Auction Design without Quasilinear Preferences Brian Baisa ⇤† August 6, 2013 Abstract Ianalyzeprivatevalueauctiondesignandassumeonlythatbiddersarerisk averse and have positive wealth effects (i.e. the good is normal). I show removing the standard quasilinearity restriction leads to qualitatively different solutions to the auction design problem with respect to both efficiency or revenue maximiza- tion. On efficiency, I show that probabilistic allocations of the good can Pareto dominate the second price auction; and there is no dominant-strategy mechanism that is both Pareto efficient and individually rationality. On revenue, I construct a probability demand mechanism with greater ex- pected revenues than standard auctions when there are sufficiently many bidders. In addition, I take a new approach to studying bid behavior when types are mul- tidimensional. Instead of characterizing bidder’s interim incentive constraints, Iplaceboundsontheirbids,andIshowthattheseboundsaresufficientfor obtaining revenue comparisons. Keywords: Auctions; Multidimensional Mechanism Design; Wealth Effects. JEL Classification: C70, D44, D82. ⇤[email protected], Amherst College, Department of Economics. †I gratefully acknowledge financial support received from the Cowles Foundation Fellowship, the Yale School of Management, the International Center for Finance and the Whitebox Advisors Fellowship. I received excellent comments on this work from seminar and conference participants and faculty at Yale, Notre Dame, UNC, Johns Hopkins, Haverford College, Amherst College, Cambridge, the University of Michigan, Tel Aviv University, USC, Stony Brook and Lund. I am especially grateful to Dirk Bergemann, Benjamin Polak, Larry Samuelson and Johannes Hörner for numerous conversations and encouragement while advising me with this project. I would also like to thank Cihan Artunç, Lint Barrage, Yeon-Koo Che, Eduardo Faingold, Amanda Gregg, Yingni Guo, Vitor Farinha Luz, Drew Fudenberg, Adam Kapor, Phil Haile, Sofia Moroni, David Rappoport, Peter Troyan and Kieran Walsh for helpful comments and conversations. 1 1Introduction In the auction design literature, it is standard to assume that bidders have quasilinear preferences. Yet there are many well-known environments in which this restriction is violated: bidders may be risk averse, have wealth effects, face financing constraints or be budget constrained. In this paper, I study the canonical independent private value auction setting for a single good and drop the quasilinearity restriction by assuming only that bidders are risk averse and have positive wealth effects (i.e., the indivisible good for sale is a normal good). I show that the auction design problem leads to qualitatively different prescriptions relative to those of the quasilinear benchmark. Instead of using standard auctions where the good is given to the highest bidder with probability one, the auctioneer prefers mechanisms where she can allocate the good to one of many different bidders, each with strictly positive probability. For auctioneers concerned with efficiency, such probabilistic allocations can Pareto dominate the second price auction. And for auctioneers concerned with maximizing expected revenues, Iconstructaprobabilitydemandmechanismthatgeneratesstrictlygreaterexpectedrevenues than standard auction formats when there are sufficiently many bidders. There are many examples of auctions where the quasilinearity restriction does not hold. One case is housing auctions. Housing auctions have developed into a multi-billion dollar industry in the United States. In Melbourne, Australia an estimated 25 50% of homes − are sold via auction (Mayer (1998)). In a housing auction, a buyer’s bidding strategy is influenced by factors like how much wealth she has (wealth effects) and the terms of the mortgage offered by her bank (financing constraints). These factors are not included in the quasilinear model, where a bidders payofftype is described only by a single dimensional variable - her valuation for the good. For another example, consider firms bidding on spectrum rights or oil tracts. The cor- porate finance literature shows that many firms have an internal spending hierarchy (see Fazzari, Hubbard and Petersen (1988)). Specifically, it is more expensive for the firm to use external financing than internal financing, because firms pay higher interest on money borrowed from third parties. A firm may be able to place a relatively low bid in an auction without needing external financing, but in order to place a relatively high bid, the firm may need to obtain external financing and pay a higher interest rate on this debt. Increasing its bid by one dollar using external financing is more costly to the firm than increasing the bid by one dollar via internal financing. Even if the firm is risk neutral, this financing con- straint makes them behave as though they have declining marginal utility of money. Further evidence of risk aversion in auctions is discussed in the related literature section. I first study the problem of an auctioneer concerned with efficiency. In the single good en- 2 vironment with private values and quasilinear preferences, the second price auction is popular because it implements a Pareto efficient allocation in dominant strategies. I show that with- out quasilinearity, there are probabilistic allocations of the good that Pareto dominate the dominant strategy equilibrium outcome of the second price auction (Proposition 1). Is there another dominant strategy mechanism that is efficient in this more general environment? I show that the answer is no: there is no individually rational dominant strategy mechanism that satisfies budget balance and implements a Pareto efficient allocation (Proposition 3). While I obtain an impossibility result on efficiency, I show more positive results for revenue maximization. Specifically, I show that the auctioneer can use randomization to increase revenues. This may seem counterintuitive because bidders are risk averse, but the intuition follows directly from the assumption that the good is normal. Since the good is normal, a bidder’s willingness to pay for it increases as her wealth increases. Similarly, her willingness to pay for any giving probability of winning the good increases as her wealth increases. It follows that the bidder is willing to pay the highest price for her first marginal ‘unit’ of probability of winning, when she has still yet to spend any money and her wealth is the highest. Thus, the bidder is willing to buy a small probability of winning the good at a price per unit of probability that exceeds her willingness to pay for the entire good. Standard auctions that allocate the good to the highest ‘bidder’ do not exploit this property of bidder risk preferences. Iconstructaprobability demand mechanism that uses lotteries to better exploit this feature of bidder preferences. The mechanism sells probabilities of winning the good like a divisible good that is in net supply one. Bidders report a demand curve over probabilities of winning. The curve reports the probability of winning the bidder demands (Q) for a given price per unit price of probability (P ). The auctioneer uses an algorithm similar to that of the Vickrey auction for a divisible good to determine each bidder’s probability of winning and expected transfer. Without quasilinearity, it becomes more difficult to characterize bidder behavior. Now, a bidder’s private information is described by a utility function instead of a single dimensional valuation. This multidimensionality complicates mechanism design problems. Armstrong and Rochet (1999) show that even in the simplest of principal-agent models with multidi- mensional types, explicitly solving for equilibria can be difficult. I take a different approach to characterizing bid behavior in my probability demand mechanism. Instead of explicitly solving for equilibria, I show that we can bound what the bidder reports to the auctioneer by eliminating dominated strategies. In particular, I show that it is a dominated strategy for a bidder to underreport her type (Proposition 4). Iusethisboundonabidders’reportstoconstructalowerboundonexpectedrevenues 3 in the probability demand mechanism. With enough bidders, this lower bound on revenues strictly exceeds an analogously constructed upper bound on the revenues from any standard auction. That is, with enough bidders the probability demand mechanism has higher ex- pected revenues than any standard auction (Propositions 6 and 7). This class of standard auctions includes the first price, second price and all pay auctions, as well as modifications of these formats to allow for entry fees and/or reserve prices. The rest of the paper proceeds as follows. The remainder of the introduction relates my work to the current literature on auction design. Section 2 describes the model and specifies the assumptions I place on bidders’ preferences. Section 3 motivates the use of probabilistic allocations and provides an example in which the dominant strategy equilibrium outcome of the second price auction is Pareto dominated by a probabilistic allocation of the good. Section 4 shows there is no symmetric mechanism that respects individual rationality and implements a Pareto efficient allocation in dominant strategies. Section 5 outlines the con- struction of the probability demand mechanism. Section 6 focuses on revenue comparisons between the probability demand mechanism and standard auction formats. Section 7 pro- vides a numerical example illustrating the practical applicability of my results. Section 9 concludes.