the surface, the problem may appear straightforward, as a simple convex combination of trusting or discarding predictions achieves satisfactory consistency and robustness bounds. Specifically, a probability $\beta$ is used to decide whether to discard or fully trust the predictions, ensuring 1,$\beta$ consistency and robustness for social welfare and $(1−\beta),\beta$ consistency and robustness for revenue. However, the paper highlights that the mechanism's revenue significantly decreases when predictions are even slightly inaccurate. Therefore, the main objective of the paper is to generalize the robustness/consistency approach and analyze the rate of revenue degradation as predictions become invalid. The principal contribution of the paper lies in its novel mechanism that goes beyond the binary decision of discarding or trusting predictions. Instead, it introduces a third option of randomly expanding the predictions. The mechanism starts from the predicted type space and considers all types $\theta \in \Theta_i$ within the $l_{\infty}$ ball around $\theta$ with a radius r, where r is randomly chosen from a discretization of the ambient type space's diameter. The paper's main result is achieved through an appropriate convex combination of these three options. Additionally, the authors explore a special case where agent types lie on the line or subspaces and derive corresponding guarantees.

Strengths

The paper is well-written and effectively presents its contributions. The introduction of the side information generalize the types of predictions in the standard consistency/robustness framework.

Weaknesses

While the expansion of predictions is interesting, the results might not be particularly surprising. Note that the work focusing on achieveing optimal revenue conditioning on maximizing welfare. This imposes significant restrictions on the revenue benchmark used for deriving theoretical guarantees.

Questions

Is there any guarantees on revenue if the benchmark is the optimal revenue instead of the optimal revenue with respect to welfare-maximizing mechanisms?

Rating

5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

3 good

Presentation

4 excellent

Contribution

3 good

Limitations

Not applicable

Thank you!

We would like to thank the authors for addressing our questions. We have no other questions at this point.