SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Theorem 3.3. Let  $\Theta = [0,H]^{\Gamma}$  and let  $\widetilde{\Theta} \subseteq \Theta$  be closed. Then  $\overline{\mathsf{WC}}(\widetilde{\Theta}) = \{\theta \in \widetilde{\Theta} : \{\theta' \in \widetilde{\Theta} : \theta' \prec \theta\} = \emptyset\}$ ,  $\mathsf{WC}(\widetilde{\Theta}) = \{\theta \in \widetilde{\Theta} : \{\theta' \in \widetilde{\Theta} : \theta' \preccurlyeq \theta\} = \emptyset\}$ , and  $\mathsf{WCH}(\widetilde{\Theta}) = \{\theta \in \Theta : \exists \theta' \in \widetilde{\Theta} \text{ s.t. } \theta \succeq \theta'\}$  is the upwards closure of  $\widetilde{\Theta}$ .
We now present our meta-mechanism, which we denote by  $\mathcal{M}$ . It uses the efficient allocation, but that allocation is enjoyed only by the subset of agents able to compete with the weakest competitors in the side information set.  $\mathcal{M}$  then implements the weakest-competitor payments on those agents.
Figure 1: Left: Example WC and WCH when  $|\Gamma| = 2$ .  $\overline{WC}(\widetilde{\Theta})$  is depicted in solid and dashed blue,  $\mathsf{WC}(\widetilde{\Theta})$  is depicted in solid blue, and  $\mathsf{WCH}(\widetilde{\Theta})$  is the region enclosed by  $\overline{WC}(\widetilde{\Theta})$ . Right: The prediction  $\widetilde{\Theta}_1$  (depicted in gray) is valid ( $\gamma^V = 0$ ), but is highly inaccurate. The prediction  $\widetilde{\Theta}_2$  (depicted in light blue) is invalid ( $\gamma^V > 0$ ), but is more accurate than  $\widetilde{\Theta}_1$ . A small expansion of  $\widetilde{\Theta}_2$  would yield a valid and highly accurate prediction.
The input subsets  $\widetilde{\Theta}_1, \ldots, \widetilde{\Theta}_n$  represent the side information/predictions given to the mechanism designer that postulate that  $\theta_i \in \widetilde{\Theta}_i$ .
Meta-mechanism  $\mathcal{M}$
Input: subsets  $\widetilde{\Theta}_1,\ldots ,\widetilde{\Theta}_n\subseteq \Theta$  given to mechanism designer.
- Based on  $\widetilde{\Theta}_1, \ldots, \widetilde{\Theta}_n$ , come up with  $\widehat{\Theta}_1, \ldots, \widehat{\Theta}_n$ .
Agents asked to reveal types  $\theta_{1},\ldots ,\theta_{n}$
- Let  $\alpha^{*} = \operatorname{argmax}_{\alpha \in \Gamma} \sum_{i=1}^{n} \theta_{i}[\alpha]$  and for each  $i$  let
$$
p _ {i} = \min _ {\widetilde {\theta} _ {i} \in \mathsf {W C} (\widehat {\Theta} _ {i})} \left(\max _ {\alpha \in \Gamma} \sum_ {j \neq i} \theta_ {j} [ \alpha ] + \widetilde {\theta} _ {i} [ \alpha ]\right) - \sum_ {j \neq i} \theta_ {j} [ \alpha^ {*} ].
$$
- Let  $\mathcal{I} = \{i : \theta_i[\alpha^*] - p_i \geq 0\}$ . If agent  $i \notin \mathcal{I}$ ,  $i$  is excluded and receives zero utility (zero value and zero payment). If agent  $i \in \mathcal{I}$ ,  $i$  enjoys allocation  $\alpha^*$  and pays  $p_i$ .
<sup>a</sup>One practical consideration is that this step might require a more nuanced implementation of an "outside option" for agents to be indifferent between participating and being excluded versus not participating at all. (We do not pursue this highly application-specific issue in this work.) In auction and matching settings this step is standard and innocuous; the agent simply receives no items.
Meta-mechanism  $\mathcal{M}$  generates welfare equal to  $\sum_{i\in \mathcal{I}}\theta_i[\alpha^* ]$  and revenue equal to  $\sum_{i\in \mathcal{I}}p_i$ .  $\mathcal{M}$  does not specify how to set  $\widehat{\Theta}_1,\ldots ,\widehat{\Theta}_n$  based on  $\widetilde{\Theta}_{1},\dots ,\widetilde{\Theta}_{n}$  (hence the "meta" label). This challenge is the subject of the later sections where we will describe, based on the setting, how to set the  $\widehat{\Theta}_i$  in order to generate high welfare and high revenue. We now establish the incentive properties of  $\mathcal{M}$ .
Theorem 3.4.  $\mathcal{M}$  is IC and IR.
Next we show that WCH precisely captures the set of agent types that never violate IR. This consideration does not arise in WCVCG since in that setting misreporting is limited to the set used in the weakest-competitor minimization, and hence IR is never violated. In our setting, we make no assumptions on the veracity of the sets  $\widehat{\Theta}_i$  and must therefore reckon with the possibility that an agent is unable to compete with the weakest competitors in  $\mathsf{WC}(\widehat{\Theta}_i)$ .
Theorem 3.5. Let  $\theta_{i}$  denote the true type of agent  $i$  and let  $\widehat{\Theta}_1,\ldots ,\widehat{\Theta}_n$  denote the side information sets used by  $\mathcal{M}$ . Then  $i\in \mathcal{I}$  for all  $\theta_{-i}\iff \theta_{i}\in \mathsf{WCH}(\widehat{\Theta}_{i})$ .
Theorem 3.5 shows that  $i$  is guaranteed to participate in  $\mathcal{M}$  regardless of other agents' types if and only if  $\theta_{i} \in \mathrm{WCH}(\widehat{\Theta}_{i})$ . We capitalize on this observation when we derive revenue guarantees for  $\mathcal{M}$ , since the welfare of  $\mathcal{M}$  is at least  $\sum_{i:\theta_{i} \in \mathrm{WCH}(\widehat{\Theta}_{i})} \theta_{i}[\alpha^{*}]$  and its revenue is at least  $\sum_{i:\theta_{i} \in \mathrm{WCH}(\widehat{\Theta}_{i})} p_{i}$ .
Before we proceed to our main analyses of the key properties and guarantees of  $\mathcal{M}$ , we briefly discuss its computational complexity. We consider the special case where the side-information sets are polytopes. Let  $size(\widehat{\Theta}_i)$  denote the encoding size of the constraints defining  $\widehat{\Theta}_i$ .
Theorem 3.6. Let  $\widehat{\Theta}_i$  be a polytope. Payment  $p_i$  in  $\mathcal{M}$  can be computed in poly  $(|\Gamma|, size(\widehat{\Theta}_i), n)$  time. Furthermore, determining membership in  $\mathsf{WCH}(\widehat{\Theta}_i)$  can be done in poly  $(|\Gamma|, size(\widehat{\Theta}_i))$  time.
4 Main guarantees of the mechanism in terms of prediction quality
In this section we prove our main guarantees on our meta-mechanism  $\mathcal{M}$  in terms of the quality of the side information  $\widetilde{\Theta}_1, \ldots, \widetilde{\Theta}_n$ . We will largely refer to the side information as predictions in this section to emphasize that  $\widetilde{\Theta}_i$  could be wildly incorrect/inaccurate. To state our results we need the following notation which will be used throughout the remainder of the paper. Given agent types  $\theta_1, \ldots, \theta_n$ , let  $\alpha_{\mathrm{opt}}$  denote the efficient allocation among the  $n$  agents and let  $\mathrm{OPT} = \max_{\alpha \in \Gamma} \sum_{i=1}^{n} \theta_i[\alpha] = \sum_{i=1}^{n} \theta_i[\alpha_{\mathrm{opt}}]$  denote the welfare of the efficient allocation (also called the total social surplus). Let  $\mathrm{VCG}$  denote VCG revenue on the  $n$  agents.
The following lemma shows that payment  $p_i$  in  $\mathcal{M}$  can be related to agent  $i$ 's value for  $\alpha_{\mathrm{opt}}$  if  $i$  has a valid side information set  $(\theta_i \in \mathsf{WCH}(\widehat{\Theta}_i))$ . We incur a loss term equal to the  $\ell_{\infty}$ -Hausdorff distance from the true type  $\theta_i$  to  $\mathsf{WC}(\widehat{\Theta}_i)$ , defined as  $d_H(\theta_i, \mathsf{WC}(\widehat{\Theta}_i)) := \max_{\widehat{\theta}_i \in \mathsf{WCH}(\widehat{\Theta}_i)} \| \theta_i - \widehat{\theta}_i \|_{\infty}$ .
Lemma 4.1. Run  $\mathcal{M}$  with  $\widehat{\Theta}_i$ . If  $i$  is such that  $\theta_i \in \mathsf{WCH}(\widehat{\Theta}_i)$ , then  $p_i \geq \theta_i[\alpha_{\mathrm{opt}}] - d_H(\theta_i, \mathsf{WC}(\widehat{\Theta}_i))$ .
Measuring the error of a prediction. Before instantiating  $\mathcal{M}$  with specific rules to determine the  $\widehat{\Theta}_i$  from the  $\widetilde{\Theta}_i$ , we define our notions of prediction error, which are motivated by Lemma 4.1.
Definition 4.2. The invalidity of a prediction  $\widetilde{\Theta}_i$ , denoted by  $\gamma_i^V$ , is the distance from the true type  $\theta_i$  of agent  $i$  to  $\mathsf{WCH}(\widetilde{\Theta}_i)$ :  $\gamma_i^V := d(\theta_i, \mathsf{WCH}(\widetilde{\Theta}_i)) = \min_{\widetilde{\theta}_i \in \mathsf{WCH}(\widetilde{\Theta}_i)} \| \theta_i - \widetilde{\theta}_i \|_\infty$ .
Definition 4.3. The inaccuracy of a prediction  $\widetilde{\Theta}_i$  is the quantity  $\gamma_i^A := d_H(\theta_i, \mathsf{WC}(\widetilde{\Theta}_i))$ .
We say that a prediction  $\widetilde{\Theta}_i$  is valid if  $\gamma_i^V = 0$ , that is,  $\theta_i \in \mathsf{WCH}(\widetilde{\Theta}_i)$ . We say that a prediction is perfect if  $\gamma_i^A = 0$  or, equivalently,  $\mathsf{WC}(\widetilde{\Theta}_i) = \{\theta_i\}$ . If a prediction is perfect, then it is also valid. Our main results will depend on these error measures. See Figure 1 for an illustration.
Consistency and robustness. We say a mechanism is  $(a, b)$ -consistent and  $(c, d)$ -robust if when predictions are perfect it satisfies  $\mathbb{E}[\text{welfare}] \geq a \cdot \text{OPT}$ ,  $\mathbb{E}[\text{revenue}] \geq b \cdot \text{OPT}$ , and satisfies  $\mathbb{E}[\text{welfare}] \geq c \cdot \text{OPT}$ ,  $\mathbb{E}[\text{revenue}] \geq d \cdot \text{VCG}$  independent of the prediction quality. Consistency demands near-optimal performance when the side information is perfect, and therefore we compete with the total social surplus OPT on both the welfare and revenue fronts. Robustness deals with the case of arbitrarily bad side information, in which case we would like our mechanism's performance to be competitive with vanilla VCG, which already obtains welfare equal to OPT. High consistency and robustness ratios are in fact trivial to achieve, and we will thus largely not be too concerned with these measures—our main goal is to design high-performance mechanisms that degrade gracefully as the prediction errors increase. In Appendix B we show that the trivial mechanism that discards all side information with probability  $\beta$  and trusts the side information completely with probability  $1 - \beta$  is  $(1, 1 - \beta)$ -consistent and  $(\beta, \beta)$ -robust, but suffers from huge discontinuous drops in performance even when predictions are nearly perfect.
Random expansion mechanism. Our guarantees will depend on  $H$ ; an upper bound on any agent's values. This is the only problem-domain-specific parameter in our results (examples are in Section 2).  $H = \max_{\theta_1,\theta_2\in \Theta}\| \theta_1 - \theta_2\|_\infty$  is the  $\ell_{\infty}$ -diameter of  $\Theta$ . For a point  $\theta$ , let  $\mathcal{B}(\theta,r) = \{\theta':\| \theta -\theta '\|_{\infty}\leq r\}$  be the closed  $\ell_{\infty}$ -ball centered at  $\theta$  with radius  $r$ . For a set  $\widetilde{\Theta}$ , let  $\mathcal{B}(\widetilde{\Theta},r) = \cup_{\widetilde{\theta}\in \widetilde{\Theta}}\mathcal{B}(\widetilde{\theta},r)$  denote the  $\ell_{\infty}$ -expansion of  $\widetilde{\Theta}$  by  $r$ . For  $\zeta_i\geq 0,\lambda_i > 0$ , and  $K_{i}\coloneqq \lceil \log_{2}((H - \zeta_{i}) / \lambda_{i})\rceil$ , let  $\mathcal{M}_{\zeta ,\lambda}$  denote the mechanism that for each  $i$  independently sets
$$
\widehat {\Theta} _ {i} = \mathcal {B} (\widetilde {\Theta} _ {i}, \zeta_ {i} + 2 ^ {k _ {i}} \cdot \lambda_ {i}), \text {w h e r e} k _ {i} \sim_ {\text {u n i f .}} \{0, 1, \dots , K _ {i} \}.
$$
We now state and discuss our main welfare and revenue guarantees on  $M_{\zeta, \lambda}$ . Define  $\log^{+}: \mathbb{R} \to \mathbb{R}_{\geq 0}$  by  $\log^{+}(x) = 0$  if  $x \leq 0$  and  $\log^{+}(x) = \max \{0, \log(x)\}$  if  $x > 0$ .
Theorem 4.4.  $\mathbb{E}[welfare] \geq \max \left\{(1 - \max_i \frac{\lceil \log_2^+((\gamma_i^V - \zeta_i) / \lambda_i)\rceil}{1 + \lceil \log_2((H - \zeta_i) / \lambda_i)\rceil}), \frac{1}{1 + \lceil \max_i \log_2((H - \zeta_i) / \lambda_i)\rceil} \right\}\mathrm{OPT}$ .
Theorem 4.5 (Revenue bound 1). Let  $\rho_{i} = 2(\gamma_{i}^{V} - \zeta_{i})\mathbf{1}(\zeta_{i} + \lambda_{i} < \gamma_{i}^{V}) + \lambda_{i}\mathbf{1}(\zeta_{i} + \lambda_{i}\geq \gamma_{i}^{V})$ . Then  $\mathbb{E}[revenue]\geq \frac{1}{1 + \lceil\max_i\log_2((H - \zeta_i) / \lambda_i)\rceil} (\mathsf{OPT} - \sum_{i = 1}^n (\gamma_i^A +\zeta_i + \rho_i))$
Theorem 4.6 (Revenue bound 2).  $\mathbb{E}[revenue] \geq (1 - \max_i \frac{\lceil \log_2^+((\gamma_i^V - \zeta_i) / \lambda_i) \rceil}{1 + \lceil \log_2((H - \zeta_i) / \lambda_i) \rceil}) (\mathsf{OPT} - \sum_{i=1}^n (\gamma_i^A + \zeta_i)) - \sum_{i=1}^n \frac{4H}{1 + \lceil \log_2((H - \zeta_i) / \lambda_i) \rceil}$ .
Proof sketch.  $\mathcal{M}_{\zeta, \lambda}$  in essence performs a doubling search with initial hop  $\zeta_i$  and  $\lambda_i$  controlling how fine-grained the search proceeds. The bounds are proven (Appendix A.3) by controlling participation probabilities  $\operatorname{Pr}(\theta_i \in \widehat{\Theta}_i)$  and accounting for conditional payments via Lemma 4.1.
First, consider constant  $\zeta, \lambda$ . Our welfare guarantee degrades from OPT to  $\Omega(\mathrm{OPT} / \log H)$  as the invalidity of the predictions increase. Revenue bound 1 degrades from  $\Omega(\mathrm{OPT} / \log H)$  as both the invalidity and inaccuracy of the predictions increase. Revenue bound 2 illustrates that we can obtain significantly better performance if the parameters  $\zeta, \lambda$  are chosen appropriately. In particular, for any  $\varepsilon, \zeta_i = \gamma_i^V$  and  $\lambda_i \leq O((H - \zeta_i) / 2^{H / \varepsilon})$  yields  $\mathbb{E}[revenue] \geq OPT - \sum_{i=1}^{n} (\gamma_i^A + \gamma_i^V + \varepsilon)$ , a bound that is only additively worse than the total social surplus (and recovers the total social surplus as  $\lambda_i \downarrow 0$  if the predictions are perfect). This bound degrades gradually as  $\zeta_i, \lambda_i$  deviate.
To summarize, if the side information is of very high quality, the best parameters  $\zeta, \lambda$  nearly recover the total social surplus OPT as welfare and revenue, and revenue degrades gradually as the chosen parameters  $\zeta, \lambda$  worsen. If the side information is of questionable quality, the best parameters  $\zeta, \lambda$  still obtain OPT as welfare, with revenue suffering additively by the prediction errors. As the parameter selection worsens, welfare and revenue degrade to  $\Omega(\mathrm{OPT} / \log H)$  with revenue suffering the same additive loss. Effective parameters can be, for example, learned from data [32]. We briefly discuss the consistency and robustness of  $\mathcal{M}_{\zeta, \lambda}$  in Appendix B.3, where we show the worst case performance of  $\mathcal{M}_{\zeta, \lambda}$  independent of prediction quality is not much worse than vanilla VCG.
4.1 More expressive forms of side information
In this subsection we establish two avenues for richer and more expressive side information languages. The first deals with uncertainty and the second with joint multi-agent predictions.
Uncertainty. We now show that the techniques we have developed so far readily extend to an even larger more expressive form of side information that allows one to express varying degrees of uncertainty. A side information structure corresponding to agent  $i$  is given by a partition  $(A_1^i,\ldots ,A_m^i)$  of the ambient type space  $\Theta_{i}$  into disjoint sets, probabilities  $\mu_1^i,\dots ,\mu_m^i\geq 0;\sum_j\mu_j^i = 1$  corresponding to each partition element, and for each partition element an optional probability density function  $f_{j}^{i};\int_{A_{j}^{i}}f_{j}^{i} = 1$ . The side information structure represents (1) a belief over what partition element  $A_{j}^{i}$  the true type  $\theta_{i}$  lies in and (2) if a density is specified, the precise nature of uncertainty over the true type within  $A_{j}^{i}$ . Our model of side information sets  $\widetilde{\Theta}_i$  considered earlier in the paper corresponds to the partition  $(\widetilde{\Theta}_i,\Theta_i\setminus \widetilde{\Theta}_i)$  with  $\mu (\widetilde{\Theta}_i) = 1$  and no specified densities. The richer model allows side information to convey finer-grained beliefs; for example one can express quantiles of certainty, precise distributional beliefs, and arbitrary mixtures of these.
Our notions of prediction error (invalidity and inaccuracy) can be naturally generalized. We define  $\overline{\gamma}_i^V = \sum_j\mu_j^i\gamma_i^V (A_j^i;f_j^i)$  and  $\overline{\gamma}_i^A = \sum_j\mu_j^i\gamma_i^A (A_j^i;f_j^i)$ , where  $\gamma_i^V (A_j^i;f_j^i) = d(\theta_i,\mathsf{WCH}(A_j^i))$  if  $f_{j}^{i} = \mathrm{None}$  and  $\gamma_i^V (A_j^i;f_j^i) = \mathbb{E}_{\widetilde{\theta}_i\sim f_j^i}[d(\theta_i,\mathsf{WCH}(\{\widetilde{\theta}_i\})]$  if  $f_{j}^{i}$  is a well-defined density. Similarly  $\gamma_i^A (A_j^i;f_j^i) = d_H(\theta_i,\mathsf{WC}(A_j^i))$  if  $f_{j}^{i} = \mathrm{None}$  and  $\gamma_i^A (A_j^i;f_j^i) = \mathbb{E}_{\widetilde{\theta}_i\sim f_j^i}[d(\theta_i,\widetilde{\theta}_i)]$  otherwise.
Our generalized version of  $M_{\zeta,\lambda}$  first samples a partition element  $A_j^i$  according to  $(\mu_1^i, \ldots, \mu_m^i)$ , and draws  $k_i \sim_{\mathrm{unif.}} \{0, \ldots, K_i\}$  where  $K_i$  is defined as before. If  $f_j^i = \mathrm{None}$ , it sets  $\widehat{\Theta}_i = \mathcal{B}(A_j^i, \zeta_i + 2^{k_i} \lambda_i)$ . Otherwise, it samples  $\widetilde{\theta}_i \sim f_j^i$  and sets  $\widehat{\Theta}_i = \mathcal{B}(\{\widetilde{\theta}_i\}, \zeta_i + 2^{k_i} \lambda_i)$ . Versions of Theorems 4.4, 4.5, and 4.6 carry forward with  $\overline{\gamma}_i^V$  and  $\overline{\gamma}_i^A$  as the error measures. In Appendix C we provide the derivations, and also take the expressive power one step further by specifying a probability space over  $\Theta_i$ ; its  $\sigma$ -algebra captures the granularity of knowledge being conveyed and its probability measure captures the uncertainty.
Joint side information. So far, side information has been independent across agents. Specifically, the mechanism designer receives sets  $\widetilde{\Theta}_i \subseteq \Theta_i$  for each agent  $i$  postulating that  $\theta = (\theta_1, \ldots, \theta_n) \in \widetilde{\Theta}_1 \times \dots \times \widetilde{\Theta}_n$ . We show that our techniques extend to a more expressive form of side information that allows one to express predictions involving multiple agents. Let  $\Theta = \Theta_1 \times \dots \times \Theta_n$ . The mechanism designer receives as side information a set  $\widetilde{\Theta} \subseteq \Theta$  postulating that  $\theta = (\theta_1, \ldots, \theta_n) \in \widetilde{\Theta}$ . Given
an agent  $i$ , a side information set  $\widetilde{\Theta} \subseteq \Theta$ , and  $\overline{\Theta}_{-i} \subseteq \times_{j \neq i} \Theta_j$ , let  $\mathrm{proj}_i(\widetilde{\Theta}; \overline{\Theta}_{-i}) = \{\widetilde{\theta}_i \in \Theta_i : \exists \theta_{-i} \in \overline{\Theta}_{-i}$  s.t.  $(\widetilde{\theta}_i, \theta_{-i}) \in \widetilde{\Theta}\}$  be the  $i$ th projection of  $\widetilde{\Theta}$  with respect to  $\overline{\Theta}_{-i}$ . The projection is the set of types for agent  $i$  consistent with  $\widetilde{\Theta}$  given that the realizations of the other agents' true types are contained in  $\overline{\Theta}_{-i}$ . First, if  $\widetilde{\Theta}$  is known to be a valid prediction, that is, the true type profile  $\pmb{\theta}$  is guaranteed apriori to lie in  $\widetilde{\Theta}$  (equivalently, the joint misreporting space is limited to  $\widetilde{\Theta}$ ), we generalize the weakest-competitor VCG mechanism as follows. First, agents are asked to reveal their true types  $\pmb{\theta} = (\theta_1, \dots, \theta_n)$ . The allocation used is  $\alpha^* = \operatorname{argmax}_{\alpha \in \Gamma} \sum_{i=1}^{n} \theta_i[\alpha]$  and agent  $i$  pays
$$
p _ {i} = \min _ {\widetilde {\boldsymbol {\theta}} _ {i} \in \mathrm {p r o j} _ {i} (\widetilde {\boldsymbol {\Theta}}; \boldsymbol {\theta} _ {- i})} \left(\max _ {\alpha \in \Gamma} \sum_ {j \neq i} \theta_ {j} [ \alpha ] + \widetilde {\boldsymbol {\theta}} _ {i} [ \alpha ]\right) - \sum_ {j \neq i} \theta_ {j} [ \alpha^ {*} ].
$$
This generalized weakest-competitor VCG mechanism is IC, IR, and revenue optimal subject to efficiency in the joint information setting for the same reason that weakest-competitor VCG is in the independent information setting.
More generally, given side information set  $\widetilde{\Theta}$ , our random expansion mechanism can be generalized as follows. Agents first reveal their true types  $\theta_{1},\ldots ,\theta_{n}$ . For each agent  $i$ , independently set  $\widehat{\Theta}_i = \mathcal{B}(\mathrm{proj}_i(\widetilde{\Theta},\pmb{\theta}_{-i}),\zeta_i + 2^{k_i}\lambda_i)$ . The same guarantees we derived previously hold, with appropriately modified quality measures: invalidity is  $\gamma_i^V = d(\theta_i,\mathsf{WCH}(\mathrm{proj}_i(\widetilde{\Theta},\pmb{\theta}_{-i})))$  and inaccuracy is  $\gamma_i^A = d_H(\theta_i,\mathsf{WC}(\mathrm{proj}_i(\widetilde{\Theta},\pmb{\theta}_{-i})))$ . An important idea highlighted by these mechanisms for joint side information is that the true types of all agents other than  $i$  can be heavily utilized in determining  $p_i$ . This model of side information affords significantly more expressive power than the agent-independent model with product structure considered previously. For example, the mechanism designer might know that sum of the valuations of two customers for a cup of coffee exceeds a particular threshold, but does not know who has the higher value. Joint side information enables such a belief to be precisely expressed. It allows the mechanism designer to refine his beliefs on one agent based on the realized true type of the other agent (which was not possible in our previous framework).
We conclude this section with the observation that these mechanisms can loosely be interpreted as prior-free quantitative analogues of the seminal total-surplus-extraction Bayesian mechanism of Crémer and McLean [20] for correlated agents (generalized to infinite type spaces by McAfee and Reny [37]). This is an interesting connection to explore further in future research.
5 Constant-parameter agents: types on low-dimensional subspaces
In this section we show how the theory we have developed so far can be used to derive new revenue approximation results when the mechanism designer knows that each agent's type belongs to some low-dimensional subspace of  $\mathbb{R}^{\Gamma}$  (these subspaces can be different for each agent).
This is a slightly different setup from the previous sections. So far, we have assumed that  $\Theta_{i} = \Theta$  for all  $i$ , that is, there is an ambient type space that is common to all the agents. Side information sets  $\widetilde{\Theta}_i$  are given as input to the mechanism designer, with no assumptions on quality/correctness (and our guarantees in Section 4 were parameterized by quality). Here, we assume the side information that each agent's type lies in a particular subspace is guaranteed to be valid. Two equivalent ways of stating this setup are (1) that  $\Theta_{i}$  is the corresponding subspace for agent  $i$  and the mechanism designer receives no additional prediction set  $\widetilde{\Theta}_i$  or (2)  $\Theta_{i} = \Theta$  for all  $i$ ,  $\widetilde{\Theta}_i = \Theta \cap U_i$  where  $U_{i}$  is a subspace of  $\mathbb{R}^{\Gamma}$ , and the mechanism designer has the additional guarantee that  $\theta_{i}\in U_{i}$  (so  $\widetilde{\Theta}_i$  is a valid side-information set). We shall use the language of the second interpretation.
In this setting, while the side information is valid, the inaccuracy errors  $\gamma_i^A$  of the sets  $\widetilde{\Theta}_i = \Theta \cap U_i$  can be too large to meaningfully use our previous guarantees. In this section we show how to fruitfully use the information provided by the subspaces  $U_1,\ldots ,U_n$  within the framework of our meta-mechanism. We assume  $\Theta = [1,H]^{\Gamma}$ , thereby imposing a lower bound of 1 on agent values (this choice of lower bound is not important, but the knowledge of some lower bound is needed).