Bayesian Incentive Compatibility via Matchings Jason D. Hartline∗ Robert Kleinberg† Azarakhsh Malekian‡ Abstract sider a relaxation of the performance constraints of We give a simple reduction from Bayesian incentive algorithms; we allow algorithms not to be worst-case compatible mechanism design to algorithm design in approximations. In this context we give a very gen- settings where the agents’ private types are multi- eral reduction from Bayesian mechanism design to dimensional. The reduction preserves performance algorithm design that approximately preserves or im- up to an additive loss that can be made arbitrarily proves the expected performance, i.e., social welfare, small in polynomial time in the number of agents and of the algorithm. Of course, our reduction can also be the size of the agents’ type spaces. applied to a worst-case β-approximation algorithm; in such a setting it results in a mechanism that is a 1 Introduction Bayesian β(1 + )-approximation. This approach to mechanism design sits well with Motivated by Internet applications the methodologies a standard paradigm for algorithm design wherein a of mechanism design have been adopted to consider practitioner might fine-tune their algorithm to the design of algorithms and protocols for selfish agents. actual workload they face rather than optimize for the A central problem in this area is in merging compu- worst case. Furthermore, in the Internet applications tational constraints (from approximation algorithms) that motivate this field, protocols are executed at a with incentive constraints (from mechanism design). large scale and distributions are likely to be learnable. Much of the recent literature on this problem has fo- This paper is a direct followup to work of Hartline cused on mechanisms satisfying the strongest possible and Lucier [8] that gives a such a Bayesian reduction incentive constraints and the strongest possible no- for the special case where the selfish agents’ prefer- tions of tractability. Namely, the focus has been on ex ences are single-dimensional, e.g., a value for receiv- post incentive compatibility (i.e., truthtelling in dom- ing service. While this work has elegant economic inant strategies) and worst-case approximation algo- motivation, its sampling-based blackbox reduction rithms. Positive results in this area are in the form of requires much extra machinery to achieve Bayesian new algorithms for paradigmatic problems that sat- incentive compatibility; furthermore, it has no clear isfy incentive constraints and match the worst-case generalization of its ironing-based approach to more performance bounds of the algorithmic design prob- the general (and fundamental) multi-dimensional set- lem sans incentive constraints. For ex post incentive ting, e.g., where an agent might have distinct values compatibility, no general reductions are known. for several services that are available. In contrast we In this paper we consider a relaxation of the in- consider multi-dimensional settings and give a very centive constraints to Bayesian incentive compatibil- simple reduction. Essentially: incentive constraints ity (BIC) where truthtelling is a Bayes-Nash equilib- in BIC mechanism design are as simple as solving rium, i.e., where strategies are a mutual best response a bipartite matching problem independently on the to strategies of other agents when agent preferences type space of each agent. In settings with small type are drawn from a known distribution. We also con- spaces but many agents, this gives a general polyno- mial time reduction from mechanism design to algo- ∗Department of Electrical Engineering and Computer rithm design. Science, Northwestern University, Evanston, IL. Email: The main approach is similar to the one in [8]. [email protected]. Supported in part by NSF Grant CCF-0830773 and NSF Career Award CCF-0846113. Our reduction will accept reported types from agents. †Department of Computer Science, Cornell University, It will transform each of these types to a distribution Ithaca, NY. Email: [email protected]. Supported by NSF over types with the properties: (a) the transforma- grants CCF-0643934 and AF-0910940, an Alfred P. Sloan tion applied to a random type from the distribution Foundation Fellowship, and a Microsoft Research New Faculty results in the transformed type with the same distri- Fellowship. ‡Department of Electrical Engineering and Computer bution, (b) the transformation weakly improves per- Science, Northwestern University, Evanston, IL. Email: formance, and (c) the algorithm composed with the [email protected]. transformation is Bayesian incentive compatible. It then inputs the transformed types into the original al- watnotai et al. [5], where “solved” means that the gorithm and returns its output. Finally, it computes approximation factor of the mechanism matches the appropriate payments. algorithmic lower bound. There is also a large lit- The transformation we apply to each agent’s erature on multi-dimensional combinatorial auctions type is based on computing the maximum weighted and approximation that we do not cite exhaustively. matching in a bipartite graph between the random There have been a few reductions from ex post types from the distribution (including the agent’s real incentive compatible mechanism design to algorithm type), termed “replicas”, and the set of outcomes of design for special classes of algorithms. Of course the the algorithm on another equal-sized random set of VCG mechanism reduces the IC mechanism design types drawn from the distribution, termed “surro- problem to exact algorithm design [13, 4, 7]. Lavi gates”. The transformation then outputs the surro- and Swamy [9] consider IC mechanisms for multi- gate type to which the agent’s real type is matched. parameter packing problems and give a technique for This basic replica-surrogate-matching approach can constructing a (randomized) β-approximation mech- be further refined and improved for the special cases anism from any β-approximation algorithm that ver- of single-dimensional agents and discrete explicitly ifies an integrality gap. For polynomial time approx- specified type spaces. (The latter reduction was in- imation schemes (PTAS), Briest et al. [3] solve the dependently and concurrently discovered by Bei and single-dimensional case and Dugmhi and Roughgar- Huang [2].) den [6] solve the case of multi-dimensional additive To solve the above maximum weighted matching valuations in downward-closed settings; both papers problem we need to be able to evaluate the value of can be viewed as blackbox reductions from the ex each replica for the outcome the algorithm obtains post IC mechanism design problem to the PTAS al- for each of the surrogates. If a replica is matched gorithm design problem. to a surrogate the value it obtains is the expected Finally, for Bayesian incentive compatibility, value over outcomes the algorithm produces when Hartline and Lucier [8] give a blackbox reduction run on the surrogate and random types of the other from BIC mechanism design to algorithm design in agents. We consider two computational models: single-dimensional settings. This reduction converts an ideal model (Section 3) where we are able to any algorithm into a mechanism and approximately exactly calculate the expected value an agent might preserves its expected (for the Bayesian prior) per- have for the distribution over outcomes produced formance. Our approach follows their methodol- by the algorithm, and a blackbox model (Section 4) ogy closely. We improve on their results for single- where we can only estimate this value by repeatedly dimensional settings and extend them to to multi- sampling the types of the other agents and running dimensional settings. the algorithm. Naturally, the blackbox model is more In concurrent and independent work Bei and realistic, though it presents some serious challenges. Huang [2] reduce ε-BIC mechanism design to algo- Results. In the ideal model the replica- rithm design in settings with discrete and explicitly surrogate matching and variants are BIC and we given type spaces. In contrast, our work for discrete give bounds on the number of replicas and surrogates settings uses the same basic approach but produces needed to ensure that the surplus loss of the result- a BIC mechanism, albeit in pseudo-polynomial time. ing mechanism is at most a small additive ε. Nat- Importantly, Bei and Huang give a nice application urally, these bounds degrade with a suitable notion of the approach to the paradigmatic problem of com- of the complexity of the type space. In the black- binatorial auctions with subadditive bidders. box model our results for single-dimensional settings can be extended to give BIC mechanisms. For dis- 2 Preliminaries crete explicitly specified type spaces we can correct We consider mechanisms for selfish agents. An agent errors inherent in sampling to give a BIC reduction in has a private type t from type space T . There are pseudo-polynomial time. For continuous type spaces n agents. When we wish to be specific about a our reduction is not BIC but it is ε-BIC. particular agent κ ∈ [n] we index as, e.g., tκ and Related Work. There has been extensive work T κ. The type profile of the n agents is denoted by on designing ex post incentive compatible and t = (t1, . , tn) ∈ T 1 × · · · × T n. An algorithm A tractable mechanisms that provide worst-case ap- maps a type profile t to an outcome x from outcome proximation guarantees. The paradigmatic problem space X . Agent κ with type tκ has valuation v(tκ, x) of single-minded combinatorial auctions was solved for outcome x. We assume that agent values are by Lehmann et al. [10] and related machine schedul- non-negative and come from a bounded range that, ing (to minimize makespan) was solved by Dhang- without loss of generality, is [0, 1]. A mechanism M = (A, p) maps the type profile into an outcome optimal outcome with agent κ but does not include via an algorithm A and into a payment profile p = κ’s value in the sum.