SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\boldsymbol {\lambda} = \operatorname {M L P} \left(\sum_ {i = 1} ^ {n + 1} \sum_ {j = 1} ^ {m} J _ {i j} ^ {\boldsymbol {\lambda}}\right) \in \mathbb {R} ^ {s}.
$$
After we get the output AMA parameters  $\mathcal{A}$ ,  $w$ ,  $\lambda$  from AMenuNet, we can compute the allocation result according to Equation (Allocation) and the payment result according to Equation (Payment). The computation of  $\mathcal{A}$ ,  $w$ ,  $\lambda$  only involves the public representations of bidders and items, without access to the submitted bids. Therefore, the mechanism is DSIC and IR (see Appendix A for proof):
Theorem 4.1. The mechanism induced by AMenuNet satisfies both DSIC and IR.
Moreover, As AMenuNet is built using equivariant operators such as transformers and  $1 \times 1$  convolutions, any permutation of the inputs to AMenuNet, including the submitted bids  $B$ , bidder representations  $X$ , and item representations  $Y$ , results in the same permutation of the allocation and payment outcomes. This property is known as permutation equivariance [Rahme et al., 2021a, Duan et al., 2023]:
Definition 4.2 (Permutation Equivalence). We say  $(g,p)$  is permutation equivariant, if for any  $B,X,Y$  and for any two permutation matrices  $\Pi_n\in \{0,1\}^{n\times n}$  and  $\Pi_m\in \{0,1\}^{m\times m}$ , we have  $g(\Pi_{n}B\Pi_{m},\Pi_{n}X,\Pi_{m}^{T}Y) = \Pi_{n}g(B,X,Y)\Pi_{m}$  and  $p(\Pi_nB\Pi_m,\Pi_nX,\Pi_m^T Y) = \Pi_np(B,X,Y)$ .
Permutation equivariant architectures are widely used in automated auction design [Rahme et al., 2021a, Duan et al., 2022, Ivanov et al., 2022]. Qin et al. [2022] have shown that this property can lead to better generalization ability of the mechanism model.
4.3 Optimization and Training
Since the induced AMA is DSIC and IR, we only need to maximize the expected revenue  $\mathrm{Rev}(g^{\theta},p^{\theta})$ . To achieve this, following the standard machine learning paradigm [Shalev-Shwartz and Ben-David, 2014], we minimize the negative empirical revenue by set the loss function as
$$
\ell (\theta , S) := \frac {1}{| S |} \sum_ {k = 1} ^ {| S |} \sum_ {i = 1} ^ {n} - p _ {i} ^ {\theta} (V ^ {(k)}, X ^ {(k)}, Y ^ {(k)}),
$$
where  $S$  is the training data set, and  $\theta$  contains all the neural network weights in the encode layer and the boost layer. However, the computation of  $\ell(\theta, S)$  involves finding the affine welfare-maximizing allocation scheme  $A^{*}$  (and  $A_{-k}^{*}$ ), which is non-differentiable. To address this challenge, we use the softmax function as an approximation. During training, we compute an approximate  $A^{*}$  by
$$
\widetilde {A ^ {*}} := \frac {1}{Z} \sum_ {A \in \mathcal {A}} \exp \left(\tau_ {A} \cdot \left(\sum_ {i = 1} ^ {n} w _ {i} b _ {i} (A) + \boldsymbol {\lambda} (A)\right)\right) \cdot A,
$$
where  $Z \coloneqq \sum_{A' \in \mathcal{A}} \exp \left( \tau_A \cdot \left( \sum_{i=1}^n w_i b_i(A') + \lambda(A') \right) \right)$  is the normalizer, and  $\tau_A$  is the softmax temperature. We can control the approximation level of  $\widetilde{A^*}$  by tuning  $\tau_A$ : when  $\tau_A \to \infty$ ,  $\widetilde{A^*}$  recovers the true  $A^*$ , and when  $\tau_A \to 0$ ,  $\widetilde{A^*}$  tends to a uniform combination of all the allocations in  $\mathcal{A}$ . The approximation  $\widetilde{A_{-k}^*}$  of  $A_{-k}^*$  is similar. By approximating  $A^*$  and  $A_{-k}^*$  through the differentiable  $\widetilde{A^*}$  and  $\widetilde{A_{-k}^*}$ , we make it feasible to optimize  $\ell(\theta, S)$  through gradient descent. Notice that in testing, we still follow the standard computation in Equation (Allocation) and Equation (Payment).
In this section, we present empirical experiments that evaluate the effectiveness of AMenuNet<sup>1</sup>. All experiments are run on a Linux machine with NVIDIA Graphics Processing Unit (GPU) cores. Each result is obtained by averaging across 5 different runs. In all experiments, the standard deviation of AMenuNet across different runs is less than 0.01.
Baseline Methods. We compare AMenuNet against the following baselines:
1. VCG [Vickrey, 1961], which is the most classical special case of AMA;
2. Item-Myerson, a strong baseline used in Dütting et al. [2019], which independently applies Myerson auction with respect to each item;
3. Lottery AMA [Curry et al., 2023], an AMA-based approach that directly sets the allocation menu, bidder weights, and boost variables as all the learnable weights.
4. RegretNet [Dütting et al., 2019], the pioneer work of applying deep learning in auction design, which adopts fully-connected neural networks to compute auction mechanisms.
5. CITransNet [Duan et al., 2022], a RegretNet-like transformer-based approach that supports contextual auctions.
Note that both RegretNet and CITransNet can only achieve approximate DSIC by adding a regret term in the loss function. We train both models with small regret, less than 0.005.
Hyperparameters. We train the models for a maximum of 8000 iterations, with 32768 generated samples per iteration. The batch size is 2048, and we evaluate all models on 100000 samples. We set the softmax temperature as 500 and the learning rate as  $3 \times 10^{-4}$ . We tune the menu size in  $\{32,64,128,256,512,1024\}^2$ . For the boost layer, we use a two-layer fully connected neural network with ReLU activation. Given the induced AMA parameters, our implementation of the remaining AMA mechanism is built upon the implementation of Curry et al. [2023]. Further implementation details can be found in Appendix B.
<sup>1</sup>Our implementation is available at https://github.com/Haoran0301/AMenuNet
2We explore the impact of different menu sizes in Appendix C.
Table 1: The experiment results of average revenue. For each case, we use the notation  $n \times m$  to represent the number of bidders  $n$  and the number of items  $m$ . The regret of both CITransNet and RegretNet is less than 0.005. The best revenue is highlighted in bold, and the best revenue among all DSIC methods is underlined.
(a) Contextual auctions (Setting (A) and (B)). We omit Lottery AMA and RegretNet since they are designed for classic auctions. "Randomized Allocations" shows the ratio of randomized allocations output by AMenuNet.
Method	DSIC?	2×2 (A)	2×5 (A)	2×10 (A)	3×2 (A)	3×5 (A)	3×10 (A)	4×2 (B)	5×2 (B)	6×2 (B)	7×2 (B)
CITransNet	No	0.4461	1.1770	2.4218	0.5576	1.4666	2.9180	0.7227	0.7806	0.8396	0.8997
VCG	Yes	0.2882	0.7221	1.4423	0.4582	1.1457	2.2967	0.5751	0.6638	0.7332	0.7904
Item-Myerson	Yes	0.4265	1.0700	2.1398	0.5590	1.3964	2.7946	0.6584	0.7367	0.8004	0.8535
AMenuNet	Yes	0.4398	1.1539	2.3935	0.5601	1.4458	2.9006	0.7009	0.7548	0.8127	0.8560
Randomized Allocations	1.42%	7.37%	18.01%	3.52%	22.25%	54.38%	12.41%	14.81%	10.76%	17.02%
(b) Classic auctions (Setting (C)-(F)).
Method	DSIC?	2×5(C)	3×10(C)	5×5(C)	3×1(D)	1×2(E)	1×2(F)
Optimal	-	-	-	-	2.7490	9.7810	0.1706
CITransNet	No	2.3788	5.9191	3.4759	2.7541	9.7551	0.1691
RegretNet	No	2.3390	5.5410	3.4260	2.7264	9.7340	0.1732
VCG	Yes	1.6645	5.0002	3.3319	2.4954	0	0
Item-Myerson	Yes	2.0755	5.3141	3.3566	2.7490	8.8359	0.1482
Lottery AMA	Yes	2.2354	5.3450	2.8527	2.7238	9.3139	0.1544
AMenuNet	Yes	2.2768	5.5896	3.3916	2.7382	9.6219	0.1701
Auction Settings. AMenuNet can deal with both contextual auctions and classic Bayesian auctions. We construct the following multi-item contextual auctions:
(A) We generate each bidder representations  $\boldsymbol{x}_i \in \mathbb{R}^{10}$  and item representations  $\boldsymbol{y}_j \in \mathbb{R}^{10}$  independently from a uniform distribution in  $[-1, 1]^{10}$  (i.e.,  $U[-1, 1]^{10}$ ). The valuation  $v_{ij}$  is sampled from  $U[0, \text{Sigmoid}(\boldsymbol{x}_i^T \boldsymbol{y}_j)]$ . This contextual setting is also used in Duan et al. [2022]. We choose the number of bidders  $n \in \{2, 3\}$  and the number of items  $m \in \{2, 5, 10\}$ .
(B) We set 2 items, and the representations are generated from the same way as in Setting (A). For valuations, we first generate an auxiliary variable  $v_{i}^{\prime}$  from  $U[0,1]$  for each bidder  $i$ , and then we set  $v_{i1} = v_{i}^{\prime} \cdot \mathrm{Sigmoid}(\boldsymbol{x}_{i}^{T}\boldsymbol{y}_{1})$  and  $v_{i2} = (1 - v_{i}^{\prime})\cdot \mathrm{Sigmoid}(\boldsymbol{x}_{i}^{T}\boldsymbol{y}_{2})$ . We do this to make the valuations of the 2 items highly correlated. We choose the number of bidders  $n\in \{4,5,6,7\}$
For classic auctions, we can assign each bidder and item a unique ID and embed them into a multidimensional continuous representation. We construct the following classic auction settings:
(C) For all bidders and items,  $v_{ij} \sim U[0,1]$ . Such setting is widely evaluated in RegretNet-like approaches [Dütting et al., 2019]. We select  $n \times m$  (the number of bidders and items) in  $\{2 \times 5, 3 \times 10, 5 \times 5\}$ .
(D) 3 bidders and 1 item, with  $v_{i1} \sim \mathrm{Exp}(3)$  (i.e. the density function is  $f(x) = 1 / 3e^{-1 / 3x}$  for all  $i \in \{1,2,3\}$ ). The optimal solution is given by Myerson auction.
(E) 1 bidder and 2 items, with  $v_{11} \sim U[4,7]$  and  $v_{12} \sim U[4,16]$ . The optimal auction is given by Daskalakis et al. [2015].
(F) 1 bidder and 2 items, where  $v_{11}$  has the density function  $f(x) = 5 / (1 + x)^6$ , and  $v_{12}$  has the density function  $f(x) = 6 / (1 + x)^7$ . The optimal solution is given by Daskalakis et al. [2015].
Revenue Experiments. The results of revenue experiments are presented in Table 1. We can see that in settings with unknown optimal solutions, AMenuNet achieves the highest revenue among all the DSIC approaches. The comparison between AMenuNet and VCG reveals that incorporating affine parameters into VCG leads to higher revenue. Notably, AMenuNet even surpasses the strong baseline Item-Myerson, which relies on prior distribution knowledge, an over strong assumption that may not hold especially in contextual auctions. In contrast, AMenuNet constructs the mechanism solely from sampled data, highlighting its data efficiency. Furthermore, AMenuNet demonstrates its
Table 2: The revenue results of ablation study. For each case, we use the notation  $n \times m$  to represent the number of bidders  $n$  and the number of items  $m$ . Some results of  $A_{\mathrm{dtm}}$  are intractable because the menu size is too large.
2×2(A)	2×10(A)	3×2(B)	5×2(B)	2×3(C)	3×10(C)
AMenuNet	0.4293	2.3815	0.6197	0.7548	1.3107	5.5896
With w = 1	0.4209	2.3848	0.6141	0.7010	1.2784	5.5873
With λ = 0	0.3701	2.2229	0.3892	0.3363	1.2140	5.4145
With Adtm	0.3633	-	0.5945	0.7465	1.2758	-
With FCN	0.4124	2.3416	0.5720	0.6963	1.2524	5.0262
(a) Setting (A) with fixed  $n = 2$
Figure 2: Out-of-setting generalization results. We use the notation  $n \times m$  to represent the number of bidders  $n$  and the number of items  $m$ . We train AMenuNet and evaluate it on the same auction setting, except for the number of bidders or items. For detailed numerical results, please refer to Appendix C.
(b) Setting (B) with fixed  $m = 2$
(c) Setting (C) with fixed  $n = 3$
ability to approach near-optimal solutions in known settings (Setting (D)-(F)), even when those solutions are not AMAs themselves. This underscores the representativeness of the AMAs induced by AMenuNet. Remarkably, the comparison between AMenuNet and Lottery AMA in classic auctions showcases the effectiveness of training a neural network to compute AMA parameters. This suggests that AMenuNet captures the underlying mutual relations among the allocation menu, bidder weights, and boosts, resulting in a superior mechanism after training. Lastly, while CITransNet and RegretNet may achieve higher revenue in many cases, it is important to note that they are not DSIC. Such results indicate that AMenuNet's zero regret comes at the expense of revenue. Compared to RegretNet-based approaches, AMenuNet's distinctive strength lies in its inherent capacity to ensure DSIC by design.
Ablation Study. We present an ablation study that compares the full AMenuNet model with the following three ablated versions: AMenuNet with fixed  $\boldsymbol{w} = 1$ , AMenuNet with fixed  $\lambda = 0$ , AMenuNet with  $\mathcal{A}_{\mathrm{dtm}}$  (all the  $(n + 1)^m$  deterministic allocations), and AMenuNet with the underlying architecture to be a 4-layers fully connected neural networks (FCN). We set the number of layers in FCN to be 4, each with 128 hidden nodes. The revenue results are presented in Table 2. For large values of  $n$  and  $m$ , the experiment result of AMenuNet with  $\mathcal{A}_{\mathrm{dtm}}$  can be intractable due to the large size of the allocation menu (59049 for  $2 \times 10$  and 1048576 for  $3 \times 10$ ). Such phenomenon indicates that we will face scalability issue if we consider all deterministic allocations. In contrast, the full model achieves the highest revenue in all cases except for  $2 \times 10(\mathrm{A})$ , where it also performs comparably to the best result. Notice that, in Setting (C),  $\boldsymbol{w} = 1$  is coincidentally the optimal solution due to the symmetry of the setting. The ablation results underscore the benefits of making the allocation menu, weights, and boost variables learnable, both for effectiveness and scalability. Moreover, even with significantly fewer learnable parameters, AMenuNet consistently outperforms AMenuNet with FCN, further highlighting the revenue advantages of a transformer-based architecture.
Out-of-Setting Generalization. The architecture of AMenuNet is designed to be independent of the number of bidders and items, allowing it to be applied to auctions with varying sizes. To evaluate such out-of-setting generalizability [Rahme et al., 2021a], we conduct experiments whose results are
(a)  $2 \times 5(\mathbf{C})$  with  $s = 128$ .
Figure 3: The top-10 allocations (with respect to winning rate) and the corresponding boosts among 100,000 test samples.
(b)  $5 \times 5(\mathbf{C})$  with  $s = 1024$ .
shown in Figure 2. The experimental results demonstrate that the generalized AMenuNet achieves revenue comparable to that of Item-Myerson, particularly in cases with varying items where AMenuNet often outperforms Item-Myerson. This highlights the strong out-of-setting generalizability of AMenuNet. Furthermore, The fact that AMenuNet can generalize to larger settings enhances its scalability.
Winning Allocations. We conducted experiments to analyze the winning allocations generated by AMenuNet. In order to assess the proportion of randomized allocations, we recorded the ratio of such allocations in the last row of Table 1a. Here, we define an allocation as randomized if it contains element in the range of [0.01, 0.99]. The results indicate that randomized allocations account for a significant proportion, especially in larger auction scales. For instance, in settings such as  $2 \times 10(\mathrm{A})$ ,  $3 \times 10(\mathrm{A})$ , and  $7 \times 2(\mathrm{B})$ , the proportion of randomized allocations exceeds  $17\%$ . Combining these findings with the results presented in Table 2, we observe that the introduction of randomized allocations leads to an improvement in the revenue generated by AMenuNet.
Additionally, we present the top-10 winning allocations based on their winning rates (i.e., allocations that are either  $A^{*}$  or  $A_{-k}^{*}$  for some  $k \in [n]$ ) in Figure 3, specifically for the  $2 \times 5(\mathrm{C})$  and  $5 \times 5(\mathrm{C})$  settings. Notably, the top-10 winning allocations tend to be deterministic, suggesting that AMenuNet is capable of identifying useful deterministic allocations within the entire allocation space. Furthermore, in the  $2 \times 5(\mathrm{C})$  setting, we observed instances where the winning allocation was the empty allocation with a substantial boost. This can be seen as a reserve price, where allocating nothing becomes the optimal choice when all submitted bids are too small.
In this paper, we introduce AMenuNet, a scalable neural network for the DSIC affine maximizer auction (AMA) design. AMenuNet constructs AMA parameters, including the allocation menu, bidder weights and boosts, from the public bidder and item representations. This construction ensures that the resulting mechanism is both dominant strategy compatible (DSIC) and individually rational (IR). By leveraging the neural network to compute the allocation menu, AMenuNet offers improved scalability compared to previous AMA approaches. The architecture of AMenuNet is permutation equivariant, allowing it to handle auctions of varying sizes and generalize to larger-scale auctions. Such out-of-setting generalizability further enhances the scalability of AMenuNet. Our various experiments demonstrate the effectiveness of AMenuNet, including its revenue, scalability, and out-of-setting generalizability.
As for future work, since we train AMenuNet using an offline learning approach, a potential direction is to explore online learning methods for training AMenuNet. Additionally, considering that the allocation menu size still needs to be predefined, it would be interesting to investigate the feasibility of making the allocation menu size learnable as well.
This work is supported by the National Key R&D Program of China (2022ZD0114900). We thank all anonymous reviewers for their helpful feedback.
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A Proof of Theorem 4.1
Proof. In AMenuNet, the computation of allocation menus  $\mathcal{A}$ , bidder weights  $\boldsymbol{w}$  and allocation boosts  $\lambda$  are based on public bidder representation  $X$  and item representation  $Y$ , without the chance to access the bids  $B$ . So we only have to prove that when  $\mathcal{A}$ ,  $\boldsymbol{w}$  and  $\lambda$  are fixed, the mechanism is both dominant strategy incentive compatibility (DSIC) and individually rational (IR).
The proof of DSIC. Denote by  $u_{k}(\pmb{v}_{k}, (\pmb{v}_{k}, B_{-k}); X, Y)$  the utility bidder  $k$  can get if she truthfully bids and  $u_{k}(\pmb{v}_{k}, (\pmb{b}_{k}, B_{-k}); X, Y)$  the utility bidder  $k$  can get if she bids  $\pmb{b}_{k}$ . In fact,  $u_{k}(\pmb{v}_{k}, (\pmb{v}_{k}, B_{-k}); X, Y)$  and  $u_{k}(\pmb{v}_{k}, (\pmb{b}_{k}, B_{-k}); X, Y)$  can also be written as  $u_{k}(\pmb{v}_{k}, (\pmb{v}_{k}, B_{-k}); A, \pmb{w}, \pmb{\lambda})$  and  $u_{k}(\pmb{v}_{k}, (\pmb{b}_{k}, B_{-k}); A, \pmb{w}, \pmb{\lambda})$ , respectively, since the parameters are determined by  $X$  and  $Y$ .
Let  $A^{*} \coloneqq A^{*}((\pmb{v}_{k}, B_{-k}); \mathcal{A}, \pmb{w}, \pmb{\lambda})$  and  $A_{-k}^{*} \coloneqq A_{-k}^{*}((\pmb{v}_{k}, B_{-k}); \mathcal{A}, \pmb{w}, \pmb{\lambda})$  be the affine-welfare-maximizing allocation and the affine-welfare-maximizing allocation excluding bidder  $k$ , when bidder  $k$  truthfully bids. Formally,
$$
A ^ {*} = \arg \max  _ {A \in \mathcal {A}} \left(\sum_ {i = 1} ^ {n} w _ {i} b _ {i} (A) + \lambda (A)\right), A _ {- k} ^ {*} = \arg \max  _ {A \in \mathcal {A}} \left(\sum_ {i \neq k} w _ {i} b _ {i} (A) + \lambda (A)\right).
$$
Similarly, we define  $\widehat{A^*} := A^*((\pmb{b}_k, B_{-k}); \mathcal{A}, \pmb{w}, \pmb{\lambda})$  and  $\widehat{A_{-k}^*} := A_{-k}^*((\pmb{b}_k, B_{-k}); \mathcal{A}, \pmb{w}, \pmb{\lambda})$  for bidder  $k$  and her bids  $\pmb{b}_k$ . Besides, let  $p_k((\pmb{v}_k, B_{-k}); \mathcal{A}, \pmb{w}, \pmb{\lambda})$  be bidder  $k$ 's payment defined by AMA. By definition,
$$
p _ {k} \left(\left(v _ {k}, B _ {- k}\right); \mathcal {A}, \boldsymbol {w}, \boldsymbol {\lambda}\right) = \frac {1}{w _ {k}} \left(\sum_ {i \neq k} w _ {i} b _ {i} \left(A _ {- k} ^ {*}\right) + \lambda \left(A _ {- k} ^ {*}\right)\right) - \frac {1}{w _ {k}} \left(\sum_ {i \neq k} w _ {i} b _ {i} \left(A ^ {*}\right) + \lambda \left(A ^ {*}\right)\right).
$$
First notice that  $A_{-k}^{*}$  and  $\widehat{A_{-k}^{*}}$  is the same since bidder  $k$ 's bid has no influence on others affine welfare. Hence,
$$
\begin{array}{l} u _ {k} \left(\boldsymbol {v} _ {k}, \left(\boldsymbol {v} _ {k}, B _ {- k}\right); X, Y\right) - u _ {k} \left(\boldsymbol {v} _ {k}, \left(\boldsymbol {b} _ {k}, B _ {- k}\right); X, Y\right) \\ = u _ {k} \left(\boldsymbol {v} _ {k}, \left(\boldsymbol {v} _ {k}, B _ {- k}\right); \mathcal {A}, \boldsymbol {w}, \boldsymbol {\lambda}\right) - u _ {k} \left(\boldsymbol {v} _ {k}, \left(\boldsymbol {b} _ {k}, B _ {- k}\right); \mathcal {A}, \boldsymbol {w}, \boldsymbol {\lambda}\right) \\ = v _ {k} \left(A ^ {*}\right) - p _ {k} \left(\left(v _ {k}, B _ {- k}\right); \mathcal {A}, \boldsymbol {w}, \boldsymbol {\lambda}\right) - \left(v _ {k} \left(\widehat {A} ^ {*}\right) - p _ {k} \left(\left(b _ {k}, B _ {- k}\right); \mathcal {A}, \boldsymbol {w}, \boldsymbol {\lambda}\right)\right)\left. \right. \\ = v _ {k} \left(A ^ {*}\right) - \frac {1}{w _ {k}} \left(\sum_ {i \neq k} w _ {i} b _ {i} \left(A _ {- k} ^ {*}\right) + \lambda \left(A _ {- k} ^ {*}\right)\right) + \frac {1}{w _ {k}} \left(\sum_ {i \neq k} w _ {i} b _ {i} \left(A ^ {*}\right) + \lambda \left(A ^ {*}\right)\right) \\ - v _ {k} \left(\widehat {A ^ {*}}\right) + \frac {1}{w _ {k}} \left(\sum_ {i \neq k} w _ {i} b _ {i} \left(\widehat {A _ {- k} ^ {*}}\right) + \lambda \left(\widehat {A _ {- k} ^ {*}}\right)\right) - \frac {1}{w _ {k}} \left(\sum_ {i \neq k} w _ {i} b _ {i} \left(\widehat {A ^ {*}}\right) + \lambda \left(\widehat {A ^ {*}}\right)\right) \\ \end{array}
$$
$$
\begin{array}{l} \stackrel {(1)} {=} v _ {k} (A ^ {*}) + \frac {1}{w _ {k}} \left(\sum_ {i \neq k} w _ {i} b _ {i} (A ^ {*}) + \lambda (A ^ {*})\right) - v _ {k} (\widehat {A ^ {*}}) - \frac {1}{w _ {k}} \left(\sum_ {i \neq k} w _ {i} b _ {i} (\widehat {A ^ {*}}) + \lambda (\widehat {A ^ {*}})\right) \\ = \frac {1}{w _ {k}} \left(\sum_ {i = 1} ^ {n} w _ {i} b _ {i} \left(A ^ {*}\right) + \lambda \left(A ^ {*}\right)\right) - \frac {1}{w _ {k}} \left(\sum_ {i = 1} ^ {n} w _ {i} b _ {i} \left(\widehat {A ^ {*}}\right) + \lambda \left(\widehat {A ^ {*}}\right)\right) \\ \end{array}
$$
$$
\begin{array}{c} (2) \\ \geq 0, \end{array}
$$
where (1) holds by  $A_{-k}^{*}$  and  $\widehat{A_{-k}^{*}}$  is the same, and (2) holds by the definition of  $A^{*}$ .
The proof of IR. With the same notations above, we aim to prove that for any given representations  $X,Y$  and others' bids  $B_{-k}$ , we have  $u_{k}(\pmb{v}_{k},(\pmb{v}_{k},B_{-k});X,Y)\geq 0$ . We use the same notation