Approximately Optimal Mechanism Design: Motivation, Examples, and Lessons Learned

Optimal mechanism design enjoys a beautiful and well-developed theory, and also a number of killer applications. Rules of thumb produced by the field influence everything from how governments sell wireless spectrum licenses to how the major search engines auction off online advertising. There are, however, some basic problems for which the traditional optimal mechanism design approach is ill-suited --- either because it makes overly strong assumptions, or because it advocates overly complex designs. The thesis of this paper is that approximately optimal mechanisms allow us to reason about fundamental questions that seem out of reach of the traditional theory. This survey has three main parts. The first part describes the approximately optimal mechanism design paradigm --- how it works, and what we aim to learn by applying it. The second and third parts of the survey cover two case studies, where we instantiate the general design paradigm to investigate two basic questions. In the first example, we consider revenue maximization in a single-item auction with heterogeneous bidders. Our goal is to understand if complexity --- in the sense of detailed distributional knowledge --- is an essential feature of good auctions for this problem, or alternatively if there are simpler auctions that are near-optimal. The second example considers welfare maximization with multiple items. Our goal here is similar in spirit: when is complexity --- in the form of high-dimensional bid spaces --- an essential feature of every auction that guarantees reasonable welfare? Are there interesting cases where low-dimensional bid spaces suffice?Comment: Based on a talk given by the author at the 15th ACM Conference on Economics and Computation (EC), June 201

Combinatorial Auctions Do Need Modest Interaction

We study the necessity of interaction for obtaining efficient allocations in subadditive combinatorial auctions. This problem was originally introduced by Dobzinski, Nisan, and Oren (STOC'14) as the following simple market scenario: $m$ items are to be allocated among $n$ bidders in a distributed setting where bidders valuations are private and hence communication is needed to obtain an efficient allocation. The communication happens in rounds: in each round, each bidder, simultaneously with others, broadcasts a message to all parties involved and the central planner computes an allocation solely based on the communicated messages. Dobzinski et.al. showed that no non-interactive ($1$-round) protocol with polynomial communication (in the number of items and bidders) can achieve approximation ratio better than $\Omega(m^{{1}/{4}})$, while for any $r \geq 1$, there exists $r$-round protocols that achieve $\widetilde{O}(r \cdot m^{{1}/{r+1}})$ approximation with polynomial communication; in particular, $O(\log{m})$ rounds of interaction suffice to obtain an (almost) efficient allocation. A natural question at this point is to identify the "right" level of interaction (i.e., number of rounds) necessary to obtain an efficient allocation. In this paper, we resolve this question by providing an almost tight round-approximation tradeoff for this problem: we show that for any $r \geq 1$, any $r$-round protocol that uses polynomial communication can only approximate the social welfare up to a factor of $\Omega(\frac{1}{r} \cdot m^{{1}/{2r+1}})$. This in particular implies that $\Omega(\frac{\log{m}}{\log\log{m}})$ rounds of interaction are necessary for obtaining any efficient allocation in these markets. Our work builds on the recent multi-party round-elimination technique of Alon, Nisan, Raz, and Weinstein (FOCS'15) and settles an open question posed by Dobzinski et.al. and Alon et. al

Economic Efficiency Requires Interaction

We study the necessity of interaction between individuals for obtaining approximately efficient allocations. The role of interaction in markets has received significant attention in economic thinking, e.g. in Hayek's 1945 classic paper. We consider this problem in the framework of simultaneous communication complexity. We analyze the amount of simultaneous communication required for achieving an approximately efficient allocation. In particular, we consider two settings: combinatorial auctions with unit demand bidders (bipartite matching) and combinatorial auctions with subadditive bidders. For both settings we first show that non-interactive systems have enormous communication costs relative to interactive ones. On the other hand, we show that limited interaction enables us to find approximately efficient allocations

Computational Efficiency Requires Simple Taxation

We characterize the communication complexity of truthful mechanisms. Our departure point is the well known taxation principle. The taxation principle asserts that every truthful mechanism can be interpreted as follows: every player is presented with a menu that consists of a price for each bundle (the prices depend only on the valuations of the other players). Each player is allocated a bundle that maximizes his profit according to this menu. We define the taxation complexity of a truthful mechanism to be the logarithm of the maximum number of menus that may be presented to a player. Our main finding is that in general the taxation complexity essentially equals the communication complexity. The proof consists of two main steps. First, we prove that for rich enough domains the taxation complexity is at most the communication complexity. We then show that the taxation complexity is much smaller than the communication complexity only in "pathological" cases and provide a formal description of these extreme cases. Next, we study mechanisms that access the valuations via value queries only. In this setting we establish that the menu complexity -- a notion that was already studied in several different contexts -- characterizes the number of value queries that the mechanism makes in exactly the same way that the taxation complexity characterizes the communication complexity. Our approach yields several applications, including strengthening the solution concept with low communication overhead, fast computation of prices, and hardness of approximation by computationally efficient truthful mechanisms

Descending Price Optimally Coordinates Search

Investigating potential purchases is often a substantial investment under uncertainty. Standard market designs, such as simultaneous or English auctions, compound this with uncertainty about the price a bidder will have to pay in order to win. As a result they tend to confuse the process of search both by leading to wasteful information acquisition on goods that have already found a good purchaser and by discouraging needed investigations of objects, potentially eliminating all gains from trade. In contrast, we show that the Dutch auction preserves all of its properties from a standard setting without information costs because it guarantees, at the time of information acquisition, a price at which the good can be purchased. Calibrations to start-up acquisition and timber auctions suggest that in practice the social losses through poor search coordination in standard formats are an order of magnitude or two larger than the (negligible) inefficiencies arising from ex-ante bidder asymmetries.Comment: JEL Classification: D44, D47, D82, D83. 117 pages, of which 74 are appendi

Ascending Auctions

A key question of auction design is whether to use an ascending- bid or a sealed-bid format. The critical distinction between formats is that an ascending auction provides the bidders with information through the process of bidding. This information is a two-edged sword. It may stimulate competition by creating a reliable process of price discovery, by reducing the winner's curse, and by allowing efficient aggregations of items. Alternatively, the information may be used by bidders to establish and enforce collusive outcomes. Ex ante asymmetries and weak competition favor a sealed-bid design. In other cases, an ascending auction is likely to perform better in efficiency and revenue terms. Moreover, information in an ascending auction can be tailored to limit collusion.Auctions; Multiple-Item Auctions, Spectrum Auctions

Draft Auctions

We introduce draft auctions, which is a sequential auction format where at each iteration players bid for the right to buy items at a fixed price. We show that draft auctions offer an exponential improvement in social welfare at equilibrium over sequential item auctions where predetermined items are auctioned at each time step. Specifically, we show that for any subadditive valuation the social welfare at equilibrium is an $O(\log^2(m))$-approximation to the optimal social welfare, where $m$ is the number of items. We also provide tighter approximation results for several subclasses. Our welfare guarantees hold for Bayes-Nash equilibria and for no-regret learning outcomes, via the smooth-mechanism framework. Of independent interest, our techniques show that in a combinatorial auction setting, efficiency guarantees of a mechanism via smoothness for a very restricted class of cardinality valuations, extend with a small degradation, to subadditive valuations, the largest complement-free class of valuations. Variants of draft auctions have been used in practice and have been experimentally shown to outperform other auctions. Our results provide a theoretical justification