Combinatorial clock auctions (CCAs) have been adopted in spectrum auctions (Bichler and Goeree, 2017) and in other applications for their simple price-update rules. There are single-stage and two-stage versions, which have been used in spectrum auctions worldwide. We devote a chapter to CCAs because of their practical relevance, as for chapter 6 on the SMRA. We will discuss both versions and a practical example of a market mechanism that includes a discussion of activity rules, which are sometimes ignored in theoretical treatments.

The Single-Stage Combinatorial Clock Auction

The single-stage combinatorial clock auction (SCCA) is easy to implement and has therefore found application in spectrum auctions and in procurement. In this section we provide a more detailed discussion of the SCCA, as introduced by Porter et al. (2003), and give an algorithmic description in algorithm 3.

Auction Process

In this type of action, prices for all items are initially zero or at a reserve price r. In every round bidders identify a package of items, or several packages, which they offer to buy at current prices. If two or more bidders demand an item then its price is increased by a fixed bid increment in the next round. This process iterates. The bids which correspond to the current ask prices are called standing bids, and a bidder is standing if she has at least one standing bid. In a simple scenario in which supply equals demand, the auction terminates and the items are allocated according to the standing bids.

If at some point there is an excess supply of at least one item and no item is overdemanded, the auctioneer determines the winners to find an allocation of items that maximizes his revenue by considering all submitted bids. If the solution displaces a standing bidder, the prices of items in the corresponding standing bids rise by the bid increment and the auction continues. The auction ends when no prices are increased and bidders finally pay their bid prices for the winning packages. We will analyze a version that uses an XOR bidding language.

The revelation principle (see theorem 3.3.1) suggests that if a social-choice function can be implemented by an arbitrary indirect mechanism (e.g., an open auction) then the same function can be implemented by a truthful direct revelation mechanism. Analyzing direct mechanisms is often more convenient, and the revelation principle allows one to argue that this restriction is without loss of generality. Yet there are cases where one prefers to implement and model the indirect version of a mechanism rather than its direct counterpart.

One argument used in the literature refers to interdependent valuations. The linkage principle implies that open auctions generally lead to higher expected revenue than sealed-bid auctions, with interdependent bidder valuations. Milgrom and Weber (1982) wrote: “One explanation of this inequality is that when bidders are uncertain about their valuations, they can acquire useful information by scrutinizing the bidding behavior of their competitors during the course of an (ascending) auction. That extra information weakens the winner's curse and leads to more aggressive bidding in the (ascending) auction, which accounts for the higher expected price.”

Another argument for open auctions is that the winners of an ascending auction do not need to reveal their true valuation to the auctioneer, only that it is above the secondhighest bid. With respect to multi-object auctions, Levin and Skrzypacz (2017) write that economists think of open auctions as having an advantage because bidders can discover gradually how their demands fit together. Most spectrum auctions are open auctions, largely for these reasons.

As a result, much recent research has focused on ascending multi-object auctions, i.e., generalizations of the single-object English auction where bidders can outbid each other iteratively. The models can be considered as algorithms, and we will try to understand the game-theoretical properties of these algorithms. In particular, we want to understand if there is a generalization of the English auction to ascending combinatorial auctions that also has a dominant strategy or ex post equilibrium. Unfortunately, the answer is negative for general valuations. However, there are positive results for restricted preferences.

A strong restriction of preferences is found in assignment markets, where bidders bid on multiple items but want to win at most one. This restriction allows us to formulate the allocation problem as an assignment problem, and there exists an ascending auction where truthful bidding is an ex post equilibrium.

Mechanism design studies the construction of economic mechanisms in the presence of rational agents with private information. It is sometimes called reverse game theory, as mechanism designers search for mechanisms which satisfy game-theoretical solution concepts and achieve good outcomes. For example, an auctioneer might want to design an auction mechanism which maximizes social welfare of the participants and exhibits dominant strategies for bidders to reveal their true valuations. We mostly discuss market mechanisms in this book, but mechanism design is not restricted to markets and the basic principles can be applied to various types of interactive decision making.

Mechanism design has relations to social choice theory, which is a field in economics focusing on the aggregation of individual preferences to reach a collective decision or social welfare. Social choice theory depends upon the ability to aggregate individual preferences into a combined social welfare function. Therefore, individual preferences are modeled in terms of a utility function. The ability to sum the utility functions of different individuals, as is done in auction theory in chapter 4, depends on the utility functions being comparable with each other; informally, individuals’ preferences must be measured with the same yardstick. The mechanism design literature on auctions typically assumes cardinal utility functions and interpersonal utility comparisons, as in the early social choice literature such as Bergson (1938) and Samuelson (1948). Note that Arrow (1950) and much of the subsequent literature assumes only ordinal preferences and rejects the idea of interpersonal utility comparisons. This means that utility cannot be measured and compared across individuals. The auction design literature as discussed in this book is an exception and assumes quasi-linear utility functions, where bidders have cardinal values for an allocation and maximize their payoff on the basis of prices in the market. Also, utility is comparable across agents. Cardinal preferences allow agents to express intensities, which is not possible with ordinal preferences only. For example, an agent might much prefer a diamond to a rock, but this information about intensity will be ignored if only ordinal preferences (diamond ≻ rock) are expressed.

There has been substantial progress in market design in the past few decades. The advent of the Internet and the availability of fast algorithms to solve computationally hard allocation problems has considerably extended the design space and has led to new types of markets which would have been impossible only 20 years ago. The theoretical models discussed in this book help one to understand when markets are efficient and when they are not. Many recent models for multi-object markets draw on the theory of linear programming and combinatorial optimization, at the same time adding to this theory in a fundamental way. This is one reason why market design has also found much attention in computer science and operations research lately.

Most models of markets in this book are normative and describe efficient markets with rational decision makers having independent and private values with quasilinear utility functions. In the natural sciences models are evaluated by their predictive accuracy in the laboratory and in the field. Unfortunately, the predictive accuracy of some auction models is low (see for example section 4.8). For example, Bayesian Nash equilibrium strategies require strong assumptions about a common-prior type distribution. Even if this distributional information is provided by a laboratory experiment, bidders do not always maximize payoff. Loss aversion, risk aversion, spite, or ex post regret can all affect the decision making of individuals. It is even less likely that bidders would follow a Bayesian Nash equilibrium strategy in the field, where the prior information of bidders is typically asymmetric and different between bidders.

Auction designs satisfying stronger solution concepts such as dominant strategies or ex post equilibrium strategies are promising, because they don't require a common prior assumption. However, such solution concepts are quite limiting in the design. In the standard independent-private-values model, the VCG mechanism is unique and, even if we allow the social welfare to be approximated, the scope of strategy-proof approximation mechanisms is narrow. Also, the VCG mechanism is strategy-proof for independent and private values. Jehiel et al. (2006) showed the impossibility of ex post implementation with interdependent values in multi-parameter settings. What is more, often the characteristics of market participants and the design goals are quite different from those assumed in the academic literature.

Many real-world markets require the solving of computationally hard allocation problems, and realistic problem sizes are often such that they cannot be solved optimally. For example, there have been spectrum auctions using the combinatorial clock auction with 100 licenses. The allocation problems are modeled as integer programs intended to produce an exact solution (see section 7.2), but this cannot always be guaranteed without restrictions. Actually, the number of bids in combinatorial spectrum auctions is typically restricted to a few hundred. Section 7.5 provides examples of allocation problems in other domains, sometimes using compact bid languages, which also cannot be solved optimally for larger instances. Sometimes it is acceptable to settle for suboptimal solutions. Often, heuristics are used to solve computationally hard problems. However, with heuristics we do not have worst-case bounds on the quality of the solution. In contrast, approximation algorithms provide worst-case bounds on the solution quality and a polynomial runtime. For this discussion, we expect the reader to have some basic familiarity with approximation algorithms; we provide an overview of the field and the necessary terminology and concepts in appendix B.4.

In this section, we analyze approximation mechanisms which provide good approximation ratios, which run in polynomial time, and which are also incentive-compatible. This leads to new notions of incentive compatibility beyond strategy-proofness (i.e., dominant-strategy incentive compatibility) and new types of mechanism. We often talk about truthfulness instead of incentive compatibility, both terms referring to different solution concepts setting incentives to reveal preferences truthfully.

One can think of the VCG mechanism as a black-box transformation from exact algorithms solving the allocation problem to a strategy-proof mechanism. A central question in this chapter is whether there is a similar black-box transformation, from approximation algorithms to truthful approximation mechanisms, which maintains the approximation ratios of non-truthful approximation algorithms.We show that black-box transformations for quite general allocation problems are possible with strong forms of truthfulness, when we use randomization.

Approximation algorithms are typically only available for idealized models such as knapsack problems or bin-packing problems. In practice, many allocation problems are quite messy and have many complicating side constraints that make it hard to find an approximation algorithm with a good performance guarantee.

Before we introduce various combinatorial auction formats, we will discuss a simple and widespread auction format for selling multiple objects, the simultaneous multiround auction (SMRA). It allows bidders to bid on multiple objects simultaneously and win multiple objects. The SMRA was designed for the US Federal Communications Commission in the early 1990s, and it has been the standard auction format for selling spectrum worldwide for many years. It is also easy to implement and therefore a natural candidate for many multi-object markets. However, it can also lead to substantial strategic problems for bidders. We discuss several opportunities for profitable manipulation, which will provide a motivation for the alternative auction designs discussed in the remainder of this book.

SMRA Rules

The SMRA is an extension of the English auction to more than one item. All the items are sold at the same time, each with a price associated with it, and the bidders can bid on any number of items. The auction proceeds in rounds, which are specific periods of time in which all bidders can submit bids. After a round is closed the auctioneer discloses who is winning and the price of each item, which coincides with the highest bid submitted on the item. There are differences in the level of information revealed about other bidders’ bids. Sometimes all bids are revealed after each round; sometimes only prices of the currently winning bids are published.

The bidding continues until no bidder is willing to raise the bid on any item further. In other words, if in one round no new bids are placed, the bidders receive the spectrum for which they hold the highest active bid; then the auction ends, with each bidder winning the blocks on which she has the highest bid, and paying this bid price for any item won.

The SMRA uses simple activity rules, which force bidders to be active from the start. Without such rules, bidders might be tempted to hide their demand and wait to see how prices develop before they start bidding. Monotonicity rules, where bidders cannot bid on more items in later rounds, are regularly used. This forces bidders to be active from the start. Activity rules can be considered to be a major innovation of this auction format.

We now focus on markets where multiple related and indivisible objects are sold. These objects might be homogeneous or heterogeneous. In general, a seller can decide to sell the objects in multiple auctions or in a single auction. Let's assume that the seller decides to sell the objects either one at a time in a sequence of single-object auctions or simultaneously; then she needs to decide on the auction format. If the seller decides to sell all objects in a single auction, there are open and sealed-bid auction designs, which we briefly introduce next. Note that we assume independent and private values of bidders and quasi-linear utility functions unless mentioned explicitly.

Since we have multiple objects, we need additional notation. Let K denote the set of objects to be auctioned (|K| = m) and k ∈ K (also l ∈ K) denote a specific object. Again, let I denote the set of bidders participating in the auction (|I| = n) and i ∈ I (also j ∈ I) denote a specific bidder.We distinguish between multi-unit auctions, where multiple identical objects are sold, and multi-item auctions, where heterogeneous objects are sold.

General Equilibrium Models

The supply and demand model of a single type of good is a partial equilibrium model where the clearance on the market of some specific good is obtained independently from prices and quantities in the markets for other goods. This makes analysis much simpler than in a general equilibrium model which includes multiple goods or objects. In partial equilibrium analysis the supply and demand for one type of good in markets with multiple buyers and sellers is analyzed in order to understand how prices arise in a competitive equilibrium. We will introduce the term competitive equilibrium more formally in definition 8.1.2. For now, such an equilibrium is a state where supply and demand are balanced and, in the absence of external influences, prices and allocation will not change.

The widespread graphical illustration of a partial equilibrium in figure 5.1 has price on the vertical axis and quantity on the horizontal axis, and it describes how competitive equilibrium prices on such markets arise and what the impact of a shift in either demand or supply implies. It provides a good initial intuition of the dynamics that can arise on exchanges.