SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Anuran Makur
amakur@purdue.edu
Marios Mertzanidis
mmertzan@purdue.edu
Alexandros Psomas
apsomas@cs.purdue.edu
Athina Terzoglou
aterzogl@purdue.edu
We study the problem of designing mechanisms when agents' valuation functions are drawn from unknown and correlated prior distributions. In particular, we are given a prior distribution  $\mathcal{D}$ , and we are interested in designing a (truthful) mechanism that has good performance for all "true distributions" that are close to  $\mathcal{D}$  in Total Variation (TV) distance. We show that DSIC and BIC mechanisms in this setting are strongly robust with respect to TV distance, for any bounded objective function  $\mathcal{O}$ , extending a recent result of Brustle et al. ([BCD20], EC 2020). At the heart of our result is a fundamental duality property of total variation distance. As direct applications of our result, we (i) demonstrate how to find approximately revenue-optimal and approximately BIC mechanisms for weakly dependent prior distributions; (ii) show how to find correlation-robust mechanisms when only "noisy" versions of marginals are accessible, extending recent results of Bei et. al. ([BGLT19], SODA 2019); (iii) prove that prophet-inequality type guarantees are preserved for correlated priors, recovering a variant of a result of Dütting and Kesselheim ([DK19], EC 2019); (iv) give a new necessary condition for a correlated distribution to witness an infinite separation in revenue between simple and optimal mechanisms, complementing recent results of Psomas et al. ([PSCW22], NeurIPS 2022); (v) give a new condition for simple mechanisms to approximate revenue-optimal mechanisms for the case of a single agent whose type is drawn from a correlated distribution that can be captured by a Markov Random Field, complementing recent results of Cai and Oikonomou ([CO21], EC 2021).
Mechanism design studies optimization in strategic settings. The designer's task is to create a mechanism that interacts with strategic agents, each with their own, private preferences over the mechanism's output. The inability to provide meaningful guarantees for important objectives — such as revenue — when studying mechanism design problems through the lens of worst-case analysis, has motivated the study of Bayesian mechanisms. In the Bayesian setting, there is a probability distribution, typically known to the designer, from which agents' private information — their types — is drawn, and the designer seeks to maximize an objective function in expectation over the randomness of the types, and, at the same time, incentivize the agents to report their type truthfully.
While we have greatly deepened our understanding of mechanism design under the Bayesian setting, when taking this approach to practice, it is natural to ask what happens if the designer has only partial information about the agents' type distributions. In recent years, a growing literature studies
the robustness of mechanisms with respect to inaccurate priors. The term "robustness" can mean many things in this context, such as mechanism design using only sampling access from the underlying prior [CR14, HMR15, DHP16, MR16, CD17, GHZ19, GW21], or mechanism design using only parametric knowledge of the underlying prior [AMDW13, Car17, GL18, BGLT19, GPTD23]. Arguably, the most ambitious line of work in this thread strives to, given a prior distribution  $\mathcal{D}$ , design a mechanism that provides good guarantees simultaneously for all "true distributions"  $\widehat{\mathcal{D}}$  that are close to  $\mathcal{D}$  under some notion of statistical distance. Recent, compelling positive results [BS11, CD17, DK19, LLY19, BCD20] show that this endeavor is, in fact, possible.
In this work, we demonstrate that a large class of such robustness results can be obtained from a fundamental duality property of total variation distance. We distill this key property into various forms that are directly applicable to mechanism design, and then illustrate how they imply a variety of robustness results for both Dominant Strategy Incentive Compatible (DSIC) and Bayesian Incentive Compatible (BIC) mechanisms. In particular, we recover a litany of known results, such as the revenue robustness of BIC mechanisms of [BCD20] and robustness of prophet inequalities of [DK19], as well as prove several new results, such as a new necessary condition for a prior distribution to exhibit a large gap between simple and optimal mechanisms (complementing recent results of [PSCW22]).
Our Contributions. We consider the problem of a designer that wants to design a mechanism that maximizes a bounded objective function  $\mathcal{O}$ , when allocating  $m$  items to  $n$  strategic agents. Agent  $i$  has a private type  $t_i \in \mathcal{T}_i$ , drawn from a probability distribution  $\mathcal{D}_i$ , which specifies her value for every subset of items  $S \subseteq [m]$ . We write  $\mathcal{D}$  for the joint distribution over agents' types.
We start in Section 3 by proving a simple, but crucial lemma, Lemma 2, about non-truthful mechanisms in this setting. Lemma 2 states that the expected performance, with respect to the objective  $\mathcal{O}$ , of a (possibly non-truthful) mechanism  $\mathcal{M}$  is stable with respect to small changes in total variation (TV) distance to the underlying prior distribution. This lemma follows from the argument for Kantorovich-Rubinstein duality of TV distance [LPW09, Vil09].
Appropriate applications of Lemma 2 allow us to prove our first robustness result: DSIC mechanisms are strongly robust. Specifically, given a mechanism  $\mathcal{M}$  that is ex-post individually rational (ex-post IR), DSIC, and an  $\alpha$  approximation to the optimal mechanism for a distribution  $\mathcal{D}$  (w.r.t. the objective  $\mathcal{O}$ ), we construct an ex-post IR and DSIC mechanism  $\widehat{\mathcal{M}}$  that is agnostic to  $\widehat{\mathcal{D}}$  and  $\alpha$  approximates, under distribution  $\widehat{\mathcal{D}}$ , the performance of the optimal mechanism for  $\widehat{\mathcal{D}}$  (w.r.t.  $\mathcal{O}$ ) minus a small error which depends on the TV distance,  $d_{\mathrm{TV}}\left(\mathcal{D},\widehat{\mathcal{D}}\right)$ , of  $\mathcal{D}$  and  $\widehat{\mathcal{D}}$  (Theorem 1). Notably,  $\mathcal{D}$  and  $\widehat{\mathcal{D}}$  need not be product distributions, and if they have the same support,  $\widehat{\mathcal{M}}$  is the same mechanism as  $\mathcal{M}$ .
We proceed to study the robustness of BIC mechanisms. As opposed to the DSIC case, the BIC property is not maintained under small perturbations of a prior, or even by small changes in the support of a distribution. Rather surprisingly, however, in Lemma 4 we show that a mechanism  $\mathcal{M}$  that is BIC w.r.t. a distribution  $\mathcal{D}$ , is also approximately BIC w.r.t. a distribution  $\widehat{\mathcal{D}}$ , where the approximation depends on the distance between  $\mathcal{D}$  and  $\widehat{\mathcal{D}}$ . Combined with Lemma 2, we show our main robustness result for BIC mechanisms (Theorem 2): a mechanism  $\mathcal{M}$  that is BIC w.r.t.  $\mathcal{D}$  is approximately BIC w.r.t.  $\widehat{\mathcal{D}}$ , and its performance for an objective function  $\mathcal{O}$  is similar under  $\mathcal{D}$  and  $\widehat{\mathcal{D}}$ , as long as  $\mathcal{D}$  and  $\widehat{\mathcal{D}}$  have the same support. Compared to the corresponding robustness result of [BCD20], Theorem 2 only holds if  $\mathcal{D}$  and  $\widehat{\mathcal{D}}$  have the same support. Furthermore,  $\mathcal{M}$  is not  $\varepsilon$ -BIC (as in [BCD20]), but  $(\varepsilon, q)$ -BIC (see Section 2). On the other hand, our robustness results hold for arbitrary objectives, not just revenue, and arbitrary distributions, not just product distributions.
In Section 4 we show numerous applications of our robustness framework. Specifically, we give five applications. First, in Section 4.1, we extend our strong DSIC robustness to BIC mechanisms, albeit only for the revenue objective and product distributions. That is, we show that given a distribution  $\mathcal{D}$  that is close to a product distribution  $\mathcal{D}^p$ , approximately optimal (w.r.t. revenue) and BIC mechanisms for  $\mathcal{D}^p$  can be transformed into approximately optimal and approximately-BIC mechanisms for  $\mathcal{D}$  (Theorem 3). En route, we prove a self-reduction, that constructs an  $\varepsilon'$ -BIC mechanism w.r.t.  $\mathcal{D}$ , given an  $(\varepsilon, q)$ -BIC mechanism w.r.t.  $\mathcal{D}$ , that might be of independent interest.
In Section 4.2 we study the correlation-robust framework of [BGLT19] (first introduced in [Car17]). At a high level, we are given the marginals of  $n$  dependent agents for a single item. The goal is to design a mechanism that maximizes (among all feasible mechanisms) the minimum, over all possible
joint distributions consistent with the given marginals, expected revenue. We extend our robustness results to this setting (Theorem 4), and get implications for the max-min revenue performance of sequential posted prices and the Lookahead auction when the given marginals are inaccurate.
In Section 4.3 we consider prophet inequalities. In the simplest version of this problem, we are shown  $n$  non-negative numbers,  $t_1, \ldots, t_n$ , one at a time, and upon arrival we need to decide (immediately and irrevocably) whether we should accept  $t_i$ , or keep going.  $t_i$ s are drawn independently from known distributions  $\mathcal{D}_i$ . A celebrated result, known as the prophet inequality, states that a simple threshold policy  $\tau$  ("pick the first  $x_i$  bigger than  $\tau$ ) has expected reward at least half of a prophet, who knows the values of all  $x_i$ s in advance (and can therefore get reward  $\max_i t_i$ ). Prophet inequalities have numerous applications in mechanism design, and are the main tool in the analysis of posted price mechanisms. Observing that prophet inequalities correspond to sequential posted prices, which are DSIC mechanisms, and applying Lemma 2 for the welfare objective, we get robustness of prophet inequalities (Corollary 1); as a special case of our application we recover a variant of a result of Dütting and Kesselheim [DK19] on the TV robustness of prophet inequalities.
Finally, in Section 4.4 we apply our robustness framework to study revenue gaps between simple and optimal mechanisms. Here, [BCKW15, HN19] construct correlated distributions whose revenue is infinite, but such that simple mechanisms (e.g., those with finite menus) cannot extract a lot of revenue. Recently, Psomas et al. [PSCW22] provide (arguably, complex) conditions that a distribution  $\mathcal{D}$  should satisfy in order to witness such large gaps; our framework readily provides new, simple, and necessary conditions (Corollary 2), complementing the results of [PSCW22]. On a similar note, Cai and Oikonomou [CO21] escape the negative results of [BCKW15, HN19] by considering distributions described by a Markov Random Field (MRF). Their revenue guarantees for simple mechanisms are controlled by  $\Delta$ , a parameter of the MRF, that is determined by how much the value of an item can be influenced by values of other items. We show that a bound on the TV distance to a product distribution also suffices to provide guarantees on the revenue of simple mechanisms (Proposition 2). Furthermore, under conditions on the MRF, bounds on  $\Delta$  imply bounds on this distance (Proposition 4), but the other direction is not necessarily true (Proposition 3); thus, getting bounds on the TV distance of an MRF to a product distribution is a meaningful endeavor.
Related Work. A number of recent works study robustness in mechanism design; see [Car17] for a survey. The paper (thematically and technically) closest to ours is [BCD20], which proves the robustness of the revenue objective for BIC mechanisms under various notions of statistical distance, including TV and Prokhorov, as well as Lévy and Kolmogorov for single parameter settings. Specifically, given a distribution  $\mathcal{D}$ , [BCD20] show how to construct a mechanism that is (approximately) BIC, and performs well, when executed on any distribution  $\widehat{\mathcal{D}}$  that is close to  $\mathcal{D}$ . Here, we focus on TV distance, and recover their result for this case, slightly extending it for the case of revenue, and significantly extending it under some minor assumptions. Dütting and Kesselheim [DK19] study prophet inequalities with inaccurate priors. Specifically, given a product distribution  $\times_{i\in [n]}\mathcal{D}_i$ , [DK19] study policies that perform well when executed on a product distribution  $\times_{i\in [n]}\widehat{\mathcal{D}}_i$ , such that  $\mathcal{D}_i$  is close to  $\widehat{\mathcal{D}}_i$  for all  $i\in [n]$ , under various statistical distances. As an application of our robustness for DSIC mechanisms, we recover and extend to non-product distributions their result for TV distance, for the special case of sequential posted prices.
A related, but different, approach is to assume sample access to the underlying distribution [CR14, HMR15, DHP16, MR16, CD17, GHZ19, GW21]. In this line of work, the goal is bound the number of samples necessary to design a near-optimal mechanism, or, given a fixed number of samples, design the best mechanism possible. Robustness results of the former style, i.e., the current paper or [BCD20], sometimes imply sample complexity results, e.g., by arguing that using the samples, one can learn a distribution that is close to the real distribution, and then applying robustness results [BCD20]. We conjecture that similar results can be shown using our robustness framework, combined with estimation in TV distance [HJW15, DG85]. Another line of work [AMDW13, Car17, GL18, BGLT19, GPTD23] assumes partial knowledge of the true prior distribution, e.g., its mean or the marginals of a correlated distribution. The goal is to find the mechanism that (approximately) maximizes the worst-case performance with respect to the missing details (e.g., the CDF consistent with the mean or the joint distribution that respects the given marginals). Our results have implications for such settings as well. Finally, certain works consider the robustness of pathological examples in mechanism design, e.g., constructions of distributions that have infinite revenue gaps between simple and optimal mechanisms [BCKW15, HN19]. For example, [PSW19] uses the lens of smoothed
analysis ([ST04]) to reason about the robustness of the [HN19] constructions. Our framework has implications about these constructions; specifically, we give a new necessary condition for a distribution to be "pathological," complementing a recent result of [PSCW22].
From a probability theory perspective, we heavily use coupling techniques. Couplings are a general proof technique in probability theory with several historically notable uses [Doe38, Sko56, Kan60, Str65, Dud68, Wic70, Dob70] (also see [Vil09, Kal21]). Given two marginal distributions, the basic idea of coupling is to construct a consistent joint distribution on a common probability space in order to deduce certain relationships between the marginals. The key coupling used in this work — Dobrushin's optimal coupling — minimizes the probability that two random variables with given marginal distributions are different, and has been historically utilized to develop sharper results on Markov chain ergodicity, cf. [Dob70, Dob71, Gri75]. It turns out that this optimal coupling also defines total variation distance, cf. [LPW09, Vil09]. Such distances exhibit Kantorovich-Rubinstein duality and are characterized by the maximal difference of expected values with respect to the given marginal distributions [Kan60, LPW09, Vil09]. In this work, we distill how optimal couplings and duality for total variation can be used in yet another setting: mechanism design.
We examine the problem of a central designer who seeks to create a mechanism that maximizes an objective function given some prior knowledge of the universe. Consider a set  $[n] \triangleq \{1, \dots, n\}$  of selfish agents and a finite set  $[m] \triangleq \{1, \dots, m\}$  of items. Each agent  $i \in [n]$  has a type  $t_i$  belonging to a set  $\mathcal{T}_i$  of possible types. We assume that, for all  $i \in [n]$ , there exists a special type  $\perp \in \mathcal{T}_i$ , interpreted as the option of not participating in the designer's mechanism. Let  $\mathcal{T} = \mathcal{T}_1 \times \dots \times \mathcal{T}_n$ . We use  $\mathcal{X}$  to denote the set of all possible allocations of the items. (In particular, the sets  $\mathcal{X}$  and  $\mathcal{T}_i$  are typically standard Borel spaces, e.g., finite-dimensional Euclidean spaces. Hence, our analysis with couplings and total variation distance in the sequel do not require measure theoretic arguments.)
The goal of the designer is to construct a mechanism  $\mathcal{M} = (x,p)$  which consists of (1) an allocation rule  $x:\mathcal{T}\mapsto \Delta (\mathcal{X})$ , which maps reported types  $(t_{1},\ldots ,t_{n})\in \mathcal{T}$  to a distribution over allocations, and (2) a payment function  $p:\mathcal{T}\mapsto [-H,H]^n$  which maps reported types to (bounded) payments for each agent, for some fixed constant  $H > 0$ . We say that a mechanism  $\mathcal{M} = (x,p)$  is defined on types  $\mathcal{T}$  if the domain of  $x$  and  $p$  is  $\mathcal{T}$ . We write  $\mathcal{M}(t) = (x(t),p(t))$  for the outcome, i.e., the allocation and payments, of mechanism  $\mathcal{M}$  on input  $t\in \mathcal{T}$ .
Each agent  $i$  has a valuation function  $v_{i}:\mathcal{T}_{i}\times \mathcal{X}\mapsto [0,H]$ , which specifies their value for an allocation, given their type. We assume that agents are quasi-linear, i.e., the utility of agent  $i$ , with type  $t_i$ , for an allocation  $A$  and a payment  $p_i$  is equal to  $v_{i}(t_{i},A) - p_{i}$ . We overload notation and write  $t_i(\mathcal{M}(t_i,t_{-i}))$  for the utility of agent  $i$  with type  $t_i\in \mathcal{T}_i$  for the outcome of the mechanism  $\mathcal{M}$  on input  $(t_i,t_{-i})$ . We use  $\mathcal{F}$  for the set of possible outcomes of a mechanism  $\mathcal{M}$ . We assume that there exists a probability distribution  $\mathcal{D}$  supported on  $\mathcal{T}$  from which agents' types are drawn.
Mechanism Design Considerations. When faced with a mechanism  $\mathcal{M}$ , each agent  $i$  reports a type  $b_i \in \mathcal{T}_i$  to the mechanism. We aim to design mechanisms that (approximately) incentivize agents to report their true types, so we typically have that  $b_i = t_i$ , for all  $i \in [n]$ . Given a mechanism  $\mathcal{M}$ , we write  $u_i^{\mathcal{M}}(t_i \gets t_i', t_{-i})$  for the difference in utility of agent  $i$  when she reports  $t_i' \in \mathcal{T}_i$  instead of her true type  $t_i \in \mathcal{T}_i$ , and all other agents report according to  $t_{-i} \in \mathcal{T}_{-i}$ . That is,  $u_i^{\mathcal{M}}(t_i \gets t_i', t_{-i}) = t_i (\mathcal{M}(t_i, t_{-i})) - t_i (\mathcal{M}(t_i', t_{-i}))$ .
We consider four, increasingly weaker, notions of incentive compatibility. First, we say that a mechanism  $\mathcal{M}$  is Dominant Strategy Incentive Compatible (henceforth, DSIC) if an agent is better off reporting her true type, no matter what other agents report, i.e., for all  $i\in [n]$ , every type  $t_i\in \mathcal{T}_i$ , possible misreport  $t_i'\in \mathcal{T}_i$ , and types  $t_{-i}\in \mathcal{T}_{-i}$  for the remaining agents, it holds that  $u_i^{\mathcal{M}}(t_i\gets t_i',t_{-i})\geq 0$ . Second, a mechanism  $\mathcal{M}$  is Bayesian Incentive Compatible (henceforth BIC) with respect to a distribution  $\mathcal{D}$ , if an agent is better off reporting her true type in expectation over the other agents' reports, i.e. if for all  $i\in [n]$ , every type  $t_i\in \mathcal{T}_i$  and possible misreport  $t_i'\in \mathcal{T}_i$ , it holds that  $\mathbb{E}_{t_{-i}\sim \mathcal{D}_{-i}|t_i}\left[u_i^{\mathcal{M}}(t_i\gets t_i',t_{-i})\right]\geq 0$ . Third, a mechanism  $\mathcal{M}$  is  $\varepsilon$ -BIC w.r.t.  $\mathcal{D}$  if for all  $i\in [n]$ , and all  $t_i,t_i'\in \mathcal{T}_i$ , it holds that  $\mathbb{E}_{t_{-i}\sim \mathcal{D}_{-i}|t_i}\left[u_i^{\mathcal{M}}(t_i\gets t_i',t_{-i})\right]\geq -\varepsilon$ . Fourth, a mechanism  $\mathcal{M}$  is  $(\varepsilon,q)$ -BIC w.r.t.  $\mathcal{D}$  if it is  $\varepsilon$ -BIC with probability at least  $1 - q$ , i.e., if for all  $i\in [n]$ , and all
$t_i, t_i' \in \mathcal{T}_i$ ,  $\operatorname{Pr}_{t_i \sim \mathcal{D}_i} \left[\mathbb{E}_{t_{-i} \sim \mathcal{D}_{-i}|t_i} \left[u_i^{\mathcal{M}}(t_i \gets t_i', t_{-i})\right] \geq -\varepsilon\right] \geq 1 - q$ . An  $(\varepsilon, 0)$ -BIC mechanism is simply  $\varepsilon$ -BIC, and a 0-BIC is simply BIC.
Finally, a mechanism  $\mathcal{M}$  is ex-post Individually Rational (henceforth ex-post IR) if for every agent  $i\in [n]$ , every type  $t_i\in \mathcal{T}_i$  and types  $t_{-i}\in \mathcal{T}_{-i}$  for the remaining agents, it holds that  $t_i(\mathcal{M}(t_i,t_{-i}))\geq 0$ .
The Designer's Objective. The designer has an objective function  $\mathcal{O}(:,.) : \mathcal{T} \times \mathcal{F} \mapsto [a, b]$  that takes as input agents' (reported) types  $t \in \mathcal{T}$  and mechanism outcomes  $\mathcal{M}(t) \in \mathcal{F}$  (noting that the mechanism's outcome might be a randomized allocation) and outputs a real number in the interval  $[a, b]$ . Let  $V = b - a$ . The task of the designer is to find an ex-post IR and truthful mechanism (under one of the aforementioned notions of truthfulness) that maximizes this objective function in expectation over the randomness of  $\mathcal{D}$ . We denote the optimal value of the objective  $\mathcal{O}$  under distribution  $\mathcal{D}$  by  $OPT_{\mathcal{O}}(\mathcal{D})$ . We specify in context whether this is with respect to DSIC, or BIC, or  $\varepsilon$ -BIC mechanisms. A mechanism  $\mathcal{M}_{\mathcal{D}}^{\alpha}$  is an  $\alpha$  approximation to the optimal mechanism under  $\mathcal{D}$ , with respect to  $\mathcal{O}$ , if  $\mathbb{E}_{t \sim \mathcal{D}}[\mathcal{O}(t, \mathcal{M}_{\mathcal{D}}^{\alpha}(t))] \geq a OPT_{\mathcal{O}}(\mathcal{D})$ .
Some of our results hold for arbitrary, bounded objectives  $\mathcal{O}$ . Two objectives of specific interest to us will be welfare and revenue. The welfare objective, denoted by  $Val(.)$ , is simply the sum of agents' valuations of an outcome, i.e.  $Val(t,\mathcal{M}(t)) = \sum_{i\in [n]}v_i(t_i,x_i(t))$ , where  $x_{i}(t)$  is the allocation of agent  $i$  in the outcome  $\mathcal{M}(t)$ . The revenue objective, denoted by  $Rev(.)$ , is the sum of agents' payments, i.e.  $Rev(t,\mathcal{M}(t)) = \sum_{i\in [n]}p_i(t)$ , where  $p_i(t)$  is the payment of agent  $i$  in the outcome  $\mathcal{M}(t)$ . We often overload notation and write  $Rev(\mathcal{M},\mathcal{D})$  for the expected revenue of mechanism  $\mathcal{M}$  under distribution  $\mathcal{D}$ , i.e.  $Rev(\mathcal{M},\mathcal{D}) = \mathbb{E}_{t\sim \mathcal{D}}[Rev(t,\mathcal{M}(t))]$ . To maintain consistency with the (vast) literature on mechanism design we further denote the optimal revenue as  $Rev(\mathcal{D})$ , and the optimal welfare as  $Val(\mathcal{D})$ , under distribution  $\mathcal{D}$ .
Statistical Distance. Throughout the paper, we are interested in how mechanisms behave under different distributions that are not "too far" from each other. Our notion of distance in this paper is total variation distance.
Definition 1 (Total Variation Distance). The total variation (TV) distance between any two probability distributions  $P$  and  $Q$  on a sample space  $\Omega$  is defined as
$$
d _ {\mathsf {T V}} (P, Q) \triangleq \sup  _ {E \subseteq \Omega} | P (E) - Q (E) |,
$$
where the supremum is over all Borel measurable subsets  $E \subseteq \Omega$ , and  $P(E)$  (resp.  $Q(E)$ ) denotes the probability of the event  $E$  with respect to the distribution  $P$  (resp.  $Q$ ).