Bid Behavior in the Uniform Price and Vickrey Auctions on a General Preference Domain

Brian Baisa⇤†‡ May 2014

Abstract Vickrey auctions are widely praised by economic theorists, yet rarely used in practice. I address this discrepancy by comparing bid behavior in the Vickrey auction with the more commonly used uniform price auction. I study the case where bidders have private values and multiunit demands, without the usual quasilinearity restriction on bidder preferences. I study a more general preference domain that nests quasilinearity, but also allows for budget constraints, financial constraints, risk aversion, and/or wealth effects. I show that truth-telling is not a dominant strategy in the Vickrey auction. Instead bidders truthfully report demand for their first unit and overstate demands for all other units. This result mirrors the incentive for demand reduction in uniform price auctions shown by Ausubel and Cramton (2002) and Ausubel et. al. (2014). While both auctions are generally inefficient, I show that when the auction is large, both give approximately equal allocations and revenues, and both are approximately ex-post efficient.

1Introduction

“Despite the enthusiasm that the Vickrey mechanism and its extensions generate among economists, practical applications of Vickrey’s design are rare at best.... The most novel version of Vickrey’s design which applies to sales in which diﬀerent bidders may want multiple units of homogenous goods - or packages of heterogenous goods - remain largely unused.” (Ausubel and Milgrom, 2006)

Vickrey auctions possess desirable properties sought out by market designers; namely, strat- egyproofness and efficiency. In fact, the results of Holmström (1979) and Williams (1999) show that any dominant strategy mechanism that does not run a deficit, gives losers zero payoffs and implements an efficient allocation is a Vickrey auction. A natural question then

⇤Amherst College, Department of Economics. †This is a ﬁrst draft. Any comments are welcome. ‡I thank the seminar audiences at The University of Michigan, the Stony Brook Game Theory Festival, IFORS-Barcelona and Tilman Borgers, Jun Ishii, Dan Jaqua, Natalia Lazzati, David Miller for helpful comments related to this project.

1 follows from the above quote: why are Vickrey auction so rare, especially in multiunit set- tings? In their paper, Ausubel and Milgrom describe four weakness of the Vickrey auction that limit its applicability. However, their critiques apply only to cases where the objects being sold are heterogenous goods and at least one buyer views the goods as complements. Their critiques do not apply to the sale of homogenous goods when bidders have (weakly) declining demands. This leaves half of their original question unanswered: why are Vickrey auctions rarely used to sell multiple homogenous goods? In this paper, I compare the Vickrey auction with the more frequently used uniform price auction. I show that when we consider amoregeneralpreferencedomain,bidbehaviorinVickreyauctionsmirrorsbidbehaviorin the uniform price auction. Indeed, many often cited deficiencies of uniform price auctions also apply to Vickrey auctions. I assume only that bidders have weakly positive wealth effects. Thus, the goods being sold are normal. My specification allows for multidimensional heterogeneity in bidder demands, risk preferences, budgets, financing constraints, etc. It does not make functional form restrictions on bidder preferences and it nests the benchmark quasilinear environment. Without the quasilinearity restriction, truthful reporting is not a dominant strategy. Instead, bidders have an incentive to truthfully report their demand for their first unit and overreport their demand for all other units. This incentive to overreport demands leads to inefficient outcomes. This mirrors Ausubel and Cramton’s (2002) noted results on demand reduction and inefficiency in uniform price auctions. In fact, I show there are cases where the uniform price auction is efficient and the Vickrey auction is not. Thus, there is no clear efficiency ranking between the two formats. The overbidding result yields an interesting testable implication. A greater number of bidders will win at least one object in the uniform price auction. In other words, a bidder has a higher chance of walking away empty-handed in the Vickrey auction versus the uniform price auction. However, bidders who do win objects in the Vickrey auction win a greater number of objects than they would in the uniform price auction. While Ausubel and Cramton illustrate the inefficiency of the uniform price auction, the large auctions literature shows that these inefficiencies vanish when there are many bidders. Swinkles (2001) and Jackson and Kremer (2006) show that uniform price auctions are approximately efficient with many bidders. I show analogous results for uniform price and Vickrey auctions without the quasilinearity restriction. In particular, I show that with many bidders, truthfully reporting demand is an ✏ best reply to any undominated strategies of other bidders. I use this to show that many bidders, both auctions give approximately equal revenues and allocations, and are approximately ex-post efficient. This result does not require that bidder types are independent or single dimensional as is typical in the large auctions literature. Instead, I place few restrictions on the distribution of types and allow for cases where bidder types are multidimensional and correlated. With these results we can revisit the above quote. When we restrict ourselves to studying quasilinear environments, the Vickrey auction implements an efficient allocation in dominant strategies. When we expand our analysis to a more general preference domain, this result disappears. Bidders overreport their demand in the Vickrey auction much like they underreport their demand in the uniform price auction. In a large auction setting, these inefficiencies disappear and the two auctions produce nearly identical allocations that are approximately efficient. Indeed, without quasilinearity, it is unclear whether the Vickrey auction has any

2 distinct advantages over the uniform price auction. The rest of the paper proceeds as follows. The remainder of the introduction relates my work to the auctions literature. Section 2 describes my model and provides a brief description of the uniform price auction and the Vickrey auction. Section 3 proves results on bid behavior in both auctions. Section 4 compares both auctions in a setting with many buyers and many units. Section 5 concludes.

Related Literature This paper contributes to the literature on auction where bidders have non-linear preferences and multiunit auctions. The paper considers the multiunit extension of Vickrey’s (1961) original design. The properties of the Vickrey auction have been widely studied since his original paper. A dy- namic version of the Vickrey auction, the clinching auction, was later suggested by Ausubel (2004). The clinching auction provides a simpler implementation of Vickrey’s design and can be seen as the multiunit analog of the well know English auction. In spite of this theoretical progress, we rarely see either mechanism used in practice. While much of this prior literature about auctions restricts attention to the case where bidder preferences are quasilinear, there are papers that have studied issues related to auction design without quasilinear preferences. In the single unit environment Matthews (1983) and Che and Gale (2006) show that risk aversion explains experimental finding that first price auctions have higher revenues than second price auctions. Maskin and Riley (1984) and Baisa (2014) study the auction design problem when bidders do not have quasilinear preferences. Of this prior work, only Che and Gale (2006) and Baisa (2014) allow multidimensional heterogeneity across risk preferences and wealth effects like the setting studied here. In the multiunit auctions literature, most work that studies bidders with non-quasilinear preferences looks at the case where bidders have hard budget constraints. Recently, Dobzin- ski, Lavi and Nisan (2012) showed that when bidders have private budgets, there is no dominant strategy mechanism that implements a Pareto efficient allocation and respects incentive compatibility when transfers are non-positive. In a related paper Hafalir, Ravi and Sayedi (2012) study Vickrey auctions for divisible goods when bidders have budgets. They show a result that is similar in spirit to the one presented here. In particular, they show that bidders never underreport their budgets. The budget constrained case is nested in the model I study here. When bidders have budget constraints, they have positive wealth effects. This paper shows these results can be extended to a larger preference domain using only the normal good assumption. Morimoto and Serizawa (2013) also study efficiency in multiunit auctions when bidders have non-quasilinear preferences, but in a setting where bidders have single unit demands. While much of the multiunit auctions literature shows that commonly used auctions are inefficient, the large auctions literature shows these inefficiencies become are negligible in when there are many bidders and many objects. In particular, Swinkles (1999, 2001) shows that discriminatory and uniform price auctions are approximately efficient with many bidders when types are independently distributed. Jackson and Kremer (2006) show that when bidder types are conditionally independent, uniform price auctions are approximately efficient, but discriminatory auctions are not.

3 This paper contributes to this literature in two ways. First, I relax the quasilinearity restriction on bidder preferences, and instead study a setting that nests quasilinearity, but also allows for multidimensional heterogeneity across bidder risk preferences and wealth eﬀects. Second, I do not assume that bidder types are independent or conditionally independent. Instead, I generalize Swinkles’ (1999) no asymptotic gaps condition to a setting that allows multidimensional heterogeneity and correlated types. I show that in this setting, truth-telling is an approximate best reply to any undominated strategy when there are many bidders. That is, both auctions are strategy proof in the large in the language of Azevedo and Budish (2013).

2Model

2.1 Bidder Preferences There is a seller with k indivisible homogenous goods. There are N bidders. A bidder i is described by her initial wealth wi R and her preferences ui where, 2 ui : 0, 1,...k R R. { }⇥ ! That is ui(x, w) is the utility of bidder i when she owns x objects and has wealth w.Iassume ui(x, ) is strictly increasing and continuous for any x =0, 1,...k.Inaddition,theobjects · being sold at auction are goods. Thus, ui(x, w) >ui(y, w) if and only if x>y.Imakeonly two additional assumption on bidder preferences. Assumption 1 states that if bidder i is unwilling to pay p for her xth object, then she is unwilling to pay p for her x+1st object. This ensures bidders have downward sloping (inverse) demand curves for the goods being auctioned and generalizes the declining marginal values assumption imposed in the benchmark quasilinear setting.

Assumption 1. (Weakly declining demand) If x 1,...,k 1 and 2{ } ui(x 1,w) ui(x, w p), then ui(x, w) ui(x +1,w p). Assumption 2 states that bidders have weakly positive wealth eﬀects. It can be explained easily in words. Suppose that bidder i was faced with the choice between having x goods for a total price of X or y goods at price of Y , where we assume x>y.Ifbidderi weakly prefers the option with more goods, then she will also weakly prefers the option with more goods if her wealth increases.

Assumption 2. (Weakly positive wealth effects) Suppose that bidder i has wealth w and x>ywhere x, y 0, 1,...k .Bidderi has weakly 2{ } positive wealth effects if i i i i u (x, w X) u (y, w Y )= u (x, w0 X) u (y, w0 Y ) w0 >w. ) 8 Definition 1. Ilet be the set of all utility functions that satisfy Assumptions 1 and 2. U

4 In the Vickrey and uniform price auctions, bidders compete by reporting demand curves. That is they submit k dimensional bids that represent their demands. I deﬁne a bidders (inverse) demand curve in the standard way. Deﬁnition 2. (Inverse Demand Curve) Bidder i with preferences ui and initial wealth wi has an (inverse) demand curve pi, where

i i i i i p (m):=max p : u (m, w pm) u (m0,w pm0) m0 0, 1,...,k . { }8 2{ } In words, pi(m) is the highest price at which bidder i demands m objects. Note that if bidder i has initial wealth wi R and preferences ui ,thenpi(m) is weakly decreasing in m and >pi(m) > 0 m 21,...,k . 2U 1 8 2{ } For comparisons, in the quasilinear benchmark bidder i’s preferences are described by i k i i i i avectorofhermarginalvaluationsv R+, where v1 v2 ... v . The quantity v 2 k h represents her marginal valuation for her hth object. Her utility function is then

x i i u (x, w)= vj + w. j=1 X The utility function fits assumptions 1 and 2 and thus, ui where pi(m)=vi . 2U m Idonotplaceanyotherrestrictionsonabidder’spreferencesbeyondassumptions1 and 2. Thus, my setting nests quasilinearity, but also allows for multidimensional heterogeneity across bidders, including cases where bidders have wealth effects, risk aversion and financial constraints. Budgets can be modeled as a limiting case where a bidder gets extremely high disutility of spending money beyond a certain threshold. Below I show two examples of preferences included in this framework. In the first example, bidders are financially constrained. In order to finance their expenditures, they must borrow money. They face an increasing interest rate on additional borrowing. Example 1. (Financially constrained bidders) i th Bidder i receives vm utils for her m marginal object. Her utility of wealth is additively separable. She finances her payments by selling bonds. She pays a higher interest rate as she lends more money. Her preferences are described by ui, where

x i i i u (x, w)= vj + f (w), j=1 ! X where f i is concave and strictly increasing. In the second example, the goods are shares of a risky asset. Tomorrow share price is a draw of a random variable with a commonly known distribution of returns. Bidders are risk averse and display decreasing absolute risk aversion. Thus, as bidders become wealthier, they become more risk tolerant and demand more shares of the risky asset. Example 2. (Risky Assets) Bidder i is risk averse and maximizes her expected utility over ﬁnal wealth levels. She receives utility gi(w) from wealth w, where gi is increasing and displays decreasing absolute

5 risk aversion. She is considering buying shares of a risky asset. A share of the asset will be worth s tomorrow, where s is the draw of a random variable with density f.Her(expected) utility from owning x shares and having wealth w is,

ui(x, w)= gi(w + xs)f(s)ds. ˆs R 2 An auction mechanism maps bids to feasible outcomes. A (feasible) outcome of the auction describes each bidders’ allocation and transfers. The allocation is described by x 2 0, 1,...,m n where xi is the number of objects won by bidder i.IdeﬁneX as the set of all { } n x 0, 1,...,m n such that xi k.Transfersaredescribedbyt Rn, where ti is the 2{ } i  2 transfer made by bidder i. P In what follows I will study bid behavior in the uniform price and Vickrey auctions. My description of the two auctions, and my notation are nearly identical to Krishna (2002).

2.2 The Uniform Price Auction In the uniform price auction, each bidder submits a k dimensional bid. The highest k bids (out of the total the k N submitted bids) win. Each good is sold at the same market price ⇥ that is set equal to the highest losing bid. I let represent the space of all feasible bids. It k k i i i is a subset of R+, where := b R+ >b1 b2 ...bk . IletxU (b ,b ) be the number { 2 |1 } i i N of objects bidder i wins in the uniform price auction given bids (b ,b ) .Similarly,I i i i 2 let tU (b ,b ) be the payment made by bidder i. i i Idenotethecollectionofbidsfromallbiddersexcepti as b .Letc denote the vector of these competing bids ordered from highest to lowest. Bidder i wins exactly m 0, 1,...k 2{ } objects, if she submits m bids that rank in the top k. i i i i bm >ck+1 m and ck m >bm+1. For simplicity, I assume that all ties are broken in favor of the higher numbered bidder. i i If bidder i wins m objects, she pays mpU , where pU =max bm+1,ck+1 m is the value of { } the highest losing bid.

Price i -i b c k-m uniform payment

Quantity

Figure 1: Payment in Uniform Price Auction(CORRECT DIAGRAM)

6 2.3 The Vickrey Auction The Vickrey auction has the same winning rule as the uniform price auction, however, the payment rule is diﬀerent. The price a bidder pays for objects is determined by other bidders reported demands. Speciﬁcally, a bidder’s payment is determined by a marginal price curve which is a residual demand curve formed by other bidders’ reported demand curves. Ikeepthesamenotationasabove.Bidderi reports a k dimensional bid bi .I i i i 2 let xV (b ,b ) be the number of objects bidder i wins in the Vickrey auction given bids i i N i i i (b ,b ) .Similarly,IlettV (b ,b ) be the payment made by bidder i. 2 m i If bidder i wins m objects, she pays j=1 ck+1 j. That is, bidder i faces an upward sloping i marginal price curve because ck+1 j is increasing in j. The marginal price of acquiring the P mth object is the k +1 mth highest bid made by i’s competitors.

Price i -i b c k-m

Vickrey payment

Quantity

Figure 2: Payment in Vickrey auction

3BidbehaviorandIneﬃciencies

3.1 In the uniform price auction Ausubel and Cramton (2002) show that when bidder preferences are quasilinear, there is an incentive to underreport demand in the uniform price auction. However, bidders will always truthfully report their demand for their ﬁrst unit of the good. They explain the intuition behind their result as follows: “when a bidder desires many units of the good, there is a positive probability that her bid on a second or later unit will be pivotal, thus determining the price the bidder pays on units that she wins. Given this, she has an incentive to bid less than her true value on later units in order to reduce the price she will pay on the earlier units.” I extend their result to my setting without quasilinear preferences. In particular, I show overreporting demand is a dominated strategy. In addition, misreporting demand for your ﬁrst unit is also weakly dominated. This result is summarized in Proposition 1.

7 Price bi

bi pi m

H L Quantity Figure 4: Strategy bi is dominated by bi.

Proposition 1. The bid bi =(bi ,bi ,...,bi ) is weakly dominated by the bid ˜bi = 1 2 k 2 (pi(1),bi ,...,bi ) (pi(1),pi(2),...,pi(k)) if bi = ˜bi. 2 k ^ 6 The implication of Proposition 1 is illustrated graphically in Figures 3 and 4.Ifbidder i’s preferences are such that she has a demand curve pi,thenthebidbi as seen in Figure 3 is weakly dominated.

Price

pi m

Quantity H L Figure 3: Underreporting in uniform price auction.

In particular, the bid bi is weakly dominated by a bid that is represented by the lower i i envelope of b and p . This is bid bi in Figure 4.

3.2 In the Vickrey auction Proposition 1 shows that in a uniform price auction, it is a dominated strategy for a bidder to overreport her demand and misreport her demand for her ﬁrst unit. Similarly, Proposition 2 shows that in the Vickrey auction, it is a dominated strategy for a bidder to underreport her demand and misreport her demand for her ﬁrst unit.

8 Proposition 2. The bid bi =(bi ,bi ,...,bi ) is weakly dominated by the bid ˜bi = 1 2 k 2 ((pi(1),bi ,...,bi ) (pi(1),pi(2),...,pi(k))) (pi(1),pi(1),...,pi(1)) if bi = ˜bi. 2 k _ ^ 6 Note that in the expression for ˜bi,theﬁnaltermisneedtoensurethatbidsareweakly decreasing and ˜bi . 2 While this result mirrors the Proposition 1 and Ausubel and Cramton’s demand reduction result, the intuition behind overbidding is distinct. The argument for why bidders overreport is broken into two cases. First, I consider a case where bidding truthfully is a best reply. In the Vickrey auction bidder i faces a residual demand curve that is determined by other bidders reports. This serves as her marginal price curve. Suppose that this residual demand curve was perfectly elastic. Her opponents bids are such that she pays a constant marginal price p for each unit of the good she acquires. This case is illustrated in Figure 5. When facing a constant marginal price curve, truthful reporting is a best reply. By truthfully reporting demand, bidder i wins her desired number objects when the price per unit is p. Suppose that she wins x objects in this case.

Price pi m Payment -i c k-m

H L

Quantity

Figure 5: Vickrey payment with perfectly elastic residual demand.

Now, suppose instead that the same bidder faces a relatively more inelastic residual demand curve. Thus, she pays an increasing marginal price for each unit of the good she acquires. In addition, suppose that the residual demand is such that bidder i pays p for her xth unit of the good. If bidder i reports her demand truthfully, she will again win x units. However, she does not pay the price p per unit. Instead, p is the price she pays for only xth last unit of the good. She pays a price less than p for all other units. Or equivalently, she pays p per unit and is then given a refund. This is illustrate in Figure 6.Sincebidder i has weakly positive wealth eﬀects, the refund weakly increases her demand for acquiring additional units of the good. This eﬀect gives bidder i an incentive to overreport her demand curve.

9 Price i p m -i c k-m

H L Payment

Quantity

Figure 6: Vickrey payment with more inelastic residual demand

The amount that bidder i should overreport her demand curve depends on the magnitude of the wealth eﬀects and the expected elasticity of the residual demand curve. Underreporting demand however, is weakly dominated. Propositions 1 and 2 display similar results. In both auctions bidders truthfully report their demand for their ﬁrst unit. In the Vickrey auction, bidders never underreport their demands for later units and in the uniform price auction bidders never overreport their demand for later units. They combine to give Lemma 1.

Lemma 1. Suppose that bidders play undominated strategies. For a given auction environment, the uniform price auction produces a (weakly) greater number of winners than the Vickrey auction. Winners in the Vickrey auction win a (weakly) greater number of objects.

In other words, a bidder is more likely to not win any objects in the Vickrey auction. Consider the same set of bidders participating in the uniform price and Vickrey auctions. There may exist a bidder that does not win any objects in a Vickrey auction, but does in the uniform price auction. Yet, the reverse can not be true. If a bidder does not win an object in the uniform price auction, she would not win any objects in the Vickrey auction. The intuition for Lemma 1 follows from Propositions 1 and 2. Both the auctions have the same winning rule and a bidder’s first bid is be the same in both auctions. However, the distribution of winning bids is higher in the Vickrey auction than the uniform price auction because bidders overreport demand in the former and underreport demand in the later. A bidder will win at least one object if her first bid (this is always her highest bid) ranks among the top k bids. The first bid is more likely to rank in the top k in the uniform price auction, where the overall distribution of submitted bids is lower. This means that bidders who do win objects in the Vickrey auction win a greater number of objects on average than they would in the uniform price auction. If the goods being auctioned are to be used for downstream competition, one may con- jecture that uniform price auctions would make for a more competitive downstream market. However, we would need to explicitly model downstream competition to fully understand its implications for the outcome of the two different auctions.

10 3.3 An Example: Overbidding and ineﬃciencies in a Vickrey auction IillustratehowoverbiddingcancauseineﬃcienciesintheVickreyauctionusingasimple example. Suppose that there are two bidders and two objects. Bidder 1 has an initial wealth of 100 and preferences represented by

u1(x, w)=2x + pw.

Bidder 2 has quasilinear preferences

u2(1,w)=30+w, u2(2,w)=30+v + w, where with probability .5, v =20and with probability .5, v =30. The realization of v is bidder 2’s private information. The ex-ante distribution of v is commonly known. We can determine bid behavior in this simple setting using iterative elimination of weakly dominated strategies. First note that bidders’ inverse demand curves are p1(1) = 36 p1(2) = 29.5 p2(1) = 30 p2(2) = v.

Truthful reporting is a dominant strategy for bidder 2. This is because she has quasilinear preferences and we can apply the standard argument for truthful reporting being a dominant strategy in Vickrey auctions. Thus b2 =(30,v). By Proposition 2 we know that bidder 1’s ﬁrst bid is to report 36.Sincebidder2 reports truthfully, bidder 1 will pay v to acquire her ﬁrst unit of the good. If she wins two units of 1 1 2 the good, she pays 30 + v. Thus, if bidder 1 reports the bid b =(36,b2) where b1 30 she 1 1 1  wins one unit and pays v.Ifshebidsb =(36,b2) where b2 > 30, she wins two objects and pays 30 + v.Itisthenadominantstrategyforbidder1 to report b1 =(36,b1) where b2 30 2 1 because 1 1 u1(2, 100 50) + u1(2, 100 60) u1(1, 100 20) + u1(2, 100 30) . 2 2

1 Thus bidder 1 always wins both goods. She pays 50 with probability 2 and 60 with probability 1 2 . 1 In this example there is a 2 probability that the allocation is ineﬃcient. When v =30 and bidder 1 receives a payoﬀof u1(2, 40).Sincebidder1 demands 1 object only when the price per unit is 30, u1(2, 100 60)

11 bidder 1 to overbid on her second unit. However, when bidder 1 pays the higher price for her first unit, she does not want to pay 30 for the second unit, while bidder 2 does. Yet, bidder 1 still wins the second unit for the price of 30. It is interesting to compare the outcome of the Vickrey auction with the uniform price auction. Iterative elimination of weakly dominated strategies uniquely predicts bid behavior in this example. Both bidders truthfully report their demand for their first unit. Neither bidder overreports their demand for their second unit, so each bidder wins exactly one unit. The market clearing price is the highest losing bid. Since each bidder wins one object, this 1 2 is the highest of second bid submitted by either bidder, max b2,b2 . Thus, it is a weakly 1 2 { } dominant strategy for both bidders to set b2 = b2 =0. Thus, the outcome is efficient but gives zero revenue. The purpose of this example is not to show that the uniform price auction is in general efficient or gives lower revenues than the Vickrey auction. Ausubel and Cramton (2002) show that even with the quasilinearity restriction, uniform price auctions are generally inefficient and explicit revenue characterizations are difficult to obtain. Instead, the point of this example is to show that overbidding in Vickrey auctions can lead to inefficiencies. In fact, there are cases where the uniform price auction is efficient and the Vickrey auction is inefficient, like the one illustrated here. This example is similar to results seen in the literature on multiunit auctions with budgets. For example, Dobzinski, Lavi and Nisan (2012) show that Vickrey auctions are inefficient when bidders have budgets.

4LargeAuctionComparisons

Before I formally describe the large auction setting, it is useful to note that it is without loss of generality to consider the case where all bidders have an initial wealth of zero. This is because any bidder with preferences ui and initial wealth wi will behave identically to a bidder with preferences uˆi and initial wealth zero, where

uî(x, t):=ui(x, wi + t) x 0, 1,...,k and t R. 8 2{ } 2 The large auction setting I consider follows in the style of Swinkles (1999, 2001) and Kremer and Jackson (2006). However, I remove their restrictions on quasilinear preferences and independent or conditionally independent types. n n Iassumethereisaseriesofauctionenvironments A 1 .InauctionA the seller { }n=1 has kn objects to sell to n bidders. Each bidder demands at most m objects, where m is independent of n.Iassumethatkn ✓j 2{ } 2 for all coordinates, then bidder i has a strictly higher inverse demand curve that bidder j, i j i p✓ (h) >p✓ (h) h =1,...,m. Iletp✓ represent the demand curve of bidder i with type 8 ✓i ⇥. 2 12 Astrategyforbidderi, i maps her type to a distribution over m-dimensional bids, B i :⇥ () . B ! I will assume that in auction setting An the profile of bidder types (✓1,...,✓n) is a draw n n from a random variable with density f :⇥ R+.Idonotrestrictbiddertypestobe ! independent draws as in Swinkles (1999, 2001) or conditionally independent as in Kremer n and Jackson (2006). The only restriction I place on f 1 is a generalization of Swinkles’ { }i=1 (1999) no asymptotic gaps condition that does not make any independence restrictions. In particular, I assume that when n is sufficiently large there is a high probability that at least one bidder will have a type that is near ✓,forany✓ ⇥.Toformalizethismathe- 2 matically, I will define B✏(✓) as an ✏ box around a point ✓. The epsilon box around ✓ is the set of all points that are within ✏ of ✓ for all coordinates,

B (✓):= ✓0 ✓ ✏ ✓0 ✓ + ✏, j =1,...,s . ✏ { | j  j  j 8 } The no asymptotic gaps condition states that for all ✓ ⇥, when n is suﬃciently large, there 2 i is a high probability that there is some bidder i with type ✓ B✏(✓) 2 Assumption 3. (No Asymptotic Gaps) n The sequence of densities f 1 is such that for any ✏>0, { }n=1 i lim P ( ✓ B✏(✓)) = 1, ✓ ⇥. n !1 9 2 8 2 The no asymptotic gaps condition allows for independent types like that studied by Swin- kles (1999) and Azevedo and Budish (2013); and the conditionally independent types case studied by studied by Kremer and Jackson (2006).

4.1 Bid Behavior The no asymptotic gaps condition ensures that, for any bidder i, when the auction is suﬃ- ciently large truthful reporting is an ✏ best reply to any undominated play of her opponents. This is true for both uniform price and Vickrey auctions. However, the reasons truthful reporting is an approximate best reply diﬀers between the two auctions. In the uniform price auction, truthful reporting is an approximate best reply when a bidder has a negligible impact on the market clearing price. In auction An,ifbidderi wins i nth at least one object, she pays at least ckn per unit, the k highest bid not submitted by i.Bidderi pays a higher price per unit when she reports a relatively higher demand. If i her reported demand is such that she wins m objects, the market clearing price is ckn+1 m. Thus, the most a bidder i can shift the market clearing price (conditional on winning at least i i one object) is ckn+1 m ckn . Truthful reporting is an approximate best reply when bidder i believes that her impact on the market clearing price is negligible. I show that Proposition 1 and Assumption 3 imply that bidder i has a negligible impact on the market clearing price when n is large. In other words, if all bidders play undominated strategies,

i i i ✏>0,P(ckn+1 m ckn >✏✓ ) 0 as n . 8 | ! !1 I will call this condition U.

13 In Vickrey auctions, truthful reporting is a best reply when a bidder faces a perfectly elastic residual demand curve. Thus, truthful reporting is an approximately best reply for bidder i if she believes the price she would pay for her first unit is approximately equal to the th price she would pay for mth unit. The price bidder i would pay for her first unit is the kn i th highest bid submitted by all other bidders, ckn . The price the bidder would pay for her m th i unit is the kj +1 m highest bid not submitted by herself, ck +1 m. I show that Proposition j 1 and Assumption 3 imply that this difference is negligible when n is large. In other words, if all bidders play undominated strategies, i i i ✏>0,P(ckn+1 m ckn >✏✓ ) 0 as n . 8 | ! !1 I will call the above condition V . Notice that conditions U and V are identical, yet have different interpretations. In the uniform price auction, the condition means the probability that a bidder can move the market clearing price by more than ✏ approaches zero as the auction becomes large. In the Vickrey auction, the condition means that the probability that a bidder pays at least ✏ more for her mth object than her first object approaches zero as the auction becomes large. In each auction, the conditions can be used to show that truthful reporting is an ✏ best reply when n is sufficiently large. The two proofs are nearly identical since bidders report their first bid truthfully in both auctions. n Proposition 3. Consider a series of auction environments A 1 .Forany✏>0,there { }n=1 exists a n⇤ N,suchthatforalln>n⇤,truthfulreportingisan✏ best reply to any undomi- 2 nated strategy in the uniform price and Vickrey auctions.

4.2 Approximate Efficiency Proposition 3 states that with many bidders, truthtelling is an ✏ best reply to any undominated strategy in both the Vickrey auction and the uniform price auction. Since the two auctions have the same winning rule, if bidders report truthfully in both auctions, the auction gives the same allocation. In this section, I show that the no asymptotic gaps condition (As- sumption 3) also implies that when there are many bidders, truthtelling gives approximately equal revenues in both auctions. In addition, when there are many bidders the outcome of both auctions is approximately ex-post efficient. The next Corollary follow from Assumption 3 and Conditions U and V .Itsaysthatifall bidders bid truthfully, they pay approximately equal amounts in the uniform price auction and the Vickrey auction when n is large. n Corollary 1. Consider a series of auction environments A n1=1 where all bidders report i ✓i { } their preferences truthfully, (✓)=p .Forany✏>0,thereexistsan⇤ N,suchthatfor B 2 all n>n⇤ i i i i P (ti (p✓ ,p✓ ) ti (p✓ ,p✓ ) >✏) > 1 ✏. U V Thus, we have shown that truthful reporting is an ✏ best reply to any undominated strategy in both auctions when n is sufficiently large. When n is large and bidders do report their types truthfully, bidders win the same number of goods and make approximately equal payments in uniform price and Vickrey auctions. Next, I show that this outcome is approximately efficient.

14 Definition 3. Given a profile of bidder types (✓1,...,✓n),anoutcome(x, t) X Rn is n 2 ⇥ ex-post Pareto efficient if @(˜x, t˜) X R such that 1 t˜i 1 ti and 2 ⇥ i=1 i=1 u(x ,t,✓i) u(˜x , t˜ ,P✓i) i P i i i i 8 where the above inequality is strict for some i 1,...,n . 2{ } That is, an allocation is ex-post Pareto efficient if any other feasible allocation that gives weakly greater revenues makes one bidder strictly worse off. I will call X Rn the set W⇢ ⇥ of all ex-post efficient allocations. We can define an analogous of ✏ ex-post Pareto efficient.

Definition 4. Given a profile of bidder types (✓1,...,✓n),anoutcome(x, t) X Rn is ✏ n 2 ⇥ ex-post efficient if @(˜x, t˜) X R such that 1 t˜i 1 ti and 2 ⇥ i=1 i=1 u(x ,t,✓i)+✏ Pu(˜x , t˜ ,✓iP) i. i i i i 8 That is, an allocation is ✏ ex-post efficient if any other feasible allocation that gives weakly greater revenues increases bidder payoffs by at most ✏. I will call (✏) X Rn the set of W ⇢ ⇥ all ✏ ex-post efficient allocations. Notice that when bidders report their types truthfully, the uniform price auction is ex- post Pareto efficient. This follows from the First Welfare Theorem. When bidders report their types truthfully the outcome is a Walrassian equilibrium where the market clearing price is the kn +1st highest bid.

Lemma 2. For any proﬁle of types (✓1,...,✓n) ⇥n if bidders report demand truthfully, the 2 outcome of the uniform price auction is ex-post Pareto eﬃcient.

Lemma 2 shows that the uniform price auction is eﬃcient when bidders report their types truthfully. Since Corollary 1 shows that the uniform price auction and the Vickrey auction provide nearly identical outcomes when n is large, it follows that truthful reporting is approximately eﬃcient in the Vickrey auction when n is large. To formalize this, I let V,n : n X Rn represent the Vickrey auction’s mapping of bids to outcomes. ! ⇥ n Lemma 3. Fix ✏>0.Consideraseriesofauctionenvironments A 1 .Thereexistsan { }n=1 n⇤ N,suchthatforalln>n⇤, 2 1 2 n P V,n(p✓ ,p✓ ,...,p✓ ) (✏) > 1 ✏. 2W ⇣ ⌘ 5Conclusion

In this paper, I compare bid behavior in the uniform price auction and the Vickrey auction. I replace the standard quasilinearity restriction with the more general assumption that the goods being sold are (weakly) normal. Without the quasilinearity restriction, the Vickrey and uniform price auctions have similar virtues and deﬁciencies. Neither auction gives bidders incentives to truthfully report their preferences. Instead, bidders overreport demand in the Vickrey auction and underreport demand in the uniform price auction. The intuition for the two results is distinct. Ausubel and Cramton’s intuition

15 for underbidding in uniform price auctions holds, even without the quasilinearity restriction. The reason bidders overreport in the Vickrey auction is due to the pricing rule. In the Vickrey auction, a bidder faces an upward sloping supply curve defined by other bidders’ report demands. If she pays a relatively low price for her first unit, positive wealth effects increase her demand for additional units. This increase in demand for additional units gives bidders an incentive to overreport their demands. Indeed, the incentives to misreport demands leads to inefficiencies in both auctions. These two results yield a testable implication. A bidder has a greater chance of winning at least one object in a uniform price auction versus a Vickrey auction. However, bidders who do win objects in the Vickrey auction win a greater number of objects than they would in the uniform price auction. While both auctions are inefficient, these inefficiencies disappear in a large auction setting. Ishowthisinamoregeneralsettingthantheliteraturetypicallyassumes.Inparticular, Iallowformultidimensionalheterogeneityacrossbidderpreferencesanddemands,aswell as correlated types. The previous literature focuses primarily on the quasilinear or budget constrained bidders and for independent or conditionally independent types. I show truthful reporting is an ✏ best reply to any undominated strategy. In addition, truthful reporting is approximately efficient and gives approximately equal revenues and allocations in both auctions. The findings of the paper give additional perspective to Ausubel and Milgrom’s question of why Vickrey auctions are so rarely used to sell multiple homogenous goods. The Vickrey auction has many desirable properties when we restrict ourselves to quasilinear preferences. Without quasilinearity, the Vickrey auction has few noticeable advantages over the more frequently used uniform price auction.

6Appendix

Proposition 1 i i i Proof. For this proof, I will simplify notation by letting U (b ,b ) be the utility of bidder i i i when she bids b and her opponents bid b in the uniform price auction. I want to show i i i i i i i N 1 that U (˜b ,b ) U (b ,b ) b .Supposethatthe(N 1)k competing bids are i 8 2 i i i described by the vector c , where c1 c2 ...c(N 1)k. Case 1:bidderi wins the same number of objects if she bids bi and ˜bi. i i i i i Suppose i wins m objects from bidding either b or ˜b .Ifm =0then U (˜b ,b )= i i i i i ˜i i i i ˜i U (b ,b )=u (0, 0).Ifm 1,...k ,thenbm, bm ck m+1 and ck m bm+1, bm+1.Ifitis i i 2˜i { } i the case that ck m bm+1, bm+1,thenthemarketclearingpriceisck m for either bid. Thus, i ˜i i i i i i i i i ˜i i i U (b ,b )=U (b ,b )=u (m, w mck m). If bm+1, bm+1 ck m,thenbm+1 is the market i ˜i i i i i i i i clearing price for either bid. Thus, U (b ,b )=U (b ,b )=u (m, w mbm+1).Finallyif i i ˜i i ˜i i bm+1 ck m > bm+1,thenck m is the market clearing price when bidder i bids b and bm+1 is the market clearing price when bidder i bids bi. Thus, i ˜i i i i i i i i i i U (b ,b ) u (m, w mck m) >u(m, w mbm+1)=U (b ,b ). Case 2:bidderi wins more objects by bidding ˜bi instead of bi.

16 This implies that, pi(1) = ˜bi >bi . If not, then bi ˜bi j 1,...,k and i wins a 1 1 j j 8 2{ } (weakly) greater number of objects by bidding bi instead of ˜bi. Thus, ˜bi strictly exceeds bi only in the ﬁrst dimension, as pi(1) = ˜bi >bi bi ˜bi j 2,...k .Ifi wins more 1 1 j j 8 2{ } objects by bidding ˜bi instead of bi, she then must win exactly one object by bidding ˜bi and zero goods by bidding bi, since she only submits a higher bid for her ﬁrst good when bidding ˜bi instead of bi. Thus,

i i i i i i i i i i i i i i U (˜b ,b )=u (1,w max ˜b ,c ) u (1,w p (1)) = u (0,w )=U (b ,b ), { 2 k } i i i i where the ﬁrst inequality follows because p (1) = ˜b c if i wins one object and the equality 1 k ui(1,wi pi(1)) = ui(0, 0) follows from the construction of pi. Case 3:bidderi wins fewer objects by bidding ˜bi instead of bi. Let m be the number of objects i wins when bidding bi and m˜ the number she wins ˜i i ˜i i when bidding bidding b .Byassumptionm>m˜ . This implies that bn > bn = p (n) n ˜i i i 8 2 m˜ +1,...,m .Ifnot,biddingb wins more than m˜ objects. Thus ck n bn = p (n) for all { } i i i i n m˜ +1,...,m .Sinceck n is weakly increasing in n and p (n),thenck m+1 p (˜m+1).If 2{˜i } ˜i i i bids b and wins m˜ objects, she pays pU˜ =max bm˜ +1,ck (˜m+1) . Thus p (˜m) pU˜ p (˜m+1). i { } i If she bids b and wins m objects, she pays pU where pU p ˜ . By the construction of p ,it U then follows that

i i i i i i i i i i i i U (˜b ,b )=u (˜m, w mp˜ ˜ ) u (m, w mp ˜ ) u (m, w mp )=U (b ,b ). U U U

Proposition 2 i i i Proof. For this proof, I will simplify notation by letting U (b ,b ) be the utility of bidder i i i when she bids b and her opponents bid b in the Vickrey auction. I want to show that i i i i i i i N 1 U (˜b ,b ) U (b ,b ) b .Supposethatthe(N 1)k competing bids are described i 8 i 2 i i by the vector c , where c1 c2 ...c(N 1)k. Case 1:bidderi wins the same number of objects if she bids bi and ˜bi. In each case, the bidder pays the same amount. This is because the marginal price of an additional unit is based on other bidders reports. Case 2: bidder i wins fewer objects by bidding ˜bi instead of bi. This implies that pi(1) = ˜bi pi(1) = ˜bi ˜bi bi j 2,...k .Ifi wins more objects by bidding bi 1 1 j j 8 2{ } instead of ˜bi, she then must win exactly one object by bidding bi and zero goods by bidding ˜bi, since she only submits a higher bid for her first good when bidding ˜bi instead of bi. Thus i i i b c p (1) and 1 k i i i i i i i i i i i i i i U (˜b ,b )=u (0,w )=u (1,w p (1)) u (1,w c )=U (b ,b ). k Case 3:bidderi wins more objects by bidding ˜bi instead of bi. Let m be the number of objects i wins when bidding bi and m˜ the number she wins when bidding bidding ˜bi.Byassumptionm>m˜ . This implies that ˜bi = pi(n) >bi n n n 8 2 17 i m i m +1,...,m˜ .Ifnot,biddingb wins more than m objects. Thus, i pays j=1 ck+1 j when { } i m˜ i ˜i ˜i i i i she bids b and j=1 ck+1 j when she bids b .Noticealsothatbm˜ = p (˜m) ck+1 m˜ ck+1 j P for all j 1 ...m˜ . By the definition of pi, 2{ P } ui(˜m, wi mp˜ i(˜m)) ui(m, wi mpi(˜m)). i m i i i Note that mp (˜m) j=1 ck+1 j 0 since p (˜m) ck+1 j for all j 1,...,m˜ . Thus, 2{ } positive wealth effects imply that P m m i i i i i i i i i i u (˜m, w mp˜ (˜m)+mp (˜m) ck+1 j) u (m, w mp (˜m)+mp (˜m) ck+1 j). j=1 j=1 X X Which can be rewritten as m m i i i i i i i i i u (˜m, w ck+1 j (˜m m)p (˜m)) u (m, w ck+1 j)=U (b ,b i). (1) j=1 j=1 X X i i i m˜ i Recalling again that p (˜m) ck+1 j j 1,...,m˜ ,impliesthat(˜m m)p (˜m) j=m+1 ck+1 j. i m i 8 2{m˜ i } Thus, (˜m m)p (˜m)+ j=1 ck+1 j j=1 ck+1 j. This implies that, P P m˜ P m i ˜i i i i i i i i U (b ,b i)=u (m, w ck+1 j) u m,˜ w ck+1 j +(˜m m)p (˜m) . (2) j=1 j=1 !! X X i i i i Combining 1 and 2 implies that U (˜b ,b i) U (b ,b i). Proposition 3 Proof. First, I will show that conditions U, V hold in both auctions. Let p✓(1) be the highest price that a bidder with type ✓ is willing to pay for her first unit. Note that p✓(1) is continuous ✓ in ✓ since u(x, w, ✓) is continuous in ✓ for all x =0, 1,...,hand w R.Letp := max✓ ⇥ p (1) ✓ i 2 n 2 and p := min✓ ⇥ p (1). Consider bidder i with preferences ✓ supp(f ).Supposethatall 2 2 bidders j = i play some undominated strategy j. Propositions X and Y imply all bidders 6 B n i report their first bid truthfully (in either auction). Since k 0. Consider any interval [p, p + ✏1] [p, p]. Continuity and monotonicity of ⇢ ✓0 preferences in ✓ implies that there exists an ✏>0 and ✓ T such that p (1) (p, p+✏1) ✓0 2 2 8 2 B✏(✓).Asn assumption 3 implies that with probability approaching 1,thereexists !1j i j abidderj with ✓ = ✓ such that ✓ B✏(✓). Since ✏1 is arbitrary, I can equivalently 6 2 look at m intervals of the form (p + x ✏1 ,p+(x +1)✏1 ) for x =0, 1,...m 1.Foreach m m interval, as n , with probability approaching 1,thereexistsabidderj with ✓j = ✓i and j !1✏1 ✏1 6 p (1) (p + x m ,p+(x +1)m ). Thus, as n ,theprobabilitythatthereareatleastm 2 ✓ !1 bidders who have types ✓ such that p (1) (p, p + ✏1) approaches 1.Sincebiddersreport 2 their first bid truthfully, this implies there are at least m bids in this interval with probability approaching 1.

18 i Recall that ckn [p, p]. In addition, for any ✏1 > 0,asn ,theprobabilitythere 2 !1 nth are at least m bids in an interval (p, p + ✏1) approaches 1. Thus, the k highest bid and n th the k +1 m highest bid can be at most ✏1 apart with probability approaching 1.Or equivalently, i i i P (ckn+1 m ckn >✏✓ ) 0 ✏>0 as n . | ! 8 !1 This is Condition U,V. Given this condition, I show that when n is sufficiently large, truthful reporting is an ✏ best reply in both auctions. I begin with the uniform price auction. By Proposition 1, I only need to check the bidder’s incentive to underreport her demand. Suppose that in auction n i i i,n i,n A ,bidderi has type ✓ and inverse demand p .Ilet (✓ ) denote the strategies of U B bidders j = i in auction j.Iassumeallbiddersj = i play undominated strategies. I let n i i,n6 i 6 n i U (b ,b ,✓ ) denote the utility of bidder i in auction AU when she bids b ,heropponents i,n i bid b and she has type ✓ . First, I will show that for any ✏>0, when n is sufficiently large, there is a 1 ✏ probability that bidder i n i i,n i,n i n i,n i,n i i U (p , (✓ ),✓ )+✏>U (b, (✓ ),✓ ), b ,✓ ⇥. B B 8 2 2 Suppose that in auction An,bidderi reports pi and wins x 1,...,k objects. She pays 2{ } p p per unit. Her utility is then u(x, xp, ✓i), where  u(x, xp, ✓i) u(y, yp, ✓i) y = x. 8 6 If she underreports her preferences she wins y x objects and pays a price p˜ p per unit.   Condition U,V implies that when n is sufficiently large, the market clearing price drops by at most ✏ with probability of at least 1 ✏. Thus, with probability at least 1 ✏,bidderi gets utility u(y, yp,˜ ✓i) u(y, y(p ✏),✓i) u(x, x(p ✏),✓i).   Thus, with a probability of at least 1 ✏,theupperboundonthebenefitofmisreportingis u(x, x(p ✏),✓i) u(x, xp, ✓i).Since✏ is arbitrary and u(x, x(p ✏),✓i) u(x, xp, ✓i) 0 ! as ✏ 0, I can equivalently state that when n is sufficiently large, there is at least a 1 ✏ ! probability,

n i i,n i,n i n i,n i,n i i U (p , (✓ ),✓ )+✏>U (b, (✓ ),✓ ), b ,✓ ⇥. B B 8 2 2 Since the benefit of misreporting is bounded for any n, the above condition implies that when n is sufficiently large bidding pi is an ✏ best reply to any undominated strategies of bidders j = i. 6 Next, I show an equivalent result for the Vickrey auction. By Proposition 2, I only need n to check a bidder’s incentive to overreport her demand. Suppose that in auction AV ,bidder i i i,n i,n i has type ✓ and inverse demand p .Ilet (✓ ) denote the strategies of bidders j = i B n i i,n 6 i in auction j.Iassumeallbiddersj = i play undominated strategies. I let V (b ,b ,✓ ) 6 n i i,n denote the utility of bidder i in auction AV when she bids b ,heropponentsbidb and she has type ✓i. I will show that for any ✏>0, when n is sufficiently large, there is a 1 ✏ probability that bidder i

n i i,n i,n i n i,n i,n i i V (p , (✓ ),✓ )+✏>V (b, (✓ ),✓ ), b ,✓ ⇥. B B 8 2 2 19 Suppose that in auction n,ifbidderi reports pi, she wins x 1,...,m objects. She 2{i } i pays a total of P for the the x objects. Thus, her utility is u(x, P,✓ ). Bidder i pays ckn x+1 for her xth object. Thus, x 1 i i P = ckn j xckn x+1.  j=0 X Since bidder i reports her demand truthfully, the number of objects she wins is the number i of objects she demands when the price per unit is ckn x+1. This means x is such that, i i i i u(x, xckn x+1,✓ ) u(y, yckn x+1,✓ ) y 0, 1,...,m . 8 2{ } If bidder i over reports her demand and still wins x objects, her payoffis unchanged. If bidder i i over-reports her demand and wins y>xobjects she pays P 0 where P 0 P +(y x)ckn x+1. This is because bidder i faces an pays an increasing marginal price for all units. Thus, she i gets utility u(y, P 0,✓ ) from when she overreports her demand. Condition V implies that for any ✏>0,thereisasufficientlylargen such that there is i i greater than a 1 ✏ probability that ckn x+1 ckn <✏for any x 0, 1,...,m .Whenthis 2{ } holds true, x 1 i i i xckn x+1 (x 1)✏ P = ckn j xckn x+1.   j=0 X i i i i i Recall that P 0 P +(y x)ckn x+1.Leta(✏)=u(x, (xckn x+1 (x 1)✏),✓ ) u(x, xckn x+1,✓ ) i i i i and b(✏)=u(y, (yckn x+1 (x 1)✏),✓ ) u(y, yckn x+1,✓ ). Both a and b are continuous i i in ✏ and a(0) = b(0) = 0. When ckn x+1 ckn <✏,thebenefitofmisreportingis i i i i i i u(y, P 0,✓ ) u(x, P,✓ ) u(y, (yckn x+1 (x 1)✏),✓ ) u(x, xckn x+1,✓ ).  Rewriting the right hand side of the inequality, we get

i i i i b(✏)+a(✏)+u(y, yckn x+1,✓ ) u(x, (xckn x+1 (x 1)✏),✓ ). i i i i Notice that we can say u(y, yckn x+1,✓ ) u(x, (xckn x+1 (x 1)✏),✓ ) 0 since we i i i i  have already know u(x, xckn x+1,✓ ) u(y, yckn x+1,✓ ).Wecanusethistosaythatthe beneﬁt of misreporting is bounded above by b(✏)+a(✏),

i i u(y, P 0,✓ ) u(x, P,✓ ) a(✏)+b(✏).  Thus when n is sufficiently large, there is a 1 ✏ probability that the benefit of misreporting is at most a(✏)+b(✏).Sincethisholdsforanarbitrary✏>0 and both a and b are continuous with a(0) = b(0) = 0, then we can equivalently say that when n is sufficiently large, there is at least a 1 ✏ probability that the benefit of misreporting is at most ✏.Orequivalently, when n is sufficiently large, there is at least a 1 ✏ probability that, n i i,n i,n i n i,n i,n i i V (p , (✓ ),✓ )+✏>V (b, (✓ ),✓ ), b ,✓ ⇥. B B 8 2 2 Since the benefit of misreporting is bounded for any n,thenforany✏>0 when n is sufficiently large, the expected benefit of misreporting is at most ✏. Thus, when n is sufficiently large bidding pi is an ✏ best reply to any undominated strategies of bidders j = i. 6

20 Corollary 1 Proof. This follows directly from conditions U, V . Consider bidder i.Fix✏>0. Pick an n suﬃciently large such that

i i ✏ i i P (ckn+1 m ckn < ✓ ) > 1 ✏ ✓ ⇥. m| 8 2

i i ✏ Suppose that ckn+1 m ckn < m . Since both auctions have identical winning rules and we assume bidders bid truthfully in both auctions, bidder i wins the same number of objects in each auction. If bidder i wins 0 objects, she makes no transfers in each auction. Suppose i i i ✓ ✓ that bidder i wins x 1,...,m objects in each auction. She pays tU (p ,p )=xpU in i i 2{ } i i i ✓ ✓ x i the uniform price auction, where ckn pU ckn+1 m.ShepaystV (p ,p )= i=1 ckn+1 x   in the Vickrey auction. Thus, P x i i i i i ✓ ✓ i ✓ ✓ i tU (p ,p ) tV (p ,p )= pU ckn+1 x . i=1 X i ✏ i i ✏ Yet pU ckn+1 x < m because we assume ckn+1 m ckn < m . Thus, x i i i i ✏ i ✓ ✓ i ✓ ✓ i tU (p ,p ) tV (p ,p )= pU ckn+1 x x ✏.  m  i=1 X i i ✏ This holds whenever ckn+1 m ckn < m . Thus, when n is suﬃciently large, i i i i P (ti (p✓ ,p✓ ) ti (p✓ ,p✓ ) >✏) > 1 ✏. U V

Lemma 3 Proof. Fix ✏>0.Supposen is suﬃciently large such

i i i i P (ti (p✓ ,p✓ ) ti (p✓ ,p✓ ) <✏) > 1 ✏. U V i i i i i i i ✓ ✓ i ✓ ✓ i i ✓ ✓ Suppose that tU (p ,p ) tV (p ,p ) <✏ i . For ease of notation I write tV = tV (p ,p ) i i i i ✓ ✓ 8 and xV = xV (p ,p ). ˜ n ˜ i i Consider any other outcome (˜x, t) X R s.t. (xV ,tV ) =(˜x, t).Ifx˜ = xV i and 2 ⇥ 6 8 t˜i ti ,thent˜i ti . Thus, some bidder i is made worse oﬀin allocation (˜x, t˜).She V V receives the same number of goods, yet makes a higher transfer. P P ˜ n i i Suppose instead that (˜x, t) X R and x˜ = xV for some i.Supposealsothat ˜i i 2 ⇥ 6 t tV and u(˜xi, t˜i,✓i) u(xi , ti ,✓i) i. P P V V 8 This implies that if x˜i xi , then t˜i ti . Since truthtelling is eﬃcient in the uniform price  V  V auction and both auctions have the same winning rule,

u(xi , p xi ,✓i) u(y, p y, ✓i) y 0,...,m, V U V U 8 2 21 where pU is the market clearing price in the uniform price auction. Since we assume i i i i i ✓ ✓ i ✓ ✓ i i tU (p ,p ) tV (p ,p ) <✏ i ,wecansaythatforalli, ✏i <✏such that tV = tU + ✏i. i i 8 9 Thus if x˜ xV i i i i i i i i implies that t˜ t + pU (˜x x ) since t˜ t .Supposethati is such that x˜ >x . V V V V Note that u(xi , p xi ,✓i) Pu(y, pPy, ✓i) y 0,...,m. V U V U 8 2 i i i i i i i i Let a (s)=u(x ,s pU x ,✓ ) u(˜x ,s pU x˜ ,✓ ). Note that a (s) is continuous in s and V V ai(0) 0. Thus  u(xi ,✏ p xi ,✓i)+ai(✏ )=u(˜xi,✏ p x˜i,✓i) u(˜xi, t˜i,✓i). V i U V i i U Or equivalently, ai(✏ ) u(˜xi, t˜i,✓i) u(xi ,✏ p xi ,✓i)=u(˜xi, t˜i,✓i) u(xi ,ti ,✓i). i V i U V V V i i Since a is continuous with a (0) 0,and✏i <✏, it follows that for any ✏˜ >0,thereisan✏  suﬃciently small such that ✏˜ u(˜xi, t˜i,✓i) u(xi ,ti ,✓i). V V

Thus, thus (xV ,tV ) (˜✏) with probability greater than 1 ✏. 2W References

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