All Nash Equilibria of the Multi-Unit Vickrey Auction

This paper completely characterizes the set of Nash equilibria of the Vickrey auction for multiple identical units when buyers have non-increasing marginal valuations and there at least three potential buyers. There are two types of equilibria: In the first class of equilibria there are positive bids below the maximum valuation. In this class, above a threshold value all bidders bid truthfully on all units. One of the bidders bids at the threshold for any unit for which his valuation is below the threshold; the other bidders bid zero in this range. In the second class of equilibria there are as many bids at or above the maximum valuation as there are units. The allocation of these bids is arbitrary across bidders. All the remaining bids equal zero. With any positive reserve price equilibrium becomes unique: Bidders bid truthfully on all units for which their valuation exceeds the reserve price

All Nash Equilibria of the Multi-Unit Vickrey Auction

This paper completely characterizes the set of Nash equilibria of the Vickrey auction for multiple identical units when buyers have non-increasing marginal valuations and there at least three potential buyers. There are two types of equilibria: In the first class of equilibria there are positive bids below the maximum valuation. In this class, above a threshold value all bidders bid truthfully on all units. One of the bidders bids at the threshold for any unit for which his valuation is below the threshold; the other bidders bid zero in this range. In the second class of equilibria there are as many bids at or above the maximum valuation as there are units. The allocation of these bids is arbitrary across bidders. All the remaining bids equal zero. With any positive reserve price equilibrium becomes unique: Bidders bid truthfully on all units for which their valuation exceeds the reserve price.Vickrey auction; Multi-unit auction; ex-post equilibrium; reserve price; uniqueness

Modeling Electricity Auctions

The recent debates over discriminatory versus uniform-price auctions in the UK and elsewhere have revealed an incomplete understanding of the limitations of some popular auction models when applied to real-world electricity markets. This has led certain regulatory authorities to prefer discriminatory auctions on the basis of reasoning from models which are not directly applicable to any existing electricity market. Vickrey auctions, although often recommended by economists, have also been ignored in these debates. This article describes the approach which we believe should be taken to analyzing these issues

Three Essays On Online Ratings and Auction Theory

This work includes three papers, focusing on online ratings and auction theory. The first chapter, "Why Do People Write Reviews? Theory and Evidence on Online Ratings" examines what motivates consumers to provide uncompensated ratings for products on in the internet. It finds that both a desire to inform other buyers and to punish or reward sellers are at play.The second chapter "All Equilibria of the Multi-Unit Vickrey Auction" characterizes all Nash Equilibria for the Vickrey auction with three or more bidders and any number of units. It shows that all equilibria fall into one of two basic families, and that there cannot be equilibria of any other type.The third chapter "Optimal Availability of Online Ratings" looks at what role consumer ratings have in determining market outcomes. It shows that more information for buyers does not always lead to increased buyer welfare, and that in some cases it may be preferable to keep some buyers uninformed of existing ratings

Auctioning Securities

Treasury debt and other divisible securities are traditionally sold in either a pay-your-bid (discriminatory) auction or a uniform-price auction. We compare these auction formats with a Vickrey auction and also with two ascending-bid auctions. The Vickrey auction and the alternative ascending-bid auction (Ausubel 1996) have important theoretical advantages for sellers. In a setting without private information, these auctions achieve the maximal revenue as a unique equilibrium in dominant strategies. In contrast, the pay- your-bid, uniform-price, and standard ascending-bid auction admit a multiplicity of equilibria that yield low revenues for the seller. We show how these results extend to a setting where bidders have affiliated private information. Our results question the standard ways that securities are offered to the public.Auctions; Multi-Unit Auctions, Security Auctions, Treasury Auctions

Modeling Electricity Auctions

The recent debates over discriminatory versus uniform-price auctions in the UK and elsewhere have revealed an incomplete understanding of the limitations of some popular auction models when applied to real-world electricity markets. This has led certain regulatory authorities to prefer discriminatory auctions on the basis of reasoning from models which are not directly applicable to any existing electricity market. Vickrey auctions, although often recommended by economists, have also been ignored in these debates. This article describes the approach which we believe should be taken to analyzing these issues.electricity markets, auctions, Vickrey auctions

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