An Introduction to Auctions
Total Page:16
File Type:pdf, Size:1020Kb
Journal of Industrial Organization Education Volume 1, Issue 1 2006 Article 5 An Introduction to Auctions Sofia B. Villas-Boas, University of California, Berkeley Recommended Citation: Villas-Boas, Sofia B. (2006) "An Introduction to Auctions," Journal of Industrial Organization Education: Vol. 1: Iss. 1, Article 5. DOI: 10.2202/1935-5041.1004 An Introduction to Auctions Sofia B. Villas-Boas Abstract These lecture notes provide an introduction to auctions. They can be covered in two or three lectures. This material is appropriate for the last part of a first-year graduate industrial organization course or as an introduction to an auctions course. KEYWORDS: auctions Author Notes: These sets of lectures are based on lecture notes for the course ARE 202, which is a first year graduate course in the PhD program in the Agricultural and Resource Economics Department at U.C. Berkeley. These notes have greatly benefited from material covered in several industrial organization courses offered at the Economics graduate program at Berkeley and from comments from Jeffrey Perloff, James Dearden, and Jen Brown. Villas-Boas: An Introduction to Auctions 1. INTRODUCTION An auction is a public sale of property or merchandise defined by explicit rules determining resource allocation and prices on the basis of bids from participants. Four main types of auctions are considered theoretically and in practice. In the “First-price sealed-bid auction” (FPSB) a bidder does not observe the opponents’ bids, and the object is won by the bidder submitting the highest bid. The winner pays her bid value and receives the object, and the losers pay nothing and receive nothing. The second type is the “Second-price sealed-bid auction” (SPSB), also called the Vickrey auction. The SPSB auction is like the FPSB auction, with one exception: the winner pays the second highest bid in the auction. The third auction type is the “English” or “Ascending-price drop-out auction”. It works much like the game of chicken; bidders are active in the auction until they decide to drop out. In this auction, the price increases incrementally from 0, and bidders withdraw from the auction when price exceeds their willingness to pay. The auction ends when only one active bidder remains. This winner pays a price equal to the current price when second-to-last person exits. Finally, the fourth type is the “Dutch” or “Descending price auction”, which is similar to the game of chicken in reverse. The price starts high and decreases until the first bidder agrees to pay the current price. In each of these auctions, the winner’s surplus is her value minus the price she pays. Each loser’s surplus is zero. Most auction studies focus on the FPSB and English auctions, although recent studies have examined online auctions such as eBay. Typically, eBay auctions are modeled as SPSB auctions. Other recent work has studied multi-unit auctions and sequential auctions (see, for example, Cramton, Shoham and Steinberg (2005)). The text is structured as follows: First, we analyze participation strategies and auction house revenue under several auction formats. Second, we discuss the main issues in auction design. We then discuss the main positive and normative goals of empirical work in auctions, and also study reduced form and structural approaches that have been developed and applied to empirical work. The last part of these notes presents a formal outline of the main issues in structural estimation of auctions in applied work. Here, we provide a primer on structural auction estimation—for two surveys, see Laffont et al. (1991) and Hendricks and Paarsch (1995). We also introduce the issue of the strategic behavior of buyers and collusion in bidding. We conclude with a paper identifying collusion in procurement auctions by Porter and Zona (1993). Some of the many topics we are not covering formally here are multi-unit issues (see Hansen (1985) for an empirical paper on the topic), the introduction of 1 Journal of Industrial Organization Education, Vol. 1 [2006], Iss. 1, Art. 5 entry, risk aversion1, budget constraints (see Klemperer’s (1998) Wallet Game), asymmetries (Hendricks and Porter (1988), Wilson (1994) and Porter (1995)), affiliation and common values (Milgrom and Weber (1982), and Laffont (1996)), and auction design. A (far from complete) list of cited and suggested references is provided in the end of the text. 2. PARTICIPATION STRATEGIES AND REVENUE EQUIVALENCE The Vickrey (Second-Price) Auction is a very popular format, perhaps due to the simple bidding strategy. Bidding one’s true valuation is a (weakly) dominant strategy—the amount a bidder pays is independent of her bid and values are also independent among bidders. The winner’s surplus is equal to her valuation minus the second-highest bid submitted, and decreases as the number of bidders increases. In the first-price auction formats, potential bidders face trade-offs—if a bidder bids her valuation then she has no surplus, and lowering the bid from her valuation increases the surplus but reduces the chances of winning. The optimal bidding strategy is more complicated than in the second-price auction and will consist of bid shading. The degree of bid shading depends upon several factors including the number of bidders and information availability. To motivate auction design, one may ask: Given the optimal bidding strategies of the potential bidders, which type of auction will the seller prefer? For a risk-neutral seller, the answer depends on the expected revenue of each type of auction. Under some assumptions (i.e., if item goes to the person with highest valuation and bidders with lowest possible valuation expect zero surplus), the seller may be indifferent between the four types of auctions because they yield the same expected revenue. What about competition? Does a seller prefer more bidders? More bidders lead to higher prices and less surplus for winners. Figure 1 plots data generated through simulations to illustrate this relationship. The y-axis presents expected price, with bidders’ valuations drawn uniformly from [20, 40] and the x-axis indicates numbers of bidders (Source: author’s calculations). 1 In Vickrey and English auctions with risk aversion, it is still a dominant strategy to bid one’s valuation. However, in Dutch and FPSB auctions, bidders may be more aggressive to increase the probability of winning, as in Holt (1979). DOI: 10.2202/1935-5041.1004 2 Villas-Boas: An Introduction to Auctions Expected Price 40 35 30 25 20 Number 1 10 100 1000 of Bidders We will define another dimension upon which auction theory has developed, namely, the dimension of uncertainty. We will define private and common value auctions. Informally, if you see the signals of the other bidders (or their bids) and it causes you to revise your own valuation of the object, then it is a common value auction. Otherwise, it is a private value auction. In a private value auction, the object is for personal use. Moreover, the object will not be resold and no bidder is speculating about future value. If there are inherent differences among bidders, then any single bidder can have a valuation that differs from the others’ valuations, and each bidder’s value will be private. In contrast, in common value auctions, the item has a single, true value which is unknown at time of bidding. An example of common value auctions are offshore oil leases; the value of oil is the same for every participant, no bidder knows her value with certainty, and each bidder has some information about the value due to exploratory drilling. Let r be the reserve price, n is the number of potential bidders, and m be the actual number of bidders. Consider an example in which bidder i’s utility = σ function is Ui U ( i ,v) , where v is the value of the good that bidder i may or σ may not observe, and i is the private signal of bidder i. Let F be the cumulative σ σ σ ≠ distribution function of ( 1,..., N ,v) . Finally, let Yi be the max{ j : j i} and W be the winning bid. We make the following assumptions: (i) Each bidder wants only one object; (ii) U is non-negative, continuous and increasing; (iii) the signals σ ∈ are one-dimensional, i R ; (iv) F is exchangeable across bidders, because no 3 Journal of Industrial Organization Education, Vol. 1 [2006], Iss. 1, Art. 5 σ σ bidder has better information; (v) ( 1,..., N ,v) are positively correlated, and (vi) F, n, U are common knowledge. Independent Private Values (IPV) =σ Ui i for each i=1, …, N and σi are all independently and identically distributed (iid), so that: N F(σ ,...,σ ) = ∏Fσ (σ ). 1 N i i=1 Bidder i’s signal is her payoff. She receives no signal about the other bidders’ values. Pure Common Value (CV) σ σ = σ Ui= v, and the cdf is now F( 1,..., N ,v) . We can v ∑ j and there is j uncertainty about v. Signals of other bidders cause you to revise your valuation, σ i . Bidding Equilibria in a Nutshell: Private Values In a Dutch auction (FPSB), your bid should be lower than your signal—that is, you should shade your bid—and your bid should be increasing in N. Let your σ = σ [ σ] signal be i . Suppose the valuations have support 0, and are uniformly distributed. If we ignore risk aversion, then (N −1) E[next highest bid]= σ N and the equilibrium bid is equal to (N −1) w = σ < σ . N In an English auction with second-price open bidding, each bidder should be willing to pay as much as her valuation.