SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Here,  $k(b_{-i})$  represents the option maximizing the weighted boosted welfare when buyer  $i$  is omitted, i.e.,  $k(b_{-i}) \in \arg \max_{k \in \mathcal{K}} \sum_{\ell \neq i} w_{\ell} b_{\ell}^{\top} x_{\ell}^{(k)} + \beta^{(k)}$ . It is known that AMA is DSIC and IR. Hence, we can assume that the submitted bids  $b_{i}$  represent the true valuations  $v_{i}$ . We also assume the ties are broken in favor of maximizing the total payment. In case of unit weights, this is equivalent to choosing the smallest  $\beta^{(k)}$  values, see (2) in Section 4. Given the menu  $M$ , the revenue of the AMA is
$$
\operatorname {R e v} (M) = \mathbb {E} _ {v \sim F} \left[ \sum_ {i} p _ {i} (v) \right].
$$
In this paper, we focus on the case when  $w_{i} = 1$  for all buyers. This is also used in the experiments in Curry et al. [2022b]. For this case, AMA can be implemented by a three layer neural network similar to RochetNet, with  $m \times n$  input neurons. For the more general case when the weights  $w_{i}$  can also be adjusted, one can include an additional layer that combines the buyers' allocations.
Note that for a single buyer and  $w_{1} = 1$ , AMA corresponds to *RochetNet*, with price  $p^{(k)} = -\beta^{(k)}$  for each menu option. Indeed, in the formula defining the price  $p_{i}(b)$ , the first term is 0, as well as the sum in the second term.
Similarly to RochetNet, the loss function, which is maximized via stochastic gradient ascent, is a softmax-approximation of the revenue  $\mathrm{Re}\nu (M)$ , in order to avoid the discontinuities introduced by the argmax. We bound the difference in the revenue in Appendix D.3, concluding that it decreases with large parameter  $Y$  as in the RochetNet case.
2.3 Mode Connectivity
One can view the revenue as a function of the menus, i.e., the parameters in the mechanism:  $(i)$  in RochetNet,  $\{(x^{(k)},p^{(k)})\}_{k\in \mathcal{K}}$ ; (ii) in AMA,  $\{(x^{(k)},\beta^{(k)})\}_{k\in \mathcal{K}}$ . We use  $\mathcal{M}$  to denote the set of all possible menus.
Definition 3. (Mode connectivity) Two menus  $M_1, M_2 \in \mathcal{M}$  are  $\varepsilon$ -mode-connected if there is a continuous curve  $\pi : [0,1] \to \mathcal{M}$  such that (i)  $\pi(0) = M_1$ ; (ii)  $\pi(1) = M_2$ ; and (iii) for any  $t \in [0,1]$ ,  $\text{Rev}(\pi(t)) \geq \min\{ \text{Rev}(M_1), \text{Rev}(M_2) \} - \varepsilon$ .
3 Mode Connectivity for the RochetNet
In this section we present and prove our main results for the RochetNet. For some statements, we only include proof sketches. The detailed proofs can be found in Appendix A in the supplementary material. The following definition plays an analogous role to  $\varepsilon$ -dropout stability in Kuditipudi et al. [2019].
Definition 4. A menu  $M$  with  $|\mathcal{K}| = K + 1$  options is called  $\varepsilon$ -reducible if there is a subset  $\mathcal{K}' \subseteq \mathcal{K}$  with  $0 \in \mathcal{K}'$ ,  $|\mathcal{K}'| \leq \sqrt{K + 1}$  such that, with probability at least  $1 - \varepsilon$  over the distribution of the valuation of the buyer, the active option assigned to the buyer is contained in  $\mathcal{K}'$ .
As noted in the Introduction, such a property can be observed in the experimental results in Dütting et al. [2019]. The motivation behind this definition is that if a menu satisfies this property, then all but  $\sqrt{K + 1}$  options are more or less redundant. In fact, if a menu is  $\varepsilon$ -reducible, then dropping all but the at most  $\sqrt{K + 1}$  many options in  $\mathcal{K}'$  results in a menu  $M'$  with  $\operatorname{Rev}(M') \geq \operatorname{Rev}(M) - \varepsilon$  because the price of any selected option is bounded by  $\| v \|_1 \leq 1$ .
As a first step towards showing the mode connectivity results, we show that 0-reducibility implies 0-mode-connectivity. We will then use this to derive our two main results, namely that two  $\varepsilon$ -reducible menus are always  $\varepsilon$ -mode-connected and that two large menus are always  $\varepsilon$ -mode-connected.
Proposition 5. If two menus  $M_1$  and  $M_2$  for the RochetNet are 0-reducible, then they are 0-model-connected. Moreover, the curve transforming  $M_1$  into  $M_2$  is piecewise linear with only three pieces.
To prove Proposition 5, we introduce two intermediate menus  $\widehat{M}_1$  and  $\widehat{M}_2$ , and show that every menu in the piecewise linear interpolation from  $M_1$  via  $\widehat{M}_1$  and  $\widehat{M}_2$  to  $M_2$  yields a revenue of at least  $\min \{\operatorname{Rev}(M_1), \operatorname{Rev}(M_2)\}$ . Using that menu  $M_1$  has only  $\sqrt{K + 1}$  non-redundant options, menu  $\widehat{M}_1$  will be defined by repeating each of the  $\sqrt{K + 1}$  options  $\sqrt{K + 1}$  times. Menu  $\widehat{M}_2$  will be derived from  $M_2$  similarly. A technical lemma makes sure that this copying can be done in such a way that each pair of a non-redundant option of  $M_1$  and a non-redundant option of  $M_2$  occurs exactly for one index in  $\widehat{M}_1$  and  $\widehat{M}_2$ .
To make this more formal, we first assume without loss of generality that  $K + 1$  is a square, such that  $\sqrt{K + 1}$  is an integer. It is straightforward to verify that the theorem is true for non-squares  $K + 1$ , too. Suppose the options in  $M_1$  and  $M_2$  are indexed with  $k \in \mathcal{K} = \{0, 1, \dots, K\}$ . Since  $M_1$  is 0-reducible, there is a subset  $\mathcal{K}_1 \subseteq \mathcal{K}$  with  $0 \in \mathcal{K}_1$ ,  $|\mathcal{K}_1| = \sqrt{K + 1}$  such that an option with index in  $\mathcal{K}_1$  is selected with probability 1 over the distribution of the possible valuations. Similarly, such a set  $\mathcal{K}_2$  exists for  $M_2$ . To define the curve that provides mode connectivity, we need the following technical lemma, which is proven in Appendix A.
Lemma 6. There exists a bijection  $\varphi \colon \mathcal{K} \to \mathcal{K}_1 \times \mathcal{K}_2$  such that for all  $k \in \mathcal{K}_1$  we have that  $\varphi(k) \in \{k\} \times \mathcal{K}_2$ , and for all  $k \in \mathcal{K}_2$  we have that  $\varphi(k) \in \mathcal{K}_1 \times \{k\}$ .
With this lemma, we can define  $\widehat{M}_1$  and  $\widehat{M}_2$ . Let  $\varphi$  the bijection from Lemma 6 and suppose  $M_1 = \{(x^{(k)},p^{(k)})\}_{k\in \mathcal{K}}$ . We then define  $\widehat{M}_1 = \{(x^{(\varphi_1(k))},p^{(\varphi_1(k))})\}_{k\in \mathcal{K}}$ , where  $\varphi_{1}(k)$  is the first component of  $\varphi (k)$ . Similarly,  $\widehat{M}_2$  is derived from  $M_2$  by using the second component  $\varphi_{2}(k)$  of  $\varphi (k)$  instead of  $\varphi_{1}(k)$ . It remains to show that all menus on the three straight line segments from  $M_1$  via  $\widehat{M}_1$  and  $\widehat{M}_2$  to  $M_2$  yield a revenue of at least min  $\{\mathrm{Rev}(M_1),\mathrm{Rev}(M_2)\}$ , which is established by the following two propositions; their proofs can be found in Appendix A.
Proposition 7. Let  $M = \lambda M_1 + (1 - \lambda) \widehat{M_1}$  be a convex combination of the menus  $M_1$  and  $\widehat{M_1}$ . Then  $\text{Rev}(M) \geq \text{Rev}(M_1)$ . Similarly, every convex combination of the menus  $M_2$  and  $\widehat{M_2}$  has revenue at least  $\text{Rev}(M_2)$ .
The idea to prove Proposition 7 is that, on the whole line segment from  $M_1$  to  $\widehat{M}_1$ , the only active options are those in  $\mathcal{K}'$ , implying that the revenue does not decrease.
Proposition 8. Let  $M = \lambda \widehat{M}_1 + (1 - \lambda) \widehat{M}_2$  be a convex combination of the menus  $\widehat{M}_1$  and  $\widehat{M}_2$ . Then,  $\text{Rev}(M) \geq \lambda \text{Rev}(\widehat{M}_1) + (1 - \lambda) \text{Rev}(\widehat{M}_2)$ .
The idea to prove Proposition 8 is that, due to the special structure provided by Lemma 6, a linear interpolation between the menus also provides a linear interpolation between the revenues. Note that without the construction of Lemma 6, such a linear relation would be false; such an example is shown in Appendix C.
Proposition 5 directly follows from Proposition 7 and Proposition 8. Based on Proposition 5, we can show our two main theorems for the RochetNet. The first result follows relatively easily from Proposition 5.
Theorem 9. If two menus  $M_1$  and  $M_2$  for the RochetNet are  $\varepsilon$ -reducible, then they are  $\varepsilon$ -mode-connected. Moreover, the curve transforming  $M_1$  into  $M_2$  is piecewise linear with only five pieces.
Proof. We prove this result by showing that every  $\varepsilon$ -reducible menu  $M$  can be linearly transformed into a 0-reducible menu  $\widetilde{M}$  such that each convex combination of  $M$  and  $\widetilde{M}$  achieves a revenue of at least  $\operatorname{Rev}(M) - \varepsilon$ . This transformation converting  $M_1$  and  $M_2$  to  $\widetilde{M}_1$  and  $\widetilde{M}_2$ , respectively, yields the first and the fifth of the linear pieces transforming  $M_1$  to  $M_2$ . Together with Proposition 5 applied to  $\widetilde{M}_1$  and  $\widetilde{M}_2$  serving as the second to fourth linear piece; the theorem then follows.
To this end, let  $M$  be an  $\varepsilon$ -reducible menu with options indexed by  $k \in \mathcal{K}$ . By definition, there is a subset  $\mathcal{K}' \subseteq \mathcal{K}$  of at most  $\sqrt{K + 1}$  many options such that, with probability at least  $1 - \varepsilon$ , the assigned active option is contained in  $\mathcal{K}'$ . Let  $\widetilde{M}$  consist of the same allocations as  $M$ , but with modified prices. For an option  $k \in \mathcal{K}'$ , the price  $\tilde{p}^{(k)} = p^{(k)}$  in  $\widetilde{M}$  is the same as in  $M$ . However, for an option  $k \in \mathcal{K} \setminus \mathcal{K}'$ , we set the price  $\tilde{p}^{(k)} > 1$  in  $\widetilde{M}$  to be larger than the largest possible valuation of any option  $\| v \|_1 \leq 1$ . It follows that such an option will never be selected and  $\widetilde{M}$  is 0-reducible.
To complete the proof, let us look at the reward of a convex combination  $M' = \lambda M + (1 - \lambda) \widetilde{M}$ . If for a particular valuation  $v$  the selected option in  $M$  was in  $\mathcal{K}'$ , then the same option will be selected in  $M'$ . This happens with probability at least  $1 - \varepsilon$ . In any other case, anything can happen, but the revenue cannot worsen by more than the maximum possible valuation, which is  $\| v \|_1 \leq 1$ . Therefore,  $\operatorname{Re} v(M) - \operatorname{Re} v(M') \leq \varepsilon \cdot 1 = \varepsilon$ , completing the proof.
Theorem 10. If two menus  $M_1$  and  $M_2$  for the RochetNet have size at least  $\left\lceil \frac{4}{\varepsilon^2} \right\rceil^{2n}$ , then they are  $\varepsilon$ -connected. Moreover, the curve transforming  $M_1$  into  $M_2$  is piecewise linear with only five pieces.
Proof Sketch. The full proof can be found in Appendix A.2. The intuition behind this theorem is that if menus are large, then they should contain many redundant options. Indeed, as in the previous theorem, the strategy is as follows. We show that every menu  $M$  of size at least  $\left\lceil \frac{4}{\epsilon^2} \right\rceil^{2n}$  can be linearly transformed into a 0-reducible menu  $\widetilde{M}$  such that each convex combination of  $M$  and  $\widetilde{M}$  achieves a revenue of at least  $\operatorname{Re}\mathbf{v}(M) - \varepsilon$ . This transformation converting  $M_1$  and  $M_2$  to  $\widetilde{M}_1$  and  $\widetilde{M}_2$ , respectively, yields the first and the fifth of the linear pieces transforming  $M_1$  to  $M_2$ . Together with Proposition 5 applied to  $\widetilde{M}_1$  and  $\widetilde{M}_2$  serving as the second to fourth linear piece, the theorem then follows.
However, this time, the linear transformation of  $M$  to  $\widetilde{M}$  is much more intricate than in the previous theorem. To do so, it is not sufficient to only adapt the prices. Instead, we also change the allocations of the menu options by rounding them to discretized values. This technique is inspired by Dughmi et al. [2014], but non-trivially adapted to our setting. Since the rounding may also modify the active option for each valuation, we have to carefully adapt the prices in order to make sure that for each valuation, the newly selected option is not significantly worse than the originally selected one. Finally, this property has to be proven not only for  $\widetilde{M}$ , but for every convex combination of  $M$  and  $\widetilde{M}$ .
After the above rounding, the number of possible allocations for any option is bounded by  $\left\lceil \frac{4}{\epsilon^2} \right\rceil^n$ . Out of several options with the same allocation, the buyer would always choose the cheapest one, implying that the resulting menu  $\widetilde{M}$  is 0-reducible.
4 Mode Connectivity for the Affine Maximizer Auctions
Throughout this section, we focus on AMAs with fixed weights  $w_{i} = 1$  for all buyers  $i$ . Similarly tochetNet, we have the following definition for AMAs.
Definition 11. A menu  $M$  with  $K + 1$  options is  $\varepsilon$ -reducible if and only if there exists a subset  $\mathcal{K}' \subseteq \mathcal{K}$ ,  $0 \in \mathcal{K}'$ ,  $|\mathcal{K}'| \leq \sqrt{K + 1}$  such that, with probability at least  $1 - \frac{\varepsilon}{m}$  over the distribution of the valuation of the buyers, (i)  $k(v_{-i}) \in \mathcal{K}'$  for any buyer  $i$ ; and (ii)  $k(v) \in \mathcal{K}'$ .
Such phenomena are observed in the experiments in [Curry et al., 2022b, Section 6.3].
Our two main results, namely that two  $\varepsilon$ -reducible menus are always  $\varepsilon$ -connected and two large menus are always  $\varepsilon$ -connected, are based on the following proposition, in which we show that 0-reducibility implies 0-connectivity.
Proposition 12. If two menus  $M_1$  and  $M_2$  are 0-reducible, then they are 0-connected. Moreover, the curve transforming  $M_1$  into  $M_2$  is piecewise linear with only three pieces.
The proof idea is similar to the proof of Proposition 5 in *RochetNet*, but requires additional arguments due to the more intricate price structure (see Appendix D.1 for more details). Based on this proposition, now, we are able to show our two main results. First, we achieve  $\varepsilon$ -connectivity from  $\varepsilon$ -reducibility.
Theorem 13. If two AMAs  $M_1$  and  $M_2$  are  $\varepsilon$ -reducible, then they are  $\varepsilon$ -mode-connected. Moreover, the curve transforming  $M_1$  to  $M_2$  is piecewise linear with only five pieces.
Before the proof, we recall how the total payment is calculated for a valuation profile  $v$ . We choose  $k(v)$  as the option which maximizes the boosted welfare,  $\sum_{i} v_{i}^{\top} x_{i}^{(k)} + \beta^{(k)}$ . According to (1), the total revenue can be written as
$$
\sum_ {i} p _ {i} (v) = \sum_ {i} \underbrace {\left(\sum_ {\ell \neq i} v _ {\ell} ^ {\top} x _ {\ell} ^ {(k (v - i))} + \beta^ {(k (v - i))}\right)} _ {\text {b o o s t e d w e l f a r e o f} v _ {- i}} - (m - 1) \underbrace {\left(\sum_ {i} v _ {i} ^ {\top} x _ {i} ^ {(k (v))} + \beta^ {(k (v))}\right)} _ {\text {b o o s t e d w e l f a r e o f} v} - \beta^ {(k (v))}. \tag {2}
$$
Proof of Theorem 13. Similar to the proof of Theorem 9, it is sufficient to show that every  $\varepsilon$ -reducible menu  $M$  can be linearly transformed into a 0-reducible menu  $\widetilde{M}$  such that each convex combination of  $M$  and  $\widetilde{M}$  achieves a revenue of at least  $\operatorname{Rev}(M) - \varepsilon$ . This can then be used as the first and fifth linear piece of the curve connecting  $M_1$  and  $M_2$ , while the middle three pieces are provided by Proposition 12.
We construct  $\widetilde{M}$  by (i) keeping all options in  $\mathcal{K}'$  unchanged; (ii) for the options  $k \in \mathcal{K} \setminus \mathcal{K}'$ , we decrease  $\beta^{(k)}$  to be smaller than  $-m$ , which implies such an option will never be selected (recall that  $0 \in \mathcal{K}'$  is assumed, and the option  $(0,0)$  is better than any such option). Consequently,  $\widetilde{M}$  is 0-reducible.
To complete the proof, let us look at the revenue of  $M' = \{(x'^{(k)},\beta'^{(k)})\}_{k\in \mathcal{K}}$ , which is a convex combination of  $M$  and  $\widetilde{M}$ :  $M' = \lambda M + (1 - \lambda)\widetilde{M}$  for  $0\leq \lambda < 1$ . Let  $k^{\prime}(v) = \arg \max_{k}\sum_{i}v_{i}^{\top}x^{\prime}_{i}^{(k)} + \beta^{\prime (k)}$ . As we decrease  $\beta^{(k)}$  for  $k\notin \mathcal{K}'$ ,  $k(v)\in \mathcal{K}'$  implies  $k^{\prime}(v)\in \mathcal{K}^{\prime}$  and, additionally, option  $k^{\prime}(v)$  and option  $k(v)$  achieve the same boosted welfare and same  $\beta$ . Therefore, since  $M$  is  $\varepsilon$ -reducible, with probability at least  $1 - \frac{\varepsilon}{m}$ , the boosted welfare of  $v$  as well as the boosted welfare of  $v_{-i}$  for all buyers  $i$  is the same for  $M$  and for  $M^{\prime}$ . According to the formula (2), the total payment for the profile  $v$  is the same for  $M$  and  $M^{\prime}$ . Therefore, the loss on the revenue can only appear with probability at most  $\frac{\varepsilon}{m}$ , and the maximum loss is at most  $m$ , which implies an  $\varepsilon$  loss in total.
Second, we show that mode connectivity also holds for those AMAs with large menu sizes, namely for  $K + 1 \geq \lceil \frac{16m^3}{\epsilon^2} \rceil^{2nm}$ .
Theorem 14. For any  $0 < \epsilon \leq \frac{1}{4}$ , if two AMAs  $M_1$  and  $M_2$  have at least  $K + 1 \geq \lceil \frac{16m^3}{\epsilon^2} \rceil^{2nm}$  options, then they are  $\varepsilon$ -mode-connected. Moreover, the curve transforming  $M_1$  to  $M_2$  is piecewise linear with only five pieces.
Proof Sketch. The full proof can be found in Appendix D.2. Similar to *RochetNet*, the idea of proving Theorem 14 is to discretize the allocations in the menu, then one can use Proposition 12 to construct the low-loss transformation from  $M_1$  to  $M_2$  by five linear pieces. To do this, one wants the loss of revenue to be small during the discretization. Consider the formula (2) of the total payment. The first two terms do not change much by a small change of the discretization. However, the last term  $\beta^{(k(v))}$  might be significantly affected by discretization, which may cause a notable decrease in the total payment. To avoid this, we perform a proportional discount on  $\beta$ , incentivizing the auctioneer to choose an allocation with a small  $\beta$ . By this approach, the original revenue will be approximately maintained. Furthermore, we show a linear path, connecting the original menu and the menu after discretizing, which will suffer a small loss.
We have given theoretical evidence of mode-connectivity in neural networks designed to learn auction mechanisms. Our results show that, for a sufficiently wide hidden layer,  $\varepsilon$ -mode-connectivity holds in the strongest possible sense. Perhaps more practically, we have shown  $\varepsilon$ -mode-connectivity under  $\varepsilon$ -reducibility, i.e., the assumption that there is a sufficiently small subset of neurons that preserve most of the revenue. There is evidence for this assumption in previous work in differentiable economics. A systematic experimental study that verifies this assumption under various distributions and network sizes is left for future work.
Our results make a first step in providing theoretical arguments underlying the success of neural networks in mechanism design. Our focus was on some of the most basic architectures. A natural next step is to extend the arguments for AMA networks with variable weights  $w_{i}$ . Such a result will need to analyze a four layer network, and thus could make headway into understanding the behaviour of deep networks. Besides RochetNet, Dütting et al. [2019] also proposed RegretNet, based on minimising a regret objective. This network is also applicable to multiple buyers, but only provides approximate incentive compatibility, and has been extended in subsequent work, e.g., Feng et al. [2018], Golowich et al. [2018], Duan et al. [2022]. The architecture is however quite different from RochetNet: it involves two deep neural networks in conjunction, an allocation and a payment network, and uses expected ex post regret as the loss function. We therefore expect a mode-connectivity analysis for RegretNet to require a considerable extension of the techniques used by us. We believe that such an analysis would be a significant next step in the theoretical analysis of neural networks in differentiable economics.
All three authors gratefully acknowledge support by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (for all three authors via grant agreement ScaleOpt-757481; for Christoph Hertrich additionally via grant agreement ForEFront-615640). Yixin Tao also acknowledges the Grant 2023110522 from SUFE.
Z. Allen-Zhu, Y. Li, and Z. Song. A convergence theory for deep learning via over-parameterization. In International Conference on Machine Learning, pages 242-252. PMLR, 2019.
S. Arora, R. Ge, B. Neyshabur, and Y. Zhang. Stronger generalization bounds for deep nets via a compression approach. In International Conference on Machine Learning, pages 254-263. PMLR, 2018.
Z. Feng, H. Narasimhan, and D. C. Parkes. Deep learning for revenue-optimal auctions with budgets. In Proceedings of the 17th International Conference on Autonomous Agents and Multiagent Systems, pages 354-362, 2018.
T. Garipov, P. Izmailov, D. Podoprikhin, D. P. Vetrov, and A. G. Wilson. Loss surfaces, mode connectivity, and fast ensembling of DNNs. Advances in Neural Information Processing Systems, 31, 2018.
Y. Giannakopoulos and E. Koutsoupias. Duality and optimality of auctions for uniform distributions. In Proceedings of the fifteenth ACM Conference on Economics and Computation, pages 259-276, 2014.
Y. Giannakopoulos and E. Koutsoupias. Duality and optimality of auctions for uniform distributions. SIAM Journal on Computing, 47(1):121-165, 2018.
N. Golowich, H. Narasimhan, and D. C. Parkes. Deep learning for multi-facility location mechanism design. In International Joint Conference on Artificial Intelligence, pages 261-267, 2018.
T. Groves. Incentives in teams. *Econometrica: Journal of the Econometric Society*, pages 617-631, 1973.
R. Kuditipudi, X. Wang, H. Lee, Y. Zhang, Z. Li, W. Hu, R. Ge, and S. Arora. Explaining landscape connectivity of low-cost solutions for multilayer nets. Advances in Neural Information Processing Systems, 32, 2019.
K. Kuo, A. Ostuni, E. Horishny, M. J. Curry, S. Dooley, P.-y. Chiang, T. Goldstein, and J. P. Dickerson. Proportionnet: Balancing fairness and revenue for auction design with deep learning. arXiv preprint arXiv:2010.06398, 2020.
A. M. Manelli and D. R. Vincent. Multidimensional mechanism design: Revenue maximization and the multiple-good monopoly. Journal of Economic Theory, 137(1):153-185, 2007.
R. B. Myerson. Optimal auction design. Mathematics of Operations Research, 6(1):58-73, 1981.
Q. Nguyen. On connected sublevel sets in deep learning. In International Conference on Machine Learning, pages 4790-4799. PMLR, 2019.
J.-C. Rochet. A necessary and sufficient condition for rationalizability in a quasi-linear context. Journal of Mathematical Economics, 16(2):191-200, 1987.
J.-C. Rochet and L. A. Stole. The economics of multidimensional screening. *Econometric Society Monographs*, 35:150-197, 2003.
T. Sandholm and A. Likhodedov. Automated design of revenue-maximizing combinatorial auctions. Operations Research, 63(5):1000-1025, 2015.
W. Shen, P. Tang, and S. Zuo. Automated mechanism design via neural networks. In Proceedings of the 18th International Conference on Autonomous Agents and MultiAgent Systems, pages 215-223, 2019.
A. Shevchenko and M. Mondelli. Landscape connectivity and dropout stability of sgd solutions for over-parameterized neural networks. In International Conference on Machine Learning, pages 8773-8784. PMLR, 2020.
W. Vickrey. Counterspeculation, auctions, and competitive sealed tenders. The Journal of Finance, 16(1):8-37, 1961.
S. Zheng, A. Trott, S. Srinivasa, D. C. Parkes, and R. Socher. The AI economist: Taxation policy design via two-level deep multiagent reinforcement learning. Science Advances, 8(18), 2022.