SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Proof of Theorem 4.5. Let  $k_{i}^{*}$  be defined as in the proof of Theorem 4.4. We compute expected revenue by computing  $\mathbb{E}[p_i]$  for each agent  $i$ . Let  $S = \{i : \theta_{i} \in \widehat{\Theta}_{i}\}$  be the (random) set of agents with valid predictions post expansion. We have
$$
\mathbb {E} [ p _ {i} ] \geq \mathbb {E} [ p _ {i} \mid k _ {i} = k _ {i} ^ {*} ] \cdot \Pr (k _ {i} = k _ {i} ^ {*}) = \frac {1}{1 + K _ {i}} \cdot \mathbb {E} [ p _ {i} \mid k _ {i} = k _ {i} ^ {*} ].
$$
Now  $k_{i} = k_{i}^{*} \Rightarrow i \in S$ , so we may apply the payment bound of Lemma 4.1:
$$
\begin{array}{l} \mathbb {E} [ p _ {i} \mid k _ {i} = k _ {i} ^ {*} ] \geq \mathbb {E} \left[ \theta_ {i} [ \alpha_ {\mathrm {o p t}} ] - d _ {H} (\theta_ {i}, \mathsf {W C} (\mathcal {B} (\widetilde {\Theta} _ {i}, \zeta_ {i} + 2 ^ {k _ {i} ^ {*}} \lambda_ {i}))) \mid k _ {i} = k _ {i} ^ {*} \right] \\ = \theta_ {i} \left[ \alpha_ {\text {o p t}} \right] - d _ {H} \left(\theta_ {i}, \mathsf {W C} \left(\mathcal {B} \left(\widetilde {\Theta} _ {i}, \zeta_ {i} + 2 ^ {k _ {i} ^ {*}} \lambda_ {i}\right)\right)\right). \\ \end{array}
$$
Next, we bound  $d_{H}(\theta_{i},\mathsf{WC}(\mathcal{B}(\widetilde{\Theta}_{i},\zeta_{i} + 2^{k_{i}^{*}}\lambda_{i})))$ . Let  $\widetilde{\theta}_i\in \mathsf{WC}(\mathcal{B}(\widetilde{\Theta}_i,\zeta_i + 2^{k_i^*}\lambda_i))$  be arbitrary. By Lemma A.1 (the statement and proof are at the end of Appendix A.3),  $\widetilde{\theta}_i\in \mathsf{WC}(\mathcal{B}(\mathsf{WC}(\widetilde{\Theta}_i),\zeta_i + 2^{k_i^*}\lambda_i))$ , so there exists  $\theta_i^\prime \in \mathsf{WC}(\widetilde{\Theta}_i)$  such that  $\| \widetilde{\theta}_i - \theta_i'\|_{\infty}\leq \zeta_i + 2^{k_i^*}\lambda$ . Moreover,  $\| \theta_{i} - \theta_{i}^{\prime}\|_{\infty}\leq \gamma_{i}^{A}$  by definition of  $\gamma_i^A$ . The triangle inequality therefore yields
$$
\left\| \theta_ {i} - \widetilde {\theta} _ {i} \right\| _ {\infty} \leq \left\| \widetilde {\theta} _ {i} - \theta_ {i} ^ {\prime} \right\| _ {\infty} + \left\| \theta_ {i} - \theta_ {i} ^ {\prime} \right\| _ {\infty} \leq \gamma_ {i} ^ {A} + \zeta_ {i} + 2 ^ {k _ {i} ^ {*}} \lambda_ {i},
$$
so, as  $\widetilde{\theta}_i\in \mathsf{WC}(\mathcal{B}(\widetilde{\Theta}_i,\zeta_i + 2^{k_i^*}\lambda_i))$  was arbitrary,  $d_H(\theta_i,\mathsf{WC}(\mathcal{B}(\widetilde{\Theta}_i,r_i^*)))\leq \gamma_i^A +\zeta_i + 2^{k_i^*}\lambda_i$ . Now, we claim  $2^{k_i^*}\lambda_i\leq \rho_i$ . To show this, consider two cases. If  $\zeta_{i} + \lambda_{i}\geq \gamma_{i}^{V}$ , that is,  $k_{i}^{*} = 0$ , then  $2^{k_{i}^{*}}\lambda_{i} = \lambda_{i} = \rho_{i}$ . If  $\zeta_{i} + \lambda_{i} < \gamma_{i}^{V}$ , then  $k_{i}^{*} > 0$  and we have  $\zeta_i + 2^{k_i^* -1}\lambda_i < \gamma_i^V\leq \zeta +2^{k_i^*}\lambda_i$ , so  $2^{k_{i}^{*}}\lambda < 2(\gamma_{i}^{V} - \zeta_{i}) = \rho_{i}$ . So
$$
d _ {H} \left(\theta_ {i}, \mathsf {W C} \left(\mathcal {B} \left(\widetilde {\Theta} _ {i}, \zeta_ {i} + 2 ^ {k _ {i} ^ {*}} \lambda_ {i}\right)\right)\right) \leq \gamma_ {i} ^ {A} + \zeta_ {i} + \rho_ {i}.
$$
$$
\mathbb {E} [ p _ {i} ] \geq \frac {1}{1 + K _ {i}} \left(\theta_ {i} [ \alpha_ {\mathrm {o p t}} ] - (\gamma_ {i} ^ {A} + \zeta_ {i} + \rho_ {i})\right),
$$
$$
\begin{array}{l} \mathbb {E} [ \text {r e v e n u e} ] = \mathbb {E} \left[ \sum_ {i = 1} ^ {n} p _ {i} \right] = \sum_ {i = 1} ^ {n} \mathbb {E} [ p _ {i} ] \geq \sum_ {i = 1} ^ {n} \frac {1}{1 + K _ {i}} \left(\theta_ {i} \left[ \alpha_ {\text {o p t}} \right] - \left(\gamma_ {i} ^ {A} + \zeta_ {i} + \rho_ {i}\right)\right) \\ \geq \frac {1}{1 + \lceil \max _ {i} \log_ {2} (\frac {H - \zeta_ {i}}{\lambda_ {i}}) \rceil} \left(\mathsf {O P T} - \sum_ {i = 1} ^ {n} (\gamma_ {i} ^ {A} + \zeta_ {i} + \rho_ {i})\right), \\ \end{array}
$$
Proof of Theorem 4.6. Let  $k_{i}^{*}$  be defined as in Theorems 4.4 and 4.5. We bound  $\mathbb{E}[p_i]$  similarly to the approach in the proof of Theorem 4.5, but account for all possible values of  $k_{i}$  (rather than only conditioning on  $k_{i} = k_{i}^{*}$ ). If  $k_{i} < k_{i}^{*}$ , then agent  $i$  does not participate and pays nothing. We have
$$
\begin{array}{l} \mathbb {E} [ p _ {i} ] = \sum_ {k = k _ {i} ^ {*}} ^ {K _ {i}} \mathbb {E} [ p _ {i} | k _ {i} = k ] \cdot \operatorname * {P r} (k _ {i} = k) \geq \frac {1}{1 + K _ {i}} \sum_ {k = k _ {i} ^ {*}} ^ {K _ {i}} \Big (\theta_ {i} [ \alpha_ {\mathrm {o p t}} ] - d _ {H} (\theta_ {i}, \mathsf {W C} (\mathcal {B} (\widetilde {\Theta} _ {i}, \zeta_ {i} + 2 ^ {k} \cdot \lambda_ {i}))) \Big) \\ \geq \frac {1}{1 + K _ {i}} \sum_ {k = k _ {i} ^ {*}} ^ {K _ {i}} \left(\theta_ {i} [ \alpha_ {\mathrm {o p t}} ] - (\gamma_ {i} ^ {A} + \zeta_ {i} + 2 ^ {k} \cdot \lambda_ {i})\right) \\ = \left(1 - \frac {k _ {i} ^ {*}}{1 + K _ {i}}\right) \left(\theta_ {i} [ \alpha_ {\mathrm {o p t}} ] - \gamma_ {i} ^ {A} - \zeta_ {i}\right) - \frac {\lambda_ {i}}{1 + K _ {i}} \sum_ {k = k _ {i} ^ {*}} ^ {K _ {i}} 2 ^ {k} \\ \geq \left(1 - \frac {k _ {i} ^ {*}}{1 + K _ {i}}\right) \left(\theta_ {i} \left[ \alpha_ {\text {o p t}} \right] - \gamma_ {i} ^ {A} - \zeta_ {i}\right) - \frac {\lambda 2 ^ {K _ {i} + 1}}{1 + K _ {i}} \\ \end{array}
$$
where in the second inequality we have used the bound  $d_H(\theta_i, \mathsf{WC}(\mathcal{B}(\widetilde{\Theta}_i, \zeta_i + 2^k \lambda_i))) \leq \gamma_i^A + \zeta_i + 2^k \lambda_i$ , which was derived in the proof of Theorem 4.5. We have  $2^{K_i + 1} \leq \lambda_i (4(H - \zeta_i) / \lambda_i) \leq 4H$ . Substituting and summing over agents yields the desired revenue bound.
Lemma A.1.  $\mathsf{WC}(\mathcal{B}(\widetilde{\Theta},r)) = \mathsf{WC}(\mathcal{B}(\mathsf{WC}(\widetilde{\Theta}),r)).$
Proof. We first prove the forwards containment. For the sake of contradiction suppose there exists  $\widetilde{\theta} \in \mathsf{WC}(\mathcal{B}(\widetilde{\Theta}, r))$  such that  $\widetilde{\theta} \notin \mathsf{WC}(\mathcal{B}(\mathsf{WC}(\widetilde{\Theta}), r))$ . Then, there exists  $\theta' \in \mathcal{B}(\mathsf{WC}(\widetilde{\Theta}), r)$  such that  $\theta' \preccurlyeq \widetilde{\theta}$ . But
$$
\mathsf {W C} (\widetilde {\Theta}) \subseteq \widetilde {\Theta} \Rightarrow \mathcal {B} (\mathsf {W C} (\widetilde {\Theta}), r) \subseteq \mathcal {B} (\widetilde {\Theta}, r),
$$
so  $\theta^{\prime}\in \mathcal{B}(\widetilde{\Theta},r)$  and  $\theta^{\prime}\preccurlyeq \widetilde{\theta}$  which contradicts the assumption that  $\widetilde{\theta}\in \mathsf{WC}(\mathcal{B}(\widetilde{\Theta},r))$
We now prove the reverse containment. For the sake of contradiction suppose there exists  $\widetilde{\theta} \in \mathsf{WC}(\mathcal{B}(\mathsf{WC}(\widetilde{\Theta}), r))$  such that  $\widetilde{\theta} \notin \mathsf{WC}(\mathcal{B}(\widetilde{\Theta}, r))$ . Then, there exists  $\theta' \in \mathcal{B}(\widetilde{\Theta}, r)$  such that  $\theta' \preccurlyeq \widetilde{\theta}$ . Furthermore, if  $\theta' \notin \mathsf{WC}(\mathcal{B}(\widetilde{\Theta}, r))$ , there exists  $\theta'' \in \mathsf{WC}(\mathcal{B}(\widetilde{\Theta}, r))$  such that  $\theta'' \preccurlyeq \theta' \preccurlyeq \theta'$  (if  $\theta' \in \mathsf{WC}(\mathcal{B}(\widetilde{\Theta}, r))$ , set  $\theta'' = \theta'$ ). From the forward inclusion, we have
$$
\mathsf {W C} (\mathcal {B} (\widetilde {\Theta}, r)) \subseteq \mathsf {W C} (\mathcal {B} (\mathsf {W C} (\widetilde {\Theta}), r)) \subseteq \mathcal {B} (\mathsf {W C} (\widetilde {\Theta}), r),
$$
so  $\theta'' \in \mathcal{B}(\mathsf{WC}(\widetilde{\Theta}), r)$  and  $\theta'' \preccurlyeq \widetilde{\theta}$  which contradicts the assumption that  $\widetilde{\theta} \in \mathsf{WC}(\mathcal{B}(\mathsf{WC}(\widetilde{\Theta}), r))$ .
A.4 Omitted proofs from Section 5
To prove Theorem 5.1 we need the following more direct bound on  $p_i$  in terms of the weakest competitor's value.
Lemma A.2. Run  $\mathcal{M}$  with  $\widehat{\Theta}_i$ . If  $i$  is such that  $\theta_i \in \mathsf{WCH}(\widehat{\Theta}_i)$ , then  $p_i \geq \widetilde{\theta}_i[\alpha_{\mathrm{opt}}]$  where  $\widetilde{\theta}_i$  is the weakest competitor in  $\widehat{\Theta}_i$  relative to  $\theta_{-i}$ .
Proof of Lemma A.2. Truncating the proof of Lemma 4.1 yields the desired statement.
Proof of Theorem 5.1. We have  $\mathbb{E}[\mathrm{welfare}] \geq \sum_{i=1}^{n} \theta_i[\alpha_{\mathrm{opt}}] \cdot \operatorname*{Pr}(\theta_i \in \widehat{\Theta}_i) \geq \sum_{i=1}^{n} \theta_i[\alpha_{\mathrm{opt}}] \cdot \operatorname*{Pr}(\ell_i = \log_2 H) = \frac{1}{\log_2 H} \cdot \mathrm{OPT}$  (since  $\theta_i \succeq \widetilde{\theta}_i (\log_2 H, \ldots, \log_2 H)$ ). The proof of the revenue guarantee relies on the following key claim: for each agent  $i$ , there exists  $\ell_{i,1}^*, \ldots, \ell_{i,k}^* \in \{1, \ldots, \log_2 H\}$  such that  $\widetilde{\theta}(\ell_{i,1}^*, \ldots, \ell_{i,k}^*) \succeq \frac{1}{2} \theta_i$ . To show this, let  $\theta_i^j$  denote the projection of  $\theta_i$  onto  $u_j$ , so  $\theta_i = \sum_{j=1}^{k} \theta_i^j$  since  $\{u_{i,1}, \ldots, u_{i,k}\}$  is an orthonormal basis. Let  $\ell_{i,j}^* = \min\{\ell : \theta_i^j \succeq z_{i,j}^\ell\}$ . Then,  $z_{i,j}^{\ell_{i,j}^*} \succeq \frac{1}{2} \theta_i^j$ , so
$$
\widetilde {\theta} \left(\ell_ {i, 1} ^ {*}, \dots , \ell_ {i, k} ^ {*}\right) = \sum_ {j = 1} ^ {k} z _ {i, j} ^ {\ell_ {i, j} ^ {*}} \succeq \sum_ {j = 1} ^ {k} \frac {1}{2} \theta_ {i} ^ {j} = \frac {1}{2} \theta_ {i}.
$$
We now bound the expected payment of agent  $i$  as in the previous results. Let  $\ell_{i}^{*} = \min_{j}\ell_{i,j}^{*}$ . We have
$$
\begin{array}{l} \mathbb {E} [ p _ {i} ] \geq \mathbb {E} \left[ p _ {i} \mid (\ell_ {i, 1}, \dots , \ell_ {i, k}) = (\ell_ {i, 1} ^ {*}, \dots , \ell_ {i, k} ^ {*}) \right] \cdot \Pr \left((\ell_ {i, 1}, \dots , \ell_ {i, k}) = (\ell_ {i, 1} ^ {*}, \dots , \ell_ {i, k} ^ {*})\right) \\ = \frac {1}{\left| W _ {\ell_ {i} ^ {*}} \right| \log_ {2} H} \cdot \mathbb {E} \left[ p _ {i} \mid (\ell_ {i, 1}, \dots , \ell_ {i, k}) = (\ell_ {i, 1} ^ {*}, \dots , \ell_ {i, k} ^ {*}) \right] \\ \geq \frac {1}{\log_ {2} H ((\log_ {2} H) ^ {k} - (\log_ {2} H - 1) ^ {k})} \cdot \mathbb {E} \left[ p _ {i} \mid (\ell_ {i, 1}, \dots , \ell_ {i, k}) = (\ell_ {i, 1} ^ {*}, \dots , \ell_ {i, k} ^ {*}) \right] \\ \geq \frac {1}{k (\log_ {2} H) ^ {k}} \cdot \mathbb {E} \left[ p _ {i} \mid (\ell_ {i, 1}, \ldots , \ell_ {i, k}) = (\ell_ {i, 1} ^ {*}, \ldots , \ell_ {i, k} ^ {*}) \right] \\ \end{array}
$$
since the probability of obtaining the correct weakest competitor  $\tilde{\theta} (\ell_{i,1}^{*},\dots ,\ell_{i,k}^{*})$  can be written as the probability of drawing the correct "level"  $\ell_i^*\in \{1,\ldots ,\log_2H\}$  times the probability of drawing the correct weakest competitor within the correct level  $W_{\ell_i^*}$ . We bound the conditional expectation with Lemma A.2,
$$
\mathbb {E} \left[ p _ {i} \mid (\ell_ {i, 1}, \dots , \ell_ {i, k}) = (\ell_ {i, 1} ^ {*}, \dots , \ell_ {i, k} ^ {*}) \right] \geq \widetilde {\theta} _ {i} \left(\ell_ {i, 1} ^ {*}, \dots , \ell_ {i, k} ^ {*}\right) [ \alpha_ {\mathrm {o p t}} ] \geq \frac {1}{2} \cdot \theta_ {i} [ \alpha_ {\mathrm {o p t}} ].
$$
$$
\mathbb {E} [ \text {r e v e n u e} ] = \sum_ {i = 1} ^ {n} \mathbb {E} [ p _ {i} ] \geq \frac {1}{2 k (\log_ {2} H) ^ {k}} \cdot \sum_ {i = 1} ^ {n} \theta_ {i} [ \alpha_ {\text {o p t}} ] = \frac {1}{2 k (\log_ {2} H) ^ {k}} \cdot \text {O P T},
$$
B Consistency and robustness
In this section we show that it is trivial to obtain high consistency and robustness ratios with an otherwise undesirable mechanism that yields poor revenue even if predictions are nearly perfect. We also discuss the consistency and robustness of our main mechanism  $\mathcal{M}_{\zeta, \lambda}$ .
For  $S \subseteq \{1, \dots, n\}$ , let  $\mathrm{OPT}_S = \sum_{i \in S} \theta_i[\alpha_{\mathrm{opt}}]$  be the welfare generated by the efficient allocation restricted to agents in  $S$ . Let  $\mathrm{VCG}(S)$  denote the revenue of the vanilla VCG mechanism when run among the agents in  $S$ . Let  $\mathrm{VCG}(\beta) = \mathbb{E}[\mathrm{VCG}(S)]$  where  $S \subseteq \{1, \dots, n\}$  is sampled by including each agent in  $S$  independently with probability  $\beta$ . In general,  $S \subseteq T \Rightarrow \mathrm{VCG}(S) \leq \mathrm{VCG}(T)$  [44], so  $\mathrm{VCG}(\beta)$  need not be increasing in  $\beta$ , but there are various sufficient conditions when revenue monotonicity does hold [6, 29, 54].
B.1 First approach: trust predictions completely
The first basic instantiation of our meta mechanism  $\mathcal{M}$  is the following: the mechanism designer simply sets  $\widehat{\Theta}_i = \mathsf{WCH}(\widetilde{\Theta}_i)$  for all  $i$ . Let  $V = \{i : \theta_i \in \mathsf{WCH}(\widetilde{\Theta}_i)\} = \{i : \gamma_i^V = 0\}$  denote the set of agents with valid predictions. The welfare of this mechanism is simply  $\mathsf{OPT}_V$  and its revenue is bounded by Lemma 4.1. If all predictions are valid and perfect, that is,  $\mathsf{WC}(\widetilde{\Theta}_i) = \{\theta_i\}$  for all  $i$ , both welfare and revenue are equal to  $\mathsf{OPT}$ . However, if all predictions are such that  $\theta_i \notin \mathsf{WCH}(\widetilde{\Theta}_i)$ , both welfare and revenue potentially drop to 0. So this mechanism is (1, 1)-consistent and (0, 0)-robust.
B.2 Second approach: discard predictions randomly
The issue with the above mechanism is that if all predictions are invalid, it generates no welfare and no revenue. We show how randomization can quell that issue. One trivial solution is to discard all predictions with probability  $\beta$ , and trust all predictions completely with probability  $(1 - \beta)$ . That is, with probability  $\beta$  set  $(\widehat{\Theta}_1,\ldots ,\widehat{\Theta}_n) = (\Theta ,\ldots ,\Theta)$  and with probability  $1 - \beta$  set  $(\widehat{\Theta}_{1},\ldots ,\widehat{\Theta}_{n}) = (\mathsf{WCH}(\widetilde{\Theta}_{1}),\ldots ,\mathsf{WCH}(\widetilde{\Theta}_{n}))$ , and then run  $\mathcal{M}$ . This mechanism achieves strong consistency and robustness ratios. Let  $V = \{i:\theta_{i}\in \mathsf{WCH}(\widetilde{\Theta}_{i})\}$  be the set of valid predictions. From Lemma 4.1, we have  $\mathbb{E}[\text{welfare}] = \beta \cdot \mathsf{OPT} + (1 - \beta)\cdot \mathsf{OPT}_V$  and  $\mathbb{E}[\text{revenue}] \geq$
$\beta \cdot \mathsf{VCG}(1) + (1 - \beta)\cdot \left(\mathsf{OPT}_V - \sum_{i\in V}\gamma_i^A\right)$ , and thus obtain  $(1,1 - \beta)$ -consistency and  $(\beta ,\beta)$ -robustness.
This approach suffers from a major issue: its revenue drops drastically the moment predictions are invalid  $(\gamma_{i}^{V} > 0)$ . In particular, if predictions are highly accurate but very slightly invalid (such as the blue prediction in Figure 1), this approach completely misses out on any payments from such agents and drops to the revenue of VCG (which can be drastically smaller than OPT). But, a tiny expansion of these predictions would have sufficed to increase revenue significantly and perform competitively with OPT. One simple approach is to set  $\widehat{\Theta}_i$  to be an expansion of  $\widetilde{\Theta}_i$  by a parameter  $\eta_{i}$  with some probability, and discard the prediction with complementary probability. If  $\gamma_i^V\leq \eta_i$  for all  $i$ , then such a mechanism would perform well. The main issue with such an approach is that the moment  $\gamma_{i}^{V} > \eta_{i}$ , our expansion by  $\eta_{i}$  fails to capture the true type  $\theta_{i}$  and the performance drastically drops. Our main mechanism  $\mathcal{M}_{\zeta ,\lambda}$  essentially selects the  $\eta_{i}$  randomly from a suitable discretization of the ambient type space to be able to capture  $\theta_{i}$  with reasonable probability.
B.3 Consistency and robustness of  $\mathcal{M}_{\zeta, \lambda}$
In this discussion we assume constant  $\zeta, \lambda > 0$ . Assuming revenue monotonicity, since  $\theta_i \in \widehat{\Theta}_i$  with probability at least  $\Omega(1 / \log H)$ , the revenue of our mechanism is never worse than  $\mathsf{VCG}(\Omega(1 / \log H))$ . Thus, in the language of algorithms-with-predictions,  $\mathcal{M}_{\zeta, \lambda}$  is  $(1, \Omega(1 / \log H))$ -consistent and, assuming revenue monotonicity,  $(\Omega(1 / \log H), \mathsf{VCG}(\Omega(1 / \log H))) / \mathsf{VCG}(1))$ -robust. If VCG revenue is submodular, the robustness ratio is  $\geq \Omega(1 / \log H)$  (but in general revenue can shrink by more than this ratio [12]). In contrast to the trivial approach that either trusted the side information completely or discarded predictions completely, our random expansion approach does not suffer from large discontinuous drops in welfare nor revenue.
Furthermore, the previous approaches had no way of gracefully dealing with invalid prediction sets. In particular, if  $\widetilde{\Theta}_i$  is an invalid prediction, even if a tiny expansion of  $\widetilde{\Theta}_i$  would have captured  $\theta_i$  (such as the blue set in Figure 1), we gave up on getting any meaningful revenue from agent  $i$ . When all predictions were invalid (that is,  $\gamma_i^V > 0$  for all  $i$ ), our guarantee dropped to  $\beta \cdot \mathrm{VCG}(1)$ . The random expansion of predictions remedies these issues. Its revenue is nearly  $\log H$ -competitive with OPT as long as  $\sum_{i} \gamma_i^A + 2 \gamma_i^V$  is not too large. In particular, if we have high-accuracy but invalid predictions that are just a small expansion away from capturing  $\theta_i$ , the mechanism in this section is nearly  $\log H$ -competitive with OPT whereas the mechanism from the previous section is compared to vanilla VCG due to invalid predictions.
C An expressive language for side information
A side information structure is a probability space  $(\Theta, \mathcal{F}, \mu)$  where the ambient type space  $\Theta$  is the sample space,  $\mathcal{F}$  is a  $\sigma$ -algebra on  $\Theta$ , and  $\mu$  is a probability measure. (We suppress the agent index for brevity.)
$\mathcal{F}$  induces a partition of  $\Theta$  into equivalence classes where  $\theta \equiv \theta'$  if  $\mathbf{1}(\theta \in A) = \mathbf{1}(\theta' \in A)$  for all  $A \in \mathcal{F}$  (so the side-information structure does not distinguish between  $\theta$  and  $\theta'$ ). Let  $A_{\theta} = \{\theta': \theta \equiv \theta'\} \in \mathcal{F}$  be the equivalence class of  $\theta$ . In this way the  $\sigma$ -algebra determines the granularity of knowledge being conveyed by the side information structure, and the probability measure  $\mu: \mathcal{F} \to [0,1]$  establishes uncertainty over this knowledge.
We define invalidity and inaccuracy of a side information structure in the natural way. Define random variables  $X_{i}^{V}, X_{i}^{A}: \Theta \to \mathbb{R}_{\geq 0}$  by
$$
X _ {i} ^ {V} (\theta) = \gamma_ {i} ^ {V} (A _ {\theta}) = d (\theta_ {i}, \mathsf {W C H} (A _ {\theta}))
$$
$$
X _ {i} ^ {A} (\theta) = \gamma_ {i} ^ {A} (A _ {\theta}) = d _ {H} \left(\theta_ {i}, \mathsf {W C} (A _ {\theta})\right).
$$
$X_{i}^{V}$  and  $X_{i}^{A}$  are  $\mathcal{F}$ -measurable since they are (by definition) constant on all atoms of  $\mathcal{F}$  (sets  $A\in \mathcal{F}$  such that no nonempty  $B\subsetneq A$  is in  $\mathcal{F}$ ). The invalidity/inaccuracy distributions on  $\mathbb{R}_{\geq 0}$  are given by
$$
\operatorname * {P r} (a \leq X _ {i} ^ {V} \leq b) = \mu (\{\theta \in \Theta : a \leq \gamma_ {i} ^ {V} (A _ {\theta}) \leq b \}) = \mu (\cup \{A _ {\theta}: a \leq \gamma_ {i} ^ {V} (A _ {\theta}) \leq b \}).
$$
The generalized version of  $M_{\zeta, \lambda}$  that receives as input a side information structure for each agent  $i$  given by  $(\Theta_i, \mathcal{F}_i, \mu_i)$  works as follows. It samples  $\widetilde{\theta}_i \sim \Theta_i$  according to  $(\mathcal{F}_i, \mu_i)$  and draws  $k_i \sim_{\mathrm{unif.}} \{0, \dots, K_i\}$  where  $K_i$  is defined as before. It then sets
$$
\widehat {\Theta} _ {i} = \mathcal {B} \left(A _ {\widetilde {\theta} _ {i}}, \zeta_ {i} + 2 ^ {k _ {i}} \lambda_ {i}\right).
$$
Executing the same analysis in the proofs of Theorems 4.4, 4.5, and 4.6 for a fixed  $\widetilde{\theta}_i$ , then taking expectation over the draw of  $\widetilde{\theta}_i$  yields similar guarantees with  $\gamma_i^V$  and  $\gamma_i^A$  replaced by  $\mathbb{E}[X_i^V]$  and  $\mathbb{E}[X_i^A]$ , respectively. To show this, we loosen the bounds in Theorems 4.4, 4.5, and 4.6 slightly to make the multiplicative terms convex. As  $X_i^V \geq 0$ , we have
$$
\left. \right. \left\lceil \log_ {2} ^ {+} \left(\frac {X _ {i} ^ {V} - \zeta_ {i}}{\lambda_ {i}}\right)\right\rceil = \left\lceil \max  \left(0, \log_ {2} \left(\frac {X _ {i} ^ {V} - \zeta_ {i}}{\lambda_ {i}}\right)\right)\right\rceil \leq 1 + \log_ {2} \left(1 + \frac {X _ {i} ^ {V} - \zeta_ {i}}{\lambda_ {i}}\right)
$$
$$
\begin{array}{l} \mathbb {E} \left[ 1 - \frac {\lceil \log_ {2} ^ {+} ((X _ {i} ^ {V} - \zeta_ {i}) / \lambda_ {i}) \rceil}{1 + \lceil \log_ {2} ((H - \zeta_ {i}) / \lambda_ {i}) \rceil} \right] \geq \mathbb {E} \left[ 1 - \frac {1 + \log_ {2} (1 + (X _ {i} ^ {V} - \zeta_ {i}) / \lambda_ {i})}{1 + \lceil \log_ {2} ((H - \zeta_ {i}) / \lambda_ {i}) \rceil} \right] \\ \geq 1 - \frac {1 + \log_ {2} (1 + (\mathbb {E} [ X _ {i} ^ {V} ] - \zeta_ {i}) / \lambda_ {i})}{1 + \lceil \log_ {2} ((H - \zeta_ {i}) / \lambda_ {i}) \rceil} \\ \end{array}
$$
by Jensen's inequality. The corresponding versions of Theorems 4.4, 4.5, and 4.6 follow.
Theorem C.1.  $\mathbb{E}[welfare] \geq \max \{(1 - \max_i \frac{1 + \log_2(1 + (\mathbb{E}[X_i^V] - \zeta_i) / \lambda_i)}{1 + \lceil \log_2((H - \zeta_i) / \lambda_i) \rceil}), \frac{1}{1 + \lceil \max_i \log_2((H - \zeta_i) / \lambda_i) \rceil} \}\mathrm{OPT}$ .
Theorem C.2 (Revenue bound 1). Let  $\rho_{i} = \mathbb{E}[2(X_{i}^{V} - \zeta_{i})\mathbf{1}(\zeta_{i} + \lambda_{i} < X_{i}^{V}) + \lambda_{i}\mathbf{1}(\zeta_{i} + \lambda_{i}\geq X_{i}^{V})]$  Then  $\mathbb{E}[revenue]\geq \frac{1}{1 + \lceil\max_i\log_2((H - \zeta_i) / \lambda_i)\rceil} (\mathsf{OPT} - \sum_{i = 1}^n (\mathbb{E}[X_i^A ] + \zeta_i + \mathbb{E}[\rho_i]))$
Theorem C.3 (Revenue bound 2).  $\mathbb{E}[revenue] \geq (1 - \max_i \frac{1 + \log_2(1 + (\mathbb{E}[X_i^V] - \zeta_i) / \lambda_i)}{1 + [\log_2((H - \zeta_i) / \lambda_i)]})$  (OPT -  $\sum_{i=1}^{n} (\gamma_i^A + \zeta_i)) - \sum_{i=1}^{n} \frac{4H}{1 + [\log_2((H - \zeta_i) / \lambda_i)]}$ .
D Beyond the VCG mechanism: affine maximizers
Given agent-specific multipliers  $\omega = (\omega_{1},\dots ,\omega_{n})\in \mathbb{R}_{\geq 0}$  and an allocation-based boost function  $\lambda :\Gamma \to \mathbb{R}_{\geq 0}$ , we define the following meta-mechanism  $\mathcal{M}(\omega ,\lambda)$  which is a generalization of our meta-mechanism  $\mathcal{M}$ . The mechanism designer receives as input  $\widetilde{\Theta}_1,\ldots ,\widetilde{\Theta}_n$ , and based on these decides on prediction sets  $\widehat{\Theta}_1,\ldots ,\widehat{\Theta}_n$ . The agents are then asked to reveal their true types  $\theta_{1},\ldots ,\theta_{n}$ . The allocation used is
$$
\alpha_ {\omega , \lambda} = \operatorname * {a r g m a x} _ {\alpha \in \Gamma} \sum_ {i = 1} ^ {n} \omega_ {i} \theta_ {i} [ \alpha ] + \lambda (\alpha).
$$
$$
p _ {i} = \frac {1}{\omega_ {i}} \left[ \min  _ {\widetilde {\theta} _ {i} \in \widehat {\Theta} _ {i}} \left(\max  _ {\alpha \in \Gamma} \sum_ {j \neq i} \omega_ {j} \theta_ {j} [ \alpha ] + \omega_ {i} \widetilde {\theta} _ {i} [ \alpha ] + \lambda (\alpha)\right) - \left(\sum_ {j \neq i} \omega_ {j} \theta_ {j} [ \alpha_ {\omega , \lambda} ] + \lambda (\alpha_ {\omega , \lambda})\right) \right].
$$
$$
\mathcal {I} = \left\{i: \theta_ {i} [ \alpha_ {\omega , \lambda} ] - p _ {i} \geq 0 \right\}.
$$
Agents in  $i$  enjoy allocation  $\alpha_{\omega, \lambda}$  and pay  $p_i$ . Agents not in  $i$  do not participate and receive zero utility.