Auction Design Without Quasilinear Preferences

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 maximization. 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 expected revenues than standard auctions when there are sufficiently many bidders. In addition, I take a new approach to studying bid behavior when types are multidimensional. 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 corporate 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 constraint 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 without 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 multidimensional 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 expected 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 construction of the probability demand mechanism. Section 6 focuses on revenue comparisons between the probability demand mechanism and standard auction formats. Section 7 provides a numerical example illustrating the practical applicability of my results. Section 9 concludes.

Related literature

This paper builds on the private value auction literature pioneered by Vickrey (1961) and later by Myerson (1981) and Riley and Samuelson (1981). Each of these papers addresses an auction design problem under the restriction that bidders’ preferences are quasilinear. Myerson studies the problem of an expected revenue maximizing auctioneer, while Vickrey studies the problem of an auctioneer concerned with efficiency. Their results show the auction design problem is solved by a mechanism where the good is assigned to a bidder only if she has the highest valuation among all bidders.1 However, there is evidence that in many auction settings, bidders do not have quasilinear preferences. Prior work has argued that risk aversion can explain deviations in bidders’ behavior, relative to the predictions of the quasilinear benchmark. Bajari and Hortaçsu (2005) argue that risk aversion serves as a reasonable explanation of experimentally observed bid behavior. Specifically, they show that it serves as a better explanation than three other competing theories - one of which is the quasilinear benchmark. Similarly, Goeree, Holt and Palfrey (2002) show risk aversion provides a parsimonious explanation of experimentally observed overbidding in first price auctions. Buddish and Takeyama (2001) cite risk aversion

1In Myerson (1981), the good is assigned to the bidder with the highest ‘virtual’ valuation. Under the monotone likelihood ratio property, this corresponds to the bidder with the highest valuation.

4 as an explanation for why sellers use buy-it-now prices. Li and Tan (2000) cite risk aversion as an explanation for why sellers use secret reserve prices. In the theoretical auctions literature, while most work focuses on the quasilinear case, there is a smaller literature that considers auctions without quasilinearity. One of the first works in this literature is by Maskin and Riley (1984). Their paper characterizes certain properties of expected revenue maximizing auctions when a bidder’s type is a single dimensional variable, ✓ [0, 1]. They show that the exact construction depends on the common 2 prior over the distribution of types and the functional form of the bidders’ utility functions. Matthews (1983, 1987) and Hu, Matthews and Zhou (2010) also study auctions without quasilinearity. However, they focus on a more structured setting where all bidders have identical risk preferences and there are no wealth effects. Within this framework, they make comparisons between auction formats. Che and Gale (2006) consider a payoffenvironment that is closer to the one studied here. They assume that bidders have multidimensional types and use this to show that when bidders are risk averse, the first price auction generates greater revenues than the second price auction. This builds on their earlier work, which studies standard auction formats with more specific departures from the quasilinear environment (see Che and Gale (1996, 1998, 2000)). My work differs from this prior work in two respects: (1) I consider a very general setting that allows for heterogeneity in bidders across many dimensions; and (2) I study the problem from an auction design perspective. This contrasts with prior work that focuses on comparisons between standard auction formats. My solution to the auction design problem uses probabilistic allocations of the good. Randomization has been advocated in other auction design settings. Celis, Lewis, Mobius and Nazeradeh (2012) construct a randomized mechanism they call the buy-it-now or take- a-chance mechanism. They study a setting where bidder preferences are quasilinear. They show that using randomization can induce more aggressive bidding in ‘thinner’ markets. For the case of budget-constrained bidders, Pai and Vohra (2010), Maskin (2000) and Laffont and Robert (1996) show that the expected revenue-maximizing mechanism may employ probabilistic allocations. Baisa (2012) studies an auction design problem in which bidders use a non-linear probability weighting function to evaluate risks (as described by Kahneman and Tversky (1979, 1992)) and finds that the solution to the auction design problem uses probabilistic allocations of the good to exploit features of bidders’ probability weighting functions. In this paper, I construct a mechanism that uses randomization and is almost revenue maximizing with many bidders. My results can be seen as analogous to those shown of Armstrong (1999). Armstrong

5 studies a diﬀerent problem, that of a multi-product monopolist selling to a representative consumer. He derives an almost optimal solution for the monopolists when there are many products. Outside of auction design, Morgan (2000) advocates using lotteries to raise money for public goods. The intuition for using lotteries in Morgan’s model is distinct from the intuition used here. Morgan shows that lottery tickets can help to overcome the free rider problem when people’s preferences are linear in money. No such free rider problem is present in my model.

2Themodel

2.1 The payoﬀenvironment

I consider a private value auction setting with a single risk neutral seller and N 2 buyers, indexed by i 1,...,N} . There is a single indivisible object for sale. Bidder i’s preferences 2{ are described by utility function ui, where

ui : 0, 1 R R. { }⇥ !

Iletui(1,wi) denote bidder i’s utility when she owns the object and has wealth wi.Similarly, ui(0,wi) denotes bidder i’s utility when she does not own the object and has wealth wi. The object is a “good” and not a “bad”; it is better to have the object than to not for any fixed wealth level, i.e. u (1,w) >u(0,w), w. i i 8 I additionally assume that bidders preferences are strictly increasing and twice continuously differentiable in wealth. This environment differs from the quasilinear case where a bidder’s preferences are com- pletely described by a one-dimensional signal, her ‘valuation’ of the object. This valuation is the most i is willing to pay for the object. In this setting, I define k(ui,wi) as bidder i’s willingness to pay for the good when she has an initial wealth wi and a utility function ui. This is the highest price bidder i will accept as a take-it-or-leave-it offer for the good.

Formally k(ui,wi) is deﬁned as,

ui(1,wi k)=ui(0,wi). (2.1)

I place only two assumptions on bidder preferences. The ﬁrst is that the good is a normal

6 good and the second is that bidders are risk averse. I assume that the good being sold is a normal good (i.e. positive wealth eﬀects). My notion of positive wealth eﬀects is analogous to the notion in the divisible goods case, where abidder’sdemandforthegoodincreasesasherwealthincreasesforaconstantpricelevel.

Assumption 1. (Positive wealth eﬀects) Bidder i has positive wealth eﬀects:

@k(w, u ) i > 0, @w for all w.

Second, I assume that bidders have declining marginal utility from money.

Assumption 2. (Risk Averse) Bidder i has declining marginal utility of money:

2 @ ui(x, w) < 0 for x =0, 1. @w2

IdefineAnd, as the set of all utility functions which satisfy Assumptions 1 and 2.Notice U 2 @k(w,ui) @ ui(x,w) that quasilinear preferences are not included in since =0and 2 =0. U @w @w Note that this is a very general setting. The only two assumptions I place on bidder preferences are positive wealth effects and risk aversion. I do not place any functional form restrictions on preferences. I also allow for multidimensional heterogeneity across bidders. This differs from most of the literature on risk aversion and auctions, where bidders preferences are described by specific functional forms and there is only single dimensional heterogeneity between bidders (see Maskin and Riley (1984), Mathews (1983, 1987)). There are many different functional forms of u that fit the risk aversion and normal good assumptions. Two examples are given below.

Example 1. If bidder i has preferences of the form ui(x, w)=gi(viIx=1 + fi(w)) where both fi and gi are concave functions and I is an indicator function, then ui .Asimpleexample 2U would be the case where gi is linear. We can then interpret this as saying bidder i’s gets a certain number of utils from owning the good, irrespective of her wealth level. She also has declining marginal utility of money, which is unaﬀected by whether or not she owns the good.

7 Example 2. The good being sold is a risky asset. It has a distribution of (monetary) returns g : R+ R+.Bidderi gets utility from wealth of fi.Shehasdecreasingabsolute ! risk aversion - for example, CRRA utility. Her expected utility of owning the asset when she starts with wealth w is ui(1,w) where

u (1,w)= f (w + x)g(x)dx. i ˆ i

Without the asset, she consumes her wealth,

ui(0,w)=fi(w).

In this example, bidder i could be a speculator bidding for a home. She knows this distribution of resale prices for the house is given by g.Asshebecomesricher,herdegreeofabsolute risk aversion decreases. This implies that she demands a smaller risk premium when buying the home. Thus, as she becomes richer her willingness to pay for the home increases.

2.2 Allocations and mechanisms

By the revelation principle, I can limit attention to direct revelation mechanisms. A mechanism describes how the good is allocated and how transfers are made. I deﬁne A as the set of all feasible assignments, where

N N A := a a 0, 1 and ai 1 , { | 2{ }  } i=1 X where ai =1if bidder i is given the object. A feasible outcome speciﬁes both transfers and a feasible assignment: A RN . Ideﬁne:=A RN as the set of feasible outcomes. 2 ⇥ ⇥ A (probabilistic) allocation is a distribution over feasible outcomes. Thus, an allocation ↵ is an element of ().

IusethenotationE↵ [ui,wi] to denote the expected utility of bidder i under allocation

↵ () when she has preferences ui and initial wealth wi.Irefertotheauctioneeras 2 person 0 and use E↵ [u0] to denote the expected transfers she receives under allocation ↵. Note that the auctioneer does not value owning the good. In Section 4,IstudymechanismsthatimplementParetoefficientallocations.Inthe quasilinear benchmark, this problem simplifies to ensuring that the bidder with the highest valuation receives the good. Without quasilinearity, the description of a Pareto efficient allocation is more complicated. Icallanallocationposterior Pareto efficient if for a given profile of preferences, increasing

8 one person’s expected utility necessarily decreases another person’s expected utility (includ- ing the auctioneer). If a benevolent social planner knew each bidder’s private information, she would select a posterior Pareto eﬃcient allocation.

Definition 1. (Posterior Pareto efficient allocations) N Assume bidders have preferences u =(u1,...,uN ) and initial wealth levels, w = 2U (w1,...,wN ).Afeasibleallocation↵ is posterior Pareto efficient if there does not exist a

↵0 () s.t. 2 E↵ [ui,wi] E↵ [ui,wi] i =0, 1 ...N. 0 8 where the above inequality holds strictly for at least one i.

This is equivalent to the notion of an ex-post Pareto superior (efficient) allocation given in Holmström and Myerson (1983, 1991). I use the term posterior as my definition mirrors Laffont and Green’s (1987) notion of posterior implementability. In words, this means conditional upon knowing all of the bidders private information, we cannot increase one person’s expected payoffwithout decreasing another person’s. There are differences between posterior Pareto efficient allocations and the common notions of ex-ante, interim and ex-post Pareto efficiency. It is impossible to design a mechanism that implements an ex-ante or interim Pareto efficient allocation when bidders are risk averse. For example, if an allocation is ex-ante Pareto efficient (by the definition of Myerson (1991)), because the bidders are risk averse, their payments to the auctioneer can not depend on re- alized types. When the bidder’s payment is independent of her report, she has no incentive to tell the truth. At the same time, my posterior Pareto efficiency differs from the typical notion of ex- post Pareto efficiency. It allows the designer to randomly allocate the good to one of many different bidders. It does not necessarily ensure that a bidder with the highest willingness to pay for the good is given the good with certainty. Instead, using randomization can actually increase everyones payoff.

9 (ex-post) (interim) Good is allocated, Bidders report their payments are made, private information game is over.

(ex-ante) (posterior) Types are drawn Conditional on all bidders private information, auctioneer determines payments and the allocation of the good (possibly randomly).

Figure 2.1: Notions of eﬃciency AdirectrevelationmechanismM maps a proﬁle of reported preferences and initial wealth levels to an allocation. That is,

M : N RN (). U ⇥ !

The direct revelation mechanism implements an posterior Pareto efficient allocation if for any reported preferences u =(u1 ...uN ) and initial wealth levels w =(w1 ...wN ),the allocation provided by M is Pareto efficient. That is, the mechanism specifies an allocation that would be Pareto efficient in the complete information environment where all bidders’ private information is known. Amechanismisimplementableindominantstrategiesifitisabestresponseforeach bidders to truthfully report her private information to the auctioneer, for any possible report of her opponent. Definition 2. (Dominant strategy incentive compatibility) AdirectrevelationmechanismM is dominant strategy incentive compatible if for any profile N N of preferences and initial wealth levels u =(ui,u i) and w =(wi,w i) R , 2U 2

EM(u,w) [ui,wi] EM((u0,u i),(w0,w i)) [ui,wi] , for any (u0,w0) R and i =1...N. 2U⇥

3Probabilisticallocations

I now show that auctioneers concerned with eﬃciency or revenue maximization should use mechanisms that allow for probabilistic allocations of the good. Such probabilistic allocations

10 give many bidders a positive probability of winning. This contrasts with most commonly studied mechanisms that assign the good to a bidder with the highest willingness to pay with certainty. The reason for using randomization is based on the assumption that the object is a normal good. When a good is a normal good, a bidder is willing to pay the most for her ﬁrst ‘unit’ of probability of winning the good. As the bidder becomes poorer, she is willing to pay less for a marginal increase in her probability of winning. To illustrate this, I consider the following example. Suppose a bidder i has an initial wealth of wi and preferences described by the utility function u, where

u(x, w)=4Ix=1 + pw.

She receives 4 utils from being given the object and pw utils from having ﬁnal wealth w. This utility function satisﬁes the risk aversion and normal good assumptions. The bidder has declining marginal utility of money, and her willingness to pay for the good increases as her initial wealth increases.2 This is illustrated by Figure 3.1.

Utils u!1, w"

15 u!0, w"

Initial Wealth: 169; WTP: 88. 5 Wealth: 49; WTP: 40.

0 w 0 50 100 150 200

Figure 3.1: u(x, w)=4Ix=1 + pw

Assume that the bidder has an initial wealth of 100. The bidder is willing to pay 64 for the good, since u(1, 100 64) = u(0, 100) = 10. Suppose there are two identical lottery tickets that each give a 50 50 chance of winning the good. One of the two tickets is randomly selected, and that ticket holder wins the good.

2A bidder is risk averse and has positive wealth effects for any utility function which has this separable form. Specifically, any utility function of the form u(x, w)=vIx=1 + g(w) where v>0 and g is strictly increasing and concave satisfies Assumptions 1 and 2.

11 The bidder can purchase 0, 1 or both tickets. If she purchases both, she wins the good with certainty. The bidder’s willingness to pay for her ﬁrst ticket is simple to calculate. It 1 is the price that makes the bidder indiﬀerent between having a 2 probability of winning and remaining at her status quo initial wealth with no chance of winning the good. Her willingness to pay for the ticket is 36 as,

.5u(1, 100 36) + .5u(0, 100 36) = u(0, 100).

The bidders willingness to pay for the good is her willingness to pay for both tickets combined, which is 64. Thus, the bidder is willing to pay a higher price for her first ticket than her second. By the time she is deciding whether to purchase the second ticket, she is poorer and thus willing to pay less for the second ticket. This would not be true if bidder i had quasilinear preferences. If bidder i had quasilinear preferences, with a valuation of 64,she would be willing to pay an equal price for both tickets - in this case 32. This property of bidder preferences is important for an auctioneer concerned with efficiency. To see this, consider a case where there are two identical bidders who both have the same preferences as the bidder described above. If the auctioneer used a second price auction to sell the good, both bidders have a dominant strategy to bid their willingness to pay - in this case, 64. Since both bidders are identical, a winner is then randomly selected and pays 64 for the good. Thus, both bidders are indifferent between winning and losing. This is equivalent to bundling both lottery tickets together and selling them to one bidder for a price of 64. However, the auctioneer can use randomization to create an allocation that Pareto dominates this outcome. In this example, the reallocation is very simple. Recall that the loser is willing to pay more for her first ticket than she is willing to pay for her second. The loser is also ex-ante identical to the winner. Thus, the loser is willing to pay more for her first ticket than the winner is willing to pay for her second ticket. The winner could give one of her lottery tickets to the loser in exchange for 32. This is a Pareto improving reallocation. The revenue for the auctioneer is unchanged. The expected payoffof the loser increases - she is willing to pay up to 36 to obtain her first ticket, but here she only pays 32.Similarly,this trade increases the expected payofffor the winner as well - she is willing to sell her second ticket for as little as 28,buthereshegets32. This example illustrates the first qualitative difference from prescriptions of the quasilinear benchmark. When bidders’ preferences are restricted to be quasilinear, the second price auction implements a Pareto efficient allocation in dominant strategies. Without the restriction of quasilinearity, there are preferences where the auctioneer can construct a probabilistic

12 allocation of the good which Pareto dominates this allocation.

Proposition 1. a There exist probabilistic allocations of the good that Pareto dominate the dominant strategy equilibrium outcome of the second price auction.

The proposition is in contrast with the results of Sakai (2008) and Saitoh and Serizawa (2008). These authors claim the second price auction is Pareto efficient when bidders have non-quasilinear preferences. They obtain their results by restricting attention to determinis- tic mechanisms and using notions of envy-freeness in place of my notion Pareto efficiency. My example shows that when we use a notion of Pareto efficiency that allows for randomization, the second price auction does not guarantee Pareto efficiency. The proposition also relates to prior findings in the general equilibrium literature. Prescott and Townsend (1984) show that even when agents are risk averse, randomization is necessary for efficiency in a number of different general equilibrium settings. They write, “the key to this conclusion is that ex post full information inefficiencies and randomness are both con- sistent with ex ante private information optimal allocations.” Here we see that the same is true. While it is ex-post efficient to assign the good to the bidder with the highest willingness to pay, there are ex-ante gains in efficiency by using randomization, even when there is no private information. The conclusion of Proposition 1 leads to a natural question. Under the quasilinearity restriction the second price auction is the unique dominant strategy mechanism that guarantees aParetoefficientallocationofthegood(Holmstrom,1979).Withoutthequasilinearityre- striction, Proposition 1 shows this is no longer true. This leads me to ask: is it possible to construct a mechanism that retains the desirable properties of the second price auction when we drop the quasilinearity restriction?IaddressthisquestioninSection4 and I show the answer is no. However, my main focus will be on studying the properties of revenue maximizing mechanisms. Here I obtain more positive results - I show that the auctioneer can use randomization to increase expected revenues. Using randomization allows the auctioneer to exploit the fact that a bidder is willing to pay a high price for her first marginal unit of probability of winning. In fact, she is willing to buy a positive probability of winning the good at a price per unit of probability that exceeds her willingness to pay for the entire good. I show that this property holds for any bidder who is risk averse and has positive wealth effects. In particular, I show that bidders’ have downward sloping demand curves for probabilities of winning the good. This probability demand curve states the probability of winning the good a bidder demands when she pays a price of p per unit of probability. The demand

13 curve captures the fact that bidders are willing to pay a higher price for their first units of probability. In fact, a bidder is willing to buy a positive probability of winning the good at prices (per unit) that exceed her willingness to pay for the entire good. Without the quasilinearity restriction, how a bidder’s payments are structured affects her payoff. Standard auctions do not equalize a bidder’s marginal utilities across different states of the world. In particular, a bidder may value a marginal dollar more when she wins than when she loses, or visa versa. By equalizing a bidder’s marginal utility across the win and lose state, the auctioneer can increase the bidder’s expected surplus. I will call such a payment scheme efficient. That is, given a bidder’s expected transfers and probability of winning the good, the efficient payment scheme maximizes her expected utility. I will show that with enough bidders, the auctioneer can extract the surplus created by efficient payments and increase revenues. I construct a bidder’s probability demand curve under the assumption that her payments are structured efficiently. To formalize this, consider a gamble where bidder i wins the good with probability q and pays x to the auctioneer in expectation. The bidder has preferences ui and initial wealth 2U wi. The efficient payment scheme maximizes bidder i’s expected utility given that she wins with probability q and pays x in expectation. Assume that she pays pw and pl conditional on winning or losing, respectively. In the efficient payment scheme, pw⇤ and pl⇤ are such that,

(pw⇤ ,pl⇤)=argmaxqui(1,wi pw)+(1 q)ui(0,wi pl). pw,pl

s.t. x = qp +(1 q)p . w l

The payments pw⇤ and pl⇤ maximize her expected utility under the constraint that she wins the good with probability q and pays x in expectation. This induces an indirect utility function Vi that is a function of her expected payments and the probability she wins the good. V (q, x):=max qu (1,w p⇤ )+(1 q)u (0,w p⇤). i pw,pl i i w i i l s.t. x = qp⇤ +(1 q)p⇤ w l The indirect utility function gives the maximal expected utility for bidder i conditional on winning the object with probability q and paying x in expectation. If bidder i makes an eﬃcient payment, her expected utility from winning the good with probability q and paying a price of p per unit of probability is Vi(q, qp).Iformbidderi’s probability demand curve by assuming that she makes eﬃcient payments.

q(p, ui,wi):=argmaxVi(q, qp). (3.1) q [0,1] 2

14 Bidder i’s probability demand curve states the probability of winning she demands when she pays a price of p per unit of probability. I economize notation by writing bidder i’s probability demand curve as qi(p)=q(p, ui,wi). Abidder’sprobabilitydemandcurveiscontinuousandweaklydecreasinginthepricep. In addition, the bidder demands a strictly positive probability of winning the good at a per unit price that exceed her willingness to pay for the entire good.

Proposition 2. a

If bidder i has preferences ui ,initialwealthwi and willingness to pay ki,then 2U (1) qi(p) is continuous and weakly decreasing.

(2) qi(ki + ✏) > 0 for some ✏>0.

Figure 3.2 shows the probability demand curve for bidder i with initial wealth of 100 and preferences u(x, w)=4Ix=1 + pw,asintheaboveexample.

p 80 75 ki = 64 70 65 60 55 q p 0.2 0.4 0.6 0.8 1.0

Figure 3.2: A probability demand curveH L

As a basis for comparison, if a bidder has quasilinear preferences, with a valuation of v for the object, her probability demand curve is a step function where, q(p)=Ip v. Notice  that this is discontinuous and q(p)=0for all prices above the bidders willingness to pay for the good v. It is the normal good assumption that implies that any bidder is willing to buy a strictly positive probability of winning the good at a per unit price that exceeds her willingness to pay for the good. This can be illustrated graphically. Consider a bidder i with an initial wealth of wi, and a willingness to pay for the good of ki.Supposethegoodwasbeing sold for exactly ki. Thus, bidder i is exactly indifferent between buying and not buying the good. Because the good is normal, increasing or decreasing bidder i’s wealth from wi would break this indifference. This means that bidder i would strictly prefers to (not) buy the good for the fixed price of ki if and only if her wealth is greater (less) than wi. Thus, ui(1,w ki) >ui(0,w) if and only if w>wi. This is illustrated in Figure 3.3. 15 Utils Utility if buy at price ki

Utility if do not buy

wealth wi

Figure 3.3: Bidder i’s indirect utility function

The upper envelope of the two curves is bidder i’s indirect utility as a function of her initial @ @ wealth when the object costs ki.Noticethat ui(1,wi ki) > ui(0,wi), which causes @w @w the curve ui(1,w ki) to cross the curve ui(0,w) from below to above at the point w = wi. This creates a local convexity in the indirect utility curve at the point wi.Byexploitingthis diﬀerence in bidders marginal utilities of money between the win state and the lose state, the auctioneer can construct a gamble where the bidder wins the good with positive probability and pays a price per unit of probability that strictly exceeds her willingness to pay for the entire good with certainty. This property of bidder preferences is similar to results shown by Kwang (1965). He shows indivisibilities in consumption can lead to risk loving behavior for the case where the utility of the good and money are separable.

4Paretoeﬃcientmechanisms

When bidder preferences are restricted to be quasilinear, the second price auction is the unique format that implements an individually rational and Pareto eﬃcient allocation in dominant strategies (Holmström, 1979). Proposition 1 shows that, without quasilinearity, the dominant strategy equilibrium outcome of the second price auction can be Pareto dominated by probabilistic allocations of the good. This leads to the question I address in this section: Is there a dominant strategy mechanism that respects individual rationality, generates a (weak) budget balance for the seller and is Pareto eﬃcient? I show the answer to this question is no. Without the quasilinearity assumption, dominant strategy implementation imposes more restrictions on the design problem. With the quasilinearity restriction, a bidder can only

16 misreport her type by submitting another (single-dimensional) valuation. Without quasilinearity, dominant strategy incentive compatibility requires that each bidder has no incentive to misreport her preferences for any set of reports by other bidders. She can misreport her preferences by reporting another (infinite-dimensional) utility function. Since each bidder has a much larger set of possible deviations, this makes dominant strategy implementation more difficult to achieve. Without the quasilinearity assumption, requiring Pareto efficiency also imposes more restrictions on the design problem. With the quasilinearity restriction, Pareto efficiency requires that the bidder with the highest valuation is given the object. Proposition 1 illustrates that this is not true when we drop the quasilinearity restriction. Without quasilinearity, requiring Pareto efficiency places restrictions both on how transfers are made and with what probability each bidder receives the good. With the quasilinearity restriction, there is a unique mechanism that is individual rational and implements a Pareto efficient allocation in dominant strategies. However, without quasilinearity, it becomes more difficult to design such a mechanism. This leads to the impossibility result I state below.

Proposition 3. a There is no dominant strategy direct revelation mechanism that is Pareto eﬃcient and sat- isﬁes individual rationality and weak budget balance.

The proof of the proposition is developed in the appendix. I show that the necessary conditions for Pareto efficiency and dominant strategy incentive compatibility form a contradiction. I prove this result by considering only a subset of the incentive constraints required for dominant strategy implementation. I show that there is no mechanism that both satisfies this subset of incentive constraints and implements a Pareto efficient allocation. Thus, there is no mechanism that satisfies all dominant strategy incentive constraints and implements a Pareto efficient outcome. In particular, I consider a subset that includes a continuum of types that are almost quasilinear. Additionally, I consider a single type with non-linear utility of money. I use the necessary conditions for Pareto efficiency and dominant strategy incentive compatibility to show that if all players report types that are almost quasilinear, the mechanism must specify an allocation that is close to the allocation specified by the second price auction. In particular, the bidder with the highest type wins the object with certainty and pays an amount approximately equal to the second highest bidder’s willingness to pay for the good. However, when one bidder is of the non-linear type, the necessary conditions for Pareto efficiency place strict restrictions on the structure of the allocation. Under these necessary

17 conditions, if a bidder has almost quasilinear preferences, she can gain by underreporting when another bidder reports her type to be non-linear. The reason I study mechanisms that satisfy individual rationality and weak budget balance is to avoid trivializing the problem. If I removed the individual rationality condition, I could form a mechanism which is both Pareto efficient and one such mechanism would randomly select a ‘winner’. It would then take a large sum of money from all other bidders and also give this money to the winner. Now that the winner is very rich and the losers are very poor, it is efficient to assign the winner the good (recall the positive wealth effects assumption). Similarly, if I removed the budget balance requirement, I could construct a mechanism in which the auctioneer does the same thing. She randomly selects a winner and gives her enough money such that it is efficient that she also owns the object. The relatively benign assumptions of weak budget balance and individual rationality avoid these trivializations. My proof strategy is similar to that used by Jehiel, Moldavanu, Meyer-Ter-Vehn and Zame (2006). They study a related multidimensional single good allocation problem when bidders have quasilinear preferences and interdependent values. They show that ex-post implementation of non-trivial social choice functions is impossible in generic multi-dimensional type spaces. To show this, they study a two bidder setting where i’s type is described by 2 (✓i1,✓i2) [0, 1] .Hervaluationisgivenby 2

vi(✓)=✓i1 + ✏ ✓j2. j=1 Y They show that in this setting, the only ex post incentive compatible social choice functions are trivial. My proof is analogous in that I also show an impossibility result by showing that the necessary conditions for dominant strategy implementation and Pareto efficiency form a contradiction, even when studying a limited type space. The results of Jehiel et. al and Proposition 3 illustrate similar points. The former illustrates the difficulties of implementing social choice functions when bidders preferences are restricted to be quasilinear, but are unrestricted in their interdependence. The latter illustrates the difficulties of implementing efficient allocations when bidders are restricted to have private values, but have unrestricted risk preferences. In both cases, multidimensional heterogeneity between bidders makes robust implementation of specific social choice functions more difficult than in the benchmark setting.

18 5Theprobabilitydemandmechanism

The previous section illustrated the importance of probabilistic allocations for an auctioneer who is concerned with efficiency. Proposition 1 shows that probabilistic allocations of the good can improve on efficiency relative to the prescriptions of the benchmark model, though Proposition 3 provides a more negative result. Now I shift focus to the second question, the problem of an auctioneer concerned with maximizing expected revenues. I obtain more positive results and show that using randomization increases revenues. I construct a probability demand mechanism that generates strictly greater expected revenues than standard auctions when there are sufficiently many bidders. Recall that Proposition 2 shows that if a bidder is indifferent between accepting or re- jecting a take-it-or-leave-it offer for the good at a price of k,therearegamblesshestrictly prefers to enter where she wins the good with positive probability q and pays strictly greater than qk in expectation. This is not true when bidders’ preferences are restricted to be quasilinear. Any such gamble violates a bidder’s individual rationality constraint. Auction formats that are commonly studied in the benchmark setting do not exploit this feature of bidders’ preferences. For example, in the first and second price auctions, it is a dominated strategy for a bidder to submit a bid above her willingness to pay for the good. However, my probability demand mechanism uses this property of bidder preferences to generate greater revenues than standard auctions. The probability demand mechanism sells bidders probabilities of winning the good like a divisible good that is in net supply one. Instead of submitting single dimensional bids, bidders reports their probability demand curves to the auctioneer. Recall a bidder’s probability demand curve states how much probability of winning she demands when she pays a price of p per unit. The construction exploits bidders preferences towards risk - it allows the auctioneer to sell bidders probabilities of winning the good at prices per unit that exceed their willingness to pay for the good with certainty. The auctioneer uses the reported probability demand curves to calculate each bidder’s expected transfer and probability of winning. This is done by using an algorithm that resembles the Vickrey auction for a divisible good. In the Vickrey auction for a divisible good, bidders report demand curves for quantities of the good. The auctioneer uses the reported demand curves to calculate each bidder’s transfer and quantity allocation. In the probability demand mechanism, I use bidders probability demand curves to analogously determine a bidder’s expected transfers and probability of winning the good. Though my probability demand mechanism and the Vickrey auction for a divisible good

19 have similar constructions, there are important differences in bidder behavior between the two. In the latter, it is a dominant strategy for bidders to truthfully report their preferences. But removing the quasilinearity restriction poses challenges not seen in the benchmark divisible goods setting. The standard approach of solving for an equilibrium by studying bidders’ interim incentive constraints proves to be intractable in multidimensional settings. Thus, I take a different approach to characterizing bid behavior and generating revenue comparisons. Instead, I partially characterize bidder behavior by placing bounds on their reports to the auctioneer. Specifically, in my probability demand mechanism, there is a lower bound on what a bidder will want to report to the auctioneer. I show it is a dominated strategy for a bidder to underreport her demand curve to the auctioneer. I can then use a bidder’s truthful report as a lower bound on what she actually will report to the auctioneer. I use this bound on bidder behavior to obtain revenue comparisons between my probability demand mechanism and standard auctions. In this section I discuss the construction of the probability demand mechanism. Section 6 discusses revenue comparisons between the probability demand mechanism and standard auction formats.

5.1 The probability demand mechanism

In the probability demand mechanism, a bidder reports her probability demand curve q(p, ui,wi), preferences ui and initial wealth level wi. The auctioneer uses the reported demand curves to calculate each bidder’s probability of winning and expected payment. Given a bidder’s probability of winning and expected payments, her payments are then structured eﬃciently. It is equivalent to assume that each bidder reports only her preferences and initial wealth levels and that the auctioneer derives her probability demand curve directly. I work with the indirect mechanism, where bidders also report probability demand curves, to illustrate the connection to the divisible goods setting. For economy of notation I write qi(p)=q(p, ui,wi). The probability that bidder i wins the object is calculated using the reported probability demand curves. Given the reported demand curves, the auctioneer calculates the (lowest) price for probabilities of winning the good that ‘clears the market.’ That is, she ﬁnds the

(lowest) price p⇤ where the total reported demand for probabilities of winning the good equals one. N

p⇤ := inf p :1= qi(p) . (5.1) p { } i=1 X Given p⇤,theprobabilitybidderi wins the good is qi(p⇤). This is the (reported) probability of winning that she demands when she pays a price of p⇤ per unit of probability and payments

20 are made eﬃciently.

The market clearing price p⇤ is not the per unit price bidders pay for probabilities of winning the good. Instead each bidder faces a probability supply curve that represents her marginal price curve for probabilities of winning the good. The supply curve curve is the residual demand for probabilities of winning. It is analogous to the residual demand curve in Vickrey auction for a divisible goods auction. Thus, the price a bidder pays for a unit of probability is determined by her opponent’s actions.

Given a price p,abidder’sprobabilitysupplycurveSi(p) states the amount of probability of winning the good that is not demanded by the N 1 other bidders.

1 j=i qj(p) if 1 j=i qj(p) > 0 Si(p)= 6 6 (5.2) 80 otherwise < P P Bidder i’s (reported) probability: demand curve equals her probability supply curve at the price p⇤. Thus, the market clearing price sets each bidder’s probability demand curve equal to her probability supply curve. Bidder i’s expected payments are determined by treating her probability supply curve as her (expected) marginal price curve. Her expected payment to the auctioneer is Xi, where,

p ⇤ Xi = tdSi(t). (5.3) ˆ0 This payment rule can be viewed as the continuous analog to the payment rule in the second price auction. Note that I suppress notation in writing Xi;itisafunctionofthecomplete proﬁle of reported demand curves (q1 ...qN ).Figure5.1 illustrates this graphically.

p 8 qi p Si p 6

4 H L H L

2 i's expected payment 0 q 0.0 0.2 0.4 0.6 0.8 1.0

Figure 5.1: Expected payment in the probability demand mechanism.

21 Thus, the reported probability demand curves determine each bidder’s expected transfers and probability of winning the good. Notice that bidder i pays a marginal price for a unit of probability that is lower than the market clearing price p⇤ . Thus, Xi p⇤qi(p⇤).  To complete the description of the mechanism, I specify precisely how payments are structured. I assume that each bidder’s payment to the auctioneer is structured efficiently given the reports. Thus, if the profile of reported demand curves (q1 ...qN ) is such that bidder i wins with probability qi(p⇤) and pays the auctioneer Xi in expectation, she makes an efficient payment. If bidder i reports her preferences to be ui and her initial wealth to be wi,shethenpayspi,w⇤ when she wins and pi,l⇤ when she loses where,

(pi,w⇤ ,pi,l⇤ )=argmaxqi(p⇤)ui(1,wi pw)+(1 qi(p⇤)) ui(0,wi pl). (5.4) pw,pl

s.t. X = q (p⇤)p +(1 q (p⇤))p . i i w i l Deﬁnition 3. (The probability demand mechanism)

The probability demand mechanism maps reported demand curves (q1,...,qN ),preferences

(u1 ...uN ),andinitialwealthlevels(w1 ...wN ) to a probabilistic allocation described by (5.1) (5.4). The probability demand mechanism can be modiﬁed so bidders only submit a probability demand curve to the auctioneer. This is done using a two-step procedure where bidders ﬁrst submit probability demand curves and each bidder’s probability of winning and expected payments are calculated as discussed above. Then, the auctioneer tells each bidder her probability of winning qi(p⇤) and the expected payment she will make Xi.Beforethe actual winner is selected, each bidder decides how to structure her payments, given that her probability of winning is qi(p⇤) and her expected payment is Xi.Alternatively,theauc- tioneer can use an ascending-bid clinching mechanism, as described in Ausubel (2004), to determine bidders probabilities of winnings and expected payments. This would not even require bidders to submit a probability demand curve.

5.2 Behavior in the probability demand mechanism

The construction of the probability demand mechanism illustrates its similarities to the Vickrey auction for a divisible good. Yet, in the benchmark divisible good setting, bidders’ preferences are restricted to be quasilinear. Under this restriction on preferences, the Vick- rey auction for a divisible good has a dominant strategy equilibrium (see Krishna (2001)). However, in the indivisible good setting that I study, I drop the quasilinearity restriction

22 on bidders’ preferences and I find that it is no longer a dominant strategy for a bidder to truthfully report her preferences. Removing the quasilinearity restriction leads to multidimensional heterogeneity between bidders. This multidimensional heterogeneity makes characterizing bidders interim incentive constraints intractable. Thus, in order to describe bidder behavior, I take a new approach. In particular, I show that we can obtain a partial characterization of bid behavior by placing bounds on a bidder’s report to the auctioneer. I show that this partial characterization of bidder behavior will already be sufficient for making revenue comparisons between my probability demand mechanism and standard auctions. In this section, I derive the bound on a bidder’s report to the auctioneer. Specifically, I show it is a dominated strategy for a bidder to report a probability demand curve which underreports her demand for winning probabilities. The intuition for this result can be understood graphically. First, I consider a hypothetical case where it is a best response for a bidder to truthfully reveal her demand curve. Suppose bidder i faces a perfectly elastic probability supply curve, and she pays a constant marginal price for a unit of probability of winning the good. Assume this price is pE.Herbestresponse is to maximize her expected utility given she pays a price pE per unit of probability. The solution to this maximization problem is qi(pE) by definition - qi(pE) is the probability of winning the good that she desires when she pays price of pE per unit of probability. Thus, truthful reporting is a best response. By truthful reporting, she wins the good with probability qi(pE) and pays pEqi(pE) in expectation.

p 8 qi p 6 S p p* i 4 H L H L 2 i's expected payment q 0.0 0.2 0.4 0.6 0.8 1.0

Figure 5.2: Expected payment when facing a perfectly elastic supply curve.

Suppose instead bidder i faces a more inelastic (relative to perfectly elastic) probability supply curve. Suppose her residual probability demand curve still passes through the point

(pE,qi(pE)). If bidder i truthfully reports her probability demand curve, she wins the good with the probability qi(pE). Thus, her probability of winning the good is the same as it is

23 when she faces the perfectly elastic supply curve. However, she pays less when she faces the more inelastic supply curve. The marginal price she pays for all but the ﬁnal unit of probability she acquires is less than pE. Thus, she pays Xi which is less than pEqi(pE) for a qi(pE) probability of winning. It is as though she faced the same supply curve that was perfectly elastic, with constant price pE,andthenshewasgivenarefundofpEqi(pE) Xi 0. This increases her demand of the good relative to the case where she pays the price of pE per unit of probability.

p 8 qi p

Si p 6 H L p* H L 4

2 i's expected payment q 0.0 0.2 0.4 0.6 0.8 1.0

Figure 5.3: Expected payment when facing a relative inelastic supply curve. Thus, while it was a best response for bidder i to truthfully report her demand curve when she faced a constant marginal price curve, she actually has an incentive to over report her demand curve when she faces an upward sloping marginal price curve. Since bidders are restricted to reporting downward sloping demand curves, bidder i will always face an upward sloping marginal price curve. The precise amount that bidder i wants to overreport her demand curve depends on the elasticity of the supply curve she expects to face. Her incentive to misreport is greater when facing a more inelastic supply curve - she is given a larger ‘refund’ relative to the case where she faces a perfectly elastic demand curve. Thus her demand increases more relative to the perfectly elastic case. What is clear is that it is never a best reply for bidder i to ever underreport her demand curve. This observation allows us to use bidder i’s truthful report as a lower bound on her actual report. Proposition 4. a

Assume bidder i has probability demand curve qi(p).Itisaweaklydominatedstrategyto report a probability demand curve q˜i where q˜i(p)

24 Proposition 4 shows that truthful reporting can serve as a lower bound on a bidder’s possible report. This lower bound on a bidder’s report enables revenue comparisons between the probability demand mechanism and other auctions.

6Revenuecomparisons

In this section, I show that with sufficiently many bidders, the expected revenues from the probability demand mechanism exceeds the expected revenues of a large class of standard auctions. In addition, the probability demand mechanism achieves approximate revenue maximization while most standard auctions do not. Included within the class of standard auctions is the first price, second price and all pay auctions. Also included are any of these auctions modified to include reserve prices or entry fees. Ishowthisinthreesteps.First,Ideriveanupperboundontheexpectedrevenuesfrom any mechanism that respects bidders’ interim individual rationality constraints. Next, I use Proposition 4 to place a lower bound on bidders’ reports to the auctioneer in the probability demand mechanism. Using the lower bound on bidders’ reports, I obtain a lower bound on the expected revenues. As N ,thislowerboundontheexpectedrevenuefromthe !1 probability demand mechanism approaches the revenue upper bound for any individually rational mechanism. Lastly, I define a broad class of ‘highest-bid’ wins mechanisms. This class of mechanisms includes most commonly studied auctions. I show that even when there are many bidders, the expected revenues from any highest-bid wins mechanism does not approach the revenue upper bound. Thus, it attains a strictly lower bound than the one attained by my probability demand mechanism.

6.1 A revenue upper bound for all mechanisms

Iassumethatbidders’preferencesareindependentlyandidenticallydistributed.Inpartic- ular, I assume there is a distribution over possible ‘types’ of bidder preferences. I describe M bidder i’s preferences by a type ti, where ti R and M is finite. I let the first element 2 represent bidder i’s initial wealth level wi. A bidder with type ti has preferences described by the utility function u(x, w, ti) when her type is ti.Iassumethatallbidderspreferenceshave M declining marginal utility of money and positive wealth effects. That is, for any ti R , 2 u(x, w, ti) . At the same time, this setup allows for heterogeneity across multiple dimen- 2U sions. I do not place restrictions on the functional forms of bidders’ utility functions. While the standard quasilinear model allows for heterogeneity in ‘valuations’, this setup allows for heterogeneity across risk preferences, initial wealth levels and financing constraints.

25 M If two types t, t0 R are relatively close, they have similar preferences. Specifically, I 2 assume u(x, w, t) is continuous in t. Each bidder’s preference is an i.i.d. draw of a random variable with density f. The distribution f has full support over T ,acompactsubsetofRM . The density maps types to anon-negativenumber,f : T R+. ! With this notation, I will briefly talk about the existence of equilibria in the probability demand mechanism. While I do not explicitly solve for an equilibrium, I can use a recent result by Reny (2011) to show that there exists a symmetric Bayes Nash equilibrium in pure strategies. However to use Reny’s result, I must first make two more assumptions. First, I assume that f is atomless and second I assume monotonicity on types which means that if ti tj for all coordinates, then q(p, ti) q(p, tj) for all p 0.Notethatthisassumption is only necessary to use Reny’s existence result. I do not impose these two assumptions anywhere else in the paper.

Proposition 5. a Under the monotonicity and atomless distribution assumptions, there exists a unique symmetric pure strategy Bayes Nash equilibrium of the probability demand mechanism.

With this, I return to the problem of deriving a revenue upper bound. For a given distribution over types, I can define an upper bound on revenues from any mechanism that is interim individually rational. To do so, I first consider the highest per unit price where a bidder demands a positive probability of winning the good. Let pi be defined as the highest per unit price where bidder i demands a positive probability of winning. Note that pi is a function of bidder i’s type, ti. Thus I write it as p(ti), where

p(ti):=sup p : q(p, ti) > 0 . p { }

Since preferences u are continuous in t, it follows that p(ti) is continuous in ti.Giventhe distribution of preferences f,IletP denote the highest possible per unit price where a bidder demands a positive probability of winning the good.

P := sup p(t):t supp(f) . { 2 }

IassumethatP< . Thus, for a suﬃciently high per unit price, no bidder demands any 1 probability of winning the good. The price P serves as an upper bound on the expected revenues from any interim individually rational mechanism. Since no bidder is ever willing to pay a per unit price that exceeds P for any positive probability of winning the good, a

26 mechanism with expected revenues that exceed P necessarily violates some bidder’s interim individual rationality constraint. Remark 1. a All mechanisms which are interim individually rational generate expected revenues less than or equal to P . This is a generous upper bound on revenues. It only assumes that a mechanism respects interim individual rationality constraints and holds for any number of bidders.

6.2 Revenues from the probability demand mechanism

Proposition 4 states that in the probability demand mechanism, it is a weakly dominated strategy for a bidder to submit a demand curve where she underreports her demand. By assuming that bidders play only undominated strategies, I show that as N ,alower !1 bound on expected revenue from the probability demand mechanism approaches the upper bound on expected revenues from any interim individually rational mechanism, P .

Proposition 6. a As N ,ifallbiddersplayundominatedstrategies,theexpectedrevenuefromtheproba- !1 bility demand mechanism approaches P .

The formal proof of the above result is in the appendix, though I discuss the intuition here. Using the deﬁnition of P and the continuity of p in t,thereisapositiveprobability that a randomly drawn bidder has a demand curve where qi(p) > 0 for any p

u(x, w)=4Ix=1 + pw.

By assuming that bidders truthfully report their probability demand curves, I obtain a lower bound on a bidder’s residual probability demand. As N increases, the lower bound on the residual probability demand curves approaches the revenue upper bound of P =80.

27 p 80 N=5 75 N=3 70 N=2

65 WTP = 64 60 qi!p" 55 0.2 0.4 0.6 0.8 1.0 q

Figure 6.1: Bounds on residual probability demand curves.

6.3 Revenue comparisons with standard auction formats

While the probability demand mechanism approaches the expected revenue upper bound of any individually rational mechanism, this alone is not an interesting result. In the benchmark quasilinear environment there are many mechanisms that approach an analogously defined bound. Here, this is not the case. I show expected revenues from standard auction formats do not approach this upper bound, even when there are many bidders. Ibeginbylookingatthetwomostcommonlystudiedformatsintheauctionliterature; the first and second price auctions. In each format, it is a dominated strategy for a bidder to submit a bid that exceeds her willingness to pay for the good. Thus, expected revenues are bounded by the highest willingness to pay of any bidder. However, Proposition 2 shows that each bidder is willing to purchase a positive amount of probability of winning at a price per unit that exceeds her willingness to pay for the (entire) good. That is, if bidder i has preferences such that, ui(1,wi ki)=ui(0,wi),thenp(ti) >ki. The probability demand mechanism exploits this feature. As N ,expectedrevenues !1 from the probability demand mechanism approach the highest price any agent is willing to pay for positive probability of winning the good P . Thus, the expected revenues from the probability demand mechanism exceed the expected revenues from the first price or second price auction when there are sufficiently many bidders.

Corollary 1. a Assume that bidders play undominated strategies. When N is suﬃciently large, expected revenues from the probability demand mechanism exceed expected revenues of the ﬁrst or second price auctions.

28 This result generalizes beyond the ﬁrst and second price auction. It extends to a broad class of indirect mechanisms where bidders submit single dimensional bids. In particular, I focus on ‘highest bid wins’ mechanisms, where a bidder receives the object only if she submits the highest bid. Additionally, the bidder is able to leave the auction at no cost by placing a bid of 0.

Thus, I study mechanisms where each bidder reports a message mi R+. The indirect 2 mechanism M maps the N messages to a distribution over feasible outcomes:

N M : R+ (). !

If mi =0,bidderi makes no transfers and wins the good with 0 probability. This is equivalent to allowing bidders free exit from the auction.

Deﬁnition 4. (Highest bid mechanism) The indirect mechanism M is a highest bid mechanism if i is given the object only if she submits the highest bid,

max mj >mi = ai =0, j=i 6 ) and if mi =0, i pays 0 transfers and ai =0.

This class of mechanisms includes many familiar mechanisms such as the ﬁrst price, second price and all pay auctions. It also includes each of these formats with entry fees or reserve prices. Most commonly studied auction formats have the property that along the equilibrium path, the probability of a tie is zero. With a suﬃcient amount of heterogeneity in preferences, this is to be expected. For example, this property is always true of any equilibrium of an all pay auction. I will say an equilibrium of a highest bid mechanism is a “no-tie” equilibrium if in equilibrium there is a zero probability of a tie along the equilibrium path.

Deﬁnition 5. (No-tie equilibrium) ABayesianNashEquilibriumofahighestbidmechanismisa“no-tie”equilibriumifin equilibrium:

P (mi =maxmj mi > 0) = 0 i. j=i 6 | 8

Whether or not a mechanism has a “no-tie” equilibrium depends on the underlying distribution of preferences f and how the mechanism M structures payments. For a given distribution of preferences, there is an upper bound on the expected revenues in any “no-tie” equilibrium of a highest bid mechanism. The upper bound is independent

29 of the number of bidders and is strictly less than the revenue upper bound derived for any interim individually rational mechanism. This shows that any “no-tie” equilibrium of a highest bid mechanism will generate strictly less revenues than the probability demand mechanism when N is suﬃciently large.

Proposition 7. a Given a distribution of preferences f,thereexistsan↵>0 such that for any N,theexpected revenues from any no-tie Bayesian Nash Equilibrium of a highest bid mechanism are less than P ↵. A direct corollary of this result is that when N is suﬃciently large, expected revenues from the probability demand mechanism are greater than the expected revenues from any “no-tie” equilibrium of a highest bid mechanism.

Corollary 2. a Assume bidders play only undominated strategies. For a ﬁxed distribution of preferences f,thereexistsan such that for all N> ,expectedrevenuesoftheprobabilitydemand N N mechanism exceed the expected revenues of any no-tie Bayesian Nash Equilibrium of a highest bid mechanism.

The results of this section show that when there are sufficiently many bidders, an auctioneer generates higher revenue from the probability demand mechanism than from most commonly studied auction formats. These formats include first price, second price and all pay auctions. Additionally, this includes any of the three formats with reserve prices or entry fees. For the first price or second price auction, this holds by Corollary 1, and for the all pay this holds from Corollary 2 by noting that any Bayesian Nash Equilibrium of an all-pay auction is a no-tie equilibrium. In the next section, I use a numerical example to compare the expected revenues of the probability demand mechanism with the expected revenues of the first and second price auctions. In particular, I consider a setting where bidders are financially constrained. The example shows that in practice, the probability demand mechanism can generate revenues that exceed the revenues of the first or second price auction even when there are relatively few bidders.

7Anumericalexample

The results from the previous section show that expected revenues from the probability demand mechanism exceed the expected revenues from standard auction formats when there

30 are sufficiently many bidders. However, when considering implementing the probability demand mechanism, two natural questions emerge. First, how many bidders are needed for the probability demand mechanism to generate greater revenues than standard auction formats? And second, how much greater are the revenues from the probability demand mechanism than other auction formats? The answers to both questions depend on the assumed distribution of preferences. In this section, I consider a particular setting that is embedded in my model: a setting with financially constrained bidders. Each bidder must borrow money to finance her payments to the auctioneer and the more money she borrows, the higher an interest rate she must pay on her loan. This is a similar setting to that studied by Che and Gale (1998). I find that revenues from the probability demand mechanism exceed those from standard auction formats, even when there is a small number of bidders. In addition, the differences in revenues are non-negligible.

7.1 The example

Bidders are financially constrained and have private values for the good for sale. Each bidder has a valuation of the good vi, where vi [5, 15]. However, I depart from the quasilinear ⇠U environment by assuming that bidders are financially constrained. In order to make payments to the auctioneer, a bidder loans money from the bank. The interest rate paid on a loan of m dollars is r(m), where m r(m)= . 100 The cost of borrowing is marginally increasing in the amount borrowed and the total amount the bidder pays (with interest), mr(m),isconvexandcontinuouslyincreasinginm.A bidder’s payoffis given by:

ui(x, m)=viIx=1 m(1 + r(m)).

These utility functions satisfies Assumptions 2 and 1. That is, bidders have declining marginal utility of money and positive wealth effects. The financing constraint is the only departure from the quasilinear environment. For context, we can view the auctioneer selling ahouse,andthebiddersasprospectivebuyers.Ifpricesareinunitsof$10, 000 or $100, 000 this can be seen as a rough approximation such a situation. Ishowthatinthissetting,theprobabilitydemandmechanismhasexpectedrevenues that exceed the revenues of standard auction formats, even when there are few bidders. I

31 write a bidder’s probability demand curve using the methodology developed in section 5.1.

50 vi p 1 p p > 1

50 vi p 50 vi p qi(p)=8 1 >⇣ ⌘ > 0 . > p p p p <> ⇣ ⌘ ⇣ ⌘ 0 p>vi > > This is illustrated in Figure 7.1. :

q 1.0 0.8 qi p 0.6 0.4 0.2 H L p 9.0 9.5 10.0

Figure 7.1: Probability demand curve for bidder i when vi =10.

7.2 Revenue comparisons

Icomparetheexpectedrevenuesfromtheprobabilitydemandmechanismtotheexpected revenues of the ﬁrst and second price auctions. I assume bidders play undominated strategies and I use Proposition 4 to obtain a lower bound on bidders’ reports. Applying results of Che and Gale (1998) gives the equilibrium bidding function for the ﬁrst price auction,

5 N 1 bf (vi)=10 25 + + vi 50. r N N In the second price auction, it is a dominant strategy for a bidder to bid her willingness to s pay for the good. That is, bidder i bids b such that ui(1, bs)=ui(0, 0). This is given by

b (v )=10p25 + v 50. s i i

Figure 7.2 illustrates the revenue comparisons between the three formats using a Monte Carlo simulation.

32 E Rev 14

H L Ê Ê Ê Ê Ê 13 Ê Ï Ï Ï ÏÊ ÏÊ Ï Ï Ï Ï ÏÊ ‡ ‡ ‡ Ï Ï ‡ ‡ ‡ Ï Ê ‡ ‡ Ï Ê ‡ ‡ Ê ‡ Ï ‡ 12 Ê ‡ Ï Ê‡ Ê Probability Demand Mechanism 11 Ï Highest willingness to pay ‡ FPA / SPA Ï Ê‡

10 Ê N 0 5 10 15 20 25 30 35

Figure 7.2: Revenue comparisons between formats

The line marked with circles is the revenue lower bound for the probability demand mechanism. Note that the line marked with squares represents the revenues of both the first and second price auction. The results of Che and Gale (1998) show the first price has greater expected revenues than the second price auction when bidders face financing constraints. In this environment, the difference in expected revenues between first and second price auctions are relatively small when compared to the expected revenue difference between either format and the probability demand mechanism. When there are 4 or more bidders, the lower bound on expected revenues from the probability demand mechanism exceeds the expected revenues of both the first and second price auctions. The diamond marked line is the expected value of the highest bidder’s willingness to pay for the good. When there are 21 or more bidders, the lower bound on revenues from the probability demand mechanism will actually exceed any bidder’s willingness to pay for the (entire) good in expectation. The difference between the lower bound on expected revenues from the probability demand mechanism and the expected revenues from the first price auction is illustrated in Figure 7.3. The figure illustrates that the expected revenue difference between the probability demand mechanism is positive even when N is small. Also the difference in expected revenues between the two formats grows as the number of bidders increases. As the number of bidders increases, expected revenue from the probability demand mechanism approaches 15. This contrasts with the first and second price auctions. As the number of bidders increases, the expected revenues approach the highest possible willingness to pay of any bidder. Here, this is 13.24.

33 Ê Ê Ê 0.5 Ê Ê Ê 0.4 Ê Ê 0.3 Ê Ê Ê 0.2 Ê

Ê 0.1 Ê Ê Ê N Ê 5 10 15 20 25 30

Figure 7.3: Revenue diﬀerential between PDM and FPA

8Discussion

In this section, I will discuss the practical applicability of the probability demand mechanism. I will discuss four main points: the generality of the design setting, the ability to implement the mechanism in practice, the robustness of the results and the use of randomization. Setting -Unlikepriorworkintheliterature,Idonotmakespecificfunctionalform restrictions on bidder preferences. My results hold in a general setting that allows for multidimensional heterogeneity across bidders. It includes heterogeneity across bidders wealth effects, risk preferences, initial wealth levels and financing/budget constraints. All that I require is that bidders are risk averse and that the good being sold is a normal good. Practical implementation -Apractitionerlookingtoimplementtheprobabilitydemand mechanism may cite another concern: in order to implement the probability demand mechanism, a bidder must report a demand schedule for all non-negative prices. We cannot practically expect bidders to be able to report entire functions. I can avoid this problem by restricting bidders to submit a discrete number of bid points. Kastl (2010) actually shows that in similar divisible good auctions used for selling treasury bills, buyers simply submit step bid functions. That is, their bids are characterized by a finite number of bid points. In fact, bidders often submit fewer bid-points than the maximal number allowed. I can similarly discretize the probability demand mechanism. The discretetized probability demand mechanism mechanism would be analogous to a multi-unit Vickrey auction. In the simplest case, it is analogous to a multi-unit Vickrey auction for two indivisible goods, 1 where the two goods are each a lottery ticket that gives a 2 chance of winning. I can use

34 an analogous proof to show underreporting demand is a dominated strategy in the discretetized mechanism. With enough bidders, it has greater revenues than a first or second price auction. As I increase the number of bid points, the mechanism will also be approximately revenue maximizing when there are many bidders. Robustness - The mechanism design literature has previously been criticized for assuming too much common knowledge between the principal and agents. The Wilson doctrine (1987) has led to recent literature in the field that has advocated the use of mechanisms that are robust to higher-order beliefs (see Bergemann and Morris (2005)). My predictions regarding the probability demand mechanism are similarly robust. Proposition 4 shows that it is a dominated strategy for a bidder to underreport her demand curve. While the degree to which bidders overreport their demand depends on higher order beliefs, the lower bound on their report does not. I use this lower bound on bidders reports to obtain revenue comparisons. With many bidders, the probability demand mechanism is approximately revenue maximizing (Proposition 6). In fact, Corollary 1 shows that the with enough bidders the probability demand mechanism has higher revenues than a first or second price auction, as long as bidders play undominated strategies. These conclusions do not even assume that all bidders share a common prior - thus they are robust to bidders’ higher-order beliefs. Randomization -Iadvocateusingrandomizationtobetterexploitfeaturesofbidders preferences. Specifically, I show allowing for randomization can increase the auctioneer revenues relative to most commonly studied auctions that allocate the good to the highest bidder. The intuition for this is discussed in section 3.Peoplemayfindusingrandomization to sell goods - in particular, large goods like a house - to be odd. However, upon further inspection, it need not be the case. First, notice that the probability demand mechanism does allows a bidder to win the good with certainty should she report a sufficiently high demand curve. That is, bidders can win with certainty, though this may require more aggressive bidding. Inducing more aggressive bidding is exactly what a revenue-maximizing auctioneer seeks to do. Secondly, notice that all auctions expose bidders to risk. If a bidder participates in a first price auction, there is no bid that guarantees the bidder wins. If a bidder, truly can not afford to lose, she will simply bid more aggressively. The probability demand mechanism exposes bidders to risk differently to exploit features of their risk preferences. There are examples from the development and economic history literatures where randomization is used in settings that are similar to the one studied here. Development economists have extensively studied the use of Rotating Savings and Credit Associations (hereafter, rosca). Roscas are a savings device commonly used in the developing world. In arosca,agroupofpeopleallplacemoneyintoa‘pot’overthecourseofaspecifiedperiod.

35 After each period, a winner is randomly selected from the group and given the money in the pot. This process continues each period, until each person has won exactly once. Besley, Coate and Loury (1993) argue that roscas use randomization to increase welfare. The setting they study is very similar to the one studied here. They show that when the consumers demand an indivisible consumption good, using randomization devices like roscas is ex-ante welfare improving. In fact, they conclude their paper by emphasizing a point similar to Proposition 1 when they say, “we present an example in which an ex post efficient market allocation is dominated (under the ex-ante expected utility criterion) by a random Rosca.” In the economic history literature, DeMarchi (1995) shows that lotteries were successfully used in place of auctions to sell Dutch artwork during the 17th century. In fact, he states that “Prices in these lotteries were higher than those yielded by ordinary sales or derived from inventory valuations.” (pg. 217-8) DeMarchi also argues that these higher prices are evidence that lotteries created more demand. Both of these empirical observations are in line with the predictions of my model. First, lotteries achieve revenues that can exceed the revenues of standard auctions. In fact, as I show, the revenues can exceed any bidder’s willingness to pay for the good with certainty. Secondly, lotteries should increase demand for the good. Allowing lotteries would allow the art dealers to appeal to a more middle class audience. While this audience may value the art very highly, they may be unable to win an auction for a painting due to their lack of money and credit. A lottery provides these bidders with a chance to win the painting at a price that is more affordable. Results in general equilibrium also advocate using randomization to improve efficiency. Prescott and Townsend (1984) study a standard general equilibrium setting that includes models of moral hazard and adverse selection. They show we must allow for randomization in order to achieve Pareto efficiency. Just as I show here, they also state that optima may be inefficient in the ex-post full information sense. In my model, this means that Pareto efficiency need not imply that the bidder with the highest willingness to pay will own the good with certainty.

9Conclusion

In the auctions literature, it is standard to restrict bidders’ preferences to be quasilinear. The quasilinearity restriction allows for tractable analysis and provides specific prescriptions for auctioneers concerned with efficiency or maximizing revenue. However, there are many economic environments where the quasilinearity restriction is violated. This includes cases where bidders have risk aversion, wealth effects or face financing constraints.

36 In this paper, I show that there are qualitative differences in auction design problems without quasilinearity. Instead of using standard formats 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 efficiency, with the quasilinearity restriction, the dominant strategy equilibrium outcome of the second price auction is Pareto efficient. Without this restriction, I show that there are probabilistic allocations that can Pareto dominate the dominant strategy equilibrium outcome of the second price auction, and there is no mechanism that implements a Pareto efficient allocation in dominant strategies. For revenue maximization, I construct a probability demand mechanism that better exploits features of bidders’ risk preferences. The mechanism uses tools from the multiunit auctions literature - it sells the indivisible good like a perfectly divisible good that is in net supply one by selling bidders probabilities of winning. I show that with enough bidders, the probability demand mechanism has greater revenues than any standard auction. Ialsotakeanewapproachinstudyingtherevenuemaximizationproblem.Sincebidders’ types are multidimensional deriving an equilibrium and characterizing bidders’ interim incentive constraints proves intractable. Instead, I obtain a partial characterization of bidder behavior. In particular, I place bounds on what a bidder reports to the auctioneer. Using the bounds on bidders’ reports, I construct a lower bound on the expected revenues from the probability demand mechanism. With many bidders, the lower bound on revenues the probability demand mechanism exceeds an upper bound for revenues for a large class of standard auctions. This paper studies the canonical independent private values auction setting, though a natural extension would apply the methodology developed here to an interdependent values setting. The properties of bidder preferences that motivate the use of probabilistic allocations in the private values case remain present in the interdependent values case; thus, similar results should be expected. Similarly, we should expect analogous results to hold in amultiunitsetting,aswell.

37 10 Appendix

Proof of Proposition 2. The curve qi(p) is weakly decreasing. Recall that qi(p) is deﬁned as

qi(p)=argmaxV (q, qp). q [0,1] 2

Since V is continuous and increasing in both arguments, it follows that qi is weakly decreasing in p.

Next, I show that qi(p) is continuous in p.Fixp>ˆ 0 and let ql := limp pˆ+ qi(p) and ! qh =limp pˆ qi(p).Ifqi is continuous then ql = qh.Notethatsinceqi is monotone, both ! limits do exist and limp pˆ+ qi(p) limp pˆ qi(p). !  ! Let xw and xl be the eﬃcient payments when i wins with probability ql and pays pqˆ l in expectation. Similarly, let yw and yl be the eﬃcient payments when i wins with probability qh and pays pqˆ h in expectation. Since Vi is continuous in both arguments Vi(ql, pqˆ l)= 1 Vi(qh, pqˆ h)= (Vi(ql, pqˆ l)+Vi(qh, pqˆ h)). Rewriting the right hand side gives, 2 1 (q u (1,w x )+(1 q )u (0,w x )+q u (1,w y )+(1 q )u (0,w y )) , 2 l i i w l i i l h i i w h i i l

If xw = yw and/or xl = yl,thenJensen’sinequalityimplies 6 6

qlui(1,wi xw)+qhui(1,wi yw) ql + qh qlxw + qhyw < ui(1,wi ) 2 2 ql + qh and/or

(1 q )u (0,w x )+(1 q )u (0,w y ) 1 q +1 q (1 q )x +(1 q )y l i i l h i i l < l h u (0,w l l h l ). 2 2 i i 1 q +1 q ✓ ◆ l h q +q q x +q y (1 q )x +(1 q )y l h l w h w l l h l Let qm = 2 , zw = q +q and zl = 1 q +1 q .Notethatpqˆ m = qmzm +(1 qm)zl l h l h and 1 V (q , q pˆ) > (V (q , pqˆ )+V (q , pqˆ )) = V (q , pqˆ ). i m m 2 i l l i h h i l l

Yet ql =argmaxq [0,1] V (q, qpˆ). Since this contradicts, it then must be the case that both 2 xw = yw and xl = yl. This implies that qh = ql.

Finally I show that qi(ki + ✏) > 0 for some ✏>0. Given that I have shown that qi is continuous and weakly decreasing, it suﬃces to show that qi(ki) > 0.Recallthefunction k(w, u) deﬁnes the willingness to pay of an agent with preferences u and initial wealth w.

Here ki = k(wi,ui). For simplicity, I use the following notation: G(w):=ui(1,w) and

B(w):=ui(0,w).Sinceui(x, w) is strictly increasing in w,bothB and G are invertible.

38 1 Since G and B are continuously diﬀerentiable, ki is continuously diﬀerentiable. Solving 1 @k(w,ui) for k gives k(w, ui)=w G (B(w)).Bytheinversefunctiontheorem, is, @w

@k(w, ui) 1 0 =1 1 B (w). @w G(G (B(w))

@k(w,ui) 1 Positive wealth eﬀects imply @w > 0. Thus, G0(G (B(w)) >B0(w). Recall:

1 1 k(w, u )=w G (B(w)) = G (B(w)) = k(w, u ). i ) i

Thus, 1 G0(G (B(w)) >B0(w)= G0(w k(w, u )) >B0(w). ) i

At w = wi this implies that

@ @ u (1,w k ) > u (0,w). @w i i i @w i i

By deﬁnition 1 u (1,w k )=u (0,w)= (u (1,w k )+u (0,w)) . i i i i i 2 i i i i i For a small ✏>0 1 (u (1,w k + ✏)+u (0,w ✏)) 2 i i i i i 1 @ @ (u (1,w k )+u (0,w)) + ✏ u (1,w k ) u (0,w) . ⇡ 2 i i i i i @w i i i @w i i ✓ ◆ @ @ Since ui(1,wi ki) > ui(0,wi) it follows that @w @w 1 1 (u (1,w k + ✏)+u (0,w ✏)) > (u (1,w k )+u (0,w)) = u (0,w). 2 i i i i i 2 i i i i i i i

1 Note that if bidder i enters a gamble where she wins with probability 2 and pays ki ✏ 1 when winning and ✏ when losing, her utility is (ui(1,wi ki + ✏)+ui(0,wi ✏)).Her 2 1 ki maximal expected utility from any gamble where she wins with probability 2 and pays 2 in 1 ki expectation is Vi( , ). Thus, 2 2 1 k 1 V ( , i ) (u (1,w k + ✏)+u (0,w ✏)) >u(0,w)=V (0, 0). i 2 2 2 i i i i i i i i

Recall

qi(ki)=argmaxVi(q, qki). q [0,1] 2

39 1 ki Since Vi( , ) >Vi(0, 0), it follows that qi(ki) =0.Sinceqi(ki) [0, 1] we have qi(ki) > 0. 2 2 6 2 It follows that qi(ki + ✏) > 0 for some ✏>0 by continuity of qi. ⇤ Proof of Proposition 3. I need only consider a case with 2 bidders. It is without loss of generality to normalize each bidder to have an initial wealth of 0.3 Ionlyconsiderasubsetoftheincentiveconstraintsnecessaryfordominantstrategy implementation. I show that there is no individually rational mechanism that implements a Pareto efficient allocation and respects this limited set of incentive constraints. Thus, there is no individually rational mechanism that implements a Pareto efficient allocation and respects all possible incentive constraints. In particular I study the incentives of two different classes of types. First, there is acontinuumoftypesindexedbyv, where v R+. A bidder who is type v R+ has 2 2 preferences described by the utility function u, where

u(x, w, v)=vIx=1 + ✏f(w)+w, where ✏>0 is small. I assume that f is strictly increasing, twice continuously diﬀerentiable and f 00 < 0. A bidder who is type v R+ has preferences that are almost identical to a 2 bidder with quasilinear preferences and valuation v. Additionally, there is a single other type, b. A bidder who is type b has preferences described by the utility function u, where

u(x, w, b)=BIx=1 + f(w), where B>0 is ﬁxed. A bidder who is type b receives B>0 utils from being given the object and f(w) utils from having a ﬁnal wealth of w.

The space of possible preferences is T , where T = R+ b. I will restrict bidder i to have [ type ti T . M represents a direct revelation mechanism which maps reported types to a 2 distribution over feasible outcomes. Let ⇡i(t1,t2) be the probability i is given the good when reported types are t1,t2 R+ b.Similarly,xi(t1,t2) gives i’s expected transfers. I show 2 [ that the necessary conditions for M to implement a Pareto eﬃcient allocation in dominant strategies will yield a contradiction. The proof proceeds according to the following steps. First, I show that if bidders 1 and 2 both have almost quasilinear preferences, the outcome of M is essentially that of the second price auction. Next, I show that when both bidders are type b,thereexistsatypecalled

3This is without loss of generality. If a bidder has initial wealth w and preferences u,herpreferences over lotteries are equivalent to those of a bidder with initial wealth w and preferences uˆ, where uˆ(x, d)= u(x, w + d) d R,x 0, 1 . 8 2 2{ }

40 vEq R+,suchthatifeitherbidderinsteadreportedvEq they would win with the same 2 probability and pay the same amount. That is, it is equivalent to report type b or vEq when you opponent reports type b.IthenusethenecessaryconditionsforParetoefficiencytoshow vEq 1 that if both bidders are type b,theyeachpaytheauctioneer 2 and win with probability 2 . A bidder would make the same payment and win with the same probability, if she instead reported her type to be vEq. Finally, I look at the case where 1 is type vEq and 2 is type b. I show that the previously shown necessary conditions form a contradiction. In particular, I show that 1 has an incentive to deviate and report a lower type. But first, I will state three necessary conditions for Pareto efficiency.

The object is always allocated to a bidder. That is ⇡1(t1,t2)+⇡2(t1,t2)=1,forany • t1,t2 T . 2

Payments are eﬃcient. Thus, given t1,t2 T ,bidderi pays a constant transfer, • 2 whether or not she is given the object via Jensen’s inequality.

The mechanism M is then fully described by ⇡1(t1,t2), x1(t1,t2) and x2(t1,t2), where • t1,t2 T .IcanthenuseVi to represent the indirect utility of i who wins with 2 probability q and pays x with certainty.

qv + ✏f( x) x if ti = v R+ V (q, x, t )= 2 i 8 qb + f( x) if ti = B. < : For a ﬁxed level of payments to the auctioneer, X,Paretoeﬃciencyrequiresthattheallo- cation maximizes a weighted sum of expected utilities. That is, there are m1,m2 > 0 such that:

(⇡1(t1,t2),x1(t1,t2),x2(t1,t2)) arg max m1V (q, x1,t1)+m2V (1 q, X x1,t2). 2 q [0,1],x1,x2 2 The solution to this maximization problem ensures that if each agent receives a positive level of probability, each has an equal marginal willingness to pay for an additional unit of probability. A bidder’s marginal willingness to pay given that she wins with probability q and pays x with certainty is:

@ @qV (q, x, ti) G(q, x, ti):= . @ V (q, x, t ) @x i

41 The necessary conditions for Pareto eﬃciency from the maximization problem imply:

⇡ (t ,t )=1 = G(1, x (t ,t ),t ) G(0, 0,t ). 1 1 2 ) 1 1 2 1 2

⇡ (t ,t ) (0, 1) = G(q(t ,t ), x (t ,t ),t )=G(1 q(t ,t ), x (t ,t ),t ) 1 1 2 2 ) 1 2 1 1 2 1 1 2 2 1 2 2 ⇡ (t ,t )=0 = G(0, 0,t ) G(1, x (t ,t ),t ). 1 1 2 ) 1  2 1 2 2 With these necessary conditions, I now show that if bidders 1 and 2 both have almost quasilinear preferences, the outcome of M is essentially that of the second price auction.

Thus, assume both players report types v1,v2 R+. 2 @ V (q, x, ti) v @q G(q, x, ti)= @ = v. V (q, x, t ) 1+✏f 0( x) ⇡ @x i

If v1 >v2 then v G(1, x, v ) >G(0, 0,v ) v , i ⇡ i j ⇡ j for any x that does not violate i’s individual rationality condition. Thus, Pareto eﬃciency condition requires ⇡1(v1,v2)=1.

If ⇡1(v1,v2)=0,thenindividualrationalityrequiresthatx1(v1,v2)=0.Ifq(v1,v2)=1, individual rationality requires that x(v1,v2) v2, 1 has an incentive to deviate to ⇡ be a type that wins with probability 1 yet pays less than x1(v1,v2). If x1(v1,v2)

If v1 = v2,thenitgivesthatx1(v1,v2)=x2(v1,v2). Recalling that x1(v1,v2) v1⇡1(v1,v2), ⇡ then ⇡1(v1,v2) .5. ⇡ Next, I explore the case where at least one bidder reports type b.Ifbothbiddersreport type b,efficiencyrequiresx1(b, b)=x2(b, b).Ifinsteadbidder1 reports type v R+ and 2 2 reports type b,both⇡1(v, b) and x1(v, b) are weakly increasing in v by standard incentive arguments. When v is sufficiently large, Pareto efficiency requires ⇡1(v, b)=1, and when v is sufficiently small, ⇡1(v, b)=0. Additionally, there is a range where each bidder wins

42 with a strictly positive probability. The following lemma shows there is an equivalent type vEq R+ such that it is equivalent for 1 to report vEq or b when 2 reports b. 2 Lemma. a The necessary conditions for Pareto eﬃciency and dominant strategy incentive compatibility imply that there exist a vEq R+ such that x1(vEq,b)=x1(b, b) and ⇡1(vEq,b)=⇡1(b, b) and 2 both x1( ,b) and ⇡1( ,b) are continuous at vEq. · ·

Proof. Suppose bidder 1 reports type v R+ and 2 reports type b.Eﬃciency requires 2 B v v R+. f ( x (v, b)) ⇡ 8 2 0 2

Since f 0 is continuous, it follows that x2(v, b) is continuous in v. v 1 1 Call vEq the v such that x2(v, b)= . Recall that ⇡i(vEq,vEq) and xi(vEq,vEq) . 2 ⇡ 2 ⇡ 2 1 vEq 1 Inextshowthat⇡1(vEq,b)=⇡2(vEq,b)= 2 and x2(vEq,b)= 2 . If ⇡2(vEq,b) > 2 , then when both players are type vEq,twohasanincentivetoreporttypeB: she wins with 1 higher probability and pays the same price. Similarly if ⇡2(vEq,b) < 2 and 1 is type vEq and 2 is type B,then2 has an incentive to deviate and report herself to be type vEq. Thus, 1 vEq vEq ⇡2(vEq,b)= and x2(vEq,b)= . Individual rationality requires x1(vEq,b) .Similarly 2 2  2 1 vEq ⇡1(b, vEq)= 2 and x1(b, vEq)= 2 . Suppose ⇡1(b, b) >.5.Ifbothplayersaretypeb eﬃciency requires that,

B B = . f ( x (b, b)) f ( x (b, b)) 0 1 0 2

vEq Thus, x1(b, b)=x2(b, b). If x1(b, b)=x2(b, b) 2 ,thenplayer2 has an incentive to deviate 1 and report type vEq. She will then win with higher probability ( )andpayless(x2(b, vEq) 2  vEq vEq 2 ). If x1(b, b)=x2(b, b) < 2 . Then it must be the case that x1(vEq,b)

Note x1(vEq,b)=x2(b, vEq) as shown above. Since x1(vEq,b) x2(b, vEq).Ifweassumed,⇡1(b, b) >.5,then,⇡2(b, b) <.5.Ifthisisso,thenifbothplayers are type b,playertwoshouldreporttypevEq.Bydoingso,sheincreasesherprobabilityof winning to .5 and pays less, since x2(b, vEq)

The symmetric argument shows we can not have ⇡2(b, b) >.5. Thus, ⇡1(b, b)=.5.If player 2 is type b,thensince⇡1(vEq,b)=⇡1(b, b)=.5,theincentiveconstraintsoftypevEq and b require that x1(b, b)=x1(vEq,b).

vEq Next, I will show that this implies that x1(b, b)=x1(vEq,b)= 2 .Bythelemmawe

43 know that x1(b, b)=x1(vEq,b).Noticethatabidderoftypeb is indiﬀerent between reporting b and vEq when her opponent reports type b.Also,typeb’s marginal willingness to pay for an additional unit of probability is exactly vEq. That is G(.5, x1(b, b),b)=vEq. This is because standard incentive constraint arguments require that for v, v0 R+, where v

⇡1(v0,b) ⇡1(v, b) v<

If G(.5, x1(b, b),b) >vEq,thenbidder1 should deviate and report a v>vEq.Shewould acquire more probability at a price per unit that is below her willingness to pay for a marginal unit. A similar argument holds if G(.5, x1(b, b),b)

B 1 v = = G(q, v ,b). Eq f ( 1 v ) 2 Eq 0 2 Eq 1 Combining these two conditions then gives that G(.5, x1(b, b),b)=G(.5, 2 vEq,b).Note 1 that G(q, x, b) is constant in q and strictly decreasing in x. Thus, x1(b, b)= 2 vEq = 1 x1(vEq,b).Recallalsothat⇡1(b, b)=⇡1(vEq,b)= 2 .Notethatif1 is type vEq and 2 is type 1 1 b,then1 expected utility from truthfully reporting is V ( , vEq,vEq) 0. 2 2 ⇡ By continuity, there exists a v 0,itisthecase that x1(v, b) v⇡1(v, b).If1 is type vEq and 2 is type b,onecanobtainaproﬁtable  deviation by reporting type v instead of type vEq. Thus, M is not dominant strategy incentive compatible.⇤

Proof of Proposition 4. I prove this by showing that if stating the demand curve q˜i instead of qi does change bidder i’s payoff, she receives a lower payoff. Thus, I need to only focus on cases where her payoffis changed by reporting q˜i instead of qi.Iformalizethe intuition discussed in the text. First consider a case where i faces a perfectly elastic residual probability demand curve (i.e. a constant marginal price per unit of probability). It is a best response for her to truthfully reveal her demand curve. Let pE > 0 be the constant marginal price for units of probability. If her payoffis changed by reporting q˜i,thenq˜i(pE) < qi(pE).Bythedefinition of qi,thenqi(pE)=qi(pE). That is, if reporting q˜i does change her payoff, it is the case that, q˜i is strictly below her demand for probability at pE.Bytheconstructionoftheprobability demand curve, truthful reporting is a best response to a perfectly elastic supply curve. Thus, she decreases her payoffby reporting type q˜i. Recalling that Vi is her indirect utility function

44 under eﬃcient payments, it follows that

Vi(qi(pE), pEqi(pE)) Vi(q, pEq) for any q (0, 1). 2

Now consider instead that bidder i faces a more inelastic (relatively to perfectly elastic) residual probability demand curve. Assume that the supply curve is such that Si(pE)= qi(pE). Once again if her payoﬀis changed by reporting q˜i,thenq˜i(pE) < qi(pE)=qi(pE), using the same argument as before. Thus, she wins with a lower probability by reporting q˜.Assumethatifshereportsq˜i,shepaysX˜ in expectation and wins with probability q˜(˜p), where q˜i(˜p)=Si(˜p).SincehermarginalpriceisstrictlybelowpE, X

V (q (p ), X) V (˜q (˜p), X˜). i i E i i

Since X X˜ + pE (qi(pE) q˜i(˜p⇤)),itissuﬃcienttoshow 

V q (p ), X˜ + p (q (p ) q˜ (˜p⇤)) V (˜q (˜p), X˜). i i E E i E i i i ⇣ ⇣ ⌘⌘

Recall X˜ 0. Thus, the above expression can be rewritten as:

V (q (p ), p q (p )+Y ) V (˜q (˜p), p q˜ (˜p⇤)+Y ). i i E E i E i i E i I have already shown that

V (q (p ), p q (p )) V (˜q (˜p), p q˜ (˜p⇤)), i i E E i E i i E i where qi(pE) > q˜i(˜p). Notice also that pEqi(pE)+Y> pEq˜i(˜p⇤)+Y . Thus it suﬃces to show that @ @ V (q (p ), p q (p )+Y ) V (˜q (˜p), p q˜ (˜p⇤)+Y ), @Y i i E E i E @Y i i E i for any Y>0. Idothisbyusingtheenvelopetheorem.Recallthattheindirectutility function is the maximal expected utility conditional on winning with probability q and paying x in expectation. Thus the marginal (expected) utility from an increase in Y equals

@ @ V (q, X + Y )= u (0,w x ). @Y i @w i i l

45 where xl is the losing payment in the eﬃcient payment scheme where the bidder wins with probability q and pays X + Y in expectation. Note that for any Y>0,undereﬃcient payments, bidder i pays less when losing by reporting q˜i instead of qi. This is because she pays a greater amount in the latter and wins with higher probability. Recalling that @ @ Vi(q, X + Y )= ui(0,wi xl),theconcavityofanagent’sutilityfunctionthenimplies @Y @w @ @ V (q (p ), p q (p )+Y ) V (˜q (˜p), p q˜ (˜p⇤)+Y ), Y>0. @Y i i E E i E @Y i i E i 8

This is what we wanted to show. ⇤

Proof of Remark 1. Consider a bidder i with type ti supp(f). Consider a gamble 2 where i wins the good with probability q and pays x in expectation. If x>qP ,thenthe gamble violates the bidder’s individual rationality constraint. Note that qi(p)=0for any p>P by deﬁnition. Thus,

V (0, 0) V (q, qP ) >V(q, x). i i i

Since Vi(q, x) is i’s maximal expected utility from any gamble of the form, win with probability q and pay x in expectation, it follows that the gamble necessarily violates her individual rationality when x>qP .

Now consider a mechanism .LetQi(ti,t i) represent the probability i is given the object given she reports type ti and all other bidders report types t i.SimilarlyXi(ti,t i) is i’s expected payment given all other bidders report types t i.LetQi(ti) be i’s interim probability of winning. Thus, i(ti)= Qi(ti,t i)f(ti,t i)dt i.Ianalogouslydeﬁne Q t i ´ Xi(ti). Interim individual rationality requires P Qi(ti) Xi(ti).  Thus the expected revenues from bidder i ex-ante are:

Xi(t)f(t)dt P Qi(t)f(t)dt. ˆt T  ˆt T 2 2 Feasibility requires the object is allocated to any bidder with probability less than or equal to 1,ex-ante. N Qi(t)f(t)dt 1. ˆ  i=1 t T X 2

46 Ex-ante expected revenues are:

N N Xi(t)f(t)dt P Qi(t)f(t)dt. P. ˆ  ˆ  ⇤ i=1 t T i=1 t T X 2 X 2

Proof of Proposition 6. Iassumeallbidderstruthfullyreporttheirprobability demand curves. Since it is a dominated strategy for bidders to underreport demand, this can serve as a lower bound on bidder’s reports. This proof shows that with many bidders, the marginal price of a unit of probability is near P with a high probability. Thus, the one unit of probability is sold at a price per unit that approaches P .

Iusethenotation,q(p, ti) to represent the probability demand curve for a bidder with type ti.Recall, P := sup p(t): t supp(f) . { 2 } Fix ✏>0.Let⌧(✏,):= t : q(P ✏, t) >, t supp(f) . This is the set of all types that { 2 } demand at least a probability of winning at price P ✏.BythedeﬁnitionofP ,theset ⌧(✏,) is non-empty when >0 is suﬃciently small. Note that: N N qi(P ✏, ti) I . ti v(✏,) i=1 1=i 2 X X This states that the total demand at price P ✏ is greater than the times the number of bidders whose demand strictly exceeds at P . N Suppose that > 2. It follows that Si(P ✏)=0for all i. That is, if 1=i Iti v(✏, 2 ) at price P ✏, all bidders demands for probabilities of winning exceed 2,thenthereiszero P residual demand for each bidder at price P ✏. Thus, if i wins with positive probability, she pays a marginal price per unit that exceeds P ✏. Thus, Xi qi(p⇤) P ✏ for all i.Since qi(p⇤)=1, this implies that total expected transfers exceed P ✏:

P N X > P ✏. i i=1 X When >0 is suﬃciently small, there is a strictly positive probability a randomly drawn bidder has type t ⌧(✏,). That is, there is a positive probability that a randomly drawn 2 bidder demands at least a probability of winning the good when she pays a price of P ✏.

47 Let ⌫(✏,) represent this probability:

⌫(✏,)= f(t)dt. ˆt ⌧(✏,) 2 Suppose >0 is such that ⌫(✏,) > 0. Then, by the law of large numbers, as N !1 the probability that the sum N exceeds 2 approaches 1. The intuition is 1=i Iti v(✏, 2 ) straightforward. There is a ⌫(✏,P) > 0 chance a randomly drawn bidder i has type ti such 2 that qi(P ✏, ti) >. Thus, with enough bidders, the probability that there are at least such bidders approaches 1. Thus, for any ↵ (0, 1),thereisafiniteN(↵) such that there is a 1 ↵ probability that 2 N > 2.Since↵ and ✏ are arbitrary, let them be arbitrarily close to 0 when 1=i Iti v(✏, 2 ) N is sufficiently large. Thus with a sufficiently large N,totalpaymentsexceedP ✏ with P probability 1 ↵, where both ↵ and ✏ are arbitrarily small. This yields the desired result. ⇤

Proof of Proposition 7. First I consider a bidder’s preferences over gambles. First, I will place a bound on how much a bidder is willing to pay for a gamble where she wins the good with an arbitrary probability q. I will then use this bound on a bidder’s willingness to pay for a gamble to obtain a bound on the revenue from any highest bid wins mechanism.

Recall that Vi(q, x) represents the maximal expected utility of bidder i from any gamble where she wins the good with probability q and pays x in expectation. Let V (q, x, ti) be the indirect utility function of type ti T .NotethatV is continuously increasing in q (the 2 probability i wins) and x (her expected payment). IdeﬁnethefunctionR(q, ti) as

R(q, t ): V (q, R(q, t ),t)=V (0, 0,t). i i i i

That is, R(q, ti) is the most that bidder i will pay (in expectation) for a gamble where she wins the good with probability q. Any gamble where i wins with probability q and pays greater than R(q, ti) in expectation will violate her individual rationality constraint. Since V is continuous in q and x, it follows that R is continuous in q.

First I show that R(q, ti) is strictly concave in q.ToshowthatR is strictly concave, I show that if qm = ↵qL +(1 ↵)qH , where ↵, qL,qH (0, 1),then 2

R(q ,t) >↵R(q ,t)+(1 ↵)R(q ,t). m i L i H i

48 There exists payments pL,w and pL,l where R(qL,ti)=qLpL,w +(1 qL)pL,l and

V (0, 0,t)=q u(1,w p ,t)+(1 q )u(0,w p ,t). i L i L,w i L i L,l i

Similarly there are analogous payments pH,w and pH,l where R(qH ,ti)=qH pH,w +(1 qH )pH,l and

V (0, 0,t)=q u(1,w p ,t)+(1 q )u(0,w p ,t). i H i H,w i H i H,l i Thus, V (0, 0,t)=↵V (q , R(q ,t),t)+(1 ↵)V (q , R(q ,t),t) i L L i i H H i i where V (q , R(q ,t),t)=q u(1,w p ,t)+(1 q )u(0,w p ,t) L L i i L i L,w i L i L,l i and V (q , R(q ,t),t)=q u(1,w p ,t)+(1 q )u(0,w p ,t). H H i i H i H,w i H i H,l i

Since both u(x, w, ti) is strictly concave in w for x =0, 1,then q u (1,w (↵p +(1 ↵)p ) ,t)+(1 q )u (0,w (↵p +(1 ↵)p ) ,t) >V(0, 0,t). m i i L,w H,w i m i i L,l H,l i i

Thus, V (qm, (↵R(qL,ti)+(1 ↵)R(qH ,ti)) ,ti) >V(0, 0,ti). This holds, because the above inequality shows there is a gamble where i wins with probability qm and pays ↵R(qL,ti)+

(1 ↵)R(qH ,ti) expectation, and has expected utility that exceeds V (0, 0,ti). Recall that V (q, x, ti) is strictly decreasing in x and I deﬁne the function R(q, ti) as

R(q, t ): V (q, R(q, t ),t)=V (0, 0,t). i i i i

Thus, R(qm,ti) >↵R(qL,ti)+(1 ↵)R(qH ,ti), which shows that R is strictly concave in q. In addition, R(0,ti)=0(i.e. no bidder is willing to pay any money for a zero probability of R(q,ti) winning). Thus, the function q is strictly decreasing in q over (0, 1). I will deﬁne the function R(q)=maxt T R(q, t). This is the most any bidder is willing 2 R(q,t) to pay for a q probability of winning the good (in expectation). Since q is decreasing R(q) in q and preferences are continuous in t, it follows that q is strictly decreasing in q.In R(q) R(q) addition, q P . Thus, q [0,1] q < P .  2 Recall that any bidder´ where i wins with probability q and pays X>qP in expectation R(q,ti) gives an expected payoﬀbelow V (0, 0,ti). Thus, qP R(q, ti). Since is strictly q decreasing in q over (0, 1),thenP>R(q,ti) for all q (0, 1). q 2

49 Inowusethisresulttoplacearevenueupperboundonanyno-tieBayesNashequilibrium of a highest bid mechanism.

Let i(ti) represent i0s strategy in a highest bid mechanism. That is, i : T (R+).Let ! ⇤ denote a no-tie Bayes Nash equilibrium of this highest bid mechanism, ⇤ = ⇤ ...⇤ . { 1 N } Given ⇤, there is a distribution of winning bid. Let i(b) denote the cumulative distribution F function of i’s bids from the ex-ante point of view, given that she plays strategy i⇤.

If i submits a bid of b she wins with probability j=i j(b).LetXi(b) be the expected 6 F payment of i if she bids b. The above result shows that Xi(b) R( j=i j(b)). Q  6 F Let G denote the ex-ante distribution of the highest bid submitted.Q Note that 1

N G(b)= (b). Fi i=1 Y By the no-tie restriction, G is continuous and increasing. Thus, it is diﬀerentiable almost everywhere and feasibility requires the object is never given away with probability that exceeds 1. That is,

G0(b)db 1, ˆ b R+  2 where N G0(b)= 0(b) (b) . Fi Fj i=1 j=i ! X Y6 Let bc be such that G(bc)=.5. That is, there is a .5 probability that the winning bid is below .5.Deﬁnethefunctionwi(b)= j=i j(b). This is the probability i wins with bid b. 6 R(q) F Note that wi(bc) .5 for all i.Since is strictly decreasing and bounded above by P , Qq then there exists a >0 where R(q) < P for all q .5. Thus, q

R(wi(b)) < P if b bc. wi(b)

This means that if you submit a bid where you win with probability .5 or greater, you must pay a price per unit of probability that is strictly below P . The expected transfers from i is,

R(wi(b)) Xi(b) i0(b)db j(b) i0(b)db. ˆ F  ˆ wi(b) F F b R+ b R+ j=i ! 2 2 Y6

50 Total transfers are then,

N N R(wi(b)) Xi(b) i0(b)db j(b) i0(b)db. ˆ F  ˆ wi(b) F F i=1 b R+ b R+ i=1 j=i ! X 2 2 X Y6

Since R(q) P and R(q) P when q>.5, I can then write q  q 

N N

Xi(b) i0(b)db j(b) i0(b) P Ib

N Note that G0(b)= i=1 j=i j(b) i0(b) and G(bc)=.5. Thus, 6 F F P ⇣Q ⌘ N 1 X (b) 0(b)db P . ˆ i Fi  2 i=1 b R+ X 2 Since >0 is ﬁxed for all N, we have that revenues from a highest bid wins mechanism are bound by P 1 . 2 ⇤

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