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.
1
Introduction
“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, strategyproofness 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 settings? 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
a more general preference domain, bid behavior in Vickrey auctions mirrors bid behavior in
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 dynamic 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, Dobzinski, 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).
2
2.1
Model
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 2 R and her preferences ui where,
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. I assume
ui (x, ·) is strictly increasing and continuous for any x = 0, 1, . . . k. In addition, the objects
being sold at auction are goods. Thus, ui (x, w) > ui (y, w) if and only if x > y. I make only
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 2 {1, . . . , k 1} and
ui (x 1, w) ui (x, w
then
ui (x, w)
ui (x + 1, w
p),
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. If bidder i 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 > y where x, y 2 {0, 1, . . . k}. Bidder i has weakly
positive wealth effects if
ui (x, w
X)
ui (y, w
Y ) =) ui (x, w0
X)
ui (y, w0
Y ) 8w0 > w.
Definition 1. I let U be the set of all utility functions that satisfy Assumptions 1 and 2.
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
pi (m) := max{p : ui (m, wi
pm)
ui (m0 , wi
pm0 )} 8m0 2 {0, 1, . . . , k}.
In words, pi (m) is the highest price at which bidder i demands m objects. Note that if bidder
i has initial wealth wi 2 R and preferences ui 2 U, then pi (m) is weakly decreasing in m and
1 > pi (m) > 0 8m 2 {1, . . . , k}.
For comparisons, in the quasilinear benchmark bidder i’s preferences are described by
a vector of her marginal valuations v i 2 Rk+ , where v1i
v2i
...
vki . The quantity vhi
represents her marginal valuation for her hth object. Her utility function is then
ui (x, w) =
x
X
vji + w.
j=1
i
The utility function fits assumptions 1 and 2 and thus, ui 2 U where pi (m) = vm
.
I do not place any other restrictions on a bidder’s preferences beyond assumptions 1 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
Bidder i receives vm
utils for her mth 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
X
ui (x, w) =
vji + f i (w),
j=1
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 g i (w) from wealth w, where g i 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,
ˆ
i
u (x, w) =
g i (w + xs)f (s)ds.
s2R
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 P
number of objects won by bidder i. I define X as the set of all
n
x 2 {0, 1, . . . , m} such that ni xi  k. Transfers are described by t 2 Rn , where ti is the
transfer made by bidder i.
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
is a subset of Rk+ , where := {b 2 Rk+ |1 > b1 b2 . . . bk }. I let xiU (bi , b i ) be the number
of objects bidder i wins in the uniform price auction given bids (bi , b i ) 2 N . Similarly, I
let tiU (bi , b i ) be the payment made by bidder i.
I denote the collection of bids from all bidders except i as b i . Let c i denote the vector of
these competing bids ordered from highest to lowest. Bidder i wins exactly m 2 {0, 1, . . . k}
objects, if she submits m bids that rank in the top k.
i
bim > ck+1
m
and ck i m > bim+1 .
For simplicity, I assume that all ties are broken in favor of the higher numbered bidder.
i
If bidder i wins m objects, she pays mpU , where pU = max{bim+1 , ck+1
m } is the value of
the highest losing bid.
Price
bi
c-i 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.
I keep the same notation as above. Bidder i reports a k dimensional bid bi 2 . I
let xiV (bi , b i ) be the number of objects bidder i wins in the Vickrey auction given bids
(bi , b i ) 2 N . Similarly, I let tiV (bi , b iP
) be the payment made by bidder i.
i
If bidder i wins m objects, she pays m
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
th
th
m object is the k + 1 m highest bid made by i’s competitors.
Price
bi
c-i k-m
Vickrey
payment
Quantity
Figure 2: Payment in Vickrey auction
3
3.1
Bid behavior and Inefficiencies
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
pi HmL
bi
Quantity
Figure 4: Strategy bi is dominated by bi .
Proposition 1. The bid bi = (bi1 , bi2 , . . . , bik ) 2
is weakly dominated by the bid b̃i =
(pi (1), bi2 , . . . , bik ) ^ (pi (1), pi (2), . . . , pi (k)) if bi 6= b̃i .
The implication of Proposition 1 is illustrated graphically in Figures 3 and 4. If bidder
i’s preferences are such that she has a demand curve pi , then the bid bi as seen in Figure 3
is weakly dominated.
Price
bi
pi HmL
Quantity
Figure 3: Underreporting in uniform price auction.
In particular, the bid bi is weakly dominated by a bid that is represented by the lower
envelope of bi and pi . 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 = (bi1 , bi2 , . . . , bik ) 2
is weakly dominated by the bid b̃i =
((pi (1), bi2 , . . . , bik ) _ (pi (1), pi (2), . . . , pi (k))) ^ (pi (1), pi (1), . . . , pi (1)) if bi 6= b̃i .
Note that in the expression for b̃i , the final term is need to ensure that bids are weakly
decreasing and b̃i 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 HmL
Payment
c-i k-m
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. Since bidder
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
pi HmL
c-i k-m
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 conjecture 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
I illustrate how overbidding can cause inefficiencies in the Vickrey auction using a simple
example. Suppose that there are two bidders and two objects. Bidder 1 has an initial wealth
of 100 and preferences represented by
p
u1 (x, w) = 2x + w.
Bidder 2 has quasilinear preferences
u2 (1, w) = 30 + w,
u (2, w) = 30 + v + w,
2
where with probability .5, v = 20 and 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
p2 (1) = 30
p1 (2) = 29.5
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. Since bidder 2 reports
truthfully, bidder 1 will pay v to acquire her first unit of the good. If she wins two units of
the good, she pays 30 + v. Thus, if bidder 1 reports the bid b1 = (36, b12 ) where b21  30 she
wins one unit and pays v. If she bids b1 = (36, b12 ) where b12 > 30, she wins two objects and
pays 30 + v. It is then a dominant strategy for bidder 1 to report b1 = (36, b12 ) where b21 30
because
1 1
u (2, 100
2
50) + u1 (2, 100
1 1
u (1, 100
2
60)
20) + u1 (2, 100
30) .
Thus bidder 1 always wins both goods. She pays 50 with probability 12 and 60 with probability
1
.
2
In this example there is a 12 probability that the allocation is inefficient. When v = 30
and bidder 1 receives a payoff of u1 (2, 40). Since bidder 1 demands 1 object only when the
price per unit is 30, u1 (2, 100 60) < u1 (1, 100 30). Thus, both bidders are made weakly
better off by an allocation which assigns each bidder 1 unit and each bidder pays 30. This
allocation gives the same revenue, strictly increases bidder 1’s payoff and leaves bidder 2
equally well off.
The intuition behind this observation follows naturally from Proposition 2. Bidder 1 is
unsure of the shape of the residual demand curve that she faces. She may pay a low price for
her first unit (20) or a high price (30). If she expects to pay a lower price for her first unit,
she is wealthier and willing to bid more for her second unit. Thus, it is a best response for
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
is the highest of second bid submitted by either bidder, max{b12 , b22 }. Thus, it is a weakly
dominant strategy for both bidders to set b12 = b22 = 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.
4
Large Auction Comparisons
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 ûi and initial wealth zero, where
ûi (x, t) := ui (x, wi + t) 8x 2 {0, 1, . . . , k} and t 2 R.
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
I assume there is a series of auction environments {An }1
n=1 . In auction A the seller
has k n objects to sell to n bidders. Each bidder demands at most m objects, where m is
independent of n. I assume that k n < n. That is, the supply of objects is fewer than one per
bidder.
To accommodate multidimensional heterogeneity across bidders, I describe a bidder i’s
preferences are described by a type ✓i . I assume that ✓i 2 ⇥ where ⇥ is a compact subset of
Rs where s is a finite number. A bidder i with type ✓i has preferences u(., ., ✓i ) 2 U.
I assume preferences are continuous and monotonic in ✓. Continuity means that u(x, w, .)
is continuous in ✓ for all x 2 {0, 1, . . . , m} and w 2 R. Monotonicity means that if ✓i > ✓j
for all coordinates, then bidder i has a strictly higher inverse demand curve that bidder j,
i
j
i
p✓ (h) > p✓ (h) 8h = 1, . . . , m. I let p✓ represent the demand curve of bidder i with type
✓i 2 ⇥.
12
A strategy for bidder i, B i maps her type to a distribution over m-dimensional bids,
Bi : ⇥ ! ( ) .
I will assume that in auction setting An the profile of bidder types (✓1 , . . . , ✓n ) is a draw
from a random variable with density f n : ⇥n ! R+ . I do not restrict bidder types to be
independent draws as in Swinkles (1999, 2001) or conditionally independent as in Kremer
and Jackson (2006). The only restriction I place on {f n }1
i=1 is a generalization of Swinkles’
(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 ✓, for any ✓ 2 ⇥. To formalize this mathematically, 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 |✓j
✏  ✓j0  ✓j + ✏, 8j = 1, . . . , s}.
The no asymptotic gaps condition states that for all ✓ 2 ⇥, when n is sufficiently large, there
is a high probability that there is some bidder i with type ✓i 2 B✏ (✓)
Assumption 3. (No Asymptotic Gaps)
The sequence of densities {f n }1
n=1 is such that for any ✏ > 0,
lim P (9 ✓i 2 B✏ (✓)) = 1, 8✓ 2 ⇥.
n!1
The no asymptotic gaps condition allows for independent types like that studied by Swinkles (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 sufficiently 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 , if bidder i wins
th
at least one object, she pays at least ckni per unit, the k n highest bid not submitted by
i. Bidder i pays a higher price per unit when she reports a relatively higher demand. If
her reported demand is such that she wins m objects, the market clearing price is ckni+1 m .
Thus, the most a bidder i can shift the market clearing price (conditional on winning at least
one object) is ckni+1 m ckni . 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,
8✏ > 0, P (ckni+1
m
ckni > ✏|✓i ) ! 0 as n ! 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 k n
highest bid submitted by all other bidders, ckni . The price the bidder would pay for her mth
unit is the kj +1 mth highest bid not submitted by herself, ckji+1 m . I show that Proposition
1 and Assumption 3 imply that this difference is negligible when n is large. In other words,
if all bidders play undominated strategies,
8✏ > 0, P (ckni+1
m
ckni > ✏|✓i ) ! 0 as n ! 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.
Proposition 3. Consider a series of auction environments {An }1
n=1 . For any ✏ > 0, there
⇤
⇤
exists a n 2 N, such that for all n > n , truthful reporting is an ✏ best reply to any undominated 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 (Assumption 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 . It says that if all
bidders bid truthfully, they pay approximately equal amounts in the uniform price auction
and the Vickrey auction when n is large.
Corollary 1. Consider a series of auction environments {An }1
n=1 where all bidders report
i
✓i
their preferences truthfully, B (✓) = p . For any ✏ > 0, there exists a n⇤ 2 N, such that for
all n > n⇤
i
i
i
i
P (tiU (p✓ , p✓ ) tiV (p✓ , p✓ ) > ✏) > 1 ✏.
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 , . . P
. , ✓n ), an P
outcome (x, t) 2 X ⇥ Rn is
1
1
ex-post Pareto efficient if @(x̃, t̃) 2 X ⇥ Rn such that i=1 t̃i
i=1 ti and
u(xi , ti , ✓i )
u(x̃i , t̃i , ✓i ) 8i
where the above inequality is strict for some i 2 {1, . . . , n}.
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 W ⇢ X ⇥ Rn the set
of all ex-post efficient allocations. We can define an analogous of ✏ ex-post Pareto efficient.
Definition 4. Given a profile of bidder types P
(✓1 , . . . , ✓nP
), an outcome (x, t) 2 X ⇥ Rn is ✏
1
1
ex-post efficient if @(x̃, t̃) 2 X ⇥ Rn such that i=1 t̃i
i=1 ti and
u(xi , ti , ✓i ) + ✏
u(x̃i , t̃i , ✓i ) 8i.
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 W(✏) ⇢ X ⇥ Rn the set of
all ✏ ex-post efficient allocations.
Notice that when bidders report their types truthfully, the uniform price auction is expost 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 k n + 1st highest bid.
Lemma 2. For any profile of types (✓1 , . . . , ✓n ) 2 ⇥n if bidders report demand truthfully, the
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.
Lemma 3. Fix ✏ > 0. Consider a series of auction environments {An }1
n=1 . There exists an
n⇤ 2 N, such that for all n > n⇤ ,
⇣
⌘
V,n ✓ 1
✓2
✓n
P
(p , p , . . . , p ) 2 W(✏) > 1 ✏.
5
Conclusion
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.
I show this in a more general setting than the literature typically assumes. In particular,
I allow for multidimensional heterogeneity across bidder preferences and demands, as well
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.
6
Appendix
Proposition 1
Proof. For this proof, I will simplify notation by letting U i (bi , b i ) be the utility of bidder
i when she bids bi and her opponents bid b i in the uniform price auction. I want to show
that U i (b̃i , b i )
U i (bi , b i ) 8b i 2 N 1 . Suppose that the (N 1)k competing bids are
described by the vector c i , where c1 i c2 i . . . c(Ni 1)k .
Case 1: bidder i wins the same number of objects if she bids bi and b̃i .
Suppose i wins m objects from bidding either bi or b̃i . If m = 0 then U i (b̃i , b i ) =
U i (bi , b i ) = ui (0, 0). If m 2 {1, . . . k}, then bim , b̃im ck i m+1 and ck i m bim+1 , b̃im+1 . If it is
the case that ck i m bim+1 , b̃im+1 , then the market clearing price is ck i m for either bid. Thus,
U i (b̃i , b i ) = U i (bi , b i ) = ui (m, wi mck i m ). If bim+1 , b̃im+1 ck i m , then bim+1 is the market
clearing price for either bid. Thus, U i (b̃i , b i ) = U i (bi , b i ) = ui (m, wi mbim+1 ). Finally if
bim+1 cik m > b̃im+1 , then ck i m is the market clearing price when bidder i bids b̃i and bim+1
is the market clearing price when bidder i bids bi . Thus,
U i (b̃i , b i )
ui (m, wi
mck
m)
> ui (m, wi
mbim+1 ) = U i (bi , b i ).
Case 2: bidder i wins more objects by bidding b̃i instead of bi .
16
This implies that, pi (1) = b̃i1 > bi1 . If not, then bij
b̃ij 8 j 2 {1, . . . , k} and i wins a
(weakly) greater number of objects by bidding bi instead of b̃i . Thus, b̃i strictly exceeds bi
only in the first dimension, as pi (1) = b̃i1 > bi1
bij
b̃ij 8j 2 {2, . . . k}. If i wins more
objects by bidding b̃i instead of bi , she then must win exactly one object by bidding b̃i and
zero goods by bidding bi , since she only submits a higher bid for her first good when bidding
b̃i instead of bi . Thus,
U i (b̃i , b i ) = ui (1, wi
max{b̃i2 , ck i })
ui (1, wi
pi (1)) = ui (0, wi ) = U i (bi , b i ),
where the first inequality follows because pi (1) = b̃i1 ck i if i wins one object and the equality
ui (1, wi pi (1)) = ui (0, 0) follows from the construction of pi .
Case 3: bidder i wins fewer objects by bidding b̃i instead of bi .
Let m be the number of objects i wins when bidding bi and m̃ the number she wins
when bidding bidding b̃i . By assumption m > m̃. This implies that bin > b̃in = pi (n) 8n 2
{m̃ + 1, . . . , m}. If not, bidding b̃i wins more than m̃ objects. Thus ck n bin = pi (n) for all
n 2 {m̃+1, . . . , m}. Since ck i n is weakly increasing in n and pi (n), then ck i m+1 pi (m̃+1). If
bids b̃i and wins m̃ objects, she pays pŨ = max{b̃im̃+1 , ck (m̃+1) }. Thus pi (m̃) pŨ pi (m̃+1).
If she bids bi and wins m objects, she pays pU where pU
pŨ . By the construction of pi , it
then follows that
U i (b̃i , b i ) = ui (m̃, wi
m̃pŨ )
ui (m, wi
mpŨ )
ui (m, wi
mpU ) = U i (bi , b i ).
Proposition 2
Proof. For this proof, I will simplify notation by letting U i (bi , b i ) be the utility of bidder
i when she bids bi and her opponents bid b i in the Vickrey auction. I want to show that
U i (b̃i , b i ) U i (bi , b i ) 8b i 2 N 1 . Suppose that the (N 1)k competing bids are described
by the vector c i , where c1 i c2 i . . . c(Ni 1)k .
Case 1: bidder i wins the same number of objects if she bids bi and b̃i .
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 b̃i instead of bi .
This implies that pi (1) = b̃i1 < bi1 . If not, then b̃ij
bij 8j 2 {1, . . . , k} and i wins
(weakly) fewer objects by bidding bi instead of b̃i . Thus, bi strictly exceeds b̃i only in the first
dimension, as bi1 > pi (1) = b̃i1 b̃ij bij 8j 2 {2, . . . k}. If i wins more objects by bidding bi
instead of b̃i , she then must win exactly one object by bidding bi and zero goods by bidding
b̃i , since she only submits a higher bid for her first good when bidding b̃i instead of bi . Thus
bi1 ck i pi (1) and
U i (b̃i , b i ) = ui (0, wi ) = ui (1, wi
pi (1))
ui (1, wi
ck i ) = U i (bi , b i ).
Case 3: bidder i wins more objects by bidding b̃i instead of bi .
Let m be the number of objects i wins when bidding bi and m̃ the number she wins
when bidding bidding b̃i . By assumption m̃ > m. This implies that b̃in = pi (n) > bin 8n 2
17
{m + 1, . . . , m̃}. If not, bidding bi wins more than m objects. Thus, i pays
P
i
i
i
i
she bids bi and m̃
j=1 ck+1 j when she bids b̃ . Notice also that b̃m̃ = p (m̃)
for all j 2 {1 . . . m̃}. By the definition of pi ,
ui (m̃, wi
m̃pi (m̃))
Pm
i
Note that mpi (m̃)
j=1 ck+1 j
positive wealth effects imply that
i
u (m̃, w
i
i
i
m̃p (m̃) + mp (m̃)
m
X
ui (m, wi
i
ck+1
i
ck+1
j)
i
u (m, w
i
j=1 ck+1 j when
i
i
ck+1
ck+1
m̃
j
mpi (m̃)).
0 since pi (m̃)
i
Pm
j
for all j 2 {1, . . . , m̃}. Thus,
i
i
mp (m̃) + mp (m̃)
j=1
m
X
i
ck+1
j ).
j=1
Which can be rewritten as
i
u (m̃, w
i
m
X
i
ck+1
j
(m̃
i
m)p (m̃))
i
u (m, w
i
j=1
m
X
j=1
i
i
Recalling again that pi (m̃) ck+1
j 8j 2 {1, . . . , m̃}, implies that (m̃ m)p (m̃)
P
P
m̃
i
i
Thus, (m̃ m)pi (m̃) + m
j=1 ck+1 j
j=1 ck+1 j . This implies that,
U i (b̃i , b i ) = ui (m, wi
(1)
i
i i
ck+1
j ) = U (b , b i ).
m̃
X
i
ck+1
j)
ui
m̃, wi
j=1
Combining 1 and 2 implies that U i (b̃i , b i )
m
X
j=1
i
ck+1
j
+ (m̃
Pm̃
m)pi (m̃)
i
j=m+1 ck+1 j .
!!
.
(2)
U i (bi , 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, . . . , h and w 2 R. Let p := max✓2⇥ p✓ (1)
and p := min✓2⇥ p✓ (1). Consider bidder i with preferences ✓i 2 supp(f n ). Suppose that all
bidders j 6= i play some undominated strategy B j . Propositions X and Y imply all bidders
report their first bid truthfully (in either auction). Since k n < n, this implies that p  ckni .
In addition, no bid exceeds p. If ⇥ is such that p = p, then Conditions U, V hold trivially as
p  ckni  c1 i  p = p. I now consider the nontrivial case where p < p.
Fix an ✏1 > 0. Consider any interval [p, p + ✏1 ] ⇢ [p, p]. Continuity and monotonicity of
0
preferences in ✓ implies that there exists an ✏ > 0 and ✓ 2 T such that p✓ (1) 2 (p, p+✏1 ) 8✓0 2
B✏ (✓). As n ! 1 assumption 3 implies that with probability approaching 1, there exists
a bidder j with ✓j 6= ✓i such that ✓j 2 B✏ (✓). Since ✏1 is arbitrary, I can equivalently
look at m intervals of the form (p + x ✏m1 , p + (x + 1) ✏m1 ) for x = 0, 1, . . . m 1. For each
interval, as n ! 1, with probability approaching 1, there exists a bidder j with ✓j 6= ✓i and
pj (1) 2 (p + x ✏m1 , p + (x + 1) ✏m1 ). Thus, as n ! 1, the probability that there are at least m
bidders who have types ✓ such that p✓ (1) 2 (p, p + ✏1 ) approaches 1. Since bidders report
their first bid truthfully, this implies there are at least m bids in this interval with probability
approaching 1.
18
Recall that ckni 2 [p, p]. In addition, for any ✏1 > 0, as n ! 1, the probability there
th
are at least m bids in an interval (p, p + ✏1 ) approaches 1. Thus, the k n highest bid and
the k n + 1 mth highest bid can be at most ✏1 apart with probability approaching 1. Or
equivalently,
P (ckni+1 m ckni > ✏|✓i ) ! 0 8✏ > 0 as n ! 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
AnU , bidder i has type ✓i and inverse demand pi . I let B i,n (✓ i,n ) denote the strategies of
bidders j 6= i in auction j. I assume all bidders j 6= i play undominated strategies. I let
U n (bi , b i,n , ✓i ) denote the utility of bidder i in auction AnU when she bids bi , her opponents
bid b i,n and she has type ✓i . First, I will show that for any ✏ > 0, when n is sufficiently
large, there is a 1 ✏ probability that bidder i
U n (pi , B
i,n
(✓
i,n
), ✓i ) + ✏ > U n (b, B
i,n
(✓
i,n
), ✓i ), 8b 2 , ✓i 2 ⇥.
Suppose that in auction An , bidder i reports pi and wins x 2 {1, . . . , k} objects. She pays
p  p per unit. Her utility is then u(x, xp, ✓i ), where
u(x, xp, ✓i )
u(y, yp, ✓i ) 8y 6= x.
If she underreports her preferences she wins y  x objects and pays a price p̃  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 ✏, bidder i
gets utility
u(y, y p̃, ✓i )  u(y, y(p ✏), ✓i )  u(x, x(p ✏), ✓i ).
Thus, with a probability of at least 1 ✏, the upper bound on the benefit of misreporting is
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,
U n (pi , B
i,n
(✓
i,n
), ✓i ) + ✏ > U n (b, B
i,n
(✓
i,n
), ✓i ), 8b 2 , ✓i 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 6= i.
Next, I show an equivalent result for the Vickrey auction. By Proposition 2, I only need
to check a bidder’s incentive to overreport her demand. Suppose that in auction AnV , bidder
i has type ✓i and inverse demand pi . I let B i,n (✓ i,n ) denote the strategies of bidders j 6= i
in auction j. I assume all bidders j 6= i play undominated strategies. I let V n (bi , b i,n , ✓i )
denote the utility of bidder i in auction AnV when she bids bi , her opponents bid b i,n 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
V n (pi , B
i,n
(✓
i,n
), ✓i ) + ✏ > V n (b, B
19
i,n
(✓
i,n
), ✓i ), 8b 2 , ✓i 2 ⇥.
Suppose that in auction n, if bidder i reports pi , she wins x 2 {1, . . . , m} objects. She
pays a total of P for the the x objects. Thus, her utility is u(x, P, ✓i ). Bidder i pays ckni x+1
for her xth object. Thus,
x 1
X
P =
ckni j  xckni x+1 .
j=0
Since bidder i reports her demand truthfully, the number of objects she wins is the number
of objects she demands when the price per unit is ckni x+1 . This means x is such that,
u(x, xckni
x+1 , ✓
i
u(y, yckni
)
x+1 , ✓
i
) 8y 2 {0, 1, . . . , m}.
If bidder i over reports her demand and still wins x objects, her payoff is unchanged. If bidder
i over-reports her demand and wins y > x objects she pays P 0 where P 0 P + (y x)ckni x+1 .
This is because bidder i faces an pays an increasing marginal price for all units. Thus, she
gets utility u(y, P 0 , ✓i ) from when she overreports her demand.
Condition V implies that for any ✏ > 0, there is a sufficiently large n such that there is
greater than a 1 ✏ probability that ckni x+1 ckni < ✏ for any x 2 {0, 1, . . . , m}. When this
holds true,
x 1
X
xckni x+1 (x 1)✏  P =
ckni j  xckni x+1 .
j=0
Recall that P 0 P +(y x)ckn x+1 . Let a(✏) = u(x, (xckni x+1 (x 1)✏), ✓i ) u(x, xckni x+1 , ✓i )
and b(✏) = u(y, (yckni x+1 (x 1)✏), ✓i ) u(y, yckni x+1 , ✓i ). Both a and b are continuous
in ✏ and a(0) = b(0) = 0. When ckni x+1 ckni < ✏, the benefit of misreporting is
i
u(y, P 0 , ✓i )
u(x, P, ✓i )  u(y, (yckni
u(x, xckni
1)✏), ✓i )
(x
x+1
x+1 , ✓
i
).
Rewriting the right hand side of the inequality, we get
b(✏) + a(✏) + u(y, yckni
x+1 , ✓
i
u(x, (xckni
)
x+1
(x
1)✏), ✓i ).
Notice that we can say u(y, yckni x+1 , ✓i ) u(x, (xckni x+1 (x 1)✏), ✓i )  0 since we
have already know u(x, xckni x+1 , ✓i ) u(y, yckni x+1 , ✓i ). We can use this to say that the
benefit of misreporting is bounded above by b(✏) + a(✏),
u(y, P 0 , ✓i )
u(x, P, ✓i )  a(✏) + b(✏).
Thus when n is sufficiently large, there is a 1 ✏ probability that the benefit of misreporting
is at most a(✏) + b(✏). Since this holds for an arbitrary ✏ > 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 ✏. Or equivalently,
when n is sufficiently large, there is at least a 1 ✏ probability that,
V n (pi , B
i,n
(✓
i,n
), ✓i ) + ✏ > V n (b, B
i,n
(✓
i,n
), ✓i ), 8b 2 , ✓i 2 ⇥.
Since the benefit of misreporting is bounded for any n, then for any ✏ > 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 6= i.
20
Corollary 1
Proof. This follows directly from conditions U, V . Consider bidder i. Fix ✏ > 0. Pick an n
sufficiently large such that
P (ckni+1
ckni <
m
✏ i
|✓ ) > 1
m
✏ 8✓i 2 ⇥.
Suppose that ckni+1 m ckni < 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
that bidder i wins x 2 {1, . . . , m} objects in each auction. She pays tiU (p✓ , p✓P) = xpU in
i
i
the uniform price auction, where ckni  pU  ckni+1 m . She pays tiV (p✓ , p✓ ) = xi=1 ckni+1 x
in the Vickrey auction. Thus,
i
tiU (p✓ , p✓
i
)
i
tiV (p✓ , p✓
i
)=
x
X
ckni+1
pU
x
.
i=1
Yet pU
ckni+1
x
<
✏
m
because we assume ckni+1
i
tiU (p✓ , p✓
i
)
i
tiV (p✓ , p✓
i
)=
ckni <
m
x
X
pU
✏
.
m
ckni+1
Thus,
x
i=1
This holds whenever ckni+1
m
✏
.
m
ckni <
i
i
P (tiU (p✓ , p✓ )
x
✏
 ✏.
m
Thus, when n is sufficiently large,
i
i
tiV (p✓ , p✓ ) > ✏) > 1
✏.
Lemma 3
Proof. Fix ✏ > 0. Suppose n is sufficiently large such
i
i
i
i
P (tiU (p✓ , p✓ )
i
i
i
i
tiV (p✓ , p✓ ) < ✏) > 1
✏.
i
i
Suppose that tiU (p✓ , p✓ ) tiV (p✓ , p✓ ) < ✏ 8i . For ease of notation I write tiV = tiV (p✓ , p✓ )
i
i
and xiV = xiV (p✓ , p✓ ).
any other outcome (x̃, t̃) 2 X ⇥ Rn s.t. (xV , tV ) 6= (x̃, t̃). If x̃i = xiV 8i and
P Consider
P
t̃i
tiV , then t̃i
tiV . Thus, some bidder i is made worse off in allocation (x̃, t̃). She
receives the same number of goods, yet makes a higher transfer.
instead that (x̃, t̃) 2 X ⇥ Rn and x̃i 6= xiV for some i. Suppose also that
P Suppose
P
t̃i
tiV and
u(x̃i , t̃i , ✓i ) u(xiV , tiV , ✓i ) 8i.
This implies that if x̃i  xiV , then t̃i  tiV . Since truthtelling is efficient in the uniform price
auction and both auctions have the same winning rule,
u(xiV , pU xiV , ✓i )
u(y, pU y, ✓i ) 8y 2 0, . . . , m,
21
where pU is the market clearing price in the uniform price auction. Since we assume
i
i
i
i
tiU (p✓ , p✓ ) tiV (p✓ , p✓ ) < ✏ 8i , we can say that for all i, 9✏i < ✏ such that tiV = tiU + ✏i .
Thus if x̃i < xiV , then
u(x̃i , t̃i , ✓i )
implies that t̃i  tiV
u(xiV , tiV , ✓i ) = u(xiV ,
x̃i ) since
pU (xiV
u(xiV , pU xiV , ✓i )
and,
u(xiV ,
tiU + ✏i , ✓i ),
tiU + ✏i , ✓i ) = u(xiV ,
u(y, pU y, ✓i ) 8y 2 0, . . . , m,
pU xiV + ✏i , ✓i )
(pU y + ✏i ) , ✓i ),
u(y,
where the final inequality follows from thePpositive
wealth effects assumption. Thus, x̃i > xiV
P
implies that t̃i tiV + pU (x̃i xiV ) since
t̃i
tiV . Suppose that i is such that x̃i > xiV .
Note that
u(xiV , pU xiV , ✓i ) u(y, pU y, ✓i ) 8y 2 0, . . . , m.
Let ai (s) = u(xiV , s
ai (0)  0. Thus
u(xiV , ✏i
pU xiV , ✓i )
u(x̃i , s
pU x̃i , ✓i ). Note that ai (s) is continuous in s and
pU xiV , ✓i ) + ai (✏i ) = u(x̃i , ✏i
pU x̃i , ✓i )
u(x̃i , t̃i , ✓i ).
Or equivalently,
ai (✏i )
u(x̃i , t̃i , ✓i )
u(xiV , ✏i
pU xiV , ✓i ) = u(x̃i , t̃i , ✓i )
u(xiV , tiV , ✓i ).
Since ai is continuous with ai (0)  0, and ✏i < ✏, it follows that for any ✏˜ > 0, there is an ✏
sufficiently small such that
✏˜
u(x̃i , t̃i , ✓i )
u(xiV , tiV , ✓i ).
Thus, thus (xV , tV ) 2 W(˜✏) with probability greater than 1
✏.
References
[1] Lawrence M. Ausubel. An efficient ascending-bid auction for multiple objects. American
Economic Review, 94(5):1452–1475, 2004.
[2] Lawrence M. Ausubel and Peter Cramton. Demand reduction and inefficiency in multiunit auctions. Working paper, 2002.
[3] Lawrence M. Ausubel, Peter Cramton, Marek Pycia, Marzena Rostek, and Marek
Weretka. Demand reduction and inefficiency in multi-unit auctions. Review of Economic Studies, Forthcoming, 2014.
[4] Lawrence M. Ausubel and Paul Milgrom. The lovely but lonely vickrey auction the
lovely but lonely vickrey auction the lovely but lonely vickrey auction the lovely but
lonely vickrey auction. SIEPR Discussion Paper No. 03-36 SIEPR Discussion Paper
No. 03-36, 2006.
22
[5] Eduardo M. Azevedo and Eric Budish. Strategy-proofness in the large. Working paper,
2013.
[6] Brian Baisa. Auction design without quasilinear preferences. Working paper, 2014.
[7] Yeon-Koo Che and Ian Gale. Revenue comparisons for auctions when bidders have
arbitrary types. Theoretical Economics, pages 95–118, 2006.
[8] Shahar Dobzinski, Ron Lavi, and Noam Nisan. Multi-unit auctions with budget limits.
Games and Economic Behavior, 74(2):486–503, 2012.
[9] Isa E. Hafalir, R. Ravi, and Amin Sayedi. A near pareto optimal auction with budget
constraints. Games and Economic Behavior, 74:699–708, 2012.
[10] Bengt Holmström. Groves’ scheme on restricted domains. Econometrica, 47(5):1137–
1144, 1979.
[11] Matthew Jackson and Ilan Kremer. The relevance of a choice of auction format in a
competitive environment. Review of Economic Studies, 73:961–981, 2006.
[12] Vijay Krishna. Auction Theory. Academic Press, 2002.
[13] Eric Maskin and John Riley. Optimal auctions with risk averse buyers. Econometrica,
52(6):1473–1518, November 1984.
[14] Steven A. Matthews. Selling to risk averse buyers with unobservable tastes. Journal of
Economic Theory, 30(2):370–400, 1983.
[15] Paul Milgrom. Putting Auction Theory to Work. Cambridge University Press, 2004.
[16] Hiroki Saitoh and Shigehiro Serizawa. Vickrey allocation rule with income effect. Economic Theory, 35(2):391–401, 2008.
[17] Toyotaka Sakai. Second price auctions on general preferenc domains: two characterizations. Economic Theory, 37:347–356, 2008.
[18] Jeroen Swinkles. Asymptotic efficiency for discriminatory asymptotic efficiency for discriminatory private value auctions. Review of Economic Studies, 66:509–528, 1999.
[19] Jeroen Swinkles. Efficiency of large private value auctions. Econometrica, 69(1):37–68,
2001.
[20] William Vickrey. Counterspeculation, auctions, and competitive sealed tenders. Journal
of Finance, 16(1):8–37, 1961.
[21] Steven Williams. A characterization of efficient, bayesian incentive compatible mechanisms. Economic Theory, 14(1):155–180, 1999.
23