Intuitively, one might expect that a seller’s revenue from an auction weakly increases as the number of bidders grows, as this increases competition. However, it is known that for combinatorial auctions that use the VCG mechanism, a seller can sometimes increase revenue by dropping bidders. In this paper we investigate the extent to which this problem can occur under other dominant-strategy combinatorial auction mechanisms. Our main result is that such failures of “revenue monotonicity ” are not limited to mechanisms that achieve efficient allocations. Instead, they can occur under any dominant-strategy direct mechanism that sets prices using critical values, and that always chooses an allocation that cannot be augmented to make some bidder better off, while making none worse off.

A key trend in the world—especially in electronic commerce—is a demand for higher levels of expressiveness in the mechanisms that mediate interactions, such as the allocation of resources, matching of peers, and elicitation of opinions from large and diverse communities. Intuitively, one would think that this increase in expressiveness would lead to more efficient mechanisms (e.g., due to better matching of supply and demand). However, until now we have lacked a general way of characterizing the expressiveness of these mechanisms, analyzing how it impacts the actions taken by rational agents—and ultimately the outcome of the mechanism. In this technical report we introduce a general model of expressiveness for mechanisms. Our model is based on a new measure which we refer to as the maximum impact dimension. The measure captures the number of different ways that an agent can impact the outcome of a mechanism. We proceed to uncover a fundamental connection between this measure and the concept of shattering from computational learning theory. We also provide a way to determine an upper bound on the expected efficiency of any mechanism under its most efficient Nash equilibrium which, remarkably, depends only on the mechanism’s expressiveness. We show that for any setting and any prior over agent preferences, the

Position auctions such as the Generalized Second Price (GSP) are commonly used for sponsored search, e.g., by Yahoo! and Google. We now have an understanding of the equilibria of these auctions, via game-theoretic concepts like Generalized English Auctions and the “locally envy-free ” property, as well as through a relationship to the well-known, truthful Vickrey-Clarke-Groves (VCG) mechanism. In practice, however, position auctions are implemented with additional constraints, in particular, bidder-specific minimum prices. Such minimum prices are used to control the quality of the ads that appear on the page. We study the effect of bidder-specific minimum prices in position auctions. Naïvely enforcing minimum prices in the VCG mechanism breaks the truthfulness of the auction; we describe two variants of VCG for which revealing the truth is a dominant strategy. The implications of bidder-specific minimum prices are more intricate for the GSP auction. Some properties proved for standard GSP no longer hold in this setting. For example, we show that the GSP allocation is now not always efficient (in terms of advertiser value). Also, the property of“envy-locality”enjoyed by GSP—which is essential in the prior analysis of strategies and equilibria— no longer holds. Our main result is to show that despite losing envy locality, GSP with bidder-specific minimum prices still has an envy-free equilibrium. 1.

In a combinatorial auction, a set of items is for sale, and agents can bid on subsets of these items. In a voting setting, the agents decide among a set of alternatives by having each agent rank all the alternatives. Many of the key research issues in these two domains are similar. The aim of this paper is to give a convenient side-by-side comparison that will clarify the relation between the domains, and serve as a guide to future research. 1

Using a model of agent behavior based around envy-reducing strategies, we describe an iterated combinatorial auction in which the allocation and prices converge to a solution in the core of the agents ’ true valuations. In each round of the iterative auction mechanism, agents act on envy quotes produced by the mechanism: hints that suggest the prices of the bundles they are interested in. We describe optimal methods of generating envy quotes for various core-selecting mechanisms. Prior work on core-selecting combinatorial auctions has required agents to have perfect information about every agent’s valuations to achieve a solution in the core. In contrast, here a core solution is reached even in the private information setting.

The word “auction ” generally refers to a mechanism for allocating one or more resources to one or more parties (or bidders). Generally, once the allocation is determined, some amount of money changes hands; the precise monetary transfers are determined by the auction process. While in some auction protocols, such as the English auction, bidders repeatedly increase their bids in an attempt to outbid each other, this is not an essential component of an auction. There are many other auction protocols, and we will study some of them in this chapter. Auctions have traditionally been studied mostly by economists. In recent years, com-puter scientists have also become interested in auctions, for a variety of reasons. Auctions can be useful for allocating various computing resources across users. In artificial intelli-gence, they can be used to allocate resources and tasks across multiple artificially intelligent “agents. ” Auctions are also important in electronic commerce: there are of course several well-known auction websites, but additionally, search engines use auctions to sell advertising space on their results pages. Finally, increased computing power and improved algorithms have made new types of auctions possible—most notably combinatorial auctions, in which

We prove that in many cases, a first-price sealed-bid combinatorial auction gives higher expected revenue than a sealed-bid simultaneous auction. This is the first theoretical evidence that combinatorial auctions indeed generate higher revenue, which has been a common belief for decades. We use a model with many bidders and items, where bidders are of two types: (i) single-bidders interested in only one item and (ii) synergy-bidders, each interested in one random combination of items. We provide an upper bound on the expected revenue for simultaneous auctions and a lower bound on combinatorial auctions. Our bounds are parameterized on the number of bidders and items, combination size, and synergy. We derive an asymptotic result, proving that as the number of bidders approach infinity, expected revenue of the combinatorial auction will be higher than that of the simultaneous auction. We also provide concrete examples where the combinatorial auction is revenue-superior. 1

The VCG mechanism is the gold standard for combinatorial auctions (CAs), and it maximizes social welfare. In contrast, the revenue-maximizing (aka optimal) CA is unknown, and designing one is NP-hard. Therefore, research on optimal CAs has progressed into special settings. Notably, Levin [1997] derived the optimal CA for complements when each agent’s private type is one-dimensional. (This does not fall inside the well-studied “singleparameter environment”.) We introduce a new research avenue for increasing revenue where we poke holes in the allocation space—based on the bids—and then use a welfare-maximizing allocation rule within the remaining allocation set. In this paper, the first step down this avenue, we introduce a new form of “reserve pricing ” into CAs. We show that Levin’s optimal revenue can be 2-approximated by using “monopoly reserve prices ” to curtail the allocation set, followed by welfare-maximizing allocation and Levin’s payment rule. A key lemma of potential independent interest is that the expected revenue from any truthful allocation-monotonic mechanism equals the expected virtual valuation; this generalizes Myerson’s lemma [1981] from the single-parameter environment. Our mechanism is close to the gold standard and thus easier to adopt than Levin’s. It also requires less information about the prior over the bidders ’ types, and is always more efficient. Finally, we show that the optimal revenue can be 6-approximated even if the “reserve pricing ” is required to be symmetric across bidders. 1

In combinatorial auctions using VCG, a seller can sometimes increase revenue by dropping bidders. In this paper we investigate the extent to which this counter-intuitive phenomenon can also occur under other deterministic dominant-strategy combinatorial auction mechanisms. Our main result is that such failures of “revenue monotonicity” can occur under any such mechanism that is weakly maximal—meaning roughly that it chooses allocations that cannot be augmented to cause a losing bidder to win without hurting winning bidders—and that allows bidders to express arbitrary single-minded preferences. We also give a set of other impossibility results as corollaries, concerning revenue when the set of goods changes, false-name-proofness, and the core.

Mechanisms are present everywhere in both business and social contexts. They govern the interactions people have with businesses, governments, and each other. One emerging trend over the past decade is a demand for higher levels of expressiveness in mechanisms that mediate interactions such as the allocation of resources, matching of peers, and elicitation of opinions. This trend has already manifested itself in combinatorial auctions and generalizations thereof. It is also reflected in the richness of preference expression offered by businesses as diverse as matchmaking sites, sites like Amazon and Netflix, and services like Google’s AdSense. In Web 2.0 parlance, this demand for increasingly diverse offerings is called the Long Tail. A driving force behind this trend is that greater expressiveness begets better matches, or greater efficiency of the outcomes. Yet, expressiveness does not come for free; it burdens users to specify more preference information. Today’s mechanisms have relied on empirical tweaking to determine how to deal with this and related tradeoffs. In this thesis, we propose to establish the foundations of expressiveness in mechanisms and its relationship to their efficiency,