Auction Design without Quasilinear Preferences - Amherst College

55 Pages·2013·1.44 MB·English

Auction Design without Quasilinear Preferences
Brian Baisa
⇤†
August 6, 2013
Abstract
I analyze private value auction design and assume only that bidders are risk
averse and have positive wealth effects (i.e. the good is normal). I show removing
the standard quasilinearity restriction leads to qualitatively different solutions to
the auction design problem with respect to both efficiency or revenue maximiza-
tion.
On efficiency, I show that probabilistic allocations of the good can Pareto
dominate the second price auction; and there is no dominant-strategy mechanism
that is both Pareto efficient and individually rationality.
On revenue, I construct a probability demand mechanism with greater ex-
pected revenues than standard auctions when there are sufficiently many bidders.
In addition, I take a new approach to studying bid behavior when types are mul-
tidimensional. Instead of characterizing bidder’s interim incentive constraints,
I place bounds on their bids, and I show that these bounds are sufficient for
obtaining revenue comparisons.
Keywords: Auctions; Multidimensional Mechanism Design; Wealth Effects.
JEL Classification: C70, D44, D82.
⇤[email protected], Amherst College, Department of Economics.
†I gratefully acknowledge financial support received from the Cowles Foundation Fellowship, the Yale
School of Management, the International Center for Finance and the Whitebox Advisors Fellowship. I
received excellent comments on this work from seminar and conference participants and faculty at Yale,
Notre Dame, UNC, Johns Hopkins, Haverford College, Amherst College, Cambridge, the University of
Michigan, Tel Aviv University, USC, Stony Brook and Lund. I am especially grateful to Dirk Bergemann,
Benjamin Polak, Larry Samuelson and Johannes Hörner for numerous conversations and encouragement
while advising me with this project. I would also like to thank Cihan Artunç, Lint Barrage, Yeon-Koo
Che, Eduardo Faingold, Amanda Gregg, Yingni Guo, Vitor Farinha Luz, Drew Fudenberg, Adam Kapor,
Phil Haile, Sofia Moroni, David Rappoport, Peter Troyan and Kieran Walsh for helpful comments and
conversations.
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1 Introduction
In the auction design literature, it is standard to assume that bidders have quasilinear
preferences. Yettherearemanywell-knownenvironmentsinwhichthisrestrictionisviolated:
bidders may be risk averse, have wealth effects, face financing constraints or be budget
constrained. In this paper, I study the canonical independent private value auction setting
forasinglegoodanddropthequasilinearityrestrictionbyassumingonlythatbiddersarerisk
averse and have positive wealth effects (i.e., the indivisible good for sale is a normal good).
I show that the auction design problem leads to qualitatively different prescriptions relative
to those of the quasilinear benchmark. Instead of using standard auctions where the good is
giventothehighestbidderwithprobabilityone,theauctioneerprefersmechanismswhereshe
can allocate the good to one of many different bidders, each with strictly positive probability.
For auctioneers concerned with efficiency, such probabilistic allocations can Pareto dominate
the second price auction. And for auctioneers concerned with maximizing expected revenues,
Iconstructaprobabilitydemandmechanismthatgeneratesstrictlygreaterexpectedrevenues
than standard auction formats when there are sufficiently many bidders.
There are many examples of auctions where the quasilinearity restriction does not hold.
One case is housing auctions. Housing auctions have developed into a multi-billion dollar
industry in the United States. In Melbourne, Australia an estimated 25 50% of homes
�
are sold via auction (Mayer (1998)). In a housing auction, a buyer’s bidding strategy is
influenced by factors like how much wealth she has (wealth effects) and the terms of the
mortgage offered by her bank (financing constraints). These factors are not included in the
quasilinear model, where a bidders payoff type is described only by a single dimensional
variable - her valuation for the good.
For another example, consider firms bidding on spectrum rights or oil tracts. The cor-
porate finance literature shows that many firms have an internal spending hierarchy (see
Fazzari, Hubbard and Petersen (1988)). Specifically, it is more expensive for the firm to
use external financing than internal financing, because firms pay higher interest on money
borrowed from third parties. A firm may be able to place a relatively low bid in an auction
without needing external financing, but in order to place a relatively high bid, the firm may
need to obtain external financing and pay a higher interest rate on this debt. Increasing
its bid by one dollar using external financing is more costly to the firm than increasing the
bid by one dollar via internal financing. Even if the firm is risk neutral, this financing con-
straint makes them behave as though they have declining marginal utility of money. Further
evidence of risk aversion in auctions is discussed in the related literature section.
I first study the problem of an auctioneer concerned with efficiency. In the single good en-
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vironmentwithprivatevaluesandquasilinearpreferences, thesecondpriceauctionispopular
because it implements a Pareto efficient allocation in dominant strategies. I show that with-
out quasilinearity, there are probabilistic allocations of the good that Pareto dominate the
dominant strategy equilibrium outcome of the second price auction (Proposition 1). Is there
another dominant strategy mechanism that is efficient in this more general environment? I
show that the answer is no: there is no individually rational dominant strategy mechanism
that satisfies budget balance and implements a Pareto efficient allocation (Proposition 3).
While I obtain an impossibility result on efficiency, I show more positive results for
revenue maximization. Specifically, I show that the auctioneer can use randomization to
increase revenues. This may seem counterintuitive because bidders are risk averse, but the
intuition follows directly from the assumption that the good is normal. Since the good is
normal, a bidder’s willingness to pay for it increases as her wealth increases. Similarly, her
willingness to pay for any giving probability of winning the good increases as her wealth
increases. It follows that the bidder is willing to pay the highest price for her first marginal
‘unit’ of probability of winning, when she has still yet to spend any money and her wealth
is the highest. Thus, the bidder is willing to buy a small probability of winning the good
at a price per unit of probability that exceeds her willingness to pay for the entire good.
Standard auctions that allocate the good to the highest ‘bidder’ do not exploit this property
of bidder risk preferences.
I construct a probability demand mechanism that uses lotteries to better exploit this
feature of bidder preferences. The mechanism sells probabilities of winning the good like a
divisible good that is in net supply one. Bidders report a demand curve over probabilities of
winning. The curve reports the probability of winning the bidder demands (Q) for a given
price per unit price of probability (P). The auctioneer uses an algorithm similar to that of
the Vickrey auction for a divisible good to determine each bidder’s probability of winning
and expected transfer.
Without quasilinearity, it becomes more difficult to characterize bidder behavior. Now, a
bidder’s private information is described by a utility function instead of a single dimensional
valuation. This multidimensionality complicates mechanism design problems. Armstrong
and Rochet (1999) show that even in the simplest of principal-agent models with multidi-
mensional types, explicitly solving for equilibria can be difficult. I take a different approach
to characterizing bid behavior in my probability demand mechanism. Instead of explicitly
solving for equilibria, I show that we can bound what the bidder reports to the auctioneer
by eliminating dominated strategies. In particular, I show that it is a dominated strategy
for a bidder to underreport her type (Proposition 4).
I use this bound on a bidders’ reports to construct a lower bound on expected revenues
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in the probability demand mechanism. With enough bidders, this lower bound on revenues
strictly exceeds an analogously constructed upper bound on the revenues from any standard
auction. That is, with enough bidders the probability demand mechanism has higher ex-
pected revenues than any standard auction (Propositions 6 and 7). This class of standard
auctions includes the first price, second price and all pay auctions, as well as modifications
of these formats to allow for entry fees and/or reserve prices.
The rest of the paper proceeds as follows. The remainder of the introduction relates my
work to the current literature on auction design. Section 2 describes the model and specifies
the assumptions I place on bidders’ preferences. Section 3 motivates the use of probabilistic
allocations and provides an example in which the dominant strategy equilibrium outcome
of the second price auction is Pareto dominated by a probabilistic allocation of the good.
Section 4 shows there is no symmetric mechanism that respects individual rationality and
implements a Pareto efficient allocation in dominant strategies. Section 5 outlines the con-
struction of the probability demand mechanism. Section 6 focuses on revenue comparisons
between the probability demand mechanism and standard auction formats. Section 7 pro-
vides a numerical example illustrating the practical applicability of my results. Section 9
concludes.
Related literature
This paper builds on the private value auction literature pioneered by Vickrey (1961) and
later by Myerson (1981) and Riley and Samuelson (1981). Each of these papers addresses
an auction design problem under the restriction that bidders’ preferences are quasilinear.
Myerson studies the problem of an expected revenue maximizing auctioneer, while Vickrey
studiestheproblemofanauctioneerconcernedwithefficiency. Theirresultsshowtheauction
design problem is solved by a mechanism where the good is assigned to a bidder only if she
has the highest valuation among all bidders.1
However, there is evidence that in many auction settings, bidders do not have quasilinear
preferences. Prior work has argued that risk aversion can explain deviations in bidders’
behavior, relative to the predictions of the quasilinear benchmark. Bajari and Hortaçsu
(2005) argue that risk aversion serves as a reasonable explanation of experimentally observed
bid behavior. Specifically, they show that it serves as a better explanation than three other
competing theories - one of which is the quasilinear benchmark. Similarly, Goeree, Holt
and Palfrey (2002) show risk aversion provides a parsimonious explanation of experimentally
observed overbidding in first price auctions. Buddish and Takeyama (2001) cite risk aversion
1In Myerson (1981), the good is assigned to the bidder with the highest ‘virtual’ valuation. Under the
monotone likelihood ratio property, this corresponds to the bidder with the highest valuation.
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as an explanation for why sellers use buy-it-now prices. Li and Tan (2000) cite risk aversion
as an explanation for why sellers use secret reserve prices.
In the theoretical auctions literature, while most work focuses on the quasilinear case,
there is a smaller literature that considers auctions without quasilinearity. One of the first
works in this literature is by Maskin and Riley (1984). Their paper characterizes certain
properties of expected revenue maximizing auctions when a bidder’s type is a single dimen-
sional variable, ✓ [0,1]. They show that the exact construction depends on the common
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prior over the distribution of types and the functional form of the bidders’ utility functions.
Matthews (1983, 1987) and Hu, Matthews and Zhou (2010) also study auctions without
quasilinearity. However, they focus on a more structured setting where all bidders have
identical risk preferences and there are no wealth effects. Within this framework, they make
comparisons between auction formats.
Che and Gale (2006) consider a payoff environment that is closer to the one studied here.
They assume that bidders have multidimensional types and use this to show that when
bidders are risk averse, the first price auction generates greater revenues than the second
price auction. This builds on their earlier work, which studies standard auction formats with
more specific departures from the quasilinear environment (see Che and Gale (1996, 1998,
2000)).
My work differs from this prior work in two respects: (1) I consider a very general
setting that allows for heterogeneity in bidders across many dimensions; and (2) I study the
problem from an auction design perspective. This contrasts with prior work that focuses on
comparisons between standard auction formats.
My solution to the auction design problem uses probabilistic allocations of the good.
Randomization has been advocated in other auction design settings. Celis, Lewis, Mobius
and Nazeradeh (2012) construct a randomized mechanism they call the buy-it-now or take-
a-chance mechanism. They study a setting where bidder preferences are quasilinear. They
show that using randomization can induce more aggressive bidding in ‘thinner’ markets. For
the case of budget-constrained bidders, Pai and Vohra (2010), Maskin (2000) and Laffont
and Robert (1996) show that the expected revenue-maximizing mechanism may employ
probabilistic allocations. Baisa (2012) studies an auction design problem in which bidders
use a non-linear probability weighting function to evaluate risks (as described by Kahneman
and Tversky (1979, 1992)) and finds that the solution to the auction design problem uses
probabilistic allocations of the good to exploit features of bidders’ probability weighting
functions. In this paper, I construct a mechanism that uses randomization and is almost
revenue maximizing with many bidders.
My results can be seen as analogous to those shown of Armstrong (1999). Armstrong
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studies a different problem, that of a multi-product monopolist selling to a representative
consumer. He derives an almost optimal solution for the monopolists when there are many
products.
Outside of auction design, Morgan (2000) advocates using lotteries to raise money for
publicgoods. TheintuitionforusinglotteriesinMorgan’smodelisdistinctfromtheintuition
used here. Morgan shows that lottery tickets can help to overcome the free rider problem
when people’s preferences are linear in money. No such free rider problem is present in my
model.
2 The model
2.1 The payoff environment
I consider a private value auction setting with a single risk neutral seller and N 2 buyers,
�
indexedbyi 1,...,N}. There isa singleindivisible objectforsale. Bidderi’spreferences
2{
are described by utility function u , where
i
ui : 0,1 R R.
{ }⇥ !
I let u (1,w ) denote bidder i’s utility when she owns the object and has wealth w . Similarly,
i i i
u (0,w ) denotes bidder i’s utility when she does not own the object and has wealth w . The
i i i
object is a “good” and not a “bad”; it is better to have the object than to not for any fixed
wealth level, i.e.
u (1,w) > u (0,w), w.
i i
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I additionally assume that bidders preferences are strictly increasing and twice continuously
differentiable in wealth.
This environment differs from the quasilinear case where a bidder’s preferences are com-
pletely described by a one-dimensional signal, her ‘valuation’ of the object. This valuation
is the most i is willing to pay for the object. In this setting, I define k(u ,w ) as bidder
i i
i’s willingness to pay for the good when she has an initial wealth w and a utility function
i
u . This is the highest price bidder i will accept as a take-it-or-leave-it offer for the good.
i
Formally k(u ,w ) is defined as,
i i
u (1,w k) = u (0,w ). (2.1)
i i i i
�
I place only two assumptions on bidder preferences. The first is that the good is a normal
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good and the second is that bidders are risk averse.
I assume that the good being sold is a normal good (i.e. positive wealth effects). My
notion of positive wealth effects is analogous to the notion in the divisible goods case, where
a bidder’s demand for the good increases as her wealth increases for a constant price level.
Assumption 1. (Positive wealth effects)
Bidder i has positive wealth effects:
@k(w,u )
i
> 0,
@w
for all w.
Second, I assume that bidders have declining marginal utility from money.
Assumption 2. (Risk Averse)
Bidder i has declining marginal utility of money:
@2u (x,w)
i
< 0 for x = 0,1.
@w2
AIndd,efine as the set of all utility functions which satisfy Assumptions 1 and 2. Notice
U
that quasilinear preferences are not included in since @k(w,ui) = 0 and @2ui(x,w) = 0.
U @w @w2
Note that this is a very general setting. The only two assumptions I place on bid-
der preferences are positive wealth effects and risk aversion. I do not place any functional
form restrictions on preferences. I also allow for multidimensional heterogeneity across bid-
ders. This differs from most of the literature on risk aversion and auctions, where bidders
preferences are described by specific functional forms and there is only single dimensional
heterogeneity between bidders (see Maskin and Riley (1984), Mathews (1983, 1987)).
There are many different functional forms of u that fit the risk aversion and normal good
assumptions. Two examples are given below.
Example 1. If bidder i has preferences of the form ui(x,w) = gi(viIx=1+fi(w)) where both
fi and gi are concave functions and I is an indicator function, then ui . A simple example
2U
would be the case where g is linear. We can then interpret this as saying bidder i’s gets
i
a certain number of utils from owning the good, irrespective of her wealth level. She also
has declining marginal utility of money, which is unaffected by whether or not she owns the
good.
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Example 2. The good being sold is a risky asset. It has a distribution of (monetary)
returns g : R+ R+. Bidder i gets utility from wealth of fi. She has decreasing absolute
!
risk aversion - for example, CRRA utility. Her expected utility of owning the asset when she
starts with wealth w is u (1,w) where
i
u (1,w) = f (w+x)g(x)dx.
i ˆ i
Without the asset, she consumes her wealth,
u (0,w) = f (w).
i i
In this example, bidder i could be a speculator bidding for a home. She knows this distribu-
tion of resale prices for the house is given by g. As she becomes richer, her degree of absolute
risk aversion decreases. This implies that she demands a smaller risk premium when buying
the home. Thus, as she becomes richer her willingness to pay for the home increases.
2.2 Allocations and mechanisms
By the revelation principle, I can limit attention to direct revelation mechanisms. A mecha-
nism describes how the good is allocated and how transfers are made. I define A as the set
of all feasible assignments, where
N
A := a a 0,1 N and ai 1 ,
{ | 2{ }  }
i=1
X
where a = 1 if bidder i is given the object. A feasible outcome � specifies both transfers
i
and a feasible assignment: � A RN. I define �:= A RN as the set of feasible outcomes.
2 ⇥ ⇥
A (probabilistic) allocation is a distribution over feasible outcomes. Thus, an allocation ↵ is
an element of �(�).
I use the notation E↵[ui,wi] to denote the expected utility of bidder i under allocation
↵ �(�) when she has preferences u and initial wealth w . I refer to the auctioneer as
i i
2
person 0 and use E↵[u0] to denote the expected transfers she receives under allocation ↵.
Note that the auctioneer does not value owning the good.
In Section 4, I study mechanisms that implement Pareto ffiecient allocations. In the
quasilinear benchmark, this problem simplifies to ensuring that the bidder with the highest
valuation receives the good. Without quasilinearity, the description of a Pareto efficient
allocation is more complicated.
Icallanallocationposterior Pareto efficientifforagivenprofileofpreferences, increasing
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one person’s expected utility necessarily decreases another person’s expected utility (includ-
ing the auctioneer). If a benevolent social planner knew each bidder’s private information,
she would select a posterior Pareto efficient allocation.
Definition 1. (Posterior Pareto efficient allocations)
Assume bidders have preferences u = (u ,...,u ) N and initial wealth levels, w =
1 N
2U
(w ,...,w ). A feasible allocation ↵ is posterior Pareto efficient if there does not exist a
1 N
↵ �(�) s.t.
0
2
E↵ [ui,wi] E↵[ui,wi] i = 0,1...N.
0 � 8
where the above inequality holds strictly for at least one i.
This is equivalent to the notion of an ex-post Pareto superior (efficient) allocation given
in Holmström and Myerson (1983, 1991). I use the term posterior as my definition mirrors
Laffont and Green’s (1987) notion of posterior implementability. In words, this means condi-
tional upon knowing all of the bidders private information, we cannot increase one person’s
expected payoff without decreasing another person’s.
There are differences between posterior Pareto efficient allocations and the common no-
tionsofex-ante, interimand ex-postPareto efficiency. Itisimpossibleto design a mechanism
thatimplementsanex-anteorinterimParetoefficientallocationwhenbiddersareriskaverse.
For example, if an allocation is ex-ante Pareto efficient (by the definition of Myerson (1991)),
because the bidders are risk averse, their payments to the auctioneer can not depend on re-
alized types. When the bidder’s payment is independent of her report, she has no incentive
to tell the truth.
At the same time, my posterior Pareto efficiency differs from the typical notion of ex-
post Pareto efficiency. It allows the designer to randomly allocate the good to one of many
different bidders. It does not necessarily ensure that a bidder with the highest willingness to
pay for the good is given the good with certainty. Instead, using randomization can actually
increase everyones payoff.
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(ex-post)
(interim) Good is allocated,
Bidders report their payments are made,
private information game is over.
(ex-ante) (posterior)
Types are drawn Conditional on all bidders
private information, auctioneer
determines payments and the
allocation of the good
(possibly randomly).
Figure 2.1: Notions of efficiency
AdirectrevelationmechanismM mapsaprofileofreportedpreferencesandinitialwealth
levels to an allocation. That is,
M : N RN �(�).
U ⇥ !
The direct revelation mechanism implements an posterior Pareto efficient allocation if for
any reported preferences u = (u ...u ) and initial wealth levels w = (w ...w ), the
1 N 1 N
allocation provided by M is Pareto efficient. That is, the mechanism specifies an allocation
that would be Pareto efficient in the complete information environment where all bidders’
private information is known.
A mechanism is implementable in dominant strategies if it is a best response for each
bidders to truthfully report her private information to the auctioneer, for any possible report
of her opponent.
Definition 2. (Dominant strategy incentive compatibility)
A direct revelation mechanism M is dominant strategy incentive compatible if for any profile
of preferences and initial wealth levels u = (ui,u i) N and w = (wi,w i) RN,
� 2U � 2
EM(u,w)[ui,wi] � EM((u0,u i),(w0,w i))[ui,wi],
� �
for any (u0,w0) R and i = 1...N.
2U⇥
3 Probabilistic allocations
I now show that auctioneers concerned with efficiency or revenue maximization should use
mechanismsthatallowforprobabilisticallocationsofthegood. Suchprobabilisticallocations
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