SlowGuess/NeurIPS_2023_OCR huggingface dataset:
This mechanism is the natural generalization of the affine-maximizer mechanism [45] parameterized by  $\omega, \lambda$  to our setting. The special case where agent misreporting is limited to  $\widehat{\Theta}_i$  is the natural generalization of the weakest-competitor VCG mechanism of Krishna and Perry [34] to affine-maximizer mechanisms. The following is a simple consequence of the proofs that  $\mathcal{M}$  and the affine-maximizer mechanism parameterized by  $\omega, \lambda$  are incentive compatible and individually rational.
Theorem D.1. For any  $\omega \in \mathbb{R}_{\geq 0}^n$  and  $\lambda : \Gamma \to \mathbb{R}_{\geq 0}$ ,  $\mathcal{M}(\omega, \lambda)$  is incentive compatible and individually rational.
Let  $\mathrm{OPT}(\omega, \lambda) = \sum_{i=1}^{n} \theta_i[\alpha_{\omega, \lambda}]$  be the welfare of the  $(\omega, \lambda)$ -efficient allocation. All of the guarantees satisfied by  $\mathcal{M}$  carry over to  $\mathcal{M}(\omega, \lambda)$ , the only difference being the modified benchmark of  $\mathrm{OPT}(\omega, \lambda)$ . Of course,  $\mathrm{OPT}(\omega, \lambda) \leq \mathrm{OPT}$  is a weaker benchmark than the welfare of the efficient allocation. However, the class of affine maximizer mechanisms is known to achieve much higher revenue than the vanilla VCG mechanism. We leave it as a compelling open question to derive even stronger guarantees on mechanisms of the form  $\mathcal{M}(\omega, \lambda)$  when the underlying affine maximizer is known to achieve greater revenue than vanilla VCG.