SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Formally, for each  $i$ , the mechanism designer knows that  $\theta_{i}$  lies in a  $k$ -dimensional subspace  $U_{i} = \operatorname{span}(u_{i,1}, \ldots, u_{i,k})$  of  $\mathbb{R}^{\Gamma}$  where each  $u_{i,j} \in \mathbb{R}_{\geq 0}^{\Gamma}$  lies in the non-negative orthant and  $\{u_{i,1}, \ldots, u_{i,k}\}$  is an orthonormal basis for  $U_{i}$ . For simplicity, assume  $H = 2^{a}$  for some positive integer  $a$ . Let  $\mathcal{L}_{i,j} = \{\lambda u_{i,j} : \lambda \geq 0\} \cap [0,H]^{\Gamma}$  be the line segment that is the portion of the
ray generated by  $u_{i,j}$  that lies in  $[0,H]^{\Gamma}$ . Let  $y_{i,j}$  be the endpoint of  $\mathcal{L}_{i,j}$  with  $\| y_{i,j}\|_{\infty} = H$  (the other endpoint of  $\mathcal{L}_{i,j}$  is the origin). Let  $z_{i,j}^{1} = y_{i,j} / 2$  be the midpoint of  $\mathcal{L}_{i,j}$ , and for  $\ell = 2,\ldots ,\log_2H$  let  $z_{i,j}^{\ell} = z_{i,j}^{\ell -1} / 2$  be the midpoint of  $\overline{0z_{i,j}^{\ell -1}}$ . So  $\| z_{i,j}^{\log_2H}\|_{\infty} = 1$ . We terminate the halving of  $\mathcal{L}_{i,j}$  after  $\log_2H$  steps due to the assumption that  $\theta_i\in [1,H]^{\Gamma}$ . For every  $k$ -tuple  $(\ell_1,\dots ,\ell_k)\in \{1,\dots ,\log_2H\} ^k$ , let  $\widetilde{\theta}_i(\ell_1,\dots ,\ell_k) = \sum_{j = 1}^k z_{i,j}^{\ell_j}$ . Furthermore, let  $W_{\ell} = \{(\ell_{1},\dots ,\ell_{k})\in \{1,\dots ,\log_{2}H\}^{k}:\min_{j}\ell_{j} = \ell \}$ . The sets  $W_{1},\ldots ,W_{\log_{2}H}$  partition  $\{1,\dots ,\log_2H\} ^k$  into levels, where  $W_{\ell}$  is the set of points with  $\ell_{\infty}$ -distance  $H / 2^{\ell}$  from the origin. For each agent  $i$ , our mechanism  $\mathcal{M}_k$  independently sets
$$
\boxed {\widehat {\Theta} _ {i} = \left\{\widetilde {\theta} _ {i} (\ell_ {i, 1}, \dots , \ell_ {i, k}) \right\} \text {w h e r e} \ell_ {i} \sim_ {\text {u n i f .}} \{1, \dots , \log_ {2} H \} \text {a n d} (\ell_ {i, 1}, \dots , \ell_ {i, k}) \sim_ {\text {u n i f .}} W _ {\ell_ {i}}.}
$$
Theorem 5.1.  $\mathcal{M}_k$  satisfies  $\mathbb{E}[welfare] \geq \mathrm{OPT} / \log_2 H$  and  $\mathbb{E}[revenue] \geq \mathrm{OPT} / (2k(\log_2 H)^k)$ .
As mentioned in Section 1,  $\mathcal{M}_k$  can be viewed as a generalization of the log  $H$  revenue approximation in the single-item limited-supply setting that is achieved by a second-price auction with a uniformly random reserve price from  $\{H / 2,H / 4,\dots ,1\}$  [27]. Our results apply not only to auctions but to general multidimensional mechanism design problems such as the examples presented in Section 2.
6 Conclusions and future research
We developed a versatile new methodology for multidimensional mechanism design that incorporates side information about agent types with the bicriteria goal of generating high social welfare and high revenue simultaneously. We designed a side-information-dependent meta-mechanism. This mechanism generalizes the weakest-competitor VCG mechanism of Krishna and Perry [34]. Careful instantiations of our meta-mechanism simultaneously achieved strong welfare and revenue guarantees that were parameterized by errors in the side information, and additionally proved to be fruitful in a setting where each agent's type lies on a constant-dimensional subspace (of the potentially high-dimensional ambient type space) that is known to the mechanism designer.
There are many new research directions that stem from our work. First, how far off are our mechanisms from the welfare-versus-revenue Pareto frontier? The weakest-competitor VCG mechanism is one extreme point, but what does the rest of the frontier look like? One possible approach here would be to extend our theory beyond VCG to the larger class of affine maximizers (which are known to contain high-revenue mechanisms)—we provide some initial ideas in Appendix D.
Computational complexity: An important facet that we have largely ignored is computational complexity. The computations in our mechanism involving weakest competitors scale with the description complexity of  $\widetilde{\Theta}_i$  (e.g., the number of constraints, the complexity of constraints, and so on). An important question here is to understand the computational complexity of our mechanisms as a function of the differing (potentially problem-specific) language structures used to describe the side-information sets  $\widetilde{\Theta}_i$ . In particular, the classes of side-information sets that are accurate, natural/interpretable, and easy to describe might depend on the specific mechanism design domain. Expressive bidding languages for combinatorial auctions have been extensively studied with massive impact in practice [46, 47]. Can a similar methodology be developed for side information?
Improved analysis when there is a known prior: Another direction is to improve the Bayesian revenue analysis of Krishna and Perry when there is a known prior over agents' types. Here, the benchmark would be efficient welfare in expectation over the prior. The (Bayesian) WCVCG mechanism of Krishna and Perry uses weakest competitors with respect to the prior's support to guarantee efficient welfare in expectation, but its revenue could potentially be boosted significantly by compromising on welfare as in our random expansion mechanism. Another direction here is to study the setting when the given prior might be inaccurate. Can our random expansion mechanism be used to derive guarantees that depend on the closeness of the given prior to the true prior? Such questions are thematically related to robust mechanism design [16]. Another direction along this vein is to generalize our mechanisms to depend on a known prior over prediction errors.
Finally, the WCVCG mechanism of Krishna and Perry is a strict improvement over the vanilla VCG mechanism, yet it appears to not have been further studied nor applied since its discovery. The weakest-competitor paradigm could have applications in economics and computation more broadly.
This material is based on work supported by the NSF under grants IIS-1901403, CCF-1733556, CCF-1910321, and SES-1919453, by the ARO under award W911NF2210266, and by Defense Advanced Research Projects Agency under cooperative agreement HR00112020003. S. Prasad thanks Morgan McCarthy for helpful discussions about real-world use cases of multidimensional mechanism design and Misha Khodak for detailed feedback on an earlier draft.
[1] Vineet Abhishek and Bruce Hajek. Efficiency loss in revenue optimal auctions. In 49th IEEE Conference on Decision and Control (CDC). IEEE, 2010.
[2] Gagan Aggarwal, Gagan Goel, and Aranyak Mehta. Efficiency of (revenue-)optimal mechanisms. In Proceedings of the 10th ACM Conference on Electronic Commerce (EC), 2009.
[3] Priyank Agrawal, Eric Balkanski, Vasilis Gkatzelis, Tingting Ou, and Xizhi Tan. Learning-augmented mechanism design: Leveraging predictions for facility location. In Proceedings of the 23rd ACM Conference on Economics and Computation (EC), 2022.
[4] Elliot Anshelevich, Koushik Kar, and Shreyas Sekar. Pricing to maximize revenue and welfare simultaneously in large markets. In International Conference on Web and Internet Economics (WINE). Springer, 2016.
[5] Aaron Archer and Éva Tardos. Truthful mechanisms for one-parameter agents. In IEEE Symposium on Foundations of Computer Science (FOCS), 2001.
[6] Lawrence Ausubel and Paul Milgrom. Ascending auctions with package bidding. The BE Journal of Theoretical Economics, 2002.
[7] Lawrence Ausubel and Paul Milgrom. The lovely but lonely vickrey auction. Combinatorial auctions, 2006.
[8] Maria-Florina Balcan, Avrim Blum, Jason D Hartline, and Yishay Mansour. Mechanism design via machine learning. In IEEE Symposium on Foundations of Computer Science (FOCS). IEEE, 2005.
[9] Maria-Florina Balcan, Avrim Blum, and Yishay Mansour. Item pricing for revenue maximization. In ACM Conference on Electronic Commerce (EC), 2008.
[10] Maria-Florina Balcan, Tuomas Sandholm, and Ellen Vitercik. A general theory of sample complexity for multi-item profit maximization. In ACM Conference on Economics and Computation (EC), 2018.
[11] Maria-Florina Balcan, Siddharth Prasad, and Tuomas Sandholm. Efficient algorithms for learning revenue-maximizing two-part tariffs. In Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence (IJCAI), 2020.
[12] Maria-Florina Balcan, Siddharth Prasad, and Tuomas Sandholm. Maximizing revenue under market shrinkage and market uncertainty. Advances in Neural Information Processing Systems (NeurIPS), 2022.
[13] Eric Balkanski, Vasilis Gkatzelis, and Xizhi Tan. Strategyproof scheduling with predictions. In Innovations in Theoretical Computer Science (ITCS), 2023.
[14] Santiago Balseiro, Christian Kroer, and Rachitesh Kumar. Single-leg revenue management with advice. arXiv preprint arXiv:2202.10939, 2022.
[15] Siddhartha Banerjee, Vasilis Gkatzelis, Artur Gorokh, and Billy Jin. Online nash social welfare maximization with predictions. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). SIAM, 2022.
[16] Dirk Bergemann and Stephen Morris. Robust mechanism design. Econometrica, pages 1771-1813, 2005.
[37] R Preston McAfee and Philip J Reny. Correlated information and mechanism design. *Econometrica: Journal of the Econometric Society*, pages 395–421, 1992.
[38] Andres Muñoz Medina and Sergei Vassilvitskii. Revenue optimization with approximate bid predictions. Advances in Neural Information Processing Systems (NeurIPS), 2017.
[39] Cade Metz. Facebook doesn't make as much money as it could—on purpose. Wired, 2015.
[40] Michael Mitzenmacher and Sergei Vassilvitskii. Algorithms with predictions. Communications of the ACM, 2022.
[41] Jamie Morgenstern and Tim Roughgarden. Learning simple auctions. In Conference on Learning Theory (COLT), 2016.
[42] Roger Myerson. Optimal auction design. Mathematics of operations research, 1981.
[43] Mallesh M Pai and Rakesh Vohra. Optimal auctions with financially constrained buyers. Journal of Economic Theory, 150:383-425, 2014.
[44] Baharak Rastegari, Anne Condon, and Kevin Leyton-Brown. Revenue monotonicity in deterministic, dominant-strategy combinatorial auctions. Artificial Intelligence, 2011.
[45] Kevin Roberts. The characterization of implementable social choice rules. In J-J Laffont, editor, Aggregation and Revelation of Preferences. North-Holland Publishing Company, 1979.
[46] Tuomas Sandholm. Expressive commerce and its application to sourcing: How we conducted $35 billion of generalized combinatorial auctions. AI Magazine, 28(3):45-45, 2007.
[47] Tuomas Sandholm. Very-large-scale generalized combinatorial multi-attribute auctions: Lessons from conducting $60 billion of sourcing. In Zvika Neeman, Alvin Roth, and Nir Vulkan, editors, Handbook of Market Design. Oxford University Press, 2013.
[48] Tuomas Sandholm and Anton Likhodedov. Automated design of revenue-maximizing combinatorial auctions. Operations Research, 2015.
[49] Hal R Varian and Christopher Harris. The vcg auction in theory and practice. American Economic Review, 2014.
[50] William Vickrey. Counterspeculation, auctions, and competitive sealed tenders. Journal of Finance, 1961.
[51] Rakesh V Vohra. Mechanism design: a linear programming approach, volume 47. Cambridge University Press, 2011.
[52] William E Walsh, David C Parkes, Tuomas Sandholm, and Craig Boutilier. Computing reserve prices and identifying the value distribution in real-world auctions with market disruptions. In AAAI, pages 1499-1502, 2008.
[53] Chenyang Xu and Pinyin Lu. Mechanism design with predictions. In Proceedings of the 31st International Joint Conference on Artificial Intelligence (IJCAI), 2022.
[54] Makoto Yokoo, Yuko Sakurai, and Shigeo Matsubara. The effect of false-name bids in combinatorial auctions: New fraud in internet auctions. Games and Economic Behavior, 46(1):174-188, 2004.
A Proofs of results
We provide complete proofs of all results from the main body of the paper.
A.1 Omitted proofs from Section 2
Proof of Theorem 2.1. Weakest-competitor VCG is incentive compatible for the same reason that VCG is incentive compatible: the minimization in the payment formula (the pivot term) is independent of bidder  $i$ 's reported type. Concretely, if  $\alpha'$  is the welfare-maximizing allocation when bidder  $i$  reports  $\theta_i'$ , bidder  $i$ 's utility from reporting  $\theta_i'$  is  $\sum_{j=1}^{n} \theta_j[\alpha'] - \min_{\widetilde{\theta}_i \in \Theta_i} (\max_\alpha \sum_{j \neq i} \theta_j[\alpha] + \widetilde{\theta}_i[\alpha])$ , which is maximized at  $\alpha' = \alpha^*$  (which proves incentive compatibility). Furthermore, for each  $i$ ,  $\sum_{j=1}^{n} \theta_j[\alpha^*] - \min_{\widetilde{\theta}_i \in \Theta_i} (\max_\alpha \sum_{j \neq i} \theta_j[\alpha] + \widetilde{\theta}_i[\alpha]) \geq \sum_{j=1}^{n} \theta_j[\alpha^*] - \max_\alpha \sum_{j=1}^{n} \theta_j[\alpha] = 0$ , which proves individual rationality. The proof that weakest-competitor VCG is revenue optimal follows from the revenue equivalence theorem; the necessary ingredients may be found in the monograph by Vohra [51]. Let  $p_i(\pmb{\theta})$  be the weakest-competitor VCG payment rule, and let  $p_i'(\pmb{\theta})$  be any other payment rule that also implements the efficient allocation rule. By revenue equivalence, for each  $i$ , there exists  $h_i(\pmb{\theta}_{-i})$  such that  $p_i'(\theta_i, \pmb{\theta}_{-i}) = p_i(\theta_i, \pmb{\theta}_{-i}) + h_i(\pmb{\theta}_{-i})$ . Suppose  $\pmb{\theta}$  is a profile of types such that  $p_i'$  generates strictly greater revenue than  $p_i$ , that is,  $\sum_{i=1}^{n} p_i'(\pmb{\theta}) > \sum_{i=1}^{n} p_i(\pmb{\theta})$ . Equivalently,  $\sum_{i=1}^{n} p_i(\theta_i, \pmb{\theta}_{-i}) + h_i(\pmb{\theta}_{-i}) > \sum_{i=1}^{n} p_i(\theta_i, \pmb{\theta}_{-i})$ . Thus, there exists  $i^*$  such that  $h_i^*(\pmb{\theta}_{-i^*}) > 0$ . Now, let
$$
\widetilde {\theta} _ {i ^ {*}} = \underset {\theta_ {i ^ {*}} ^ {\prime} \in \Theta_ {i ^ {*}}} {\operatorname {a r g m i n}} \underset {\alpha \in \Gamma} {\max} \sum_ {j \neq i} \theta_ {j} [ \alpha ] + \theta_ {i ^ {*}} ^ {\prime} [ \alpha ]
$$
be the weakest competitor with respect to  $\theta_{-i^*}$ . If weakest-competitor VCG is run on the type profile  $(\widetilde{\theta}_{i^*}, \theta_{-i^*})$ , the agent with type  $\widetilde{\theta}_{i^*}$  pays their value for the efficient allocation. In other words, the individual rationality constraint is binding for  $\widetilde{\theta}_{i^*}$ . Since  $h_{i^*}(\theta_{-i^*}) > 0$ ,  $p_i'$  violates individual rationality, which completes the proof.
A.2 Omitted proofs from Section 3
Proof of Theorem 3.3. Let  $i$  denote the index of the agent under consideration with type space  $\widetilde{\Theta}$ . Let  $\theta \in \widetilde{\Theta}$  be a point such that there exists  $\theta' \in \widetilde{\Theta}$  with  $\theta' \prec \theta$ . Then,
$$
\max  _ {\alpha \in \Gamma} \sum_ {j \neq i} \theta_ {j} [ \alpha ] + \theta^ {\prime} [ \alpha ] <   \max  _ {\alpha \in \Gamma} \sum_ {j \neq i} \theta_ {j} [ \alpha ] + \theta [ \alpha ]
$$
for all  $\theta_{-i} \in \Theta_{-i}$ . So  $\theta \notin \overline{\mathsf{WC}}(\widetilde{\Theta})$ , which shows that  $\overline{\mathsf{WC}}(\widetilde{\Theta}) \subseteq \{\theta \in \widetilde{\Theta} : \{\theta' \in \widetilde{\Theta} : \theta' \prec \theta\} = \emptyset\}$ . To show the reverse containment, let  $\theta \in \widetilde{\Theta}$  be such that  $\{\theta' \in \widetilde{\Theta} : \theta' \prec \theta\} = \emptyset$ . Consider any  $\theta_{-i} = (\theta_1, \ldots, \theta_{i-1}, \theta_{i+1}, \ldots, \theta_n)$  such that
$$
\sum_ {j \neq i} \theta_ {j} [ \alpha_ {1} ] + \theta [ \alpha_ {1} ] = \sum_ {j \neq i} \theta_ {j} [ \alpha_ {2} ] + \theta [ \alpha_ {2} ] = \dots = \sum_ {j \neq i} \theta_ {j} [ \alpha_ {| \Gamma |} ] + \theta [ \alpha_ {| \Gamma |} ].
$$
The existence of such a  $\theta_{-i}$  can be shown explicitly as follows. Let  $j\neq i$  be arbitrary. For all  $k\notin \{i,j\}$  set  $\theta_{k} = (0,\dots ,0)$ . Without loss of generality relabel the allocations in  $\Gamma$  such that  $\theta [\alpha_1]\geq \theta [\alpha_2]\geq \dots \geq \theta [\alpha_{|\Gamma |}]$ . Then, set
$$
\theta_ {j} = \left(0, \theta [ \alpha_ {1} ] - \theta [ \alpha_ {2} ], \ldots , \theta [ \alpha_ {1} ] - \theta [ \alpha_ {| \Gamma |} ]\right) \in [ 0, H ] ^ {\Gamma}.
$$
Then,  $\theta$  minimizes
$$
\max  \left\{\sum_ {j \neq i} \theta_ {j} [ \alpha_ {1} ] + \theta [ \alpha_ {1} ], \sum_ {j \neq i} \theta_ {j} [ \alpha_ {2} ] + \theta [ \alpha_ {2} ], \dots , \sum_ {j \neq i} \theta_ {j} [ \alpha_ {| \Gamma |} ] + \theta [ \alpha_ {| \Gamma |} ] \right\}
$$
since any  $\theta'$  that attains a strictly smaller value must satisfy  $\theta' \prec \theta$  (and no such  $\theta'$  exists, by assumption). So  $\theta \in \overline{\mathsf{WC}}(\widetilde{\Theta})$ , which proves the reverse containment. The characterizations of WC and WCH follow immediately.
Proof of Theorem 3.4.  $\mathcal{M}$  is incentive compatible for the exact same reason weakest-competitor VCG is incentive compatible (Theorem 2.1). Individual rationality is an immediate consequence of how  $\mathcal{M}$  is defined: all agents with potential individual-rationality violations (those not in  $\mathcal{I}$ ) do not participate and receive zero utility.
Proof of Theorem 3.5. Let  $\theta_{i}$  denote the true type of agent  $i$  and let  $\theta_{-i} = (\theta_{1},\ldots ,\theta_{i - 1},\theta_{i + 1},\ldots ,\theta_{n})$  denote the reported types of the other agents. Suppose  $\theta_{i}\notin \mathsf{WCH}(\widehat{\Theta}_{i})$ . Then, there exists  $\widetilde{\theta}_i\in \overline{\mathsf{WC}} (\widehat{\Theta}_i)$  such that  $\widetilde{\theta}_i\succ \theta_i$ , and there exists  $\theta_{-i}$  such that  $\widetilde{\theta}_i\in \mathrm{argmin}_{\widehat{\theta}_i\in \widehat{\Theta}_i}\max_{\alpha \in \Gamma}\sum_{j\neq i}\theta_j[\alpha ] + \widehat{\theta}_i[\alpha ]$  is a weakest competitor relative to  $\theta_{-i}$  (the existence of  $\theta_{-i}$  follows from the same reasoning as in the proof of Theorem 3.3). As  $\widetilde{\theta}_i\succ \theta_i$ , agent  $i$  's overall utility will be negative. The utility is unchanged and remains negative if  $\widetilde{\theta}_i$  is replaced by  $\theta_{i}^{*}\in \mathsf{WC}(\widehat{\Theta}_{i})$  that is also a weakest competitor relative to  $\theta_{-i}$ . So we have shown there exists  $\theta_{-i}$  such that  $i\notin \mathcal{I}$ .
Conversely suppose  $\theta_{i}\in \mathsf{WCH}(\widehat{\Theta}_{i})$ . Then, there exists  $\theta_i^\prime \in \mathsf{WC}(\widehat{\Theta}_i)$  such that  $\theta_{i}\succeq \theta_{i}^{\prime}$ . Let  $\pmb{\theta}_{-i}$  be arbitrary. Agent  $i$  's utility is  $\sum_{j = 1}^{n}\theta_{j}[\alpha^{*}] - \min_{\widetilde{\theta}_{i}\in \mathsf{WC}(\widehat{\Theta}_{i})}(\max_{\alpha}\sum_{j\neq i}\theta_{j}[\alpha ] + \widetilde{\theta}_{i}[\alpha ])\geq$ $\sum_{j = 1}^{n}\theta_{j}[\alpha^{*}] - (\max_{\alpha}\sum_{j\neq i}\theta_{j}[\alpha ] + \theta_{i}^{\prime}[\alpha ])\geq \sum_{j = 1}^{n}\theta_{j}[\alpha^{*}] - (\max_{\alpha}\sum_{j\neq i}\theta_{j}[\alpha ] + \theta_{i}[\alpha ]) = 0,$  so  $i\in \mathcal{I}$ , as desired.
Proof of Theorem 3.6. The weakest competitor in  $\widehat{\Theta}_i$  relative to  $\theta_{-i}$  is the solution  $\widetilde{\theta}_i\in \mathbb{R}^\Gamma$  to the linear program
$$
\min  \left\{\gamma : \begin{array}{l} \widetilde {\theta} _ {i} [ \alpha ] + \sum_ {j \neq i} \theta_ {j} [ \alpha ] \leq \gamma \forall \alpha \in \Gamma , \\ \widetilde {\theta} _ {i} \in \widehat {\Theta} _ {i}, \gamma \geq 0 \end{array} \right\}
$$
with  $|\Gamma| + 1$  variables and  $|\Gamma| + size(\widehat{\Theta}_i)$  constraints. Generating the first set of constraints requires computing  $\sum_{j \neq i} \theta_j[\alpha]$  for each  $\alpha \in \Gamma$ , which takes time  $\leq n|\Gamma|$ .
Checking membership of  $\theta_{i}$  in  $\mathsf{WCH}(\widehat{\Theta}_i)$  is equivalent to checking feasibility of a polytope
$$
\theta_ {i} \in \mathsf {W C H} (\widehat {\Theta} _ {i}) \iff \left\{\widetilde {\theta} _ {i}: \widetilde {\theta} _ {i} \in \widehat {\Theta} _ {i}, \theta_ {i} [ \alpha ] \geq \widetilde {\theta} _ {i} [ \alpha ] \forall \alpha \in \Gamma \right\} \neq \emptyset
$$
defined by  $size(\widehat{\Theta}_i) + |\Gamma|$  constraints.
More generally, the complexity of the above two mathematical programs is determined by the complexity of constraints needed to define  $\widetilde{\Theta}_i$ : for example, if  $\widetilde{\Theta}_i$  is a convex set then they are convex programs. Naturally, a major caveat of this brief discussion on computational complexity is that  $|\Gamma|$  can be very large (for example,  $|\Gamma|$  is exponential in combinatorial auctions).
A.3 Omitted proofs from Section 4
Proof of Lemma 4.1. Let  $p_i$  denote the payment collected from agent  $i$ , where  $i$  is an agent such that  $\theta_i \in \widehat{\Theta}_i$ . Let  $\theta_i^*$  be the weakest competitor in  $\widehat{\Theta}_i$  with respect to  $\theta_{-i}$ . The utility for agent  $i$  under  $\mathcal{M}$  is
$$
\begin{array}{l} \theta_ {i} \left[ \alpha_ {\text {o p t}} \right] - p _ {i} = \sum_ {j = 1} ^ {n} \theta_ {j} \left[ \alpha_ {\text {o p t}} \right] - \min  _ {\widehat {\theta} _ {i} \in \widehat {\Theta} _ {i}} \left(\max  _ {\alpha \in \Gamma} \sum_ {j \neq i} \theta_ {j} [ \alpha ] + \widehat {\theta} _ {i} [ \alpha ]\right) \\ = \sum_ {j = 1} ^ {n} \theta_ {j} [ \alpha_ {\mathrm {o p t}} ] - \left(\max  _ {\alpha \in \Gamma} \sum_ {j \neq i} \theta_ {j} [ \alpha ] + \theta_ {i} ^ {*} [ \alpha ]\right) \\ \leq \sum_ {j = 1} ^ {n} \theta_ {j} [ \alpha_ {\mathrm {o p t}} ] - \left(\sum_ {j \neq i} \theta_ {j} [ \alpha_ {\mathrm {o p t}} ] + \theta_ {i} ^ {*} [ \alpha_ {\mathrm {o p t}} ]\right) \\ = \theta_ {i} [ \alpha_ {\text {o p t}} ] - \theta_ {i} ^ {*} [ \alpha_ {\text {o p t}} ] \\ \leq \max _ {\widehat {\theta} _ {i} \in \mathsf {W C} (\widehat {\Theta} _ {i})} \left\| \theta_ {i} - \widehat {\theta} _ {i} \right\| _ {\infty} = d _ {H} (\theta_ {i}, \mathsf {W C} (\widehat {\Theta} _ {i})), \\ \end{array}
$$
Proof of Theorem 4.4. For each agent  $i$ , let  $k_{i}^{*}$  be the smallest  $k \in \{0, \dots, K_{i}\}$  such that  $\theta_{i} \in \mathsf{WCH}(\mathcal{B}(\widetilde{\Theta}_{i}, \zeta_{i} + 2^{k} \cdot \lambda_{i}))$ . Equivalently,  $k_{i}^{*}$  is the minimal  $k$  such that  $\gamma_{i}^{V} \leq \zeta_{i} + 2^{k} \lambda_{i}$ . So  $k_{i}^{*} = \lceil \log_{2}^{+}((\gamma_{i}^{V} - \zeta_{i}) / \lambda_{i}) \rceil$ . We have, using the fact that  $\theta_{i} \in \widehat{\Theta}_{i} \Rightarrow i \in \mathcal{I}$  (Theorem 3.5),
$$
\mathbb {E} [ \text {w e l f a r e} ] = \mathbb {E} \left[ \sum_ {i = 1} ^ {n} \theta_ {i} \left[ \alpha_ {\text {o p t}} \right] \cdot \mathbf {1} (i \in \mathcal {I}) \right] \geq \mathbb {E} \left[ \sum_ {i = 1} ^ {n} \theta_ {i} \left[ \alpha_ {\text {o p t}} \right] \cdot \mathbf {1} \left(\theta_ {i} \in \widehat {\Theta} _ {i}\right) \right] = \sum_ {i = 1} ^ {n} \theta_ {i} \left[ \alpha_ {\text {o p t}} \right] \cdot \Pr (\theta_ {i} \in \widehat {\Theta} _ {i})
$$
$$
\Pr (\theta_ {i} \in \widehat {\Theta} _ {i}) = \Pr (k _ {i} \geq k _ {i} ^ {*}) = 1 - \Pr (k _ {i} <   k _ {i} ^ {*}) = 1 - \frac {k _ {i} ^ {*}}{1 + K _ {i}} = 1 - \frac {\lceil \log_ {2} ^ {+} ((\gamma_ {i} ^ {V} - \zeta_ {i}) / \lambda_ {i}) \rceil}{1 + \lceil \log_ {2} ((H - \zeta_ {i}) / \lambda_ {i}) \rceil}.
$$
$$
\mathbb {E} [ \text {w e l f a r e} ] \geq \left(1 - \max  _ {i} \frac {\lceil \log_ {2} ^ {+} ((\gamma_ {i} ^ {V} - \zeta_ {i}) / \lambda_ {i}) \rceil}{1 + \lceil \log_ {2} ((H - \zeta_ {i}) / \lambda_ {i}) \rceil}\right) \cdot O P T.
$$
If all predictions are valid, we get  $\mathbb{E}[\mathrm{welfare}] = \mathrm{OPT}$ . The other term of the bound follows from  $\operatorname*{Pr}(\theta_i\in \widehat{\Theta}_i)\geq \operatorname*{Pr}(k_i = k_i^*) = \frac{1}{1 + K_i}$ .