SlowGuess/NeurIPS_2023_OCR huggingface dataset:
We note that  $\Omega$  is either discrete or a measurable subset of a finite-dimensional Euclidean space in our analysis, and hence, it is always a standard Borel space. For any probability distributions  $P$  and  $Q$  on  $\Omega$ , let  $\Pi(P, Q)$  be the (non-empty) set of all couplings of  $P$  and  $Q$ , i.e., all joint probability distributions  $\gamma = P_{X,Y}$  of two random variables  $X, Y \in \Omega$  such that the marginal distributions are  $P_X = P$  and  $P_Y = Q$ , respectively. The following definition will be of utility in the sequel [Dob70, LPW09].
Definition 2 (Optimal Coupling). For any two probability distributions  $P$  and  $Q$  on  $\Omega$ , we define the optimal coupling of  $P$  and  $Q$  as the joint distribution  $\gamma^{*} = \arg \min_{\gamma \in \Pi(P, Q)} \mathbb{E}_{(X, Y) \sim \gamma}[\mathbb{1}\{X \neq Y\}]$  of two random variables  $(X, Y)$  that has marginal distributions  $P$  and  $Q$  of  $X$  and  $Y$ , respectively, and minimizes the probability that  $X$  is different to  $Y$ .
We note that such an optimal coupling always exists [Dob70, Vil09]. The next lemma presents several useful characterizations of TV distance including an optimal coupling representation (which demonstrates how TV distance is a Wasserstein distance with respect to the discrete metric [Vil09]); see [Mak19, Section 2.2.1] and [PW22, Theorem 7.7] for a compilation of other characterizations.
Lemma 1 (Equivalent Characterizations of TV Distance [Str65, Dob70, LPW09, Vil09]). For any two probability distributions  $P$  and  $Q$  on  $\Omega$ , we have
$$
\begin{array}{l} d _ {\mathrm {T V}} (P, Q) = \frac {1}{2} \| P - Q \| _ {1} \\ = \min  _ {\gamma \in \Pi (P, Q)} \Pr_ {(X, Y) \sim \gamma} [ X \neq Y ] \\ = \max _ {f: \Omega \to [ - \frac {1}{2}, \frac {1}{2} ]} \mathbb {E} _ {X \sim P} \left[ f (X) \right] - \mathbb {E} _ {Y \sim Q} \left[ f (Y) \right], \\ \end{array}
$$
where  $\| P - Q \|_1$  is the  $\mathcal{L}^1$ -distance between  $P$  and  $Q$ , the second equality is the optimal coupling characterization which minimizes  $\operatorname*{Pr}_{(X,Y) \sim \gamma} [X \neq Y] = \mathbb{E}_{(X,Y) \sim \gamma} [\mathbb{1}\{X \neq Y\}]$  over all couplings  $\gamma$  of  $P$  and  $Q$ , and the third equality is the Kantorovich-Rubinstein dual characterization which takes the maximum over all (measurable) functions  $f: \Omega \to \mathbb{R}$  bounded by  $\sup_{t \in \Omega} |f(t)| \leq \frac{1}{2}$ .
3 Robustness of Mechanisms Under Total Variation Distance
In this section, we prove our robustness results for DSIC and BIC mechanisms. Missing proofs throughout the section are deferred to Appendix A.
At the heart of our approach lies the following lemma, which shows that, even for correlated prior distributions, assuming truthful bidding, a mechanism's performance with respect to an arbitrary objective function  $\mathcal{O}$  is stable under small perturbations to the prior.
Lemma 2. Let  $P$  and  $Q$  be two arbitrary probability distributions supported on  $\mathcal{T}$  and let  $\mathcal{M}$  be any mechanism. Assuming truthful bidding, for all objective functions  $\mathcal{O}(.,.) \in [a,b]$ , letting  $V = b - a$ , it holds that  $\mathbb{E}_{t\sim P}\left[\mathcal{O}(t,\mathcal{M}(t))\right] - \mathbb{E}_{t'\sim Q}\left[\mathcal{O}(t',\mathcal{M}(t'))\right] \leq Vd_{\mathsf{TV}}(P,Q)$ .
We note that  $\mathcal{M}$  in the statement of Lemma 2 may not be DSIC or BIC, for neither  $P$  nor  $Q$ .
Proof of Lemma 2. The objective function is lower bounded by  $a$  and upper bounded by  $b$ ; thus, we have that for any  $t, t' \in \mathcal{T}$ ,  $\mathcal{O}(t, \mathcal{M}(t)) - \mathcal{O}(t', \mathcal{M}(t')) \leq V \mathbb{1}\{t \neq t'\}$ . Since this inequality holds for all  $t, t' \in \mathcal{T}$ , it also holds after taking an expectation with respect to any coupling  $\gamma$  of  $P$  and  $Q$ , and specifically for the optimal coupling  $\gamma^*$  between  $P$  and  $Q$  (see Definition 2):
$$
\mathbb {E} _ {(t, t ^ {\prime}) \sim \gamma^ {*}} \left[ \mathcal {O} (t, \mathcal {M} (t)) - \mathcal {O} \left(t ^ {\prime}, \mathcal {M} \left(t ^ {\prime}\right)\right) \right] \leq V \mathbb {E} _ {(t, t ^ {\prime}) \sim \gamma^ {*}} \left[ \mathbb {1} \{t \neq t ^ {\prime} \} \right]. \tag {1}
$$
Using the second form of Lemma 1, the RHS of (1) is equal to  $V d_{\mathrm{TV}}(P, Q)$ . For the LHS of (1), using linearity of expectation, the fact that  $\mathcal{O}(t, \mathcal{M}(t))$  does not depend on  $t'$ , and the fact that  $\mathcal{O}(t', \mathcal{M}(t'))$  does not depend on  $t$ , we have that  $\mathbb{E}_{(t, t') \sim \gamma^*}[\mathcal{O}(t, \mathcal{M}(t)) - \mathcal{O}(t', \mathcal{M}(t'))] = \mathbb{E}_{t \sim P}[\mathcal{O}(t, \mathcal{M}(t))] - \mathbb{E}_{t' \sim Q}[\mathcal{O}(t', \mathcal{M}(t'))]$ . Putting everything together, we get the desired inequality.
In the remainder of this section, we show that this lemma can be used to prove strong robustness results for DSIC and BIC mechanisms.
3.1 DSIC Mechanisms
Our main robustness result for DSIC mechanisms is stated as follows.
Theorem 1 (Robustness for DSIC). Let  $\mathcal{D}$  and  $\widehat{\mathcal{D}}$  be two arbitrary distributions supported on  $\mathcal{T}$ , respectively, such that  $d_{\mathrm{TV}}(\mathcal{D},\widehat{\mathcal{D}})\leq \delta$ , and let  $\mathcal{O}(.,.)\in [a,b]$  be an objective function. Let  $\mathcal{M}_{\mathcal{D}}^{\alpha}$  be an ex-post IR, DSIC, and  $\alpha$ -approximate mechanism (under  $\mathcal{D}$ ), with respect to the optimal (under  $\mathcal{D}$ ) ex-post IR and DSIC mechanism for  $\mathcal{O}$ . Then, letting  $V = b - a$ , it holds that
$$
\mathbb {E} _ {t \sim \widehat {\mathcal {D}}} \left[ \mathcal {O} (t, \mathcal {M} _ {\mathcal {D}} ^ {\alpha} (t)) \right] \geq \alpha O P T _ {\mathcal {O}} (\widehat {\mathcal {D}}) - (1 + \alpha) V \delta ,
$$
where  $OPT_{\mathcal{O}}(\widehat{\mathcal{D}})$  is the performance of the optimal (for  $\mathcal{O}$ ) DSIC mechanism.
Proof. We apply Lemma 2 twice. First, choosing  $P = \mathcal{D}$ ,  $Q = \widehat{\mathcal{D}}$ , and  $\mathcal{M} = \mathcal{M}_{\mathcal{D}}^{\alpha}$ , Lemma 2 implies
$$
\mathbb {E} _ {t \sim \mathcal {D}} \left[ \mathcal {O} (t, \mathcal {M} _ {\mathcal {D}} ^ {\alpha} (t)) \right] - \mathbb {E} _ {t ^ {\prime} \sim \widehat {\mathcal {D}}} \left[ \mathcal {O} (t ^ {\prime}, \mathcal {M} _ {\mathcal {D}} ^ {\alpha} (t ^ {\prime})) \right] \leq V d _ {\sf T V} \left(\mathcal {D}, \widehat {\mathcal {D}}\right) \leq V \delta .
$$
Now, let  $\mathcal{M}_{\widehat{\mathcal{D}}}^{*}$  be the optimal (under  $\widehat{\mathcal{D}}$ ) ex-post IR and DSIC mechanism for  $\mathcal{O}$ . Using the definition of  $\mathcal{M}_{\mathcal{D}}^{\alpha}$ , and re-arranging we have
$$
\begin{array}{l} \mathbb {E} _ {t ^ {\prime} \sim \widehat {\mathcal {D}}} \left[ \mathcal {O} (t ^ {\prime}, \mathcal {M} _ {\mathcal {D}} ^ {\alpha} (t ^ {\prime})) \right] \geq \mathbb {E} _ {t \sim \mathcal {D}} \left[ \mathcal {O} (t, \mathcal {M} _ {\mathcal {D}} ^ {\alpha} (t)) \right] - V \delta \\ \geq \alpha \mathbb {E} _ {t \sim \mathcal {D}} \left[ \mathcal {O} (t, \mathcal {M} _ {\mathcal {D}} ^ {*} (t)) \right] - V \delta \\ \geq \alpha \mathbb {E} _ {t \sim \mathcal {D}} \left[ \mathcal {O} (t, \mathcal {M} _ {\hat {\mathcal {D}}} ^ {*} (t)) \right] - V \delta , \\ \end{array}
$$
where the last inequality is because  $\mathcal{M}_{\widehat{\mathcal{D}}}^{*}$  is feasible (i.e. ex-post IR and DSIC) for  $\mathcal{D}$ . Second, choosing  $P = \widehat{\mathcal{D}}$ ,  $Q = \mathcal{D}$ , and  $\mathcal{M} = \mathcal{M}_{\widehat{\mathcal{D}}}^{*}$ , by Lemma 2, plus re-arranging, we have that  $\mathbb{E}_{t' \sim \mathcal{D}}\left[\mathcal{O}(t', \mathcal{M}_{\widehat{\mathcal{D}}}^{*}(t'))\right] \geq \mathbb{E}_{t \sim \widehat{\mathcal{D}}}\left[\mathcal{O}(t, \mathcal{M}_{\widehat{\mathcal{D}}}^{*}(t))\right] - V\delta$  Combining these inequalities we have:
$$
\mathbb {E} _ {t \sim \widehat {\mathcal {D}}} \left[ \mathcal {O} (t, \mathcal {M} _ {D} ^ {\alpha} (t)) \right] \geq \alpha \mathbb {E} _ {t \sim \widehat {\mathcal {D}}} \left[ \mathcal {O} (t, \mathcal {M} _ {\widehat {\mathcal {D}}} ^ {*} (t)) \right] - (1 + \alpha) V \delta .
$$
Intuitively, Theorem 1 states that, if a mechanism is approximately optimal for  $\mathcal{D}$ , then it is also approximately optimal for all  $\widehat{\mathcal{D}}$  that are close in total variation distance, paying a small additive error. Note that neither  $\mathcal{D}$  nor  $\widehat{\mathcal{D}}$  needs to be a product distribution. That is, if  $\mathcal{D}$  and  $\widehat{\mathcal{D}}$  have the same support, a DSIC mechanism  $\mathcal{M}_{\mathcal{D}}^{\alpha}$  performs approximately-optimally under  $\widehat{\mathcal{D}}$ . Finally, notice that in the above theorem, we assume that the  $\mathcal{D}$  and  $\widehat{\mathcal{D}}$  share the same support; in Appendix A.1 we relax this assumption, and show how to modify the mechanism  $\mathcal{M}_{\mathcal{D}}^{\alpha}$ , in a way that is agnostic to  $\widehat{\mathcal{D}}$ , and provide exactly the same guarantee.
3.2 Bayesian Incentive Robustness
In this section, we study BIC mechanisms. As opposed to DSIC mechanisms, arguing about whether a BIC mechanism remains BIC after perturbing the prior distribution is a lot more involved. Our goal in this section is to prove that the BIC property degrades gracefully as a function of the TV distance, even for arbitrary objectives, albeit, with two small technical caveats (compared to the DSIC robustness): (1)  $\mathcal{D}$  and  $\widehat{\mathcal{D}}$  must share the same support, and (2) the incentive guarantees of our mechanisms also degrade. The second requirement is necessary; in Section 4.1 we show how to bypass the first requirement for the revenue objective, recovering a slightly stronger version of a TV robustness result of Brustle et al. [BCD20].
First, we need the following Markov-like technical lemma that when two joint distributions  $P_{X,Y}$ ,  $Q_{X,Y}$  are close in TV distance, then, with high probability, the conditional distributions  $P_{Y|X=x}$ ,  $Q_{Y|X=x}$  are also close in TV distance.
Lemma 3. Let  $P_{X,Y}$ ,  $Q_{X,Y}$  be two probability distributions for the (possibly multivariate) random variables  $X$  and  $Y$ . Let  $P_{Y|X=x}$  (resp.  $Q_{Y|X=x}$ ) be the probability distribution of  $P_{X,Y}$  (resp.  $Q_{X,Y}$ ) conditioned on  $X = x$ , and let  $Q_{X}$  be the marginal probability distribution of  $X$  as dictated by  $Q_{X,Y}$ . Then, for all  $q \in [0,1]$ ,  $\operatorname{Pr}_{x \sim Q_X}\left[d_{\mathsf{TV}}(P_{Y|X=x}, Q_{Y|X=x}) > \frac{2d_{\mathsf{TV}}(P_{X,Y}, Q_{X,Y})}{q}\right] \leq q$ .
Next, we prove that a BIC mechanism for  $\mathcal{D}$  is  $(\varepsilon, q)$ -BIC for  $\widehat{\mathcal{D}}$ , assuming that  $\mathcal{D}$  and  $\widehat{\mathcal{D}}$  have the same support and small TV distance.
Lemma 4. Let  $\mathcal{D}$  and  $\widehat{\mathcal{D}}$  be two probability distributions supported on  $\mathcal{T}$ , with  $d_{\mathrm{TV}}\left(\mathcal{D},\widehat{\mathcal{D}}\right)\leq \delta$ . If  $\mathcal{M}$  is an ex-post IR and BIC mechanism w.r.t.  $\mathcal{D}$  then it is also an ex-post IR and  $(\frac{8H\delta}{q},q)$ -BIC mechanism w.r.t.  $\widehat{\mathcal{D}}$ , for all  $q\in [0,1]$ .
To prove the above lemma we leverage Lemma 3. We know that with high probability the perception of each agent over the distributions of the rest of the agents is very close under  $\mathcal{D}$  and  $\widehat{\mathcal{D}}$ . That is why, if agent  $i$  cannot gain by misreporting under  $\mathcal{D}$ , then with high probability, she cannot significantly increase her utility by misreporting under  $\widehat{\mathcal{D}}$ .
As a direct implication of Lemma 2 and Lemma 4 we get our main robustness result for BIC.
Theorem 2 (Robustness for BIC). Let  $\mathcal{D}$  and  $\widehat{\mathcal{D}}$  be two arbitrary distributions supported on  $\mathcal{T}$ , such that  $d_{\mathrm{TV}}\left(\mathcal{D},\widehat{\mathcal{D}}\right)\leq \delta$ , and let  $\mathcal{O}(:,.)\in [a,b]$  be an objective function. Let  $\mathcal{M}_{\mathcal{D}}$  be a mechanism that is ex-post IR and BIC w.r.t.  $\mathcal{D}$ . Then  $\mathcal{M}_{\mathcal{D}}$  is also ex-post IR and  $\left(\frac{8H\delta}{q},q\right)$ -BIC w.r.t.  $\widehat{\mathcal{D}}$ , for all  $q\in [0,1]$ . Also, letting  $V = b - a$ , it holds that  $\mathbb{E}_{t\sim \widehat{\mathcal{D}}}[\mathcal{O}(t,\mathcal{M}_{\mathcal{D}}(t))]\geq \mathbb{E}_{t\sim \mathcal{D}}[\mathcal{O}(t,\mathcal{M}_{\mathcal{D}}(t))] - V\delta$ .
Comparing to the corresponding result of Brustle et al. [BCD20], Theorem 2 only holds if  $\mathcal{D}$  and  $\widehat{\mathcal{D}}$  have the same support. Also, the guarantee on incentives is weaker:  $\mathcal{M}_{\mathcal{D}}$  is not  $\varepsilon$ -BIC (as in [BCD20]), but  $(\varepsilon, q)$ -BIC. However, our robustness results hold for arbitrary objectives, not just revenue, and arbitrary distributions, not just product distributions.
On Tightness of Robustness Results. Regarding Lemma 2, it is known that  $\mathbb{E}_{X\sim P}[f(X)] - \mathbb{E}_{X\sim Q}[f(X)]\leq d_{\mathsf{T}\mathsf{V}}(P,Q)$  for all functions  $f$  bounded by  $1 / 2$  (see Lemma 1), and equality holds for the function  $f^{*}(x) = 1 / 2$  if  $P(x)\geq Q(x)$  and  $f^{*}(x) = -1 / 2$  otherwise. We can use this to show that equality holds for Lemma 2 when the objective function and the mechanism, combined, look like this function, i.e.,  $f^{*}(x) = \mathcal{O}(x,\mathcal{M}(x))$  (with appropriate re-scaling when  $V$  is not 1). This yields a sufficient condition for tightness of Lemma 2.
Regarding Theorem 1, it is straightforward to construct a tight example for the case of revenue and welfare. For instance, for the case of a single agent, letting  $\mathcal{T} = [0,V]$ , consider the case that the distribution  $P$  is a point mass at  $V$ , and distribution  $Q$  takes the value  $V$  with probability  $1 - \delta$ , and zero otherwise. The TV distance between  $P$  and  $Q$  is  $\delta$ . Consider the simple mechanism  $\mathcal{M}$  that posts a price of  $V$ . Its revenue/welfare under  $P$  is  $V$ , and its revenue/welfare under  $Q$  is  $(1 - \delta)V$ . The main issue with generalizing to arbitrary objectives is that a worst-case arbitrary objective can do something uninteresting, e.g., take the value  $c$  no matter what, where naturally our result is not tight.
Regarding Theorem 2, tightness of the revenue objective follows from the aforementioned tightness of Theorem 1 for revenue. Tightness for the BIC guarantee follows from the following example. Consider the single item case where two identical bidders have valuation 1 with probability 0.5 and valuation 2 with probability 0.5 independently from each other. Now consider the mechanism where the first bidder always takes the item if he bids 2 and pays 1.5. When he bids 1 and the second bidder bids 1, he again takes the item and and now pays 1. Finally, if the first bidder bids 1 and the second bidder bids 2, then the second bidder takes the item and pays 2. For the specific distribution we selected, it is easy to check that this mechanism is BIC. Now assume that the second bidder's distribution changes to having valuation 1 with probability  $0.5 + \varepsilon$  and 2 with probability  $0.5 - \varepsilon$ . The TV distance between the two distributions is exactly  $\varepsilon$ . However our mechanism is no longer BIC. Whenever the first bidder's valuation is 2, if he reports truthfully, he will always get 0.5 utility. If he instead reports 1, he will make  $0.5 + \varepsilon$  utility (on expectation). Taking into account this observation, we can see that our mechanism is now  $\varepsilon$ -BIC.
In this section, we show a number of applications of Lemma 2, and Theorems 1 and 2.
4.1 BIC Mechanisms
We start by showing applications of our robustness results for the revenue objective of BIC mechanisms. Our goal is to extend our DSIC robustness result (Theorem 1) to BIC mechanisms. That is, we'd like, given an approximately optimal BIC mechanism for a distribution  $\mathcal{D}$ , to get an approximately optimal and approximately BIC mechanism for distribution  $\widehat{\mathcal{D}}$ . We will achieve this goal for the revenue objective and product distributions; missing proofs can be found in Appendix B.
Towards our main result for this section, we prove the following lemma, which might be of independent interest. Intuitively, the lemma shows that one can turn an  $(\varepsilon, q)$ -BIC mechanism into a  $O(\varepsilon + nqH)$ -BIC mechanism, paying a small loss in revenue.
Lemma 5. For any product distribution  $\mathcal{D} = \times_{i\in [n]}\mathcal{D}_i$ , given a mechanism  $\mathcal{M}$  that is ex-post IR and  $(\varepsilon, q)$ -BIC w.r.t.  $\mathcal{D}$ , we can design a mechanism  $\widehat{\mathcal{M}}$  that is ex-post IR and  $O(\varepsilon + nqH)$ -BIC w.r.t.  $\mathcal{D}$ , such that  $\operatorname{Rev}(\widehat{\mathcal{M}}, \mathcal{D}) \geq \operatorname{Rev}(\mathcal{M}, \mathcal{D}) - nqV$ .
Our main theorem for this stage is stated as follows.
Theorem 3. Let  $\mathcal{D}$  be a probability distribution supported on  $\mathcal{T}$ , and let  $\mathcal{D}^p$  be a product distribution such that  $d_{\mathrm{TV}}(\mathcal{D},\mathcal{D}^p)\leq \delta$ . Let  $\mathcal{M}_{\mathcal{D}^p}^\alpha$  be an ex-post IR, BIC, and  $\alpha$ -approximate mechanism (under  $\mathcal{D}^p$ ), with respect to the revenue optimal (under  $\mathcal{D}^p$ ) ex-post IR and BIC mechanism. Then,  $\mathcal{M}_{\mathcal{D}^p}^\alpha$  is ex-post IR and  $(\frac{8H\delta}{q},q)$ -BIC with respect to  $\mathcal{D}$ , for all  $q\in (0,1]$ . Furthermore,  $\operatorname{Rev}(\mathcal{M}_{\mathcal{D}^p}^\alpha ,\mathcal{D})\geq \alpha OPT(\mathcal{D}) - O\left((1 + \alpha)V\sqrt{n\sqrt{\delta}}\right)$ .
In order to get the full benefits of Theorem 3, one needs the product distribution  $\mathcal{D}^p$  that is the closest (in TV distance) to the original distribution  $\mathcal{D}$ . Let  $\varepsilon$  be this optimal distance. The following proposition shows that, for every distribution  $\mathcal{D}$ , the distance of  $\mathcal{D}$  to the product of its marginals is bounded by  $(n + 1)\varepsilon$ , and therefore, Theorem 3 holds for every distribution  $\mathcal{D}$  for  $\delta = (n + 1)\varepsilon$ .
Proposition 1. Let  $\mathcal{D} = \times_{i\in [n]}\mathcal{D}_i$  be a product distribution supported on  $\times_{i\in [n]}\mathcal{T}_i$ , and let  $\widehat{\mathcal{D}}$  be a joint distribution supported on  $\times_{i\in [n]}\mathcal{T}_i$ , whose marginal over  $\mathcal{T}_i$  is exactly  $\mathcal{D}_i$ . Then, if there exists a product distribution  $\mathcal{D}^p$  such that  $d_{\mathsf{TV}}\left(\widehat{\mathcal{D}},\mathcal{D}^p\right)\leq \varepsilon$ , it holds that  $d_{\mathsf{TV}}\left(\mathcal{D},\widehat{\mathcal{D}}\right)\leq (n + 1)\varepsilon$ .
4.2 Marginal Robustness
As a second application, we consider the setting of Bei et al. [BGLT19]. In this problem, we want to sell one item to  $n$  dependent agents, and we only know the marginal distribution of each agent  $i$ . Overloading notation, let  $t_i \in \mathcal{T}_i \subseteq \mathbb{R}_+$  be the valuation of agent  $i$  for the item. There is a distribution  $\mathcal{D}$ , supported on  $\times_{i \in [n]} \mathcal{T}_i$ , from which agents' valuations are sampled from. For the sake of simplicity, we only argue about discrete distributions; however, our results easily extend to the continuous case. Let  $\mathcal{D}_i$  be the marginal distribution of agent  $i$ , i.e.,  $\operatorname{Pr}_{t_i \sim \mathcal{D}_i} [t_i = v_i] = \sum_{v_{-i} \in \mathcal{T}_{-i}} \operatorname{Pr}_{t \sim \mathcal{D}} [t = (v_i, v_{-i})]$ . The designer knows marginal distributions  $\mathcal{D}_i$  for each  $i$  but not  $\mathcal{D}$ . Given a set of marginal distributions  $(\mathcal{D}_1, \dots, \mathcal{D}_n)$  let  $\Pi(\mathcal{D}_1, \dots, \mathcal{D}_n)$  be the set of all distributions consistent with such marginals, i.e.  $\Pi(\mathcal{D}_1, \dots, \mathcal{D}_n) = \{\mathcal{D}' | \operatorname{Pr}_{t_i \sim \mathcal{D}_i} [t_i = v_i] = \sum_{v_{-i} \in \mathcal{T}_{-i}} \operatorname{Pr}_{t \sim \mathcal{D}'} [t = (v_i, v_{-i})], \forall i \in [n], \forall t_i \in \mathcal{T}_i\}$ . Our goal is to find a mechanism  $\mathcal{M}$  such that  $\mathcal{M} = \arg \max_{\mathcal{M}'} \min_{\mathcal{D}' \in \Pi(\mathcal{D}_1, \dots, \mathcal{D}_n)} \mathbb{E}_{t \sim \mathcal{D}'}[\mathcal{O}(t, \mathcal{M}'(t))]$  where  $\mathcal{M}'$  is taken over all possible ex-post IR and DSIC mechanisms. We can prove the following theorem.
Theorem 4. Given a set of marginals  $(\mathcal{D}_1, \dots, \mathcal{D}_n)$  and a DSIC and ex-post IR mechanism  $\mathcal{M}^\alpha$  such that  $\min_{\mathcal{D} \in \Pi(\mathcal{D}_1, \dots, \mathcal{D}_n)} \mathbb{E}_{t \sim \mathcal{D}}[\mathcal{O}(t, \mathcal{M}^\alpha(t))] \geq \alpha \max_{\mathcal{M}'} \min_{\mathcal{D} \in \Pi(\mathcal{D}_1, \dots, \mathcal{D}_n)} \mathbb{E}_{t \sim \mathcal{D}}[\mathcal{O}(t, \mathcal{M}'(t))]$ , then for any set of marginals  $(\mathcal{D}_1', \dots, \mathcal{D}_n')$  such that for all  $i \in [n]$ ,  $d_{\mathrm{TV}}(\mathcal{D}_i, \mathcal{D}_i') \leq \varepsilon$ , it holds that  $\min_{\mathcal{D}' \in \Pi(\mathcal{D}_1', \dots, \mathcal{D}_n')} \mathbb{E}_{t \sim \mathcal{D}'}[\mathcal{O}(t, \mathcal{M}^\alpha(t))] \geq \alpha \max_{\mathcal{M}'} \min_{\mathcal{D}' \in \Pi(\mathcal{D}_1', \dots, \mathcal{D}_n')} \mathbb{E}_{t \sim \mathcal{D}'}[\mathcal{O}(t, \mathcal{M}'(t))] - (1 + \alpha)n\varepsilon V$ .
The input to our problem is a set of marginals; however, we do not sample from these marginals to compute our objective. Instead, these marginals are used in order to derive a new distribution from which we will sample. Therefore, we cannot immediately "black box" the results we have shown until now. To prove Theorem 4 we first relate, and bound the "distance" between  $\Pi(\mathcal{D}_1, \dots, \mathcal{D}_n)$  and  $\Pi(\mathcal{D}_1', \dots, \mathcal{D}_n')$ . Then, we need to re-prove arguments equivalent to the ones used for our robustness results so far, while taking into consideration the min max nature of the problem. We postpone the formal proof of Theorem 4 to Appendix C.
[BGLT19] show that Sequential Posted Prices Mechanisms are a 4.78-approximation and there exists a Lookahead Auction that is a 2-approximation with respect to revenue for the above problem. Using Theorem 4, we can readily get implications for the robustness of those two auctions.
4.3 Prophet Inequalities
Here, we study the prophet inequality problem. Recall that in this problem, agents arrive over time; in the  $i$ -th step, we need to (immediately and irrevocably) decide on the allocation of agent  $i$ , whose type  $t_i$  is drawn from a marginal distribution  $\mathcal{D}_i$ . Our goal is to design a prophet inequality: a policy that competes with the optimal in hindsight welfare maximizing allocation. It is known that, in fairly general domains, one can achieve this goal using posted prices, i.e., set a price  $p_j$  for each item  $j$ , and let users pick their utility-maximizing subset of items [SC84, FGL14, RS17, KW19, ANSS19, DKL20, DFKL20]; see [Luc17] for a survey. Using Lemma 2 for the welfare objective we prove robustness for prophet inequalities, noting that posted price mechanisms are ex-post IR and DSIC.
Corollary 1. For a posted price mechanism  $\mathcal{M}$ , and distributions  $\mathcal{D}$ ,  $\widehat{\mathcal{D}}$  such that  $d_{\mathrm{TV}}(\mathcal{D}, \widehat{\mathcal{D}}) \leq \delta$ , it holds that  $Val(\mathcal{M}, \mathcal{D}) \geq Val(\mathcal{M}, \widehat{\mathcal{D}}) - V\delta$ .