SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Maria-Florina Balcan
ninamf@cs.cmu.edu
Siddharth Prasad
sprasad2@cs.cmu.edu
Tuomas Sandholm
Optimized Markets, Inc.
Strategy Robot, Inc.
Strategic Machine, Inc.
sandholm@cs.cmu.edu
We develop a versatile new methodology for multidimensional mechanism design that incorporates side information about agent types to generate high social welfare and high revenue simultaneously. Prominent sources of side information in practice include predictions from a machine-learning model trained on historical agent data, advice from domain experts, and even the mechanism designer's own gut instinct. In this paper we adopt a prior-free perspective that makes no assumptions on the correctness, accuracy, or source of the side information. First, we design a meta-mechanism that integrates input side information with an improvement of the classical VCG mechanism. The welfare, revenue, and incentive properties of our meta-mechanism are characterized by novel constructions we introduce based on the notion of a weakest competitor, which is an agent that has the smallest impact on welfare. We show that our meta-mechanism, when carefully instantiated, simultaneously achieves strong welfare and revenue guarantees parameterized by errors in the side information. When the side information is highly informative and accurate, our mechanism achieves welfare and revenue competitive with the total social surplus, and its performance decays continuously and gradually as the quality of the side information decreases. Finally, we apply our meta-mechanism to a setting where each agent's type is determined by a constant number of parameters. Specifically, agent types lie on constant-dimensional subspaces (of the potentially high-dimensional ambient type space) that are known to the mechanism designer. We use our meta-mechanism to obtain the first known welfare and revenue guarantees in this setting.
Mechanism design is a high-impact branch of economics and computer science that studies the implementation of socially desirable outcomes among strategic self-interested agents. Major real-world use cases include combinatorial auctions (e.g., strategic sourcing, radio spectrum auctions), matching markets (e.g., housing allocation, ridesharing), project fundraisers, and many more. The two most commonly studied objectives in mechanism design are welfare maximization and revenue maximization. In many settings, welfare maximization, or efficiency, is achieved by the classic
Vickrey-Clarke-Groves (VCG) mechanism [19, 28, 50]. Revenue maximization is a much more elusive problem that is only understood in very special cases. The seminal work of Myerson [42] characterized the revenue-optimal mechanism for the sale of a single item in the Bayesian setting, but it is not even known how to optimally sell two items. It is known that welfare and revenue are generally at odds and optimizing one can come at the great expense of the other [1, 4, 7, 23, 33].
In this paper we study how side information (or predictions) about the agents can help with bicriteria optimization of both welfare and revenue. Side information can come from a variety of sources that are abundantly available in practice such as predictions from a machine-learning model trained on historical agent data, advice from domain experts, or even the mechanism designer's own gut instinct. Machine learning approaches that exploit the proliferation of agent data have in particular witnessed a great deal of success both in theory [8, 10, 38, 41] and in practice [24, 25, 46, 52]. In contrast to the typical Bayesian approach to mechanism design that views side information through the lens of a prior distribution over agents, we adopt a prior-free perspective that makes no assumptions on the correctness, accuracy, or source of the side information. A nascent line of work (part of a larger agenda on learning-augmented algorithms [40]) has begun to examine the challenge of exploiting predictions when agents are self-interested, but only for fairly specific problem settings [3, 13-15, 26, 53]. We contribute to this area with a general side-information-dependent meta-mechanism for a wide swath of multidimensional mechanism design problems that aim for high social welfare and high revenue.
Our main contribution is a versatile meta-mechanism that integrates side information about agent types with the bicriteria goal of simultaneously optimizing welfare and revenue.
In Section 2 we formally define the components of multidimensional mechanism design with side information. The abstraction of multidimensional mechanism design is a rich language that allows our theory to apply to many real-world settings including combinatorial auctions, matching markets, project fundraisers, and more—we expand on this list of examples further in Section 2. We also present the weakest-competitor VCG mechanism introduced by Krishna and Perry [34] and prove that it is revenue-optimal among all efficient mechanisms in the prior-free setting (extending their work which was in the Bayesian setting for a fixed known prior).
In Section 3 we present our meta-mechanism for mechanism design with side information. It generalizes the mechanism of Krishna and Perry [34]. We introduce the notion of a weakest-competitor set and a weakest-competitor hull, which are constructions that are crucial to understanding the payments and incentive properties of our meta-mechanism.
In Section 4 we prove that our meta-mechanism—when carefully instantiated—achieves strong welfare and revenue guarantees that are parameterized by errors in the side information. Our mechanism works by independently expanding the input predictions, where the expansion radius for each prediction is drawn randomly from a logarithmic discretization of the diameter of the ambient type space. Our mechanism achieves the efficient welfare OPT and revenue at least  $\Omega(\mathrm{OPT} / \log H)$  when the side information is highly informative and accurate, where  $H$  is an upper bound on any agent's value for any outcome. Its revenue approaches OPT if its initialization parameters are chosen wisely. Its performance decays gradually as the quality of the side information decreases (whereas naive approaches suffer from huge discontinuous drops in performance). Prior-free efficient welfare OPT, or total social surplus, is the strongest possible benchmark for both welfare and revenue. Finally, we extend our methods to a more general, more expressive side information language.
In Section 5 we use our meta-mechanism to derive new results in a setting where each agent's type is determined by a constant number of parameters. Specifically, agent types lie on constant-dimensional subspaces (of the potentially high-dimensional ambient type space) that are known to the mechanism designer. For example, a real-estate agent might infer a buyers' relative property values based on value per square foot. When each agent's true type is known to lie in a particular  $k$ -dimensional subspace of the ambient type space, we show how to use our meta-mechanism to guarantee revenue at least  $\Omega (\mathrm{OPT} / k(\log H)^k)$  while simultaneously guaranteeing welfare at least  $\mathrm{OPT} / \log H$ .
Traditionally it is known that welfare and revenue are at odds and maximizing one objective comes at the expense of the other. Our results show that side information can help mitigate this difficulty.
Side information in mechanism design. Various mechanism design settings have been studied under the assumption that some form of public side information is available. Medina and Vassilvitskii [38] study single-item (unlimited supply) single-bidder posted-price auctions with bid predictions. Devanur et al. [22] study the sample complexity of (single-parameter) auctions when the mechanism designer receives a distinguishing signal for each bidder. More generally, the active field of algorithms with predictions aims to improve the quality of classical algorithms when machine-learning predictions about the solution are available [40]. There have been recent explicit connections of this paradigm to settings with strategic agents [3, 13-15, 26]. Most related to our work, Xu and Lu [53] study auctions for the sale of a (single copy of a) single item when the mechanism designer receives point predictions on the bidders' values. Unlike our approach, they focus on deterministic modifications of a second-price auction. An important drawback of determinism is that revenue guarantees do not decay continuously as prediction quality degrades. For agents with value capped at  $H$  there is an error threshold after which, in the worst case, only a  $1 / H$ -fraction of revenue can be guaranteed (this is not even competitive with a vanilla second-price auction). Xu and Lu [53] prove that such a revenue drop is unavoidable by deterministic mechanisms. Finally, our setting is distinct from, but similar to in spirit, work that uses public attributes for market segmentation to improve revenue [8, 11].
Welfare-revenue tradeoffs in auctions. Welfare and revenue relationships in Bayesian auctions have been widely studied since the seminal work of Bulow and Klemperer [17]. Welfare-revenue tradeoffs for second-price auctions with reserve prices in the single item setting have been quantified [21, 31], with some approximate understanding of the Pareto frontier [23]. Anshelevich et al. [4] study welfare-revenue tradeoffs in large markets, Aggarwal et al. [2] study the efficiency of revenue-optimal mechanisms, and Abhishek and Hajek [1] study the efficiency loss of revenue-optimal mechanisms.
Constant-parameter mechanism design. Revenue-optimal mechanism design for settings where each agent's type space is of a constant dimension has been studied previously in certain specific settings. Single-parameter mechanism design is a well-studied topic dating back to the seminal work of Myerson [42], who (1) characterized the set of all truthful allocation rules and (2) derived the Bayesian optimal auction based on virtual values (a quantity that is highly dependent on knowledge of the agents' value distributions). Archer and Tardos [5] also characterize the set of allocation rules that can be implemented truthfully in the single-parameter setting, and use this to derive polynomial-time mechanisms with strong revenue approximation guarantees in various settings. Kleinberg and Yuan [33] prove revenue guarantees for a variety of single-parameter settings that depend on distributional parameters. Constrained buyers with two-parameter values have also been studied [36, 43].
Combinatorial auctions for limited supply. Our mechanism when agent types lie on known linear subspaces can be seen as a generalization of the well-known logarithmic revenue approximation that is achieved by a second-price auction with a random reserve price in the single-item setting [27]. Similar revenue approximations have been derived in multi-item settings for various classes of bidder valuation functions such as unit-demand [30], additive [35, 48], and subadditive [9, 18]. To the best of our knowledge, no previous techniques handle agent types on low-dimensional subspaces. Furthermore, our results are not restricted to combinatorial auctions unlike most previous research.
2 Problem formulation, example applications, and weakest-competitor VCG
We consider a general multidimensional mechanism design setting with a finite allocation space  $\Gamma$  and  $n$  agents.  $\Theta_{i}$  is the type space of agent  $i$ . Agent  $i$ 's true private type  $\theta_{i} \in \Theta_{i}$  determines her value  $v(\theta_{i}, \alpha)$  for allocation  $\alpha \in \Gamma$ . We will interpret  $\Theta_{i}$  as a subset of  $\mathbb{R}^{\Gamma}$ , so  $\theta_{i}[\alpha] = v(\theta_{i}, \alpha)$ . We use  $\pmb{\theta} \in \bigtimes_{i=1}^{n} \Theta_{i}$  to denote a profile of types and  $\pmb{\theta}_{-i} \in \Theta_{-i} := \bigtimes_{j \neq i} \Theta_{i}$  to denote a profile of types excluding agent  $i$ . We now introduce our model of side information. For each agent, the mechanism designer receives a subset of the type space predicting that the subset contains the agent's true yet-to-be-revealed type. Formally, the mechanism designer receives additional information about each agent in the form of a refinement of each agent's type space, given by  $\widetilde{\Theta}_{1} \subseteq \Theta_{1}, \ldots, \widetilde{\Theta}_{n} \subseteq \Theta_{n}$ . These refinements postulate that the true type of bidder  $i$  is actually contained in  $\widetilde{\Theta}_{i}$  (though the mechanism designer does not necessarily know whether or not these predictions are valid). We refer to the  $\widetilde{\Theta}_{i}$  as side-information sets or predictions. To simplify exposition, we assume that prior
to receiving side information the mechanism designer has no differentiating information about the agents' types, that is,  $\Theta_{1} = \dots = \Theta_{n}$ . Let  $\Theta$  denote this common ambient type space.
A mechanism with side information is specified by an allocation rule  $\alpha (\pmb {\theta};\widetilde{\Theta}_1,\dots ,\widetilde{\Theta}_n)\in \Gamma$  and a payment rule  $p_i(\pmb {\theta};\widetilde{\Theta}_1,\dots ,\widetilde{\Theta}_n)\in \mathbb{R}$  for each agent  $i$ . We assume agents have quasilinear utilities. A mechanism is incentive compatible (IC) if  $\theta_{i}\in \mathrm{argmax}_{\theta^{\prime}_{i}\in \Theta_{i}}\theta_{i}[\alpha (\theta^{\prime}_{i},\pmb{\theta}_{-i};\widetilde{\Theta}_{1},\dots ,\widetilde{\Theta}_{n})] - p_{i}(\theta^{\prime}_{i},\pmb{\theta}_{-i};\widetilde{\Theta}_{1},\dots ,\widetilde{\Theta}_{n})$  holds for all  $i,\theta_{i}\in \Theta_{i},\pmb{\theta}_{-i}\in \Theta_{-i},\widetilde{\Theta}_{1}\subseteq \Theta_{i},\dots ,\widetilde{\Theta}_{n}\subseteq \Theta_{n}$ , that is, agents are incentivized to report their true type regardless of what other agents report and regardless of the side information used by the mechanism (this definition is equivalent to the usual notion of dominant-strategy IC and simply stipulates that side information ought to be used in an IC manner). A mechanism is individually rational (IR) if  $\theta_{i}[\alpha (\theta_{i},\pmb{\theta}_{-i};\widetilde{\Theta}_{1},\dots ,\widetilde{\Theta}_{n})] - p_{i}(\theta_{i},\pmb{\theta}_{-i};\widetilde{\Theta}_{1},\dots ,\widetilde{\Theta}_{n})\geq 0$  holds for all  $i,\theta_{i},\pmb{\theta}_{-i},\widetilde{\Theta}_{1},\dots ,\widetilde{\Theta}_{n}$ . We will analyze a variety of randomized mechanisms that randomize over IC and IR mechanisms. Such randomized mechanisms are thus IC and IR in the strongest possible sense (as supposed to weaker in-expectation IC/IR). An important note: no assumptions are made on the veracity of  $\widetilde{\Theta}_i$ , and agent  $i$  's misreporting space is the ambient type space  $\Theta_{i}$ .
Our model of side information within the rich language of multidimensional mechanism design allows us to capture a variety of different problem scenarios where both welfare and revenue are desired objectives. We list a few examples of different multidimensional mechanism settings along with examples of different varieties of side information sets.
- Combinatorial auctions: There are  $m$  indivisible items to be allocated among  $n$  agents (or to one). The allocation space  $\Gamma$  is the set of  $(n + 1)^m$  allocations of the items and  $\theta_i[\alpha]$  is agent  $i$ 's value for the bundle of items she is allocated by  $\alpha$ . Let  $X$  and  $Y$  denote two of the items for sale. The set  $\widetilde{\Theta}_i = \{\theta_i : \theta_i[\{X, Y\}] \geq 9, \theta_i[\{X\}] + \theta_i[\{Y\}] \geq 10\}$  represents the prediction that agent  $i$ 's values for  $X$  and  $Y$  individually sum up to at least $10, and her value for the bundle is at least $9. Here,  $\widetilde{\Theta}_i$  is the intersection of linear constraints.
- Matching markets: There are  $m$  items (e.g., houses) to be matched to  $n$  buyers. The allocation space  $\Gamma$  is the set of matchings on the bipartite graph  $K_{m,n}$  and  $\theta_i[\alpha]$  is buyer  $i$ 's value for the item  $\alpha$  assigns her. Let  $\alpha_1, \alpha_2, \alpha_3$  denote three matchings that match house 1, house 2, and house 3 to agent  $i$ , respectively. The set  $\widetilde{\Theta}_i = \{\theta_i : \theta_i[\alpha_1] = 2 \cdot \theta_i[\alpha_2] = 0.75 \cdot \theta_i[\alpha_3]\}$  represents the information that agent  $i$  values house 1 twice as much as house 2, and 3/4 as much as house 3. Here,  $\widetilde{\Theta}_i$  is the linear space given by span  $(\langle 1, 1/2, 4/3 \rangle)$ .
- Fundraising for a common amenity: A multi-story office building that houses several companies is opening a new cafeteria on a to-be-determined floor and is raising construction funds. The allocation space  $\Gamma$  is the set of floors of the building and  $\theta_i[\alpha]$  is the (inverse of the) cost incurred by building-occupant  $i$  for traveling to floor  $\alpha$ . The set  $\widetilde{\Theta}_i = \{\theta_i : \| \theta_i - \theta_i^* \|_p \leq k\}$  postulates that  $i$ 's true type is no more than  $k$  away from  $\theta_i^*$  in  $\ell_p$ -distance, which might be derived from an estimate of the range of floors agent  $i$  works on based on the company agent  $i$  represents. Here,  $\widetilde{\Theta}_i$  is given by a (potentially nonlinear) distance constraint.
- Bidding for a shared outcome: A delivery service that offers multiple delivery rates (priced proportionally) needs to decide on a delivery route to serve  $n$  customers. The allocation space  $\Gamma$  is the set of feasible routes and  $\theta_i[\alpha]$  is agent  $i$ 's value for receiving her packages after the driving delay specified by  $\alpha$ . Let  $\alpha_t$  denote an allocation that imposes a driving delay of  $t$  on agent  $i$ . The set  $\widetilde{\Theta}_i = \{ \theta_i : \theta_i[\alpha_0] \geq 50, \theta_i[\alpha_{t+1}] \geq f_t(\theta_i[\alpha_t]) \forall t \}$  is the prediction that agent  $i$  is willing to pay $50 to receive her package as soon as possible, and is at worst a time discounter determined by (potentially nonlinear) discount functions  $f_t$ . Here, the complexity of  $\widetilde{\Theta}_i$  is determined by the  $f_t$ .
In our results we will assume that  $\Theta = [0,H]^{\Gamma}$  imposing a cap  $H$  on any agent's value for any allocation. This is the only problem-specific parameter in our results. In the above four bulleted examples  $H$  represents the maximum value any agent has for the grand bundle of items, any available house, the cafeteria opening on her floor, and receiving her packages with no delay, respectively.
The weakest-competitor VCG mechanism
The VCG mechanism can generally be highly suboptimal when it comes to revenue [7, 39, 49] (and conversely mechanisms that shoot for high revenue can be highly welfare suboptimal). However, if efficiency is enforced as a constraint of the mechanism design, then the weakest-competitor VCG (WCVCG) mechanism introduced by Krishna and Perry [34] is in fact revenue optimal (they call it the generalized VCG mechanism). While VCG payments are based on participation externalities, WCVCG payments are based on agents being replaced by weakest competitors who have the smallest impact on welfare. This approach yields a strict revenue improvement over vanilla VCG. Krishna and Perry [34] proved that the Bayesian version of WCVCG is revenue optimal among all efficient, IC, and IR mechanisms. The (prior-free) WCVCG mechanism works as follows. Given reported types  $\theta$ , it uses the efficient allocation  $\alpha^{*} = \operatorname{argmax}_{\alpha \in \Gamma} \sum_{i=1}^{n} \theta_{i}[\alpha]$ . The payments are given by  $p_{i}(\theta) = \min_{\widetilde{\theta}_{i} \in \Theta_{i}} (\max_{\alpha \in \Gamma} \sum_{j \neq i} \theta_{j}[\alpha] + \widetilde{\theta}_{i}[\alpha]) - \sum_{j \neq i} \theta_{j}[\alpha^{*}]$ . Here,  $\Theta_{i}$  is the ambient type space of agent  $i$ . If  $0 \in \Theta_{i}$ ,  $p_{i}$  is the vanilla VCG payment. Krishna and Perry [34] prove the following result in the Bayesian setting, which we reproduce in a stronger prior-free form for completeness.
Theorem 2.1. Weakest-competitor VCG is revenue-optimal subject to efficiency, IC, and IR.
Proofs of all results in this paper are in Appendix A. Our meta-mechanism (Section 3) is a generalization of WCVCG that uses side information sets rather than the ambient type space to determine payments. (Misreporting is not limited to side information sets.) Our meta-mechanism relaxes efficiency in order to use the side information to boost revenue.
3 Weakest-competitor sets and our meta-mechanism
In this section we present our meta-mechanism for mechanism design with side information. Our meta-mechanism generalizes the WCVCG mechanism. We begin by introducing some new constructions based on the concept of a weakest competitor. These constructions are the key ingredients in understanding the role of side information in our meta-mechanism. Let  $\theta \preceq \theta^{\prime}$  if  $\theta [\alpha ]\leq \theta^{\prime}[\alpha ]$  for all  $\alpha \in \Gamma$ . Let  $\theta \preccurlyeq \theta^{\prime}$  if  $\theta [\alpha ]\leq \theta^{\prime}[\alpha ]$  for all  $\alpha \in \Gamma$  and there exists  $\alpha^{\prime}\in \Gamma$  with  $\theta [\alpha^{\prime}] < \theta^{\prime}[\alpha^{\prime}]$ . Let  $\theta \prec \theta^{\prime}$  if  $\theta [\alpha ] < \theta^{\prime}[\alpha ]$  for all  $\alpha \in \Gamma$ . We assume  $\Theta_{i} = \Theta = [0,H]^{\Gamma}$  for all  $i$ , that is, all agents share a common ambient type space with no up-front differentiating information.
Definition 3.1. The extended weakest-competitor set of a closed set  $\widetilde{\Theta}_i$ , denoted by  $\overline{\mathsf{WC}} (\widetilde{\Theta}_i)$ , is the subset of all weakest competitors in  $\widetilde{\Theta}_i$  over all possible type profiles of the other agents. Formally,  $\overline{\mathsf{WC}} (\widetilde{\Theta}_i) := \{\mathrm{argmin}_{\widetilde{\theta}_i\in \widetilde{\Theta}_i}(\max_{\alpha \in \Gamma}\sum_{j\neq i}\theta_j[\alpha ] + \widetilde{\theta}_i[\alpha ]):\pmb {\theta}_{-i}\in \Theta_{-i}\}$ . The weakest-competitor set of  $\widetilde{\Theta}_i$ , denoted by  $\mathsf{WC}(\widetilde{\Theta}_i)$ , is the subset of  $\overline{\mathsf{WC}} (\widetilde{\Theta}_i)$  where ties in the argmin are broken by discarding any  $\theta^\prime$  in the argmin if there exists  $\theta$  also in the argmin with  $\theta \preccurlyeq \theta^{\prime}$ . We call members of both  $\overline{\mathsf{WC}} (\widetilde{\Theta}_i)$  and  $\mathsf{WC}(\widetilde{\Theta}_i)$  weakest competitors and say  $\widehat{\theta_i}$  is a weakest competitor relative to  $\pmb{\theta}_{-i}$  if  $\widehat{\theta}_i\in \mathrm{argmin}_{\widetilde{\theta}_i\in \widetilde{\Theta}_i}\max_{\alpha \in \Gamma}\sum_{j\neq i}\theta_j[\alpha ] + \widetilde{\theta}_i[\alpha ]$ .
The weakest-competitor set is a natural notion of a lower bound corresponding to a given predicted type set. From the perspective of WCVCG, the payment of an agent with true type in  $\widetilde{\Theta}_i$  only depends on  $\mathsf{WC}(\widetilde{\Theta}_i)$  and not on  $\widetilde{\Theta}_i$ . Motivated by this observation, we define the weakest-competitor hull, which can be viewed as a "weakest-competitor relaxation".
Definition 3.2. The weakest-competitor hull of  $\widetilde{\Theta}_i$ , denoted by  $\mathsf{WCH}(\widetilde{\Theta}_i)$ , is the maximal set  $S$  such that  $\mathsf{WC}(S) = \mathsf{WC}(\widetilde{\Theta}_i)$  (no  $T \supset S$  satisfies  $\mathsf{WC}(T) = \mathsf{WC}(\widetilde{\Theta}_i)$ ).
Weakest-competitor sets and hulls can be simply characterized without explicit reference to the mechanics of WCVCG. Figure 1 displays examples in a two-dimensional type space.