Enhancing Affine Maximizer Auctions with Correlation-Aware Payment

We thank the reviewer for the insightful feedback and constructive suggestions. We address the specific points below.

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Weakness 1: About the violation of IR. The proposed mechanism is only approximately IR by design. How do violations of IR affect the practical use of the proposed method? It seems unreasonable for a bidder to participate in an auction where they might end up with negative utility: it means a bidder could end up paying more than their actual bid, which I find unjustified.

Response: We acknowledge that the violation of IR, or IR-regret, may influence bidder trust in the system. However, we would like to clarify that the impact is relatively minimal within the framework we propose.

Firstly, as shown in our experimental results, our two-phase optimization ensures that IR-regret converges to the target regret value in most cases. This indicates that IR-regret is relatively controllable, and we can adjust its value according to different needs.

Secondly, with low IR-regret, we offer a straightforward approach to transform our trained CA-AMA into a strictly IR mechanism: we provide each bidder with the option to opt out (without payment or allocation) after seeing their allocation and payment. This ensures that only bidders with positive utility remain, while those with negative utility will choose to quit. As a result, the modified mechanism satisfies strict IR. This adjustment does not affect the optimality of truthful reporting, maintaining both DSIC and IR properties.

We will include additional experimental results to demonstrate this modification in the final version of the paper.

• 2-bidder 5-item, Symmetric Negative Correlation

• 2-bidder 5-item, Asymmetric Negative Correlation

Note that "CA-AMA (Strict IR)" refers to the mechanism after post-modification, which satisfies both DSIC and strict IR. As shown in the tables, CA-AMA continues to outperform other baseline methods in most scenarios.

Finally, as defined in Definition 2.2, our IR is stronger than Bayesian-IR and closer to ex-post IR. This means our CA-AMA may not violate Bayesian-IR. In practical scenarios, bidders may only be aware of their own valuation and compute the expected utility based on the uncertainty of others. As a result, the expected utility can still be positive, even when the realized utility is negative. Because the expected utility is positive, bidders would still participate in the auction.

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Weakness 2: About the experiments on real data.

Response: We believe that our method is also applicable to real data distributions. However, real data is often held by industrial companies and is not readily available for research purposes. Therefore, in line with previous studies, we conducted experiments using various synthetic distributions.

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Weakness 3: About the standard errors for Table 1, Figure 2, and Figure 5.

Response: Overall, the standard error is relatively small (generally less than 5% of the revenue performance). We will include this information in the final version of the paper to provide clarity.

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Weakness 4: About the comparison with RegretNet.

Response: Our primary innovation lies in improving the classic AMA within DSIC mechanisms. As such, we did not include RegretNet in our experiments. Moreover, the paper by [1] on AMenuNet already demonstrated that the classic randomized AMA achieves competitive results compared to RegretNet. Since AMenuNet is used as the baseline in our paper, this sufficiently demonstrates that our CA-AMA is also competitive with RegretNet.

We will add further discussion on the comparison between CA-AMA, which introduces IR-regret, and RegretNet, which introduces IC-regret. In theory, RegretNet has the potential to express the universal optimal auction, while CA-AMA is still limited in its expressiveness. Nonetheless, CA-AMA has several advantages:

1. IR-regret computation: Calculating IR-regret is significantly easier than computing IC-regret. IR-regret only requires the resulting payment and valuation of an auction. In contrast, IC-regret needs additional sampled bids to compute an approximate best bid aside from truthful reporting, which is time-consuming during training due to the need for more auction results.

2. Post-processing flexibility: IR-regret is easier to adjust. As noted in our response to Weakness 1, if a low IR-regret or zero regret is desired, we provide simple methods to achieve this goal. RegretNet, on the other hand, faces difficulties in adjusting the degree of IC violation without further training.

3. Interpretability: Since CA-AMA is based on AMA, it is more interpretable. The allocation menu, bidder weights, and allocation boosts are transparent.

Overall, we believe CA-AMA strikes a balance between empirical performance, theoretical properties, and implementation efficiency.

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Weakness 5: About the theoretical results.

Response: We would like to add more discussion regarding the difficulty of proving our conjecture that "CA-AMA = AMA in bidder-independent auctions" (the first part of Theorem 3.3) for multi-item auctions. We are attempting a proof by contradiction, constructing an AMA that achieves at least the same revenue as CA-AMA.

Firstly, the optimal revenue in multi-item settings is generally unknown and remains a fundamental open problem in game theory. Even for AMA, there are no existing results that characterize the optimal AMA in general valuation distributions. This makes it challenging to utilize the properties of the optimal AMA.

Secondly, in multi-item auctions, the allocation of one item can depend on the allocation of others. For example, an item might be reserved if bidders' valuations for other items are low. This introduces significant differences between the single-item and multi-item cases, resulting in more discontinuities in allocations.

Therefore, we primarily validate the performance of CA-AMA in multi-item settings empirically. We believe that further theoretical results concerning AMA and CA-AMA are valuable areas for future work.

We thank the reviewer for the insightful feedback and constructive suggestions. We address the specific points below.

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Weakness: Minor improvements over Randomized AMA except for strongly correlated (negative or positive) values.

Response: We appreciate the opportunity to clarify the contributions of CA-AMA, particularly concerning its performance in scenarios with varying degrees of correlation.

First, it is crucial to understand that in distributions with lower correlation, classic AMA and CA-AMA may indeed exhibit similar expressiveness. Our theoretical results, specifically Theorem 3.3, demonstrate that "optimal CA-AMA = optimal AMA" in single-item, bidder-independent cases. We have also observed that this property holds even in cases with correlation, provided the support of each conditional distribution remains the same. Specifically, for a bidder $i$, if the support satisfies $\mathrm{supp}(\mathcal{F}\_i(V_{-i})) \equiv S$ for all realizations of $V_{-i}$, the result continues to apply. We further conjecture that this result extends to the multi-bidder setting. Therefore, the minor improvement in such scenarios is not a weakness but a theoretically guaranteed outcome, which itself is a valuable part of our contribution.

Second, we respectfully argue that strong correlations are, in fact, plausible in real-world scenarios. Take online advertising, for example, where advertisers often have completely opposing preferences. In such contexts, CA-AMA offers a more flexible and powerful tool to leverage this correlation information compared to AMA. As is shown in the Figure 2, and Figure 4, 5 in the Appendix, CA-AMA significant improvement in performance.

Finally, from a practical implementation standpoint, the optimization process for CA-AMA is only marginally more complex than AMA's. The increase in training time is minimal (as shown in Table 2), and performance remains stable across all settings. More importantly, CA-AMA consistently outperforms AMA in nearly all scenarios. When compared to methods like RegretNet, which requires computing IC-regret and offers only approximate IC, and GemNet, which uses Integer Programming to ensure allocation feasibility, we believe that CA-AMA strikes a better balance among empirical performance, theoretical properties, and and implementation efficiency.

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Question: Which of the competing methods also have a non-zero regret for IR (and if so, how much), and which guarantees IR? Probably showing the loss (-revenue + regret) would help, I would expect CA-AMA to be optimal there, but it would be helpful to have an understanding such as "for every X unit of revenue gained, some Y united of regret must be paid for".

Response: First, we would like to clarify that all competing methods, including Item-Myerson, Item-CAN, VCG, and Randomized AMA, guarantee strict IR, meaning they do not incur IR regret. Our innovation introduces a conceptual shift, where we allow for the loss of IR during training, integrating it into the optimization goal. With different preset IR-regret targets, our implementation ensures that the achieved IR-regret is close to the target, while still maximizing revenue performance.

Additionally, to the best of our knowledge, no previous work has characterized the trade-off between IR-regret and revenue in such detail. We also offer a straightforward approach to modify our trained CA-AMA into a strictly IR mechanism: we add an additional option for each bidder, allowing them to opt out (without payment or allocation) after viewing their allocation and payment. This ensures that only bidders with positive utility will stay, while those with negative utility will choose to quit. Consequently, the modified mechanism satisfies strict IR. Since this modification does not affect the optimality of truthful reporting, the mechanism remains DSIC and IR.

We will add further experimental results to the final version of the paper to illustrate this modification:

• 2-bidder 5-item, Symmetric Negative Correlation

• 2-bidder 5-item, Asymmetric Negative Correlation

Note that the "CA-AMA (Strict IR)" refers to the mechanism after post-modification, which satisfies both DSIC and strict IR. As shown in the table, CA-AMA outperforms the baseline methods in most scenarios.

Dear Reviewer Csj4,

Thank you once again for your time and effort in providing valuable comments on our paper. As the author-reviewer discussion phase approaches its conclusion, we would like to kindly confirm whether we have sufficiently addressed all (or at least some) of your concerns. In particular, we have provided additional explanations regarding the empirical improvement of our method and the balance between IR-regret and revenue performance.

Should any questions remain or if further clarification is needed, please do not hesitate to let us know. If you are satisfied with our responses, we would greatly appreciate your consideration in adjusting the evaluation scores accordingly.

We sincerely look forward to your feedback.

We thank the reviewer for the insightful feedback and constructive suggestions. We address the specific points below.

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Weakness 1: The problem looks a bit incremental, it simply adds a payment term to AMA.

Response: We understand the concern that our proposed mechanism might appear to be an incremental addition to the classic AMA by simply incorporating a payment term. However, we respectfully clarify that this addition is a crucial enhancement that addresses a significant limitation of AMA, leading to substantial improvements in both theoretical expressiveness and empirical performance.

Our theoretical analysis rigorously demonstrates the importance of this modification. We show that in specific correlated distributions, deterministic AMA can perform arbitrarily poorly compared to our proposed CA-AMA. Furthermore, there's a strict gap in performance between any randomized AMA and CA-AMA. These theoretical findings highlight a fundamental flaw in the classic AMA that our approach rectifies.

Empirical validation further supports our claims. As illustrated in both Figure 2 and Figure 4 of our experiments, CA-AMA consistently shows significant improvement over the classic AMA. This practical superiority, combined with our theoretical guarantees, underscores the non-incremental nature and impactful contribution of our work.

Moreover, the simplicity of our modification can be a strength, as it's straightforward to implement and integrates seamlessly with existing AMA-based methods. This means that despite its apparent simplicity, our work offers an efficient and valuable extension to the AMA framework, providing novel insights into optimal multi-bidder, multi-item auction theory.

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Weakness 2: It makes the optimization problem much more complicated by introducing the IR constraint.

Response: We acknowledge that introducing the IR constraint for regret minimization adds complexity to the optimization problem. However, we firmly believe this extension is justified and offers significant advantages.

IR-regret is computationally efficient compared to other methods involving computing a regret term. For instance, computing the IC-regret, which is used in RegretNet and its successors, requires finding an approximate best-bid strategy and computing a large number of additional auctions. The feasibility regret in GemNet involves complex integer programming post-processing. Compared to these, although CA-AMA is still not universally expressive, its training time is comparable to classic AMA, which is much more efficient (as shown in Table 2). The hyperparameter for IR-regret minimization is also less sensitive under our updating rule (see equation below Line 213). Furthermore, our theoretical analysis confirms that CA-AMA is strictly more expressive than AMA, effectively addressing its key limitations.

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Weakness 3: The theoretical analyses mainly cover the single-bidder case.

Response: We conjecture that the result “CA-AMA = AMA in bidder-independent auctions” (first part of Theorem 3.3) extends to multi-item auctions. The empirical result can provide some evidence: in bidder-independent cases, $\alpha=0$ in Table 1, the performance of CA-AMA is similar to AMA. We plan to further explore this extension, but this is out of the scope of this paper. We would like to discuss the challenge arising in the multi-item case.

Firstly, the optimal revenue in multi-item settings is generally unknown and remains a fundamental open problem in game theory. Even for AMA, there are no existing results that characterize the optimal AMA in general valuation distributions. This makes it challenging to utilize the properties of the optimal AMA.

Secondly, in multi-item auctions, the allocation of one item can depend on the allocation of others. For example, an item might be reserved if bidders' valuations for other items are low. This introduces significant differences between the single-item and multi-item cases, resulting in more discontinuities in allocations.

We believe further exploration on this topic would help to tackle the fundamental challenge of optimal auction design for a general multi-item auction.

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Question 1: It seems that the correlation-aware payment is an arbitrary function of other bidders' valuations. What are the major benefits of such a big freedom? Can you limit correlation-aware payment to a simpler form?

Response: The correlation-aware payment is modeled as an arbitrary function to leverage the flexibility of neural networks for practical implementation. As derived in the paper (equation below Line 245), the target form of the correlation-aware payment is:

$p_i^{\text{OPT-core}}(V_{-i}) = \inf_{v_i \in \text{supp}(\mathcal F_i(V_{-i}))} u_i^{\text{AMA}}((v_i, V_{-i}); \mathcal A, w, \lambda).$

Then we further compute the utility under the AMA mechanism, $u_i^{\text{AMA}}$. Specifically, for a fixed menu $\mathcal A$ whose size is $K$, the utility is $u_i^{\text{AMA}} = \max\_{k \in [K]} \left(\sum\_{j=1}^N w_j v_j \cdot (A_k)_j + \lambda\_k\right) - \max\_{k \in [K]} \left(\sum\_{j=1, j\neq i}^N w_j v_j \cdot (A_k)_j + \lambda\_k\right).$ As shown in the equation, the payment is a minus of two maximums of multiple linear functions. Therefore, in theory, a structure with this property can characterize the target correlation-aware function.

Inspired by this, we conduct additional experiments that use a simpler architecture, MaxMinusMaxNet, defined as:

class MaxMinusMaxNet(nn.Module):
    def __init__(self, input_dim: int, k: int):
        super().__init__()
        self.linear_branch1 = nn.Linear(input_dim, k)
        self.linear_branch2 = nn.Linear(input_dim, k)

    def forward(self, x: torch.Tensor):
        out1 = self.linear_branch1(x)
        max1, _ = torch.max(out1, dim=1)
        out2 = self.linear_branch2(x)
        max2, _ = torch.max(out2, dim=1)
        return max1 - max2

The results demonstrate that this architecture yields results comparable to the MLP, indicating robustness to simpler forms. As the results are similar, we will maintain the experimental results in the main paper and add a discussion about this issue in the final version.