GROWRANGE: Anytime VCG-Based Mechanisms

We introduce anytime mechanisms for distributed optimization with self-interested agents. Anytime mechanisms retain good incentive properties even when interrupted before the optimal solution is computed, and provide better quality solutions when given additional time. Anytime mechanisms can solve easy instances of a hard problem quickly and optimally, while providing approximate solutions on very hard instances. In a particular instantiation, GROWRANGE, we successively expand the range of outcomes considered, computing the optimal solution for each range. Truth-revelation remains a dominant strategy equilibrium with a stage-based interruption, and is a best-response with high probability when the interruption is time-based. Introduction Designing mechanisms to solve distributed optimization problems with self-interested agents is becoming increasingly important in a wide variety of settings, from ecommerce, to the allocation of computational resources in open systems, to planning in multi-agent systems. This field, called computational mechanism design (CMD), aims to design solutions that are both incentive-compatible (with truth revelation in a game-theoretic equilibrium) and tractable. Combinatorial auctions (CAs), with agents that demand bundles of items, are a canonical problem in CMD. All previous work on tractable and strategyproof mechanisms (with truth-revelation in a dominant-strategy equilibrium) for CAs has considered restricted domains of agent preferences. For instance, Lehmann et al. (2002) describe a fast and strategyproof CA for single-minded agents that demand only one bundle. But, there are many other examples (Mu’alem & Nisan 2002; Archer et al. 2003; Bartal, Gonen, & Nisan 2003). These methods do not apply to the general CA problem. In this paper, we introduce anytime mechanisms, as a new paradigm for the design of incentive-compatible and tractable mechanisms. Anytime mechanism will solve easy instances of a hard family of problems quickly and optimally, while returning approximate solutions and retaining strategyproofness on very hard instances. Provable worstcase approximation results are dropped in favor of good performance on most problems coupled with the ability to terminate the algorithm with an approximate solution on the Copyright c 2004, American Association for Artificial Intelligence (www.aaai.org). All rights reserved. very hardest of problems.1 We address the challenge of retaining useful incentive properties, such that truthful bidding is an equilibrium whenever a mechanism is terminated. It is worth emphasizing that a naive approach, in which a Vickrey-Clarke-Groves (VCG) mechanism (see Nisan & Ronen (2000)) is coupled with an anytime winnerdetermination algorithm, would not be strategyproof unless the algorithm has enough time to solve the problem optimally. The strategyproofness of the VCG mechanism ordinarily relies on the optimality of its decision. Our solution builds on maximal-in-range VCG mechanisms (Nisan & Ronen 2000).We implement anytime mechanisms as staged mechanisms, with a new range of outcomes considered in each stage and the optimal solution computed for each new range. Importantly, the range adopted in each stage must not depend on any information reported by agents. In each stage we also compute the optimal solution to the problem without each agent. When interrupted, the anytime mechanism implements the best solution found so far, and determines VCG-based payments on the basis of the best solutions found so far without each agent. The mechanism will continue to compute a better solution when provided with more time. To understand the incentive properties we consider two different models of interruption. First, we consider stagebased interruptions, in which the mechanism is interrupted after some number of stages. Truth-revelation remains a dominant strategy equilibrium, because the final solution is optimal over the union of the ranges explored, with individual ranges chosen without regard to agent bids. Throughout the paper it is important that the interruption comes from the center, or some third-party, and not from one of the agents. A more realistic model is one in which the interruption process is time-based, for instance an answer might be required after 10 minutes. A new concern here is that an agent can indirectly affect the sequence of ranges explored by changing the difficulty of the problem through its bids, and thus influence the progress made by the algorithm before interruption. Our solution is to use consensus functions (Goldberg & Hartline 2003) to compute a conservative and agentindependent estimate of the number of stages completed by the time of an interruption, with results from any additional stages discarded. Together with additional assumpIndeed, Nisan & Ronen (2000) and Lavi et al. (2003) suggest that no worst-case polynomial time combinatorial auction can be strategyproof and provide good worst-case approximation properties, without assuming a restricted preference domain. tions about the maximal influence that an agent can have on the run time, this makes truthful bidding a best-response with high probability, whatever the bids of other agents. We illustrate our methods in the context of CAs. We define GROWRANGE, which is a particular partition-based instantiation. The empirical results illustrate encouraging performance on hard problems with high run time variance. Preliminaries Mechanism design (MD) considers a system of rational selfinterested agents and the problem of choosing an outcome from a finite set of possibilities. Let denote the number of agents. Each agent has a type, , that defines its value for each possible choice . This information is privately known to each agent. Let ! " ! # $ &%' denote a type vector and )( to denote all types except that of agent . Agents have quasilinear utility,