Rakesh V. Vohra

Combinatorial Auctions: A Survey

Many auctions involve the sale of a variety of distinct assets. Examples are airport time slots, delivery routes, network routing, and furniture. Because of complementarities or substitution effects between the different assets, bidders have preferences not just for particular items but for sets of items. For this reason, economic efficiency is enhanced if bidders are allowed to bid on bundles or combinations of different assets. This paper surveys the state of knowledge about the design of combinatorial auctions and presents some new insights. Periodic updates of portions of this survey will be posted to this journals Online Supplements web page at http://joc.pubs.informs.org/OnlineSupplements.html.

The orienteering problem

Orienteering is a sport in which start and end points are specified along with other locations. These other locations have associated scores. Competitors seek to visit, in a fixed amount of time, a subset of these locations on the way from the start point to the end point in order to maximize the total score. An effective center-of-gravity heuristic is presented that outperforms heuristics from the literature.

New algorithms for an ancient scheduling problem

We consider the on-line version of the original <italic>m</italic>-machine scheduling problem: given <italic>m</italic> machines and <italic>n</italic> positive real jobs, schedule the <italic>n</italic> jobs on the <italic>m</italic> machines so as to minimize the makespan, the completion time of the last job. In the on-line version, as soon as job <italic>j</italic> arrrives, it must be assigned immediately to one of the <italic>m</italic> machines. We present two main results. The first is a (2–ε)-competitive deterministic algorithm for all <italic>m</italic>. The competitive ratio of all previous algorithms approaches 2 as <italic>m</italic>→<inline-equation> <f> ∞</f> </inline-equation>. Indeed, the problem of improving the competitive ratio for large <italic>m</italic> had been open since 1966, when the first algorithm for this problem appeared. The second result is an optimal randomized algorithm for the case <italic>m</italic> = 2. To the best of our knowledge, our 4/3-competitive algorithm is the first specifically randomized algorithm for the original, <italic>m</italic>-machine, on-line scheduling problem.

On Ascending Vickrey Auctions for Heterogeneous Objects

Abstract We construct an ascending auction for heterogeneous objects by applying a primal-dual algorithm to a linear program that represents the efficient-allocation problem for this setting. The auction assigns personalized prices to bundles, and asks bidders to report their preferred bundles in each round. A bidders prices are increased when he belongs to a “minimally undersupplied” set of bidders. This concept generalizes the notion of “overdemanded” sets of objects introduced by Demange, Gale, and Sotomayor for the one-to-one assignment problem. Under a submodularity condition, the auction implements the Vickrey–Clarke–Groves outcome; we show that this type of condition is somewhat necessary to do so. When classifying the ascending-auction literature in terms of their underlying algorithms, our auction fills a gap in that literature. We relate our results to various ascending auctions in the literature.

Algorithmic Game Theory: Sponsored Search Auctions

One of the more visible means by which the Internet has disrupted traditional activity is the manner in which advertising is sold. Offline, the price for advertising is typically set by negotiation or posted price. Online, much advertising is sold via auction. Most prominently, Web search engines like Google and Yahoo! auction space next to search results, a practice known as sponsored search. This chapter describes the auctions used and how the theory developed in earlier chapters of this book can shed light on their properties. We close with a brief discussion of unresolved issues associated with the sale of advertising on the Internet.

Calibration with many checking rules

Each period an outcome (out of finitely many possibilities) is observed. For simplicity assume two possible outcomes, a and b. Each period, a forecaster announces the probability of a occurring next period based on the past.Consider an arbitrary subsequence of periods (e.g., odd periods, even periods, all periods in which b is observed, etc.). Given an integer n, divide any such subsequence into associated sub-subsequences in which the forecast for a is between [i/n, i+ 1/n), i ∈ {0, 1,...,n}.We compare the forecasts and the outcomes (realized next period) separately in each of these subsubsequences. Given any countable partition of [0, 1] and any countable collection of subsequences, we construct a forecasting scheme such that for all infinite strings of data, the long-run average forecast for a matches the long-run frequency of realized as.

Sequential Bandwidth and Power Auctions for Distributed Spectrum Sharing

We study a sequential auction for sharing a wireless resource (bandwidth or power) among competing transmitters. The resource is assumed to be managed by a spectrum broker (auctioneer), who collects bids and allocates discrete units of the resource via a sequential second-price auction. It is well known that a second price auction for a single indivisible good has an efficient dominant strategy equilibrium; this is no longer the case when multiple units of a homogeneous good are sold in repeated iterations. For two users with full information, we show that such an auction has a unique equilibrium allocation. The worst-case efficiency of this allocation is characterized under the following cases: (i) both bidders have a concave valuation for the spectrum resource, and (ii) one bidder has a concave valuation and the other bidder has a convex valuation (e.g., for the other useriquests power). Although the worst-case efficiency loss can be significant, numerical results are presented, which show that for randomly placed transmitter-receiver pairs with rate utility functions, the sequential second-price auction typically achieves the efficient allocation. For more than two users it is shown that this mechanism always has a pure strategy equilibrium, but in general there may be multiple equilibria. We give a constructive procedure for finding one equilibrium; numerical results show that when all users have concave valuations the efficiency loss decreases with an increase in the number of users.

Algorithmic Game Theory: Mechanism Design without Money

Despite impossibility results on general domains, there are some classes of situations in which there exist interesting dominant-strategy mechanisms. While some of these situations (and the resulting mechanisms) involve the transfer of money, we examine some that do not. Specifically, we analyze problems where agents have single-peaked preferences over a one-dimensional “public” policy space; and problems where agents must match with each other.

Market Research and Market Design

We study trading models when the distribution of signals such as costs or values is not known to traders or the mechanism designer when the profit-maximizing trading procedure is designed. We present adaptive mechanisms that simultaneously elicit this information (market research) while maintaining incentive compatibility and maximizing profits when the set of traders is large (market design). First, we study a monopoly pricing model where neither the seller nor the buyers know the distribution of values. Second, we study a model with a broker intermediating trade between a large number of buyers and sellers with private information about their valuations and costs. We show that when the set of traders becomes large our adaptive mechanisms achieve the same expected profits for the monopolist and the broker as when the distribution of signals is common knowledge.

Characterization of Revenue Equivalence

A saw blade guard for a sawing power tool, particularly a mitre saw, comprises a protective element having two side walls each formed as a part of a circle, and a peripheral wall connecting the side walls with one another, the side walls having a center and an edge spaced from the center, the protective element being formed as an integral element and provided with a member in the region of the edge for turning the protective element as a whole.