Optimal Auction Design in the Joint Advertising

Q1. We agree this is an excellent suggestion. In fact, we noticed other reviewers raised similar questions. Here we clarify our key findings:

1. Single-slot scenario:
   BundleNet achieves allocations closer to the optimal solution compared to baseline methods (as demonstrated in Table 1). Regarding the single-slot scenario, we have added a new visualization to comprehensively demonstrate BundleNet's capabilities. Based on the original experimental setup, we took two well-trained models (with average individual regret values below 0.001, where the current JRegNet even shows higher revenue than BundleNet) and made their allocation results deterministic rather than randomized. The results show that BundleNet's allocations align more closely with the optimal solution.Here is the figure:https://anonymous.4open.science/r/BundleNet-6E3E/visualpicture.jpg

2. Multi-slot scenario:
   BundleNet consistently yields the highest revenue among all compared approaches (shown in Table 2).

For the camera-ready version, we will follow conventional practice by:

• Bolding the highest-revenue method in all tables
• Adding explicit textual markers (e.g., "★best") where appropriate
• Including a footnote to explain these annotations

This will eliminate potential ambiguity while maintaining rigorous comparison standards.

Q2. I think this is a good question that is innovative enough for the article. Let me provide a more systematic response:

1. We propose two novel solutions for optimal mechanism design in joint advertising, and empirically verify the consistency between neural network results and theoretical outcomes.

2. For automated mechanism design, we present a new approach that differs from traditional neural network modifications. Our core methodology is as follows: By reformulating the problem structure and optimization process, we circumvent neural networks' limitations in handling sparse graph data, thereby improving performance.

Q1. To comprehensively validate our approach, we have conducted additional experiments across diverse settings:

Q2. We agree this is an excellent suggestion. In fact, we noticed other reviewers raised similar questions. Here we clarify our key findings:

1. Single-slot scenario:
   BundleNet achieves allocations closer to the optimal solution compared to baseline methods (as demonstrated in Fig. 4).

2. Multi-slot scenario:
   BundleNet consistently yields the highest revenue among all compared approaches (shown in Table 2).

For the camera-ready version, we will follow conventional practice by:

• Bolding the highest-revenue method in all tables
• Adding explicit textual markers (e.g., "★best") where appropriate
• Including a footnote to explain these annotations

This will eliminate potential ambiguity while maintaining rigorous comparison standards.

Q3. We sincerely appreciate the reviewer's careful reading regarding the writing details. We agree that the social welfare (SW) formulation could be more precise. In the revised version, we will correct it to:

$$\text{SW} =E_{\mathbf{v} \sim F}\left[v_0\left(\mathbf{1} -\sum_{e \in E}x^e(\mathbf{v})\right) \boldsymbol{\lambda}^T + \sum_{e=(r,s) \in E}x^e(\mathbf{v}) (v_r + v_s)\boldsymbol{\lambda}^T \right]$$

Q1. This is a very good question. I think this method may limit the space of discoverable mechanisms, but the difference is not significant.

1. Our core idea is to improve performance by limiting the space of discoverable mechanisms, avoiding the neural network's processing of sparse graph data. In the RegretFormer paper, their improvement likely relies more on expanding the discoverable mechanism space to find a better mechanism. However, for the more challenging problem of joint auctions, our experiments show that as the number of bundles increases, the performance of JRegNet gradually worsens. The main reason for this could be the sparsity of the graph. As the number of bundles increases, the sparsity of the graph significantly strengthens. Encoding this into the neural network inevitably degrades performance.
2. We argue that the inequality in Lemma 5.1 becomes an equality when the following condition holds: For every agent $i \in R \cup S$, the utility functions $u_i^e$ across all edges $e$ can be simultaneously maximized by the same strategy $v'_i$.
3. Our method does not significantly limit the space of discoverable mechanisms. In our training process, we observed that increasing the number of misreport points to 10 does not lead to a significant difference in the sum of individual regrets versus the sum of bundle regrets. Additionally, the result of BundleNet converges to approximately the theoretical optimal.
4. The inequality in Lemma 5.1 is difficult to extend into an equality.

Q2. Regarding the single-slot scenario, we have added a new visualization to comprehensively demonstrate BundleNet's capabilities. Based on the original experimental setup, we took two well-trained models (with individual average regret values below 0.001, where the current JRegNet even shows higher revenue than BundleNet) and made their allocation results deterministic rather than randomized. The results show that BundleNet's allocations align more closely with the optimal solution. Here is the figure:https://anonymous.4open.science/r/BundleNet-6E3E/visualpicture.jpg

Q3. We conducted experiments using real-world data, including 10 bundles and 5 slots. The dataset consists of 102,679 samples, with 80% used for training and 20% for validation. JRegNet and BundleNet are both used in the fine sorting phase. During preprocessing, we normalized the bidding data to facilitate neural network learning. After obtaining the final results, we scaled the values back to the original bid price range. We evaluated the revenue of the three methods, and for methods with IC Violation, $\widehat{rgt}$ during testing on normalized data was less than 0.0005. The results are presented below:

Note: In the real world data bidding data, there are actually fewer than 50% of the cases that involve more than 5 bundles. Instead of treating them separately, we use a single unified neural network to process all bundle sizes, which contributes to JRegNet’s superior performance over VCG.

Q4. BundleNet's approach is very transferable:

1. While the idea of joint auctions originated in online advertising, it can be applied to more general scenarios, such as government procurement projects where a single project requires collaboration between companies from different domains. Extending this to allow more participants to form coalitions for joint bidding presents an interesting theoretical problem. Our modeling approach could be similarly adapted for such cases.
2. As demonstrated in an AAAI 2025 paper (building upon the new scenario introduced in KDD 2024's "Ad vs Organic: Revisiting Incentive Compatible Mechanism Design in E-commerce Platforms"), JRegNet has been successfully applied to merged ranking mechanism design, achieving excellent results.
3. We particularly emphasize our core research insight: When adapting RegretNet to broader mechanism design problems where neural networks struggle with certain features, our method of constraining the discoverable mechanism space can effectively improve performance.

Q5. I consider this an open problem in automated mechanism design. Existing methods for multi-agent, multi-item auctions face various limitations:

• RegretNet can only guarantee approximate IC
• GemNet may suffer from over-allocation issues
• AMenuNet might not sufficiently approximate theoretical solutions

From my perspective, the key may lie in characterizing feasible auctions through utility function properties (analogous to Rochet’s theorem for single-item cases). Similar to RochetNet’s approach, we could:

• Formalize such properties for multi-bidder settings
• Integrate them into neural architecture design
• Develop a new automated mechanism design framework

We are currently investigating concrete implementations, though technical details remain under study.

We would like to thank you for the constructive comments and suggestions! Below are our responses to your questions.

Q1. We conducted experiments using real-world data from a leading Internet company. In practice, advertisements will go through the stages of recall, rough sorting, and fine sorting. A maximum of 10 advertisements and 5 advertising spaces will be reserved after rough sorting and will go into the fine sorting stage. JregNet and BundleNet are both used in the fine sorting phase. The dataset consists of 102,679 samples, with 80% used for training and 20% for validation. We compared the revenue differences among RVCG, JRegNet, and BundleNet. During preprocessing, we normalized the bidding data to facilitate neural network learning. After obtaining the final results, we scaled the values back to the original bid price range. We evaluated the revenue of these three methods, and for methods with IC Violation, $\widehat{rgt}$ during testing on normalized data was less than 0.0005. The results are presented below：

Note: In the real world data bidding data, in the fine sorting stage, there are actually less than 50% of the cases that involve more than 5 bundles. Instead of treating them separately, we use a single unified neural network to process all bundle sizes, which contributes to JRegNet’s superior performance over VCG.

Q2. In our experiments, the hyperparameter settings are consistent with the open-source implementations of JRegNet and RegretNet.

For both JRegNet and BundleNet, we use a neural network with 6 hidden layers, each containing 100 neurons, for training and testing. We initialize our Lagrange parameters as $\forall e \in E, \lambda_e = 2.0$. In each gradient ascent step, the update parameter for the Lagrange multiplier and the quadratic penalty term is set to $\rho = 2.0$. The learning rate for searching the optimal misreport value is $\gamma = 0.1$, and the neural network learning rate is $\eta = 0.001$. We use the Adam optimizer for all updates. The primary modification we made was adjusting the batch size for training both networks to 1024. We will release our code upon the acceptance of this paper to facilitate reproduction and further research.

Q3. Based on the aforementioned hyperparameters, we present a sensitivity analysis of the batch size and the initial value of the Lagrange multiplier in the $U_2$ scenario：

• Sensitivity Analysis of Batch Size：

  	128	256	512	1024	2048	4096
  BundleNet	0.4993	0.4971	0.5174	0.5262	0.5286	0.5295
  JRegNet	0.5535	0.5594	0.5641	0.5638	0.5685	0.5658

• Sensitivity Analysis of Initial Lagrange Multiplier

  	1.0	2.0	3.0	4.0	5.0	6.0
  BundleNet	0.1480	0.5330	0.5373	0.5267	0.5229	0.5240
  JRegNet	0.5669	0.5659	0.5653	0.3837	0.1386	0.1165

Q4. Thank you for your feedback. We will provide a more detailed description of the joint advertisement in the label for Figure 1. The image on the left illustrates traditional advertising, where the retailer submits a bid for the ad, while the supplier (brand owner) does not participate in the auction. In this case, the platform only receives the bid from the retailer. In contrast, as shown in the right image, for a joint advertisement, both the retailer and the supplier participate in the auction. They submit bids simultaneously, and the platform receives bids from both parties for the same ad.

Thank you for recognizing our work. Here we re-emphasize the key contributions and findings of our work:

1. Novel solution: We propose a novel solution for automated mechanism design in joint advertising markets, and empirically demonstrate the alignment between neural network outputs and theoretical predictions.
2. Better performance: For automated mechanism design, we introduce a fundamentally different approach from conventional neural network adaptations. Our core methodology works by reformulating the problem structure and redesigning the optimization process, thereby overcoming neural networks' inherent limitations in processing sparse graph data while achieving performance improvements.
3. Interesting Findings : We theoretically and empirically show that directly applying RegretNet-like methods to new problem domains may fail to discover optimal mechanisms, as significant gaps can persist between learned mechanisms and theoretically optimal solutions. Our analysis demonstrates that successful application requires problem-specific modifications with rigorous theoretical validation.