In combinatorial auctions, multiple goods are sold simultaneously and bidders may bid for arbitrary combinations of goods. Determining the outcome of such an auction is an optimization problem that is NP-complete in the general case. We propose two methods of overcoming this apparent intractability. The first method, which is guaranteed to be optimal, reduces running time by structuring the search space so that a modified depth-first search usually avoids even considering allocations that contain conflicting bids. Caching and pruning are also used to speed searching. Our second method is a heuristic, market-based approach. It sets up a virtual multi-round auction in which a virtual agent represents each original bid bundle and places bids, according to a fixed strategy, for each good in that bundle. We show through experiments on synthetic data that (a) our first method finds optimal allocations quickly and offers good anytime performance, and (b) in many cases our second method, despite lacking guarantees regarding optimality or running time, quickly reaches solutions that are nearly optimal. 1 Combinatorial Auctions Auction theory has received increasing attention from computer scientists in recent years. 1 One reason is the explosion of internet-based auctions. The use of auctions in business-to-business trades is also increasing rapidly [Cortese and Stepanek, 1998]. Within AI there is growing interest in using auction mechanisms to solve distributed resource allocation problems. For example, auctions and other market mechanisms are used in network bandwidth allocation, distributed configuration design, factory scheduling, and operating system memory allocation [Clearwater, 1996]. Market-oriented programming has

We investigate the “generalized second-price ” (GSP) auction, a new mechanism used by search engines to sell online advertising. Although GSP looks similar to the Vickrey-Clarke-Groves (VCG) mechanism, its properties are very different. Unlike the VCG mechanism, GSP generally does not have an equilibrium in dominant strategies, and truth-telling is not an equilibrium of GSP. To analyze the properties of GSP, we describe the generalized English auction that corresponds to GSP and show that it has a unique equilibrium. This is an ex post equilibrium, with the same payoffs to all players as the dominant strategy equilibrium of VCG. (JEL D44, L81, M37) This paper investigates a new auction mechanism, which we call the “generalized secondprice” auction, or GSP. GSP is tailored to the unique environment of the market for online ads, and neither the environment nor the mechanism has previously been studied in the mechanism design literature. While studying the properties of a novel mechanism is often fascinating in itself, our interest is also motivated by the spectacular commercial success of GSP. It is the dominant transaction mechanism in a large and rapidly growing industry. For example, Google’s total revenue in 2005 was $6.14 billion. Over 98 percent of its revenue came from GSP auctions. Yahoo!’s total revenue in 2005 was $5.26 billion. A large share of Yahoo!’s revenue is derived from sales via GSP auctions. It is believed that over half of Yahoo!’s revenue is derived from sales via GSP auctions. As of May 2006, the combined market capitalization of these companies exceeded $150 billion. Let us briefly describe how these auctions work. When an Internet user enters a search

When an auction of multiple items is performed, it is often desirable to allow bids on combinations of items, as opposed to only on single items. Such an auction is often called "combinatorial ", and the exponential number of possible combinations results in computational intractability of many aspects regarding such an auction. This paper considers two of these aspects: the bidding language and the allocation algorithm. First we consider which kinds of bids on combinations are allowed and how, i.e. in what language, they are specified. The basic tradeoff is the expressibility of the language versus its simplicity. We consider and formalize several bidding languages and compare their strengths. We prove exponential separations between the expressive power of different languages, and show that one language, "OR-bids with phantom items", can polynomially simulate the others. We then consider the problem of determining the best allocation -- a problem known to be computationally intractable. We suggest an approach based on Linear Programming (LP) and motivate it. We prove that the LP approach finds an optimal allocation if and only if prices can be attached to single items in the auction. We pinpoint several classes of auctions where this is the case, and suggest greedy and branch-and-bound heuristics based on LP for other cases. 1

The routing of traffic between... this paper, we address the problem of interdomain routing from a mechanism-design point of view. The application of mechanism-design principles to the study of routing is the subject of earlier work by Nisan and Ronen [15] and Hershberger and Suri [11]. In this paper, we formulate and solve a version of the routing-mechanism design problem that is different from the previously studied version in three ways that make it more accurately reflective of real-world interdomain routing: (1) we treat the nodes as strategic agents, rather than the links; (2) our mechanism computes lowest-cost routes for all source-destination pairs and payments for transit nodes on all of the routes (rather than computing routes and payments for only one source-destination pair at a time, as is done in [15,11]); (3) we show how to compute our mechanism with a distributed algorithm that is a straightforward extension to BGP and causes only modest increases in routingtable size and convergence time (in contrast with the centralized algorithms used in [15,11]). This approach of using an existing protocol as a substrate for distributed computation may prove useful in future development of Internet algorithms generally, not only for routing or pricing problems. Our design and analysis of a strategyproof, BGP-based routing mechanism provides a new, promising direction in distributed algorithmic mechanism design, which has heretofore been focused mainly on multicast cost sharing.

One of the major achievements of mechanism design theory is the family of truthful (incentive compatible) mechanisms often called VCG (named after Vickrey, Clarke and Groves). When applying VCG mechanisms to complex mechanism design problems such as combinatorial auctions a problem emerges: even finding optimal outcomes is computationally intractable. A striking observation is that if the optimal outcome is replaced by the results of computationally tractable approximation algorithms or heuristics then the resulting mechanism (termed VCG-based) is no longer necessarily truthful! The first part of this paper considers this problem in depth and shows that it is almost universal. Specifically, we prove that essentially all reasonable approximations or heuristics for combinatorial auctions as well as a wide class of cost minimization problems yield non-truthful VCG-based mechanisms. The second part of this paper proposes a method for handling this non-truthfulness. We introduce a...

Abstract. Some important classical mechanisms considered in Microeconomics and Game Theory require the solution of a difficult optimization problem. This is true of mechanisms for combinatorial auctions, which have in recent years assumed practical importance, and in particular of the gold standard for combinatorial auctions, the Generalized Vickrey Auction (GVA). Traditional analysis of these mechanisms—in particular, their truth revelation properties—assumes that the optimization problems are solved precisely. In reality, these optimization problems can usually be solved only in an approximate fashion. We investigate the impact on such mechanisms of replacing exact solutions by approximate ones. Specifically, we look at a particular greedy optimization method. We show that the GVA payment scheme does not provide for a truth revealing mechanism. We introduce another scheme that does guarantee truthfulness for a restricted class of players. We demonstrate the latter property by identifying natural properties for combinatorial auctions and showing that, for our restricted class of players, they imply that truthful strategies are dominant. Those properties have applicability beyond the specific auction studied.

The field of economic mechanism design has been an active area of research in economics for at least 20 years. This field uses the tools of economics and game theory to design "rules of interaction " for economic transactions that will, in principle, yield some desired outcome. In this paper I provide an overview of this subject for an audience interested in applications to electronic commerce and discuss some special problems that arise in this context.

In this paper, we show how to design truthful (dominant strategy) mechanisms for several combinatorial problems where each agent’s secret data is naturally expressed by a single positive real number. The goal of the mechanisms we consider is to allocate loads placed on the agents, and an agent’s secret data is the cost she incurs per unit load. We give an exact characterization for the algorithms that can be used to design truthful mechanisms for such load balancing problems using appropriate side payments. We use our characterization to design polynomial time truthful mechanisms for several problems in combinatorial optimization to which the celebrated VCG mechanism does not apply. For scheduling related parallel machines (QjjCmax), we give a 3-approximation mechanism based on randomized rounding of the optimal fractional solution. This problem is NP-complete, and the standard approximation algorithms (greedy load-balancing or the PTAS) cannot be used in truthful mechanisms. We show our mechanism to be frugal, in that the total payment needed is only a logarithmic factor more than the actual costs incurred by the machines, unless one machine dominates the total processing power. We also give truthful mechanisms for maximum flow, Qjj P Cj (scheduling related machines to minimize the sum of completion times), optimizing an affine function over a fixed set, and special cases of uncapacitated facility location. In addition, for Qjj P wjCj (minimizing the weighted sum of completion times), we prove a lower bound of 2 p 3 for the best approximation ratio achievable by a truthful mechanism.

This paper analyzes coalitions among self-interested agents that need to solve combinatorial optimization problems to operate e ciently in the world. By colluding (coordinating their actions by solving a joint optimization prob-lem) the agents can sometimes save costs compared to operating individually. A model of bounded rationality is adopted where computation resources are costly. It is not worthwhile solving the problems optimally: solution quality is decision-theoretically traded o against computation cost. A normative, application- and protocol-independent theory of coalitions among bounded-rational agents is devised. The optimal coalition structure and its stability are signi cantly a ected by the agents ' algorithms ' performance pro les and the cost of computation. This relationship is rst analyzed theoretically. Then a domain classi cation including rational and bounded-rational agents is in-troduced. Experimental results are presented in vehicle routing with real data from ve dispatch centers. This problem is NP-complete and the instances are so large that|with current technology|any agent's rationality is bounded by computational complexity. 1

Introduction Automated negotiation systems with self-interested agents are becoming increasingly important. One reason for this is the technology push of a growing standardized communication infrastructure---Internet, WWW, NII, EDI, KQML, FIPA, Concordia, Voyager, Odyssey, Telescript, Java, etc---over which separately designed agents belonging to different organizations can interact in an open environment in realtime and safely carry out transactions. The second reason is strong application pull for computer support for negotiation at the operative decision making level. For example, we are witnessing the advent of small transaction electronic commerce on the Internet for purchasing goods, information, and communication bandwidth [29]. There is also an industrial trend toward virtual enterprises: dynamic alliances of small, agile enterprises which together can take advantage of economies of scale when available (e.g., respond to mor