*Combinatorial auctions* is excellent and exceptional in practically all attributes I would care about in this type of work. This includes the breadth and depth of the topics covered and the language employed. Additionally, my praise also extends to minor details, such as the existence of an exhaustive author and subject index, and the quality of its typesetting, especially with regard to the mathematical apparatus used in some of the chapters. For researchers and practitioners, both on the seller side and on the buyer side, who deal with (combinatorial) auctions, this book is a must-read.

Given the fact that not even Wikipedia has an entry for “combinatorial auction” (as of the time of this writing), what, then, is the subject matter of this work and what makes it (and this book) so interesting?

A combinatorial auction (CA) is an auction in which bidders can place bids on combinations of items (so-called packages) rather than on just individual items alone. The possibility of combining arbitrary items into a package realizes two advantages over single-item auctions. First, bidders can more fully express their preferences, which is especially important when some of the items in a package are complements (the utility of both items combined exceeds the sum of the individual utilities). Second, the seller benefits through the greater efficiency of a CA: auction revenue is generally higher and economic efficiency is also improved—which is especially important for the government. Furthermore, the auctioneer can add arbitrary constraints to the award process in order to avoid concentration in the hands of only one bidder, or to favor particular types of bidders over others.

Organized in five modules, the book sets out to describe the aforementioned advantages of CA, while mitigating its existing and openly acknowledged downsides. These limitations center on the two notions of complexity and bidder control over price bids from the floor.

Complexity arises not only for the bidder who has to compute the true utility over an arbitrary combination of auctioned items (given a set of *N* items, theoretically the bidder would have to appraise the utility of 2^{N} different combinations), but also for the auctioneer through the computational effort to determine who to reward with which packages at what price.

Module 1 begins by describing and analyzing various CA mechanisms from a mathematical and economic theory point of view. The starting point is the “lovely, but lonely” Vickrey (Clarke-Groves) auction (1961), which is extended to more sophisticated ways (iterative, simultaneous ascending auctions, and auctions using particular clock designs).

As a bid in a CA is an expression of a bidder’s preference for various bundles of items, the question of how to exactly formulate and communicate the set of bids to the auctioneer is rather nontrivial. Module 2, “Bidding and Efficiency,” addresses this challenge by introducing and analyzing appropriate bidding languages.

Module 3 tackles the so-called winner determination problem of how to compute the allocation of items to the individual bidders, given a set of bids. The bad news here is that the problem is NP-complete (that is, it falls in the class of problems hardest to solve); the good news is the existence of some rules of thumb for searching the space of allocations that in practice allow one to solve even large problems.

Module 4 picks up the question of how to test and implement CA mechanisms from Module 1. Solutions to the winner determination problem of Module 3 are discussed.

Module 5 presents design, experiences, best practices, and lessons learned from applying CA mechanisms to four real-word situations in the following areas: sale of airspace system resources (that is, takeoff and landing slots), truckload transportation, operation of bus routes, and industrial procurement.

An author index with 480 authors, a subject index with approximately 350 terms, and a highly useful combinatorial auction glossary conclude this remarkable work. The latter provides short definitions or explanations of key notions ranging from seemingly simple terms (“bid”) to advanced concepts (“collusion”).

Amusingly, the term “auction” is not defined, but this (pedantic and frivolous, I have to confess) comment already limits any critiques one could apply to this flawless and definitely highly recommended work.