SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Using Corollary 1 for  $\mathcal{D} = \times_{i\in [n]}\mathcal{D}_i$  and  $\widehat{\mathcal{D}} = \times_{i\in [n]}\widehat{\mathcal{D}}_i$  that are product distributions, we can get the TV robustness result of Dütting and Kesselheim [DK19] for sequential posted prices as a special case, noting that (1) if  $d_{\mathrm{TV}}\left(\mathcal{D}_i,\widehat{\mathcal{D}}_i\right)\leq \varepsilon$ , then  $d_{\mathrm{TV}}\left(\mathcal{D},\widehat{\mathcal{D}}\right)\leq n\varepsilon$ , and (2) if valuations are normalized to [0, 1] (as in [DK19]),  $V = n$ .
4.4 Gaps between Simple and Optimal Mechanisms
Here, we show applications of our robustness results to the study of simple and approximately optimal mechanisms. Missing proofs are deferred to Appendix D.
Motivated by numerous negative results for revenue optimal auctions (see [Das15] for a survey), a major research thread in mechanism design studies the performance of simple mechanisms [CHK07, CHMS10, CMS15, Yao15, RW15, CM16, CDW16, CZ17, KW19, BILW20]. Two canonical mechanisms that are considered simple in this literature are: (1) the mechanism that sells each item separately (and optimally), and (2) the mechanism that optimally sells all items as a grand bundle; let  $SRev(\mathcal{D})$  and  $BRev(\mathcal{D})$  be the expected revenue of these mechanisms, respectively, under prior  $\mathcal{D}$ . On the flip side, product distributions are known to witness simple mechanisms that are approximately optimal. For instance, Babaioff et. al. [BILW20] prove that for a single agent and a product distribution  $\mathcal{D}^p$ ,  $\max\{SRev(\mathcal{D}^p), BRev(\mathcal{D}^p)\} \geq \frac{1}{6} Rev(\mathcal{D}^p)$ , even though the individual approximation factors for  $SRev(\mathcal{D}^p)$  and  $BRev(\mathcal{D}^p)$  are  $O(1 / \log(m))$  and  $1/m$ , respectively [HN13, LY13]. Note that in the single-agent context, by product distribution, we mean with respect to items. The following observation is an immediate implication of Theorem 1.
Observation 1. Let  $\mathcal{SM}$  be a family of mechanisms, such that  $\mathbb{E}_{t\sim \widehat{\mathcal{D}}^p}\left[\mathcal{O}(t,\mathcal{SM}_{\widehat{\mathcal{D}}^p}(t))\right] \geq \alpha OPT_{\mathcal{O}}(\widehat{\mathcal{D}}^p)$  for some objective  $\mathcal{O}$ , where  $\mathcal{SM}_{\widehat{\mathcal{D}}^p}$  is a mechanism in  $\mathcal{SM}$  parameterized by the product distribution  $\widehat{\mathcal{D}}^p$ . Then, for any (possibly non-product) distribution  $\mathcal{D}$  that is close to some product distributions  $\mathcal{D}^p$ , and specifically,  $d_{\mathrm{TV}}(\mathcal{D},\mathcal{D}^p) \leq \frac{\alpha OPT_{\mathcal{O}}(\mathcal{D})}{2(1 + \alpha)V}$ , we have that  $\mathbb{E}_{t\sim \mathcal{D}}\left[\mathcal{O}(t,\mathcal{SM}_{\mathcal{D}^p}(t))\right] \geq \frac{\alpha}{2} OPT_{\mathcal{O}}(\mathcal{D})$ .
A construction of Hart and Nisan [HN13] shows that there exists a non-product distribution  $\mathcal{D}$  such that  $SRev(\mathcal{D}) \leq 2m$ ,  $BRev(\mathcal{D}) \leq 2m$  but  $Rev(\mathcal{D}) \to \infty$ . Recently, [PSCW22] show necessary and sufficient conditions for a distribution  $\mathcal{D}$  to exhibit such infinite gaps between the revenue of simple and optimal mechanisms. Unfortunately, these conditions are rather complex (namely, they are conditions on the existence of infinite sequences of points with certain properties). Here, leveraging Observation 1 for the revenue objective, we complement results of [PSCW22] by giving a new, simple necessary condition that a "pathological" construction must satisfy:
Corollary 2. For a single agent and any distribution  $\mathcal{D}$  such that  $\frac{\text{Rev}(\mathcal{D})}{\text{BRev}(\mathcal{D})} \geq 2m$  it must be the case that for any product distribution  $\mathcal{D}^p$ ,  $d_{\text{TV}}(\mathcal{D}, \mathcal{D}^p) \geq \frac{\text{Rev}(\mathcal{D})}{4V}$ . Furthermore, if  $\frac{\text{Rev}(\mathcal{D})}{\text{SRev}(\mathcal{D})} \in \Omega(\log(m))$  it must be the case that for any product distribution  $\mathcal{D}^p$ ,  $d_{\text{TV}}(\mathcal{D}, \mathcal{D}^p) \geq \frac{\text{Rev}(\mathcal{D})}{4V}$ .
As a final application of our framework, we give new positive results on simple mechanisms.
Proposition 2. Let  $\mathcal{D}$  be a distribution supported on  $\mathcal{T}$ , and let  $\mathcal{D}^p$  be a product distribution such that  $d_{\mathrm{TV}}(\mathcal{D},\mathcal{D}^p) < \delta$ . Then, we have  $\max \{SRev(\mathcal{D}),Brev(\mathcal{D})\} \geq \frac{1}{6} Re\nu (\mathcal{D}) - \frac{7}{6} H\delta$ .
Proposition 2 implies that the better of bundling and selling each item separately for a distribution  $\mathcal{D}$  is a good approximation to  $Rev(\mathcal{D})$ , as long as  $\mathcal{D}$  is close to a product distribution. Cai and Oikonomou [CO21] prove that the same mechanism is a good approximation for dependent distributions that can be captured by a Markov Random Field (MRF); see Appendix D for basic definitions regarding MRFs. The approximation ratio of [CO21] is controlled by  $\Delta$ , a parameter of the MRF that is determined by how much the value of an item can be influenced by the values of the other items. Specifically, [CO21] prove that  $\max \{SRev(\mathcal{D}), BRev(\mathcal{D})\} \geq \frac{1}{12e^{4\Delta}} Rev(\mathcal{D})$ . In Proposition 3, we prove that there exist distributions such that the object of interest for Proposition 2, i.e., the distance to a product distribution, is arbitrarily small, while the objective of interest for [CO21], the parameter  $\Delta$ , is arbitrarily large. At the same time, in Proposition 4 we show that when the MRF parameter  $\Delta$  is bounded, we can bound the distance of a distribution  $\mathcal{D}$  to a product distribution, for distributions that can be represented by MRFs that only have pairwise edges.
Proposition 3. For any  $0 < k < 1/2$ , there exists a distribution  $\mathcal{D}$  produced by an MRF with parameter  $\Delta$  and a product distribution  $\mathcal{D}^p$  such that  $d_{\mathsf{TV}}(\mathcal{D}, \mathcal{D}^p) \leq 2k^2$  and  $\Delta \geq \frac{1}{4}\log \left(\frac{1}{k}\right)$ .
Proposition 4. Let  $\mathcal{D}$  be a distribution produced by an MRF with only pairwise edges. Then, there exists a product distribution  $\mathcal{D}^p$  such that  $d_{\mathsf{TV}}(\mathcal{D},\mathcal{D}^p)\leq \min \{\sqrt{m\Delta / 4},\sqrt{1 - e^{-m\Delta / 2}}\}$ , where  $m$  is the number of items.
Alexandros Psomas is supported in part by an NSF CAREER award CCF-2144208, a Google Research Scholar Award, and a Google AI for Social Good award. Athina Terzoglou is supported in part by an NSF CAREER award CCF-2144208. Marios Mertzanidis is supported in part by a DOE award SC0022085, and NSF awards CCF-1814041, CCF-2209509, and DMS-2152687.