First-Price Auctions with Budget
Constraints*†
Maciej H. Kotowski‡
March 23, 2013
Abstract Consider a first-price sealed-bid auction
where participants have affiliated valuations
and private budget constraints; that is, bidders have private multidimensional types. This
article investigates the existence and nature of monotone equilibria in this setting. Hard
budget constraints introduce two competing effects on bidding. The direct effect
depresses bids as participants hit their spending limit. The strategic effect encourages
more aggressive bidding by participants with large budgets. Together these effects can
yield discontinuous equilibrium strategies stratifying competition along the budget
dimension. The strategic consequences of private budget constraints can be a serious
confound in interpreting bidding behavior in auctions.
Keywords: First-Price Auctions, Budget Constraints
JEL: D44
*This
draft is made available to stimulate discussion and to gather feedback. It has not benefited from a formal
peer review. Please do not (re)post this article. Please contact the author for the latest draft before citing. All
comments and suggestions are greatly appreciated.
†I thank Benjamin Hermalin, John Morgan, Chris Shannon, Adam Szeidl, Steven Tadelis, and especially
Shachar Kariv. I am grateful for the feedback of numerous seminar audiences. This paper supersedes and
corrects results and discussion originally communicated in Kotowski (2010)and included in Kotowski(2011). I
accept responsibility for any remaining errors or omissions.
‡Kennedy School of Government, Harvard University. 79 JFK Street, Cambridge, MA 02138. E-mail:
maciej_kotowski@hks.harvard.edu.
1
Budget constraints are a fundamental feature of many auctions. For example, bidders
may be buying real estate. A house’s attractiveness is partly idiosyncratic and all potential
buyers face a private spending limit determined by a financial institution. Problems securing
a large loan may prevent a buyer from placing a competitive bid on a desirable property.
Oncebiddershave multidimensional privateinformationcomposedofavalue-signalwhich
informs their preferences, and a budget, which defines their feasible strategy set, even
simple auctions feature nontrivial equilibrium behavior. For concreteness, consider a
first-price, sealed-bid auction for one item. Bidders simultaneously submit their best offer to
the seller and the highest bidder wins. Only the winner pays her own bid. In this
straightforward setting, private budgets introduce two effects on equilibrium bidding.
Budgets have a direct effect constraining bidders; however, they also introduce a strategic
effect whereby highbudget bidders deliberately outbid high-valuation opponents who have
the misfortune of being budget-constrained. The strategic effect emerges as a discontinuity
in the equilibrium strategy and it endogenously stratifies competition within the auction along
the budget dimension. These effects change the strategic interaction of the first-price
auction with consequences for its efficiency, revenue potential, and bidders’ perception of
the winner’s curse. Even the existence of a monotone equilibrium becomes a delicate
matter.
1Inthis
studywe investigate equilibrium bidding infirst-price auctionswith privatebudget
constraints. Our analysis considers both the existence of a well-behaved equilibrium and the
strategic implications of equilibrium bidding. While our general model accommodates
interdependent and affiliated values and continuously distributed types, we begin with two
examples highlighting the novelties in our environment. Examples 1 and 2 extend the
classic Vickrey (1961) model.As we argue below, the issues they raise apply generally.
Example 1. There are two risk-neutral bidders with private values. Each bidder receives a
signal Si i.i.d.~ U[0,1] regarding her own value for the item. If a bidder with value-signal
Si=siwins by bidding bi, her payoff is si-bi; otherwise, it is zero. Ties are resolved with a fair
coin flip. Suppressing bidder subscripts, the function a(s) = s/2 is the symmetric
Bayesian-Nash equilibrium bidding strategy.
Suppose, additionally, that each bidder has a private budget wi, which is independent of her
value-signal. With probability 1/2, a bidder has a budget of w= 1/4 and with probability 1/2it
is wi=3/4. We assume that a bidder of type (si,wii)cannot bid above
1Textbook
treatments and other extensions to Vickrey’s model are found in Krishna (2002) and Menezes &
Monteiro (2005). Milgrom (2004, p. 132) studies an extreme variant of Example 1 where all bidders have a low
budget with certainty.
2
ß(si, 3 )4
)
bi
12
ß(si,
14
14
0
'1 ˆsˆs
si
Figure 1: The strategy equilibrium in Example 1. High-budget bidders
increase their bid discontinuously to 1/4at ˆs. Low-budget bidders must bid
below 1/4.
irrespective of si. Hence,
a(si)=si
Figure 1 presents a sketch of the new symmetric equilibrium ß(si,wi). Its
definition is
si
i =3/4
if si ∈[0,ˆs]
,
5+9s2 i
]and w =1/4
18+18si
1+9s
'∈(ˆs,ˆs
if si
6+18si
ß(si,wi)=
if si
i
'
14
3
'
∈
(
ˆ
s
wi /2can no longer be an equilibrium strategy since a low-budget bidder cannot afford to
bid above 1/4.
2i
if si ∈(ˆs,1]and w
=1/4
i
,1]and w
2
=11/27.
w
The proof that ßis an equilibrium is very simple, but long. It is presented in
here ˆs=1/3and ˆs online appendix.
Example 1 conveys two main points concerning equilibrium behavior. First, the
equilibrium strategy features a prominent discontinuity at ˆs. The intuition for this
discontinuity rests on the strategic tradeoff made by a high-budget bidder in equilib
A high-budget bidder always has the option of placing a bid above 1/4. Exercising
option, which is worthwhile only for a bidder with a valuation greater than ˆs,
substantially increases the probability of a win as she is guaranteed to outbid a
low-budget competitor. Private budget constraints may therefore encourage more
aggressive bidding among certain auction participants than would occur in a world
without budget limits. The intuition for the discontinuity
at ˆs'is similar; however, such bidders may tie with others rather than win outright. Second,
although the discontinuities imply higher bids by some types of bidders, on
ithe
margin bids are often less brazen than if budget constraints were absent. In Figure 1,
the slope of ß(·,w) declines moving past ˆs. The intuition here hinges on the endogenous
stratification of competition along the budget dimension. For instance, bidders placing a bid
above 1/4are competing on the margin only against other high-budget opponents. The
decline in competition reduces the incentive to bid aggressively at distinct budget levels. A
similar intuition holds for low-budget bidders when si'∈(ˆs,ˆs]. Although the above lessons
appear to be specific to Example 1, or are perhaps an artifact
of the discrete budget distribution, this is not the case. The strategic effects, and the
associated discontinuities, can arise even when budgets assume a continuum of values.
While the construction is more elaborate, and we detail it below, the intuition is fully
captured by the reasoning in Example 1.
The strategic effects seen in Example 1 are just one of the new issues raised by budget
constraints. A more subtle matter concerns how the private wealth-effects associated with
budgets may interact with preferences, a topic previously unexplored in the auction
literature. Seemingly unintuitive effects may arise, with implications for equilibrium
existence. For example, consider risk-aversion. Suppose that increasing a bidder’s wealth
decreases her aversion to risk. In a first-price auction, decreasing aversion to risk
encourages a bidder to bid less (Krishna, 2002, p. 38). Intuitively, the insurance with respect
to the auction outcome from a higher bid is less valuable. Thus, a high-budget bidder may
justifiably bid less than a counterpart with the same value-signal but a lower budget. This
observation conflicts with the usual understanding of a monotone, nondecreasing strategy,
which is at the core of the vast majority of auction studies. The following example, adapting
the model of Cox et al. (1988), highlights this phenomenon.
Example 2. Consider the model of Example 1. Again, Si i.i.d.~ U[0,1]and a bidder may have a
budget of 1/4or 3/4with equal probability. Unlike Example 1, suppose a winning bidder of
type (si,wi) receives a payoff of (si-bi)wi+1 4. Payoffs following a loss are zero. While contrived
for tractability, these preferences capture the idea that bidders with a large budget are less
risk-averse. A high budget implies risk-neutrality while a low budget implies strict
4
ˆs''
bi
ˆs' 1 ˆs
ß(si, 3 )4
)
12
ß(si,
14
14
si
0
Figure 2: Equilibrium strategy in Example 2.
risk-aversion. The following strategy, sketched in Figure 2, is a symmetric
equilibrium:
v 11-3)s24(si2 i+33(19 v
if si ∈[0,ˆs]
) = 2si +24(
if si ''∈[0,ˆs]
3 11-63)+ v 11-3)2
,ˆs]
ß( si,3 4) =
,1]
si22s2 i+13 v11-42 4(si+1)
16s3 i
if si ∈(ˆs,1] , ß( si,
14
14
if si ''∈(ˆs
if si '∈(ˆs
'
and ˆsis presented in the online appendix.
When si
5
˜0.298. A verification that ßis an equilibrium and the precise definition of ˆs'is low, the
w is dominant and a high-budget bidder bids strictly less
budget’s effect on risk preferences
here ˆs= v than a low-budget opponent. When siis large, the strategic effects noted already in
Example 1 dominate. Consequently, competition becomes stratified along the budget
dimension.
The strategies exhibited in Example 2 question the existence of an equilibrium in
monotone, nondecreasing strategies in auctions with private budget constraints.
Investigating this question, albeit in an environment admitting more natural preferences
and interdependent values, forms the starting point of our analysis in Section 2. Since
monotone equilibria form the core of most applied auction studies, it is critical to
investigate their existence in this arguably prosaic setting. We address this question
with political deftness by redefining the notion of a monotone, nondecreasing strategy in
a manner appropriate for our setting. Our definition accommodates the countervailing
incentives associated with budget constraints. Using a recent equilibrium existence
result from Reny (2011), our relaxed notion of mono''11-3,
ˆs
=
3 (4 v '11-3),
tonicity allows us to conclude that an equilibrium with economically-meaningful properties
exists. Roughly, bidders with larger value-signals always bid more in equilibrium. Those with
larger budgets tend to—but need not necessarily—bid more in equilibrium. The equilibrium
in Example 2 exhibits these features and they apply more generally. Except for the
private-values case, passing the equilibrium to a continuum action space depends on an
endogenous tie-breaking rule (which we discuss below), a common feature of auctions with
multidimensional types and interdependent values.
Whereas the existence of a “monotone” equilibrium is reassuring, it is alimited conclusion
regarding equilibrium behavior. Thus, in Section 3 we specialize to a symmetric model with
interdependent and affiliated valuations, as in Milgrom & Weber (1982), and we augment it
with continuously distributed private budget constraints. Fang & Parreiras (2002) analyze
the second-price auction in this setting and it extends the analysis of Che & Gale (1998).
Example 1’s main features, such as discontinuous equilibrium strategies and endogenously
stratified competition, can appear in this benchmark environment. For example, equilibrium
bid distributions can be bimodal even when the underlying type-space is uniform and expost
valuations have a unimodal distribution. The documented strategic effects seriously
confound inference regarding model primitives as bidders often react non-monotonically to
the presence of private budgets. A discussion of applications and a brief comparison with
the second-price auction concludes our analysis.
An appendix collects proofs and an online appendix presents extensions and
discussions of a more narrow interest.
1 Literature
2Researchers
have recognized the possibility of budget constraints in auctions for over thirty
years;however, few studies treat both valuations and budgets as private information.
Indeed, many studies examine budget constraints in multi-unit or sequential settings where
a budget’s strategic importance is more transparent a priori (Pitchik & Schotter, 1988;
Benoît & Krishna, 2001;Pitchik, 2009;Brusco & Lopomo,2008,2009). In such models,
assumptions simplifying the information structure or the auction format are a practical
necessity. Budgets feature in many internet-motivated applications, such as selling
advertisements (Ashlagi et al. , 2010), and play a prominent role in spectrum auctions
(Cramton, 1995; Salant, 1997; Bulow et al. , 2009). Thus, budget constraints are an
economically-meaningful element of
2Rothkopf
6
(1977) and Palfrey (1980) are early discussions of multi-unit auctions with budgets.
many auction markets. We contend that complex or multi-unit environments, like spectrum
auctions, are not
necessary to appreciate the strategic effects posed by budgets—“classic,” single item
settings are sufficient, although surprisingly research here has been more limited. Early
research by Che & Gale (1996a,b) considers first-price, second-price, and all-pay auctions
in a commonvalue setting where only budgets are private information. In an important
paper, Che & Gale (1998) discuss first-price auctions allowing for private budget constraints
and explore some of the revenue implications. Their analysis was limited to the private
values case with risk-neutral bidders. In studying auction revenues when bidders have
multidimensional, independent types, Che & Gale (2006) develop techniques that can be
applied to the case of budgetconstraints. Fang&Parreiras (2002,2003)study equilibrium
bidding inatwo-bidder, second-price auction with private budget constraints allowing for
affiliated and interdependent valuations. Their setting forms the starting point of our analysis
in section 3. Kotowski & Li (2012) consider the all-pay auction and the war of attrition with
private budget constraints in the case of N-bidders. Textbook discussions of basic models of
auctions with budget constraints are included in Krishna (2002) and Milgrom (2004).
3The
optimal auction design problem in the spirit of Myerson (1981) incorporating budget
constraints is particularly challenging and has been considered under various guises
(Laffont & Robert, 1996; Monteiro & Page, 1998; Che & Gale, 1999, 2000; Maskin, 2000;
Malakhov & Vohra, 2008; Richter, 2011). Pai & Vohra (2011) offer the most comprehensive
description of optimal mechanisms but restrict attention to discrete types. Auction design
accounting for budget limits has also spurred a growing literature among computer
scientists.
Whereasmostoftheabovemodelsassume“hard” budgetconstraints, somepapersexplore
expostdefaultorasymmetries inbidfinancingability(Zheng,2001;Jaramillo,2004;RhodesKropf
& Viswanathan, 2005). While appropriate for some applications, such models may mask the
strategic effects of budget constraints alone. Therefore, we treat budgets as a rigid bound
for bids. Cho et al. (2002) suggest that overcoming budget constraints can encourage
collusion or the formation of bidding consortia. We assume non-collusive behavior.
3Borgs
et al. (2005) and Dobzinski et al. (2012) are representative contributions of the vast literature in this
tradition.
7
2 A General Model and Equilibrium Existence
4Recalling
the challenge posed by Example 2, we begin by investigating the existence of a
“monotone” equilibrium.Wealsointroducenotationthatwillbeusedthroughoutthesequel. Our
model rests on three assumptions. Assumption A-1 considers bidders’ preferences.
Assumptions A-2 and A-3 introduce two complementary information structures. Each places
some limit on the correlation in bidders’ private information, which is a practical necessity in
this environment. We discuss both model components in turn.
Preferences and Strategies There is a set of bidders N = {1,...,N} participating in a first-price,
sealed-bid auctionfor one item. Each player has a private type θi=(si,wi)∈Si×W i=Ti.
si=[si,¯siiis a bidder’s value-signal and it takes values in S]. A value-signal is a bidder’s
information about the item for sale. wiis a player’s budget and W i=[wi,¯wi]⊂Ri. The budget
defines an absolute spending limit above which a bidder cannot bid. Each bidder has
available a set of valid competitive bids B⊂[ri+,8)and a non-competitive null bid l<mini{ri},
which guarantees a loss in the auction. Bids are placed simultaneously and the bidder
submitting greatest competitive bid wins the item. Only the winner makes a payment to the
auctioneer.
We adopt standard notation. Signal profiles are in boldface, as in s=(s1), and θ=(θ1,...,θNN
i=1)is a profile of types. We let S =×Si,...,sNand we define W and Tsimilarly. As usual,
s-i=(s1,...,si-1,si+1,...,sN)and S-iis defined analogously. Our first assumption concerns a
bidder’s utility function.
Assumption A-1. Whenever ri =bi =wi, the payoffs of bidder isatisfy the following:
bounded, continuous and decreasing in bi, nondecreasing in1(s,wi), and
differentiable. Moreover, ∂¯ui/∂si=κi>0.
.
¯
u
i
(
s
,
w
i
,
b
i
)
:
S
×
W
i
→Ris bidder i’s payoff when she does not win the auction. ui
i
i > ¯ B,
¯ui(s,wi,b
)
<ui(wi)for all
(s,wi).
2. ui(wi):W i is bounded, nondecreasing, and differentiable.
i
3. There exists ¯ B=r such that for all
b
=0, then for all (s,wi), ¯ui(s,wi,bi)=ui(wi).
4. If bi
Establishing the existence of an equilibrium in auction games is a research program spanning several
4
decades.
deCastro&Karney(2012)presentacomprehensivesurveyofthis(now)large,specializedliterature.
8
5. For all bi >b' i =ri, ¯ui(s,wi,bi)-¯ui(s,wi
∂¯ui =sup dui <8.
6. inf
dwi
,b' i)is nondecreasing in (s,wi).
∂wi
A-1 adapts standard conditions from Maskin & Riley (2000), Reny & Zamir
(2004), or McAdams (2007), among others, to our environment. While the
differentiability requirements are stronger than typically encountered, they
greatly simplify much of the following argument. The lower bound on ∂¯uin
A-1(1) is similar in our setting to Assumption A2(v) from Athey (2001). A-1(3)
says that bidders do not wish to spend an unbounded amount and A-1(4) says
a bidder would happily take the item for free. A-1(5) is an
increasing-differences condition common to the literature. A-1(6) is a technical
condition which is not restrictive. Its role will be apparent shortly. To simplify
our notation, we write ui(s,wi,bi)=¯ui(s,wi,bi)-ui(wii/∂si). As highlighted by
Example 2, when bidders’ preferences are sufficiently general, equilibrium behavior may not conform with traditional conceptions of monotonicity. To
accommodate the behavior suggested by Example 2, we will amend the
standard notion of a monotone strategy. Reny (2011) pioneered the weakening
of the definition of a monotone strategy in games of incomplete information
and our analysis closely parallels his examination of the multi-unit auction with
risk-averse bidders. The analogy between our model of a single-item auction
with private budget constraints and a multi-unit auction is quite clear. An
auction with private budget constraints can be interpreted as an unusual
multi-unit auction: the highest bidder wins the item for sale and all bidders “win”
their budget irrespective of their feasible bid.
As a first step in (re)defining a monotone strategy, we introduce an
alternative partial order on the type-space. This ordering relates changes in a
bidder’s information to changes in her utility.
∂¯ui .
= sup dui -inf∂¯u∂w
i
Definition 1. Let ξi
inf∂s
dwi
i
i
on Si ×W i as (si,wi)=θi
9
(s' i,w' i) ⇐⇒ wi
i(wi
i
=w' i and si
i -s' i
i
i,bi
Define the partial order =θi = ξ-w' i).
Figure 3 sketches the greater- and less-than sets for type. An increase in si
) = v(s)+wi
ˆθ
increases, si
alone implies a higher type. If wmust increase commensurably to imply a
higher type. If bidders have prototypical quasilinear preferences, i.e. ¯u(s,w
-b
an
d
wi
θ' =θi ˆ θi
i
¯wi
ˆ
θ
i
wi
si
Figure 3: The =θi ordering
.
=0 and =θi reduces to the usual coordinate-wise ordering of R2. We can now defin
u (wi)=wi, then ξi
strategies of interest.
Definition 2. An admissible (pure) strategy for bidder iis a measurable f
B∪{l}suchthatßi(si,wi)=wi. Astrategyisnondecreasingifθi=θiθ' i=⇒ ßi(θii:T)
2weakens the usual definition ofamonotone, nondecreasing strategy. A
budget and a sufficiently higher value-signal imply a higher bid. Althou
outside of our general model, its equilibrium reflects this intuition.
The Information Structure Our analysis studies in parallel two distinct,
information structures.
Both have natural motivations and particular applications will dictate th
specification. The first posits independent types across bidders but allo
between a bidder’s value-signal and budget. The second allows for som
dependence among value-signals, but demands budgets be determine
Our first environment considers a setting where bidders’ types are i
However, a bidder’s own value-signal and budget may be correlated. C
conduct their analysis in a special case of this environment.
Assumption A-2. The joint density of players’ types is f(θiand θjare inde
f(θ) is strictly positive, bounded, and for all i=j, θ
10
5While
a tractable formulation, the independence of bidders’ private information is often
untenable in applications. Therefore, we propose a complementary second environment that
allows for some statistical dependence among bidders’ value-signals in the form of
affiliation.Even though affiliation is already a restrictive property (de Castro, 2010), it is often
insufficient to guarantee the existence of an equilibrium in nondecreasing strategies.
Therefore, in Assumption A-3 we introduce an additional limit on the “relative degree” of
affiliation. Similar limits have been fruitfully exploited by Fang & Parreiras (2002), Krishna &
Morgan (1997), and Lizzeri & Persico (2000), among others.
To state Assumption A-3, we first introduce some notation. If h(s)is a density function, we
write H(s-i|si) = t-i=s-ih(t-i|si)dt-iand ¯ H(s-i|si) = t-i=s-ih(t-i|s. Also, define λ(s-i|si)=h(s-i|s) ¯H(s-ii|si)and
s(s-i|si)=h(s-i|s) H(s-ii|si)i)dt-i. λand sare (conditional) hazard-rate and reverse hazard rate
functions.
Assumption A-3. The joint density of players’ types, f(θ1,...,θN)=h(s)g(w), is strictly positive,
bounded, and it satisfies the following conditions:
1. Value-signals are affiliated, but independent of budgets. Their joint density is h(s).
Moreover, if (si,wi)=θi(s' i,w' i)and ri=bi=min{ ¯ B,w' i}, then for a.e. s-i
(a) ui(si,s-i,wi,bi)λ(s-i|si)=ui(s' i,s-i,w' i,bi)λ(s-i|s' i), and (b) ui(si,s-i,wi,bi)s(s-i|si)=ui(s' i,s-i,w'
i,bi)s(s-i|s' i)
with strict inequalities if si >s' i.6
2. Budgets are mutually independentand independentof value-signals. Theirjointdensity is
g(w).
Remark 1. Assumption A-3 is satisfied if value-signals are independent. It can also be
satisfied when value-signals are strictly correlated. For example, consider N = 2, r= 0,
ui(s,wi,bi)=1 2(si+sj)-bi, and h(s1,s2)=4 5(1+s1s2iθi). By continuity, it will also hold in sufficiently
nearby models where the =ordering has bite. While assumptions akin to A-3 have appeared
before, our formulation is slightly more
restrictive owing to A-3(1b). Intuitively, A-3(1a) and A-3(1b) ensure that the direct influence
of sion preferences dominates its informational content concerning s7-i. Consequently,
interim expected utility is nondecreasing in a bidder’s own value-signal.
5Weusethepropertiesofaffiliatedrandomvariablesfreely.
Milgrom&Weber(1982)offeranintroduction.
A-3(1b) was absent from this study’s previous versions and we introduce it here.
7A-3(1a) and A-3(1b) facilitate verification of a single crossing condition when value-signals are affiliated
6Condition
11
Equilibrium Existence In sufficiently general auction models, the existence of an equilibrium
depends intimately
ion
the prevailing tie-breaking procedures (Jackson et al. , 2002). As Araujo et al. (2008)
discuss, this is particularly true when bidders’ private information is multidimensional— as in
the case of private budget constraints. We therefore offer two statements regarding
equilibrium existence. The first statement concerns a setting where the set of each bidder’s
valid bids, B, is set so as to preclude relevant ties. Subsequently, we discuss equilibrium
existence when ties can occur and tie -breaking procedures matter.
Let ϕi(bi,b-i)be the probability that bidder iwins the auction with a bid of bgiven that other
bidders bid b-i. Following tradition, we suppose ϕiiconforms to uniform tie-breaking and
ϕi(l,b-i)=0for all b-i. Thus, bidder i’s (interim) expected payoff at the bid biis
Ui(bi,ß-i|θi)=ui(wi)+ T-i
ϕi(bi,ß-i)ui(s,wi,bi)f(θ-i|θi)dθ-i.
By following the strategy , bidder i’s (ex-ante) payoff is Ui(ßi,ß-i)=E[Ui(ßi(θi),ß-i)|θ)]. The strategy profile
ßi
-i) is a Nash equilibrium if for every i, Ui(ß* i,ß* -ii) = Ui(ßi,ß* -i) for all admissible
ßi. We can now offer our first result concerning equilibrium bidding.
Theorem 1. Suppose that for each i, Bi is finite, non-empty, and ∀i=j, Bi nBj =Ø.
1. If A-1 and A-2 are satisfied, there exists an equilibrium in nondecreasing strategies.
82.
If A-1 and A-3 are satisfied, there exists an equilibrium in nondecreasing strategies.
The proof of Theorem 1 rests on ensuring that our model satisfies the conditions outlined
in Reny (2011). We already mentioned the analogy between our model of an auction with
budget constraints and his analysis of the multiunit auction. Readers familiar with that
argument will recognize the key lemmas in our analysis.
When bidders place bids from a continuum, relevant ties are a serious technical concern
andthetie-breakingprotocolmatters. Regrettably, theuniformtie-breakingruleisnoteasily
accommodating of an equilibrium. We explain why this is the case below. The following
theorem outlines some of the possibilities when bidders place bids from a continuum.
while computing an expectation conditional on T⊂T. Without these conditions, Tadditionally needs to be a
sublattice of T-i-i, which is not the typical case in our application.
8In the online appendix we show how Theorem 1(2) obtains without Assumption A-3(1b) when there are two
bidders with quasilinear preferences.
12
Theorem 2. Suppose that for all i, Bi =[ri,8). Theorem 1 continues to hold if either:
1. Bidders have private values and ties among high bidders are resolved with a
uniform randomization; or,
2. The standard allocation rule with uniform tie-breaking, ϕi, is replaced by an
“endogenous” tie-breaking rule, which depends on an announcement of bidder’s
private information to resolve ties.
9Endogenous
tie-breaking rules are the central focus of Simon & Zame (1990) and
Jackson et al. (2002). For convenience, the tie-breaking rule in part 2 calls upon the
construction of Araujo & de Castro (2009), which they originally propose for a
double-auction setting. Their construction carries over naturally to our environment as
well. Roughly, a tie-breaking rule is defined with respect to a sequence of monotone
equilibria from near-by games where the allocation rule is smoothed with slight
perturbations. The rule ensures continuity of payoffs facilitating the existence of an
equilibrium. de Castro (2011) presents a recent discussion of approximating games in
this manner and the implications for equilibrium existence. He also places such methods
in the wider context of equilibrium existence in discontinuous games.
10Jackson
(2009) discusses special tie-breaking rules in auctions and we refer the reader
to his thorough assessment. In our environment, the need for special tie-breaking
protocols stems from the indecision experienced by bidders should they tie for the
winning bid. Standard arguments regarding equilibrium existence in first-price auctions
depend critically on a bidder’s desire to bid infinitesimally more to break a hypothetical tie
in her own favor. Although obvious in a private-value setting, winner’s curse effects make
this desire not obvious once valuations are interdependent.Absent strong assumptions, it
is not generally true in a setting with budget constraints, even with two bidders, since
winner’s curse effects are more intricate.
To sketch some of the complications, suppose bidder ibids band bidders j=iare each
following a nondecreasing strategy ßj. Bidder i’s bid defeats two kinds of opponents.
First, it defeats all opponents who have a budget less than b. This event is good news
concerning the item’s true value. It is a “winner’s blessing” to defeat budget-constrained
opponents since their value-signals will be of average quality and not negatively
selected. Second, the bid defeats a subset of opponents who may have a budget greater
than bbut a relatively
9Important
contributions in this literature include Dasgupta & Maskin (1986) and Reny (1999), among
others.
10Reny & Zamir (2004) prove this as part of their analysis.
13
low value-signal. With regard to this group, winner’s curse effects are more acute and a tie
will disproportionally implicate this second class of opponents. Depending on the situation’s
details, bidder imay prefer to win or lose such a tie. The “relatively-low” value-signals may or
may not be sufficiently good to warrant a slightly higher bid.
The concerns highlighted above aretypical ofmultidimensional auctionenvironments and
are mainly due to value interdependence. However, value interdependence alone does not
preclude equilibrium existence under standard tie-breaking rules. The environment’s
asymmetries alsoplayarole. Aswe documentbelow, inabenchmark symmetric environment
with continuously-distributed types and interdependent values, we can identify an
equilibrium in nondecreasing strategies where ties are zero-probability events. Thus, the
tie-breaking rule is superfluous. More interestingly, however, the equilibrium strategies we
identify exhibit the qualitative features of Example 1—such as discontinuities and an
endogenous stratification of competition—suggesting that these economic phenomena are a
recurring feature of bidding in budget-constrained environments and deserve careful
scrutiny in applied studies.
3 A Symmetric, Two-Bidder Model
Although the existence of a well-behaved equilibrium is reassuring, a question remains to
what degree Example 1’s economic features carry over to a more general environment. In
this section we provide one extension by allowing bidders’ types to assume a continuum of
values. While our discussion is necessarily incremental, in case 3 below we construct a
discontinuous, monotone, pure strategy, symmetric equilibrium in a symmetric first-price
auction. Although unusual, its intuition is fully reflective of the equilibrium from Example 1.
Such equilibria can introduce serious confounds in empirical applications. For example,
even when the type-space is uniform and valuations are unimodal, bimodal equilibrium bid
distributions can arise.
iWe
maintain assumptions A-1 and A-3. (However, A-3(1b) is not necessary for our
subsequent analysis). Thus, our environment is one of affiliated, interdependent values. We
specialize the model further by supposing that there are two ex ante symmetric bidders,
value-signals take on values in S=[0,1], budgets are in W i=[w ,¯w], and there is no reserve
price. We briefly address reserve prices before the conclusion. Assumption A-4 summarizes
the key additional restrictions.
Assumption A-4. Suppose the environment satisfies the following conditions: 1. For all i,
¯ui(s,wi,bi)=v(si,sj)+wi-bi, ui(wi)=wi, v(0,0)=0, and v(1,1)=¯v.
14
2. The density h(s1,s2)is permutation symmetric and continuously differentiable.
3. Bidders’ budgets are distributed i.i.d. according to the c.d.f. G(·). G(·)is C 211, concave,
and its support satisfies w =¯v= ¯w.
A-4imposesadditionalsymmetryonpreferencesandontheinformationstructure. Therefore, it
builds on Milgrom & Weber (1982)’s classic setting. Like Fang & Parreiras (2002) we restrict
attention to two bidders for expositional clarity. A-4(3) is stronger than considered in Fang &
Parreiras (2002) and the concavity condition fulfills the same role as Assumption 5 from Che
& Gale (1998). It is a sufficient condition for our analysis, which can be weakened in specific
cases.
The discussion below is organized into three cases depending on w . In case 1, wis low
and the equilibrium is defined primarily by a differential equation. However, identifying the
appropriate boundary condition requires care. This case also introduces concepts that help
identify the remaining two scenarios. Second, if w is large, the equilibrium is a constrained
extension of the equilibrium strategy without budget constraints. Finally, when w is
intermediate, equilibrium bidding features a discontinuity in the spirit of Example 1. In
equilibrium, bidders endogenously separate into two groups—those with relatively large
budgets and others with small budgets. Unlike Example 1, the equilibrium bid distribution
has no mass points and has connected support. This case highlights the tension between
the direct and strategic effects introduced by private budget constraints. Bids become
depressed by budget limits; however, the chance to exploit others’ (possible) spending limit
is a strategic option encouraging more aggressive bidding. Although our cases are defined
in reference to w , we hope that our discussion makes clear that the qualitative effects we
identify can easily emerge in more general circumstances. While nontrivial, the equilibria we
study are likely among the tamest that may emerge in further generalizations of our model.
Since we focus onsymmetric equilibria, we alsoeconomize onsubscripts by presenting
our model from bidder 1’s view-point. We let (S,W)be her value-signal and budget
respectively. We denote bidder 2’s value-signal and budget pair with (Y,Z). Finally, since it
appears frequently, recall from Milgrom & Weber (1982) that in the corresponding model
without budget constraints the equilibrium bidding strategy a(s)is the solution to
a'(s)=(v(s,s)-a(s)) h(s|s)H(s|s) , a(0)=0. (1)
11To
limit subscripts, hereafter section we write g(·)for the density of the budget distribution of a single bidder.
In Section 2, g(·)was the joint density of all bidders’ budgets. No confusion should result.
15
Cas
e 1:
Lo
w
Min
ima
l
Bud
get
s
(w˜
0)
In a
bro
ad
sen
se,
the
foll
owi
ng
ana
lysi
s
app
lies
wh
ene
ver
w is
“lo
w”
(de
fine
d
bel
ow)
,
but
for
con
cret
ene
ss
ass
um
e
w=
0.
To
con
stru
ct
an
equ
ilibri
um,
our
app
roa
ch
is
to
“gu
ess
,
ana
lyze
,
and
veri
fy.”
Spe
cific
ally,
sup
pos
ea
sy
mm
etri
c
equ
ilibri
um
stra
teg
y
has
the
for
m
ß
(
s
wh
ere
¯
b(s)
—t
heb
idpl
ace
dby
abi
dde
roft
ype
(s,¯
w)
—is
diff
ere
ntia
ble
and
stri
ctlyi
ncr
eas
ing.
Thi
s
stra
,
w
)
=
m
i
n
{
¯
b
(
s
)
,
w
}
(
2
)
teg
y
imp
lies
Leo
ntie
f
“iso
bid”
cur
ves
in
(s,
w)spa
ce
with
¯
b(s)
defi
nin
g
the
loc
us
of
ki
n
kp
oi
nt
s.
Si
n
c
e
C
h
e
&
G
al
e
(1
9
9
8)
,
(2
)
h
a
s
b
e
e
n
th
e
ty
pi
c
al
st
ra
te
g
y
c
o
n
si
d
er
e
d
in
th
e
lit
er
at
ur
e.
It
is
m
o
n
ot
o
n
e
in
th
e
s
e
n
s
e
of
d
e
fi
ni
ti
o
n
2.
If
th
e
o
p
p
o
si
n
g
bi
d
d
er
is
fo
ll
o
wi
n
g
(2
),
th
e
e
x
p
e
ct
e
d
p
a
y
o
ff
of
a
bi
d
d
er
w
h
o
bi
d
s
ß(
x,
¯
w
)=
¯
b(
x)
is
(v(s,y)¯ b(x))h(y|s)g(z)dydz+
Ui0(ß(x,¯w)|s,¯w)=
¯
b(x)
0
(v(s,y)¯ b(x))h(y|s)g(z)dydz.
0 1
¯
w
¯b(x)
x
¯
b'
The first term is the contribution to expected utility from defeating all
opponents with a budget z=¯ b(x). The second term is the contribution to
expected utility from defeatingall opponents with a budget z> ¯ b(x)but
who have a value-signal y=x, and are thereforebidding less than ¯ b(x).
Computing the necessary first-order condition,∂Ui(ß(x,¯w)|s,¯w) ∂x =0, gives:
d(b,x|s)=b+ G(b)g(b) +
γ(z)=
g(z)
dy,
h(y|s)
1-H(y|s)
. (3)
v(s,y)h(y|s) , η(x|s)= x1
1-H(x|s)
H(x|s)
g(b)(1-H
1-G(z) , λ(y|s)=
¯
1
¯ b(s)-v(s,s))(s)= λ(s|s)(γ( ¯ b(s))(η(s|s)-d(
b(s),s|s)) 6
and simplifying notation gives
¯ b'
,
We make two observations. First, it is not clear that (3) has a solution satisfying the
desiderata asserted above. Notably,¯ b(s)must be strictly increasing for all swhile the sign
of (3) is unclear. Second, (3) gives little guidance concerning boundary conditions. With
interdependent values,¯ b(0)>0may be a reasonable situation. Conditional on s=0 and
on winning the auction with a small bid, the item may have a positive expected value if the
losing opponent was budget constrained with high probability.
Toaddresstheseissuesweproceedtodevelopanunderstandingofthequalitativebehavior of
solutions to (3). Specifically, rather than analyzing (3) directly, we will study instead the
closely-related plane-autonomous system
˙s =γ(b)(η(s|s)-d(b,s|s)) ˙b =λ(s|s)(b-v(s,s)) . (4)
Following Birkhoff & Rota (1978, Chapter 5) we can recover solutions of (3) by noting that
d = dbdt d =
.
b
s ˙b(s,b)
d
dt ˙s(s,b)
s
Interpreting the problem in this manner allows for the use of elementary techniques from
phase-plane analysis to discern the qualitative behavior of candidate solutions and has the
advantage of working around the inherent singularities associated with a direct analysis of
(3). We emphasize that the direction of motion as a function of time in (4) is immaterial for
our purposes. Rather, we are interested in the entire paths taken by solution orbits, which
(locally) are integral curves of (3). Our goal is to identify solutions to (4) that can be
represented by a function meeting the desiderata imposed above on¯ b(s). This section’s
12remainder
undertakes this task.
To describe the behavior of a plane-autonomous system, we are initially interested in two
related objects: the nullclines and the critical points. The nullclines of (4) are loci in
(s,b)spacewhereorbitsarehorizontalorvertical. Respectively, thesesetsareν={(s,b):˙
b(s,b)=0}and ψ={(s,b): ˙s(s,b)=0}. It will prove convenient to express νand ψas functions of
sin the following manner: ν(s)={b:(s,b)∈ν}and ψ(s)={b:(s,b)∈ψ}. By inspection, when ν(s)=Ø,
ν(s)=v(s,s). Similarly, when ψ(s)=Ø, ψ(s)is single-valued and is defined
12We note that ˙sand ˙ bare notdefined atthe boundary whereb= ¯wand s=1, respectively. This technical
point is not a concern since we will be able to extend the identified solution to the boundary via (3) using
standard results in the theory of ordinary differential equations. For expositional clarity we suppress this
technical detail.
17
13implicitly
by η(s|s)=d(ψ(s),s|s).
Moreover, if
Sψ
={s:ψ(s)=Ø}, the function
ψ(s) if s∈Sψw if s/∈Sψ
*
*) is
15
a critical point if ˙s(s*
*
b(s*,b
2. If (s*,b
1
4
i
s
˜ ψ(s)=
and the function ˜ ψ(s)will play an im
˙*)=0. When w =0, there exists at least one critical point.The follo
the behavior of the (˙s,˙ b)system around the nullclines and the
T
Lemma
h 1. Consider the system (˙s, ˙ b)defined in (4).
1. If eb>(<)ν(s), then ˙ b(s,b)>(<)0. If b>(<)ψ(s), then ˙s(s,b)<(>)0
c
o
n
t
s
i
e
n
t
u
o
S
u
ψ
s
.
* 16)is a critical point, then it is (generically) either a node or a saddle point.
Critical points occur where nullclines meet. (s ,b
,b
Aconsequence ofLemma1isthatthereexist solutionsof
(4)thattendtowardoremanate from each critical point. In the case when the critical
point is a saddle point, these solutions would correspond to the point’s stable and
unstable manifolds. In the case of a stable or unstable node, there would be
infinitely many such solutions.
Going forward, we introduce the following assumption to simplify our
presentation.
Assumption A-5. There exists at most one critical point.
The online appendix extends ouranalysis to the case ofmultiple critical points.
However, the substantive conclusions we reach carry over to the more general
case. A sufficient condition ensuring Assumption A-5 holds is that ψ(s) is
non-increasing. We are unaware of any economic interpretation to this sufficient
condition.
Leveraging Assumption A-5, we can construct a qualitatively accurate phase
portrait for (4). Figure 4 sketches a generic situation. The nullclines ψand νpartition
the space into four regions—R1,...,R4—that meet at the critical point (s*,b*). We plot
several sample solutions and the thick arrows indicate the direction of the system’s
solutions in each region. When there is one critical point, it is a saddle point. (In the
case of multiple critical points, they alternate between saddles and nodes as
sincreases.)
G(·)is not concave, ψ(s)may not be single-valued. Much of the analysis we develop carries
over to this situation, but many more cases need to be considered.
14See Lemma B9 in the
appendix.
15See Lemma B10 in the
appendix.
16See Remark B3 in the
appendix.
13When
18
b*
¯
b0
ν
R4
b
¯
b1
s1
R1
R2
R3
s*
0
ψ
Figure 4: Phase-portait of the (˙s, ˙ b) system defined in (4). There is one
critical point at
( ,b*). It is a saddle point. Together, the (thick, black) stable manifolds define ¯
s b(s).
* Figure 4 also informs our understanding of ¯ b(s). This function was hypothesized to
be
continuous, strictly increasing, and satisfying (3). If such a function exists, it
must begin in region R1, “pass through” the critical point (s*,b*)and continue
in R. Only in regions R1and R33is ˙ b/˙s>0. Our preceding analysis also
confirms that the only possible paths
consistent with these requirements correspond to the stable manifolds of
(4), illustrated in Figure 4 as paths tending toward (s*,b*) from the points
(0, ¯ b0) and (1, ¯ b1). Let ¯ bbe the
¯ b}. (5)
closure of the union of these two paths and let
¯ b(s)={b:(s,b)∈
Defined in this manner, ¯ b(s)is continuous, strictly increasing and an
integral curve of (3). It
has an endogenously determined initial condition ¯ ¯ and terminal condition ¯
b b(1)=
b(0)=
0
¯
b
1
is routine.
symmetric equilibrium.
. Having identified a function¯ b(s)that meets our requirements, proving
the following theorem
Theorem 3. Suppose w =0, then ß(s,w)=min{ ¯ b(s),w}where ¯ b(s)is defined by (5) is
a
The following example illustrates our discussion thus far. The example
corresponds (via scaling) to one used by Fang & Parreiras (2002) to
illustrate their model of the second-price auction with budget constraints.
19
Υ
1b
b
Α
0.5
Ψ
0 0.5 1 s
Figure 5: Equilibrium sketch in Example 3. There is a single
critical point at the intersection of ν(s)and ψ(s).
Example 3. Suppose v(si,sj)=si
+sj , W i i.i.d.~ U[0,2], and Si i.i.d.
¯
.
¯
b(s))
(1+s)(3s-1)-4(s-1) ¯ b(s)
b
'
+ 3s 8 v 5and ν(s) = 2s. The critical point is a
1 5We can show that ψ(s) 5 4+ 1 s-1
0
4
=(5-2 v-2 5( v
(
v
~ U[0,1]. Some algebra gives
(s)= 2(2¯
b(s))(2s-
5) ˜ 0.105. The eigenvalues of the Jacobian of (˙s, ˙ b) at the
critical point are5+ v205) ˜ -6.62 and2 5205v5) ˜ 4.83.
Therefore, the critical point is a saddle point and the stable
manifolds approach it at a slope of˜0.85. Numerical
approximations allow us to conclude that¯ b˜0.1202while ¯ b 13
v 5+v 205˜0.878. Figure 5 plots¯ b(s)and a(s)=s, which is the
equilibrium in the same model but with no
budget constraints. The nullclines and representative solutions
from the associated planeautonomous system are included for
context. This example highlights the attenuation of the winner’s
curse: low value-signal bidders with relatively large budgets bid
more than in the same environment absent budget constraints.
When w >0, two factors may complicate the preceding
analysis. First, the expected payoff from the bid¯ b(0) may
be negative—a fact inconsistent with equilibrium behavior.
0,w )for some
>0.
,b*
s0
20
Second, the manifold tending to
(s
) from below may not originate on the vertical axis as it does in Figur
its initial condition could be (s
In either case, Theorem 3 cannot apply; otherwise, we have the following definition and
corollary.
Definition 3. w is low if a critical point exists, s0=0, and η(0|0)-w=0. Corollary 1. If w is low,
Theorem 3 continues to hold. Proof. The preceding analysis continues to apply. As¯
b(s)depends on G(·), the equilibriumchanges with w , but it is still of the form ß(s,w)=min{ ¯
b(s),w}.
Analysis of the subsequent two cases builds on the preceding results. Notably, the
function¯ b(s):[s0,1]→[w ,¯w]defined as the closure of the union of the upward-sloping
(stable) manifolds passing through the critical point (s*,b*)will reappear. ¯ b(s)is reserved as
notation
for this function.
Case 2: High Minimal Budgets (w »0) If budgets are likely to be large relative to most
valuations, a strong intuition suggests that
they should matter strategically only for high-valuation bidders. In Example 1, for instance,
low-valuation bidders (s<ˆs) adopt the strategy a(s)bidding as if budgets are strategically
irrelevant. Extending this observation to the general model is relatively straightforward.
When w is sufficiently large all bidders with a value-signal below some threshold ˜swill bid
according to the strategy a(s), defined in (1). Bidders with a value-signal s>˜s, will bid min{˜
b(s),w}where˜ b(s)is an appropriate solution to (3).
Definition 4. w >0is sufficiently large if a-1(w) /∈Sψ. The nomenclature “sufficiently large” is
motivated by the limiting case w = a(1) which
ensures definition 4 is satisfied in practice. However, noting Example 4 below, w can be
sufficiently large in less-trivial instances as well. The following theorem describes
equilibrium bidding in this case.
Theorem 4. Suppose w is sufficiently large and let ˜s=a-1(w). Let ˜ b(s)be the solution to
a(s) if s=˜s min{˜
(7)
b(s),w} if s>˜s
˜ ˜ b(s)-v(s,s))(s)= λ(s|s)(γ( ˜
, ˜ b(˜s)=w . (6)
b b(s))(η(s|s)-d(˜ b(s),s|s))
'
Then
,
ß(s,w)=
2
1
is a symmetric equilibrium.
Unlike case 1, the desired solution to the main differential equation (6) bypasses the
critical point (if such a point exists at all). Abusing terminology, we can say that˜ b(s)
extends a(s) into the range of bids above w and as a result ß(s,w) is continuous. ˜ b(s)
accounts for the change in the marginal effectiveness of bidding due to budget limits. Bids
above w defeat opponents who have a low value-signal and a relatively low budget.
Locally, this encourages more aggressive bidding by bidders with relatively large budgets.
The next corollary and example make this effect precise.
i
such that for all bidders of type (s,w), with s∈(˜s,s'˜
b(s)=a(s).
Corollary 2. There exists s'
i i.i.d.
˜ b'
suppose w>0, a-1(w)∈Sψ
-1
)and w sufficiently close to ¯w, ß(s,w)=
Example 4. Again, suppose
,sj)=si +sj and Si i.i.d.
v(s
~ U[0,1]. Unlike Example 3, suppose w =1/2and thus W~ U[1/2,2]. The differential
equation describing ˜ b(s)reduces to
˜ b(s))(2- ˜ b(s)) .
(s)= 2(2s'
of Corollary 2 holds with strict inequality.
4(1-s) ˜ b(s)+s(3s+2)-2
Since a(s)=s, ˜s=1/2. Figure 6 depicts the functions ˜ b(s),a(s), and ν(s). The conclusion
Case 3: Intermediate Minimal Budgets The intermediate case is defined in opposition to the
low- and high-w cases. Therefore,
, and that neither of the above cases applies. In this situation,
the preceding theory is inadequate to describe equilibrium behavior. For instance, at (˜s,w ),
˜s= a(w ), the differential equation (6) evaluates to ˜ b<0 precluding an increasing extension
as in the high-wcase. Such features prompt consideration of discontinuous bidding
strategies similar to Example 1.
Although the equilibrium strategy we introduce in this section is the natural analogue of
the strategies we defined in the preceding cases, its definition is comparatively intricate and
we must take a brief detour to provide motivation. By analogy to Example 1, we posit that
bidders will bid a(s)when their value-signal is sufficiently low. At some critical value, ˆs,
high-budget bidders will increase their bid discontinuously. To identify the location and
22
1b
Υ
b
w
0.5
Α
0 0.5 1 s
Figure 6: Equilibrium sketch in Example 4. Given the example’s
parameters, there is no critical point.
the magnitude of this discontinuity we begin by defining the
function µ:[0,1]→[w ,¯w]as
µ(s)=
w if s<smin{ ¯ b(s),0˜ ψ(s)} if s=s0if there is a critical
point. If there is no critical point, ¯ b(s)does not exists and we
simply
let µ(s)= ˜ ψ(s). The technical motivation for µ(s) is that when
µ(s) >w the differential
equation ˆ'b
ˆ b(s)-v(s,s))(s)= λ(s|s)(γ( ˆ
b(s))(η(s|s)-d(ˆ b(s),s|s))
will admit a strictly increasing solution corresponding to the initial
condition (s,µ(s)). As beforehand, this follows by examining the
system (4) and extending a solution through the critical point if
necessary. Hence, {(s,µ(s)):µ(s) >w } is a natural set of
candidate “landing-points” for the discontinuous bid increase.
To entertain a discontinuous bid increase from a(s)to µ(s),
requires a bidder to be just
23
indifferent between the two options. To identify such cases consider the
inequality
= sUtility from bid of a(s)
(8)
(v(s,y)-µ(s))h(y|s)dy+G(µ(s)) s1
Utility from bid of µ(s)
-1
a(s), if s=ˆs b
(s) if
. (9)
s>ˆs,w<f(s) min{
1ˆ b(s),w}
if s>ˆs,w=f(s)
γ( ˆ b(s))(η(s|s)-d(ˆ b(s),s|s))
ß(s,w)=
'
subject to ˆ b(ˆs)=µ(ˆs).
1
(v(s,y)-µ(s))h(y|s)dy
and
define the set Z ={s=a
s
0
0(v(s,y)-a(s))h(y|s)dy
(w ):(8) holds}. In the above inequality, the utility
from the bid µ(s) is defined under the conjecture that the other bidder also
increases her bid discontinuously to µ(y) when her value-signal is y= sand that
higher types bid strictly more. Of course, in a symmetric equilibrium this conjecture
will prove correct.
Lemma B11 in the appendix confirms that when w is intermediate, Z =Ø. We
define ˆs=supZ. At this point, (8) holds with equality. Like in Example 1, ˆsis the
value-signal of the lowest type who increases her bid discontinuously.
With ˆsdefined, we can introduce the strategy of interest, which we sketch in
Figure 7. We provide intuition motivating this strategy after its definition. Let
The value ˆsis defined as above, a(s)is defined in (1) and:
• ˆ b(s)is the solution to
ˆ bˆ b(s)-v(s,s))(s)= λ(s|s)(
•
b 17
(s)is the solution to
(s)= (v(s,s)-b
b' 1
subject to b1(ˆs)=a(ˆs).
G(f(y))h(y|s)dy (11)
(10
)
1(s))h(s|s)G(f(s))
H(
From the arguments in the low-wcase, ˆ b(s) exists and is strictly increasing and continuous. We can
1
employ
the (˙s, ˙ b)system to define the appropriate solution as beforehand. When there exists a critical
7
point
ˆ b(ˆs)=µ(ˆs)and extends it to values s>s*
24
and ˆs=s*, ¯ b(s)solves (10) with the initial condition . The online appendix discusses constructingˆ b(s)when there are multi
b
min{ ˆ b(s),w}
sw¯w ˆs1
µ(ˆs)ˆs w'
Figure 7: ß(s,w)when wis intermediate.
• f(s):[ˆs,1]→[w ,µ(ˆs)]is a non-increasing, continuous function such that
µ(ˆs)
b1(s
)
w
a(s)
= G(f(s))(1-H(s|s))
f(s)-b1(s)
G(f(y))h(y|s)dy , (12)
η(s|s)-f(s)
1 (ˆs'
f
(
ˆ
s
)
=
µ
(
ˆ
s
)
,
a
n
d
f
(
ˆ
s
'
)
=
b
b
i
d
w
i
n
s
w
it
h
g
r
e
a
t
e
r
p
r
o
b
a
b
il
it
y
.
S
e
c
o
n
d
,
b
e
'Thedefinedstrategyhasadis
c
continuityalongtheset{(ˆs,w)
a
:w=µ(ˆs)}∪{(s,f(s)):s∈[ˆs,ˆs+u)
}. Importantly, however, s
ß(·,w ):[0,1]→Ris
e
continuous and ß’s range is
a
an interval. The intuition b
underlying this strategy i
echos Example 1. Biddersd
who increase their bid
o
discontinuously at ˆsare f
indifferent between the ˆ
higher bid µ(ˆs)and the b
bid a(ˆs). The larger bid (
ˆ
is attractive for two
reasons. First, a higher s
) bests
H many bidders with low budgets, it has the added benefit of
(
attenuating
the winner’s curse. These reasons encourage a bidder with a value-signal
ˆ than ˆsto favor the much higher bid.
greater
s
Ifbidders
| with abudgetofw= ˆ b(ˆs)bidstrictly above w , bidders with lower budgets
mustsomehow
respond. Consider the problem now facing a bidder with a value-signal
s
ˆs=ˆs+o
and
a
budget
ˆwoo= ˆ b(ˆs)-o. Such a bidder is tempted to bid her entire
)
endowment
and so asto benefit from the two positive effects noted above. However, sh
+
does not compare bidding a(ˆso)versus ˆwoto see if the high bid is indeed worthwhile. Th
fact that
) all opposing types with a budget in excess ofˆ b(ˆs) are already outbidding her
changes
= her incentives—she is
facingwless effective competition on the margin if she bids low. The same effect is seen
25
f
o
r
s
o
m
e
ˆ
s
'
s
ˆ
s
∈
(
ˆ
s
,
1
]
.
1 for bidders with a low budget and a value-signal of s∈ (ˆs,ˆs). Therefore, she
compares bidding ˆwoversus b1(ˆso)which accounts for this change in marginal incentives.
The subtle point, however, is that b1(·) is not independent of the rate at which bidders prefer
to bid their entire endowment versus b1(s). As sincreases and more types discontinuously
increase their bid above w , bidders with even lower budgets but higher value-signals have
even less incentive to bid aggressively on the margin.
'Example
Theorem 5. Suppose w is intermediate and value-signals are independent. Then (9) is a
symmetric equilibrium.
Proof. See Appendix B. Verifying that this is an equilibrium is routine, but the discontinuities
necessitate a case-by-case argument. Preparatory work involves showing that the
functionsfandb1existwiththestatedproperties. (Theyareintroducedaboveinaself-referential
manner.) The independence assumption greatly simplifies the overall argument.
The strategy ß(s,w)generates a bimodal distribution of equilibrium bids with few bids near
w. After observing such a bid distribution, an observer not accounting for the strategic
implications of budget constraints may infer erroneously that there are two distinct classes
of bidders: those with high values and those with low values. Even with a uniform type
distribution, such an endogenous segregation of bids can arise.
*
3 v 2˜
+sj , W i i.i.d.~ U[1/5,2], and Si i.i.d.
Example 5. Suppose v(si,sj)=si
5
1
ˆ b'
ˆ b(s)-2s)
26
~ U[0,1]. This is the setting of Examples 3 and 4 except now w =1/5. There is a critical
point at s
=10.151. The differential equation describing ˆ
b(s)is
ˆ
(s)= 10(2b(s))(
Figure 8 depicts the functions a,b
20 ˆ b(s)(s-1)-5s(3s+2)+7
and a(s)=s. The initial discontinuity is at ˆs˜0.181where some bidders shift to a bid of
µ(ˆs)˜0.286—an increase in bid of approximately 58 percent. Therefore, the discontinuity
can be quantitatively large.
, ˆ b,ν,ψ, and f. Several representative paths from
the associated plane-autonomous system are included to relate the example with previously
derived results.
Figure 9 presents a histogram of the resulting bid distribution. Only 24 percent of types
actually place a bid equal to their budget; nevertheless, the strategic consequences of
budget constraints are pronounced. Without budget constraints, the equilibrium bid
distribution would be uniform.
1b
Υ
b
0.5
ΨΑ
w
F
b
1
0
0
.
5
1
s
F
i
g
u
r
e
8
:
E
q
u
i
l
i
b
r
i
u
m
s
k
e
t
Dis
cus
sio
n
and
App
licat
ion
s
We
con
clu
de
by
pla
cin
g
the
pre
ced
ing
ana
lysi
s in
con
text
by
con
sid
erin
g
sev
eral
c
h
i
n
E
x
a
m
p
l
e
5
.
app
licat
ion
s.
Inp
arti
cul
ar,
we
pro
vid
e
aco
mp
aris
on
with
the
sec
ond
-pri
ce
auc
tion
’ssy
mm
etri
c
equ
ilibri
um
stra
teg
y.
We
als
o
hig
hlig
ht a
diff
ere
nce
in
the
co
mp
arat
ive
stat
ic
beh
avi
or
of
equ
ilibri
um
stra
tegi
es
bet
we
en
the
two
for
mat
s. A
disc
ussi
on
of
res
erv
e
pric
es
con
clu
des
.
Wh
en
bud
get
con
stra
ints
are
pre
sen
t,
eve
n
sm
all
res
erv
e
pric
es
ma
y
dec
rea
se
exp
ect
ed
rev
enu
es.
The
Sec
ond
-Pri
ce
Auc
tion
Fan
g&
Par
reir
as
(20
02)
stu
dy
the
sec
ond
-pri
ce
auc
tion
’s
sym
met
ric
equ
ilibri
um.
Ref
erri
ng
to
thei
r
deri
vati
on
but
ado
ptin
g
our
not
atio
n,
the
equ
ilibri
um
stra
teg
y in
the
sec
ond
-pri
ce
auc
tion
is
ß2
(s
,w
)=
v(
s,
s)
if
s
<ˆ
s
m
in
{b
22
(s
),
w
}
if
s
=ˆ
s2
where
(s)is a solution to the boundary-value
b2
problem
(s)) (s)-v(s,s)]
2(s)] , b2(1)=v(1,1).
[b2
[η(s|s)-b
27
b' 2(s)=
λ(s|s)γ(b2
4000
3000
2000
1000
0.2 0.4 0.6 0.8 Bid
Figure 9: Histogram of equilibrium bids in Example 5. The figure was generated by
evaluating ß(s,w)on a fine uniform grid.
When
>0, it is a point of discontinuity in the bidding strategy and v(ˆs2,ˆs2
ˆs2
b2
+ 2s→ˆs
2,ˆs2)up
to w at s=ˆs2, all types with value-signals s>ˆs2
If ß1 (s,w)and ß2
1
8
1(s,w)=ß2
1
1(s,w)
= ß2
18
28
) <w = lim(s).
As in the first-price auction, the discontinuity’s presence depends on model parameters.
Although discontinuities may exist, budget constraints in a second-price auction do not
introduce the same strategic implications as in the first-price auction. There is no
endogenous stratification of competition along the budget dimension. If bids increase
discontinuously from v(ˆswill bid at least w.
(s,w)are symmetric equilibrium strategies in the first- and second-price
auctions, respectively, and for simplicity w =0, it is straightforward to confirm that bidders in
a first-price auction shade their bids more: ß(s,w)(see the online appendix). When ß(s,w)<w,
the preceding weak inequality is strict. On the other hand, noting instances where ß(s,w),
we can exploit the invariance of bids to the auction format to test for the presence of budget
constraints or to draw some inferences concerning their nature. Such tests are unusual
insofar as they demand observing bidding across auction formats while maintaining the
environment otherwise constant.However, they do in principle enable some identification of
private budget constraints, as distinct from valuations, based on bidding behavior alone.
Needless to say,
For example, bidders could submit format-contingent bids.
extending existing theories of empirical identification in auctions to accommodate privateG1' (w)=w(4-w)/4. G0
budget constraints will require novel forms of variation and strong assumptions.
Otherwise, learning about two unobserved primitives (s,w)from a single bid is a difficult
proposition.
Changing Financial Constraints Budget constraints offer a new dimension along which the
auction environment may change. Our preceding discussion focused on changing the
budget distribution’s support parameterized by w . Here we consider changing the budget
distribution G(·)holding [w ,¯w]fixed. Specifically, consider two distributions of budgets,
Gand G1, with the same support, such that G0likelihood ratio dominates G10. Intuitively,
financial constraints are more severe when budgets are distributed according to Grather
than G01. For example, such differences may reflect changing macroeconomic conditions.
In the second-price auction, changing the budget distribution from G0to G1increases
unconstrained types’ bids (Fang & Parreiras, 2002). In the first-price auction, in contrast,
increasing the severity of budget constraints analogously has ambiguous effects on
bidding: an unconstrained bidder may bid more or less. As likelihood ratio dominance is
already a restrictive stochastic ordering, this ambiguity can be interpreted as a negative
conclusion from an empirical perspective.
To confirm an ambiguous relationship, we revisit the model of Example 3 but with three
alternative distributions of budgets on [0,2]: G0(w) = w/2, G1
li
screen bidders along multiple dimensions, its introduction may reduce the auction’s
expected revenue. Corollary 3 follows from Theorem 3 and we state it without proof.
Corollary 3. Suppose w is low and let ß(s,w)=min{ ¯ b(s),w} be the equilibrium
strategyabsent any reserve prices. Let r∈(w, ¯ b(0)]be a common reserve price. Then
ßr(s,w)=
l if w<r min{¯ b(s),w} if w=ris an equilibrium when the reserve price is r. The
auction’s expected revenue has decreased.
When r< ¯ b(0) the bids of all types with a budget w= rare unchanged. Expectedrevenue
declines since the reserve price reduces the probability of sale by screening only low-budget
bidders. Without budget constraints, in contrast, a small reserve price screens out
low-valuation bidders and increases expected revenue through its uplift of the
lowestvaluation participating bidder’s equilibrium bid.
When private budget constraints exist, setting the revenue-maximizing reserve price is a
delicate exercise. For instance, even in the introductory model of Example 1 the
revenuemaximizing reserve price varies discontinuously with the model’s parameters (the
budget values and their distribution). If high budgets are very likely, a reserve price of r *= 1/2
is optimal. This is also the revenue-maximizing reserve price when budget constraints are
absent from the model. If bidders are often meaningfully budget-constrained, the optimal
reserve price will be strictly lower.
4 Concluding Remarks
Private budget constraints are a key feature of many economic environments and, as
argued above, have a deep strategic effect which organizes interaction in the first-price
auction. Although monotone equilibria (in a qualified sense) exist, equilibrium strategies
demand a marked amendment of existing intuition. Private budgets have both direct and
strategic effects on equilibrium bids and they can subtly interact with preferences changing
bidder’s incentives. Increasing the wealth of a bidder may not necessarily imply a greater
bid. Budget
constraintsalsochangethenatureofcompetitionleadingtoitsstratificationalongthebudget
dimension. The option to exploit other bidders’ budget constraints introduces a strategic
effect that breaks the natural relationship between valuations and bids.
30
Our results suggest many possibilities for further research. For instance, the degree to
which the inefficiencies introduced by budget limits can be ameliorated by resale is largely
unexplored. Similarly, extending the analysis to multi-unit settings or addressing deeper
technical matters within the standard environment, such as equilibrium uniqueness or
empirical identification, are open questions. Untangling the more subtle relationship
between bids, valuations, and budgets stands as a new, and in many cases continuing,
challenge for theoretical and empirical researchers alike.
31
A Appendix: Proofs of Theorems 1 and 2
The proofs of Theorems 1 and 2 follow a standard argument, similar to those employed by
Athey (2001) and Reny & Zamir (2004). Following some preliminary lemmas, we appeal to
the equilibrium existence results in Reny (2011) to complete the argument.
=bi =w' i. If (si,wi)=θi
(s' i,w' i), then ui(si,s-i,wi,bi
Lemma A1. Fix (s-i,w-i)and ri
i(s' i,s-i,w' i,bi).
' i)= inf
∂
-s' i)+ inf i∂¯u∂
∂
(wi -w' i)
u
¯u
i
(s
¯
' i)
i
ui
¯ui(si,s-i,wi,bi)-¯ui(s' i,s-i
)= u
Proof. We can establish the following inequalities:
i
i
∂si
∂wi
i
∂wi
-w' i)
i
,w ,b
∂ i
∂wi
-w' i)+ inf
(w -w
ξ
i(w
si
∂¯ (w
= inf=
ui
sup
i
=ui(w )-ui (w'
i)
Rearranging the final inequality gives the desired result.
i >s' i. (1) θi h(s-i|s' i)
i i -i
¯H(s-i|s' i) = h(s-i|s)
(2) H(s-i|s' i)=H(s-i|si). (3) ¯ H(s-i|s' i
'i
¯H(s-ii|si)
-i
θ
H(s-i|si)
i
i(bi,ß-i|θ
1. Ui(bi,ß-i|θ' i
2. Ui(bi,ß-i|θi
i
)= ¯
L
emma A2. Let si
)-Ui
Proof. These facts about affiliation follow from Milgrom & Weber (1982).
Lemma A3. Suppose A-1 and A-3 hold. Let ßbe a profile of nondecreasing strategie
bidders N \{i}. Let θ=' i. Suppose b>bare feasible bids for a bidder of type θwhich tie
probability zero with the bids submitted by bidders j=i. Moreover, suppose U ' i)=Ui(l,ß
i|θ). Then,
(b' i,ß-i ' i|θ)=(>)0 =⇒ U
(bi,ß |θi)-Ui (b' i,ß-i|θi)=(>)0.
)-Ui
(b' i,ß-i |θi)=(<)0 =⇒ U
( ,ß-i ' i|θ)-Ui (b' i,ß-i ' i|θ)=(<)0.
b
Proof. We prove part 1. Part 2 follows similarly. It is sufficient to show that Ui(bi,ß-i
i (b'
' i|θ)=Ui(bi,ß-i|θi)-Ui
(b' i,ß-i|θi).
i,ß-i
32
|θ)U
'
i
Suppose that b' i'= land define,
= {(s-i,w-i): maxj=i'
ßj (sj,wj ' i) = b '. Then,
maxj=i
Aßj(sj,wj)=bi
-i
A'
i,ß-i ' i|θ)-Ui (b' i,ß-i ' i|θ
,w' i ' i)]h(s ds-i |s' i)g(w-i)dw-ids
-i [¯ui(s' i -i,w' i
i)-¯u (s' i,s
-i
(1)
-i,w-i):
u
(s' i,s-i
˜A '
-i
)g(w -i)dw-i ,s-i
' i,bi
,b
,b
-i|s
(2)
i(s
i
,w-i
'
' -i,w
-i
.
-i
' -i
)∈A
} and A = {(s}. Suppose that Pr[A]>0and let ˜ A=A\A
-i
-i
0=Ui(b ,s,b) =i
,b
-i
-i
-i
)∈A
-i
, if (s-i,w
[¯ ' i,s-i,w' i,bi)-¯ui(s' i,s-i,w' i
ui
,b' i)]h(s-i|s' i)g(w-i)dw-ids-i
,b
-i
-i -i
=w-i,s' i] ] ')|W -i
'i
=E
[¯u=E
=E = A'
i
-ids
→[¯ui(s' i,s ,w' )-¯ui(s' i,s ,w' ' i)]1((s-i,w-i
' 19)is
(s
i
i,b
nondecreasing.
+ i,w)h(s' i We consider term (1) first. From Assumption A-1, ¯u' i,w
i
A'
E[[
¯u
) is non-decreasing in san
sthen for all s= s, (s-i') ∈ A
(s' i,S-i,w' i,bi)-¯ui(s' i,S-i
' ,b' i)]1((S-i,W
i
' ,W -i )]1((S i ,w i(si,S-i,wi,bi)-¯ui(si,S-i
E[[
¯u
-i
[
,wi,b i(si,s-i,wi,bi)-¯ui(si,s-i
)]h(s | )g(w-i)dw
¯
s
u
Next, consider term (2). Given ˜ A⊂T-i , define the following
subsets:
˜ )=0}, ˜-A={(s-i,w-i)∈ ˜
={(s-i,w-i)∈ ˜ A:u
A+ A:uii(s(sii,s-i,s-i,w,wii,bi,bi)<0}.
19 1(x∈X)is
33
the indicator function.
˜ A+Using
A-3 we note the following inequalities: = u˜ A= i(s+˜ A= +˜ A+'
h(s' i)g(w-i|s' i) ¯H(s)
h(s-i-i|s|s' ii-i) ¯) ¯H(s)h(s-i-i|s|sii)dwH(s) ¯-iH(s)g(w-i-i-i|sds' i|s'
i-i)g(w-i)g(w)dw-ids-i-i)dw-i)dw-ids-ids-i
¯ H(s-i|s' i)
The second inequality follows from ui(si,s-i,wi,bi)= 0 and
¯H(s-i|si)
iuiuu,sii(s-i' i(s(sii,w,s-i,s-i,s-i' i,bi,w' i,w,wii)h(s,bi,bi,bi-i|s)
i(si,s-i,wi,bi)<0and
H(s-i
-i|si)
-i|s' i)g(w-i)dw-ids-i
) h(s) h(s-i|s' i) H(s-i-i|s|s' ii) H(s)
H(s-i|si-i) H(s-i|s' i|s'
i)g(w-i)g(w-i)dw-i)dw-ids-ids-i
ui
˜ A-
i)h(s-i|si)g(w-i)dw-ids-i
=
' i,s-i,w' i,bi)h(s
ui(s' i,s-i,w' i,bi
ui(si,s-i,wi,bi
ui(si,s-i,wi,b
˜
A
=
=
-
˜
A
-
˜
A
-
' i,s-i,w' i
i)h(s
-i
|θ|s' i)g(w-i
-i
i ui(si,s i
=
,wi,b )h(s
.|si)g(w-i)dw-ids
i
'i
ui(s
i
' i,ß
i
i
-i
i
i
˜A
|s' i=
1. Similarly, since u) H(s(s=1,
,b-iCombining the previous results
gives )dw
Therefore, 0=Ui(bi,ß-i'
' i)-U (
b
d
s
-i|θ )=U
˜
A
-i
( ,ß | )-U
b
θ
i
(b'
i,ß
-i
-i
| ). If instead b'
θ i
⊂
¯[ B<8.
34r
i,wi
i
,
¯
i
B
]
f
o
r
s
o
m
e
=
l or Pr[A]=0, then the above analysis applies but only considering term (2).
Remark A1. If A-1 and A-2 are satisfied, Lemma A3 continues to obtain.
The proof is analogous, but simpler.
Proof of Theorem 1 We introduce two formal modifications to our model.
First suppose that there is no upper
bound on the bid that a bidder of type θcan place, i.e. there is no budget
constraint, but bidders continue to have a two-dimensional type (s).
Noting A-1, we can assume that for each i, B
Second, toincorporatethebudgetlimit, modifyeachbidder’sutilityfunctionasfollows. If abidder
exceeds her budget, bi>wi, then bidder ireceives apenalty of-F utils irrespective of the
auction’s outcome in addition to their auction payoff. Set F such that all bids bi>wi
are strictly dominated by the bid lirrespective of the bids of other bidders. Since and ui
¯ui
are bounded, such a constant exists. Examining players’ behavior in such modified games
where agents are penalized for deviating “inappropriately” (from the analyst’s perspective)
is often seen in the auction literature (for example see Jackson & Swinkels (2005)). Also
Che & Gale (1998) note this possibility in the context of auctions with budget constraints.
let ρi(θi;ß-i
Noting this modified auction, for a profile of nondecreasing strategies ß-i
i:
UiUi(bi(bi,ß-i,ß-i|θ-i|θ-i)=wi .
if bi)-F if bi
>wi
ρi(θi;ß-i)=arg maxbi∈Bi∪{l}
i
θi i,ß
)=Ø. Since a bid above wi
-i. For all θi, ρi(θi;ß-i
i
-i, supρi(θi,ß-i
)=w
Proof. Since the set Bi
θ
) be the set of bidder i’s best-response bids when her type is θ
Lemma A4. Consider the modified auction game described above. Fix a nondecreasing
strategy profile ß)is nonempty, bounded, and increasing in the strong set order.
-i
∪{l}is nonempty and discrete, ρis strictly dominated by (
θ
lirrespective of ß
'
i(θi;ß-i) is nondecreasing, let θ
= i 'i and let b ∈ ρ ( ;ß
i
θ
'i
(b' i,ß-i|θi)=Ui(bi,ß-i
' i|θ)-U
(b ,ß |θ
∈ρ (θ ;ß ),
-i
|θ
i
35
(b' ,ß-i |θ
i
|θi
i(θ
= w' i
=
b
i
i
(
θ
i
'
i,ß
' i,ß
i)=
|θ
i
i
-i
i)
i
i
)
.
=
⇒
b
U
-i
i
i ' i)-U
(
b
-i|θ
i(b
-i
i
)-U
i
' i)
=⇒ b ∈ρ
'
i;ß
(b' i,ß-i
)-U
i(bi,ß-i
i
i i
. To show that ρ
-i) and b
'
' i;ß-i). If bi
for all such bids, we are done. Otherwise, suppose b' >bi
i
i
i an
0=Ui
' i=
>b , both bids are feasible for bidders of type θ and θ'
i
d
b
∈ ρ. Since w. By Lemma A3 or Remark A1,
0=U
)is increasing in the strong set order.
As the interim best-reply correspondence is nondecreasing, player i has at least one
nondecreasing best reply when others are following nondecreasing strategies (Topkis,
1998, Theorem 2.8.3).
Hence, ρ
i
Lemma A5. Consider the modified auction game described above. There exists an
equilibrium in nondecreasing strategies.
Proof. The conclusion follows from Reny (2011) and specifically Proposition 4.4, Theorems
4.1 and 4.3, and Remark 1. His conditions G.1–G.6 are satisfied by our game and are easy
to verify.
Remark A2. An equilibrium in the modified auction game is an equilibrium in an auction with
budget constraints where each bidder’s set of available (competitive) bids is finite.
Proof of Theorem 2 To prove Theorem 2(1) we start with the setup of Theorem 1 and we
consider a sequence
auction games, indexed by k, where the set of a bidder’s feasible bids becomes dense in
[rik i,8). In auction k, the set of feasible bids for bidder iis B=[ri, ¯ B]nPk ik iwhere Pi= {r-νi+km
2:m=1,...,Mk ik i}and M=⌈2k( ¯ B-(ri-νi))⌉. {νk ik j}are small fixed constants such that PnP=Øfor
all k. We let Bi=[ri, ¯ B]. Let ßkibe the corresponding sequence of equilibria.
LemmaA6(Theorem2(1)). Assumeprivatevalues, i.e. ui(si,s-i,wi,bi)=ui(si
,s' -i,wi,bi
i =[ri
k
+
i
1
k
-i
i:T→Bi
i
i
˜ ßk i(θi)=
l wk imax{b∈B∪{l}:b<min(ßi(θi)+As k→8, the (ex-ante)
loss to bidder ifrom using ˜ ßk i1 2k,wi)} wversus ßiii
, ¯ B].). Then there exists an equilibrium in nondecreasing strategies when ∀i, B
Proof. This is a consequence of better-reply security (Reny, 1999); see also the proof of
Corollary 5.2 in Reny (2011) and Jackson & Swinkels (2005). Briefly, for kvery large ßis an
o-equilibrium of the first-price auction where all bidders can place bids from B. Given any
monotone strategy profile ß, bidder ican approximate the payoff she would receive from
the (feasible) strategy ß∪{l}. Specifically, suppose bidder ibids
+ 1k .
<r
=r
becomes arbitrarily small
and this conclusion holds versus all strategies that the other bidders may follow.
To prove Theorem 2(2), we adopt the argument of Araujo & de Castro (2009, Theorem
2). Since the construction is very similar, forbrevity we only sketch the relevant steps. Again
take a sequence of auctions indexed by k; however, each auction has the following features:
36
k i1.
i
In auction k, B =[ri, ¯ B]. The bid lis always available and l<mini ri -2.
=wi
2. The penalty for exceeding a budget constraint is modified as follows. If , it is zero. If w<bi=wi+1/k, it
bi
+ok 1,...,bN +ok
N
k 1,...o
k. Considering the sequence of equilibria, ßk
a.e.
andfora.e. (si,wi), ˆ ß (si,wi)=wi
i
such thatß
ˆ ßi
i
i
i˜
= ˆ ßj(tj
i(θ
ϕ
(Bj :
(
×
k
i
j
i
(
ß
k
(
t
)
)
i
f
b
˜ϕi(b;t)=
limki
. The allocation rule in auction kfor bidder iis ˜ϕk
i(b)=E[ϕi(b1
k
i
→
i
)]where (ok N)is
3 a vector of i.i.d. uni
[-1/k,1/k].
The modified allocation and penalty r
a bidder has a nondecreasing best respo
strategies. For each k, an auction with th
nondecreasing strategies, ß
, and passing to a subsequence as need
Lemma A.10), there exists a nondecreas
i
∪{l}))×T→[0,1]as
ˆ
ß j)∀j ϕ(b) otherwise .
), and an announcement of her type, ˜s
ˆ
, a bidder’s strategy is a bid, ˜ . There exists aspecial tie-breaking rule, which depends on pla
ßi
ß
announcement of their private information, for which the biddin
equilibrium bidding strategy. Define this allocation rule ˜ϕ
Under ˜ϕ
). ˜
ϕ
i(θi
allocate
s the
item to
the
highest
bidder
(if
unambi
guous)
and
announ
cement
s of
private
types
are
used to
resolve
ties
among
high
bidders.
Lemma A7 (Theorem 2(2)). Suppose
Bstrategy profile where for all bidders
i, ˜ ß
i i)and ˜s ( )=θ
is an
θ
equilibrium.
i
i
i
( )= ˆ (
θ ß
θ
i
=[ri, ¯ B]and the allocation rule ˜ϕis in effect.
The
Proof. The argument mirrors the proof of Lemma 4 in Araujo & de Castro (2009).
B Appendix: Proofs from Section 3
Lemma B8. d(b,x|s) is increasing in band xand decreasing in s. η(x|s)is increasing
in both arguments.
37
Proof. Since g'(b)=0, dis increasing in b. dis increasing in xas H(x|s)is increasing in x.
Finally from affiliation, if s'>sthen H(y|s')=H(y|s)and thus,H(y|s') 1-H(y|s')=H(y|s) 1-H(y|s)'. Thus,
d(b,x|s)=d(b,x|s). The conclusions concerning ηfollow easily from its definition.
exists s* >0 such that ˜ ψ(s*
Lemma B9. ˜ ψ(s):[0,1)→[w,8)is continuous.
Proof. Suppose ˜ ψ(s) is not continuous at some s1. If ˜ ψ(s1) >w , then ˜ ψ(s) and η(s1|s1)
= d(ψ(s1),s1|s11) =ψ(s1) by definition. Hence ψ(s) = Ø for all ssufficiently close to s,
˜ψ(s)=ψ(s). Since ψ(s)is continuous when it is defined ˜ ψ(s)must be continuous at s11—
contradiction. Therefore,˜ ψ(s1) = w . Since ˜ ψ(s) is not continuous at s, we can find a
sequence tk→ s1such that for each k ˜ ψ(tk1) >w+ofor o>0 sufficiently small. Since
˜ψ(tk)=¯v, we can find a further subsequence, tk' such that { ˜ ψ(t' )}converges to some
value ˜ψ1= ¯w+o. By continuity, however, we conclude that lim kk' η(tk' |tk' )-d( ˜ ψ(tk' ),tk' |t' ) =
η(s1|s1)=d( ˜ ψ1,s1|s1). But then either ˜ ψ1∈ψ(s1)or ˜ ψ1k=w . Both of these conclusions are
contradictions. Hence˜ ψ(s)is continuous.
Lemma B10. If w=0, then there exists a critical point.
Proof. Since 0=d(0,0|0)=η(0|0)=¯v= ¯w=d(¯w,0|0), ψ(0)=Ø. If ψ(0)=0, then (0,0) is a
critical point. If ψ(0)>0, then since
)
is
n
o
n-
n
e
g
at
iv
e.
T
h
e
n,
*)
∂b (s*,b
)
=-γ(b
∂b (s*,b <0
)
*)-d(b,s*|s*
=0
∂ ˙ 2b
∂˙s ∂s-
are real-valued. The eigenvalues will be real if
∂˙s ∂b
(s*,b*)
+4
=γ' *(b)(η(s*|s* ))
∂˙s ∂ ˙ bwhen
∂b ∂s
(s*
∂b
+γ(b
*
3
8
∂
d
evaluated at
*)
∂d
,b*
*
w
h
er
e
th
e
c
o
n
cl
u
si
o
n
fo
ll
o
w
s
fr
o
m
d
st
ri
ctl
y
in
cr
e
a
si
n
g
in
b
a
n
d
γ(
b*
)=
0.
Al
s
o,
*(b
-v(s*,s*
=
∂λ∂s
∂s∂ ˙ b
(s*,b*)
(s*,b*)Therefor
e, 4
R
e
m
ar
k
B
3.
Si
n
c
e
th
e
ei
g
e
n
v
al
u
e
s
of
(B
1)
ar
e
re
al
-v
al
u
e
d,
th
e
o
ut
st
a
n
di
n
g
c
o
+λ(s*|s*)
∂v
∂s
*,b
)
=-λ(s*|s*
∂s (s*,b <0
)
∂
s
∂ ∂
˙
s ˙
∂ b
b
=0))
>0as
neede
d.
*
(
s
) ∂v
*
n
c
er
n
is
th
at
o
n
e
of
th
e
ei
g
e
n
v
al
u
e
s
is
z
er
o.
T
h
at
is,
w
h
e
n
e
v
al
u
at
e
d
at
(s
*,
b*
),
∂˙s
∂s
S
u
c
+
∂ ˙ b+
∂b
∂˙s ∂s
∂ ˙ 2b
∂b
+4
∂˙s ∂ ˙ b=0,
∂b
∂s
or
∂˙s + ∂ ˙ b
∂s
∂b
∂˙s ∂s
∂ ˙ 2b
∂b
+4
∂˙s
∂b ∂˙
b∂s
=0.
h
c
a
s
e
s
ar
e
n
o
ng
e
n
er
ic
a
n
d
a
s
m
al
l
p
er
tu
rb
at
io
n
to
th
e
m
o
d
el
re
s
ol
v
e
s
th
e
sit
u
at
io
n.
A
ty
pi
c
al
in
st
a
n
c
e
of
a
n
o
ng
e
n
er
ic
c
a
s
e
w
o
ul
d
o
cc
ur
if,
fo
r
e
x
a
m
pl
e,
ψ
(s
)a
n
d
ν(
s)
w
er
e
e
x
a
ctl
y
ta
n
g
e
nt
at
a
cr
iti
c
al
p
oi
nt
(s
*,
b*
).
Pr
o
of
of
T
h
e
or
e
m
3.
S
u
p
p
o
s
e
th
e
ot
h
er
bi
d
d
er
is
fo
ll
o
wi
n
g
th
e
pr
e
sc
ri
b
e
d
st
ra
te
g
y
ß(y,z) = min{ ¯ b(y),z}. The expected utility of a bidder of type (s,w), w =bids ¯ ¯ b(0),
b(x)=wis
who
0¯
b(x)
Di
ff
er
e
nt
ia
ti
n
g
wi
th
re
s
p
e
ct
to
x
a
n
d
c
ol
le
cti
n
¯
01
Ui(
¯
b(x)
|s,w
)=
v(s,y)-
0
¯ b(x) h(y|s)g(z)dydz+
¯
w
x
v(s,y)-
¯ b(x)
b
(
x
)
h(y|s)g(z)dydz.
g
te
r
m
s,
∂Ui( ¯
b(x)|s,w)
∂x =g(
¯ b(x))(1-H(x|s))
v(s,x)λ(x|s)γ(
¯ b(x))
To show that ¯ b(s) is optimal among all bids¯ b(x), x∈ ¯
∂Ui( ¯
b-1
[0,∂Ui
b(x)|s,w)∂x
∂x
∂Ui( ¯ b(x)|s,w)∂x =
By a similar argument, if x <s,
=
0
a
n
d
x
<
s
=
⇒
(¯
b(x
)|s,
w)
(
w
)],
it
is
s
u
ffi
ci
e
nt
to
s
h
o
w
th
at
x
>
s
=
⇒
=
i
∂Ui( ¯
b(x)|s,w)∂x
39
¯
b(·)
.
¯ b(x) ¯ b'(x) η(x|s)-d(¯ b(x),x|s) .
+
0.
S
u
p
p
o
s
e
x
>
s.
B
y
a
ffi
li
at
io
n,
η(
x|
x)
=
η(
x|
s)
a
n
d
d(
¯
b(
x)
,x
|x
)=
d(
¯
b(
x)
,x
|s
).
H
o
w
e
v
er
,
al
s
o
d
u
e
to
a
ffi
li
at
io
n
λ(
x|
x)
=
λ(
x|
s)
a
n
d
b
y
A
ss
u
m
pt
io
n
A3(
1
a)
,
v(
s,
x)
λ(
x|
s)
=
v(
x,
x)
λ(
x|
x)
.
T
h
u
s,
∂U ( ¯ b(x)|x,w)∂x =0. (B2)
=
0.
T
h
u
s,
a
bi
d
d
er
of
ty
p
e
(s
,w
)
h
a
s
n
o
in
c
e
nt
iv
e
to
d
e
vi
at
e
to
a
n
y
fe
a
si
bl
e
bi
d
in
th
e
ra
n
g
e
of
Since bids greater than ¯ b(1)are
inferior to
¯ b(1), it remains to verify that the bid
b<
¯
b(0
)
is
n
ot
a
pr
o
fit
a
bl
e
d
e
vi
at
io
n.
T
h
e
re
s
ul
ti
n
g
e
x
p
e
ct
e
d
ut
ilit
y
is
0 1
(v(s,y)-b)h(y|s)g(w)dydw=G(b)(η(0|s)-b).
Ui(b|s,w)= 0b
This expression is concave in band achieves a maximum at ˇ bwhich solves η(0|s) ˇ
=G( ˇ b)/g(ˇ b). Recall that ψ(0) satisfies η(0|0) = ψ(0)+G(ψ(0)) g(ψ(0))
b+
henceUi(b|s,w)=Ui( ¯
consi
for ssufficiently close to zero, ¯ b(s) = ψ(s),
b(0)|s,w)=Ui(ß(s,w)|s,w). Finally der a
¯ b(0) = ψ(0). Therefore, ∀b = ¯
b(0)
,
. Consequently, by
˜ b(x), x= ˜s. The expected p
this
affiliation, η(0|s) = η(0|0) =⇒ˇ b = ψ(0) because b+G(b)/g(b) is increasing. However,
bidder with a budget of w<
other bidder, she solves maxb=w
¯ b(0). Given the strategy followed by
the
G(b)(η(0|s)-b). By the argument above, ß(s,w)= wis their
optimal bid.
Proof of Theorem 4. As in Theorem 3, we need only verify that no type of bidder wishes to
deviate to an alternative bid in the range of ß(s,w). There are two cases:
1. Consider a bidder with a value-signal s<˜sand budget w. Then ß(s,w)=a(s). Since
a(s)is the equilibrium bid in a first-price auction without budget constraints, this type of
bidder will not have any profitable deviations to any bids a(x), x∈[0,˜s]. Therefore, we
need only rule out deviation to a bid
0 1
0 x
bid is
(v(s,y)˜ b(x))h(y|s)dy.
Ui( ˜ b(x)|s,w)=G(˜
b(x))
(v(s,y)˜ b(x))h(y|s)dy+(1-G(˜ b(x))
∂Ui( ˜
b(x)|s,w)∂x
i(
˜ b(x)|s,w)=Ui(w |s,w)=Ui(a(˜s)|s,w)=Ui( ˜ bf
40
y an argument parallel to (B2) we know that if x>˜s= s,
= 0. This implies
B
that U(s)|s,w).
2. Consider a bidder with a value-signal s>˜sand budge
Theorem 3 establishes that this bidder has no profitable
Consider a deviation to a bid a(x), x= ˜s. The expected
Ui(a(x)|s,w)= ∂Uix 0(v(s,y)-a(x))h(y|s)dy. Differentiating this expression gives(a(x)|s,w)
∂x=H(x|s) (v(s,x)-a(x))h(x|s) H(x|s)=H(x|s) (v(x,x)-a(x))h(x|x) H(x|x)-a'-a(x) '(x) =0.
Theinequalityfollowsfromaffiliation. Hence, Ui(a(x)|s,w)=Ui(w |s,w)=
Ui(ß(s,w)|s,w).(a(˜s)|s,w)=Ui
+s→˜
˜ b'(s). We can
Proof of Corollary 2. It is sufficient to show that a'(˜s)<lim
s
write
H(s|s)
H(s|s) (1-G(
+
(v(s,s)˜
h(s|s)
1-H(s|s)
˜ b'(s)=
-γ( ˜
b(s) H(s|s) .
1-G( ˜ b(s))
b(s))(η(s|s))
G( ˜ b(s))
˜
˜ b(s))(1-H(s|s)
b
(
s
)
)
+
The first bracketed term is positive for all s= ˜s. Moreover, since η(s|s)> ˜ b(s), g(w )=0,and
lim+s→˜s˜ b(s)=w , the limit as s→˜s+of the first bracketed term is strictly greater than 1. But,
lims→˜s+ (v(s,s)˜ b(s))h(s|s) H(s|s)=a'(˜s). Thus, lim+s→˜s˜ b(s)>a'(˜s).
Lemma B11. When wis intermediate, Z=Øand at ˆs=supZ, (8) holds with equality.
Proof. We first show that Z=Ø. There are two cases. First, suppose µ(s)=w for some s= a -1(w ).
Forsuchans, (8)reducesto s 0(v(s,y)-w )h(y|s)dy= s 0(v(s,y)-a(s))h(y|s)dy, which is true since w
>a(s). Suppose instead µ(s)>wfor all s<a-1(w). In this case, there is a critical point, s0=0, and ¯
b(s0)=w. Hence, (8) reduces to
G( ¯ b(s0)) 01(v(0,y)¯ b(0))h(y|0)dy=G(¯ b(s0))(η(0|0)¯ b(0))=0.
The inequality obtains since η(0|0)¯ b(0)=η(0|0)-w =0.To confirm that there is a value at which
(8) holds with equality, it is sufficient to show that at ˜s=a-1(w ), the inequality in (8) is reversed.
Consider the following maximization problem: maxb=w˜s 0(v(˜s,y)-b)h(y|˜s)dy+G(b) 1
˜s(v(˜s,y)-b)h(y|˜s)dy. The first-order
41
condition leads to 0=η(˜s|˜s)-d(b,˜s|˜s)or b=ψ(˜s). Since ˜s∈S0˜s, ψ(˜s)=µ(˜s)=w. Thus,
(v(˜s,y)-µ(˜s))h(y|˜s)dy+G(b) ˜s1ψ(v(˜s,y)-µ(˜s))h(y|˜s)dy= 0˜s(v(˜s,y)-a(˜s))h(y|˜s)dy.
That (8) holds with equality at ˆsfollows from continuity.
Proof of Theorem 5 The proof has two parts. The first set of lemmas concerns the functions
fand b. The final part verifies that the proposed strategy is an equilibrium. We specialize our
notation to the case of independent value-signals as stated in the theorem; thus,
H(y|s)=H(y), etc.We start by deriving some necessary relationships between b 11and f, which
we use to construct a map Λ. A fixed point of Λwill be used to define b1and f. b1and fneed to
be consistent with two facts:
f(s)describes an indifference condition. For ˆs=s=ˆs, types (s,f(s))are indifferent
between bids of b1(s)and f(s). The expected utility from bidding f(x), x=ˆs, is
'1.
Ui(f(x)|s,f(s))= 0ˆs(v(s,y)-f(x))h(y)dy (B3)+ ˆsx(v(s,y)-f(x))G(f(y))h(y)dy
(B4)+G(f(x)) x1(v(s,y)-f(x))h(y)dy. (B5)
(B3) is the contribution to utility from defeating all bidders of type y<ˆs. (B4) is the
contribution to utility from defeating bidders of type y∈[ˆs,x)but who have w<f(y). Such
bidders are bidding according to b1(·). (B5) is the contribution to utility from defeating
all bidders of type y=xwho have a budget less than f(x). In an analogous manner, the
expected utility of bidding b1(x)is
Ui(b1(x)|s,w)= 0ˆs(v(s,y)-b1(x))h(y)tdy+ ˆsx(v(s,y)-b1(x))G(f(y))h(y)dy.(B6)
Both expressions for expected payoffs assume b1is increasing and continuous and
that fis decreasing and continuous. In equilibrium, we will require that U(s)|s,f(s))=
Uii(b1(f(s)|s,f(s)). Rearranging the terms in this equality gives (12) which is reproduced
42
here: f(s)-b1(s) η(s|s)-f(s)
(·) must be optimal for all types who do not bid above w . The usua
2. b1
condition yields the differential equation (11) which is reproduced
1,c2)=sups∈[ˆs,1] 1 '=0 =⇒ b(s)= [v(s
1(x)|s,w) ∂x x=s
ˆsG(f(y))h(y)dy (B8
1
2
|c1
∂
U
i
(
b
We wish to show that there exist band fthat meet (B7) and (B8), along with the collection
of boundary and monotonicity conditions from (9). Let C[ˆs,1] be the set of continuous
functions on the interval [ˆs,1]equipped with the usual metric, d(c(s)c(s)|, and define the
following subsets:
∈ [ˆs,1], then |b(s)-b(s'
supH(ˆs)s∈[ˆs,1]
v(s,s)h(s)
')|= sups∈[ˆs,1]
F(s|b,ζ)=min arg minˆ f∈[w
'
ζ(s)
ˆ
f-b(s)η(s|
s)ˆ f
=
0,
,η(s|s)]
• B ⊂ C[ˆs,1]. If b∈ B then b(ˆs) = a(ˆs) and if s,s'
)| = |s-s'|.• Z ⊂C[ˆs,1]. If ζ∈Z then ζ(ˆs)=H(ˆs), H
|ζ(s)-ζ(sh(s) |s-s'|.B and Z are both equicontinuo
With (b,ζ)∈B ×Z given, consider the following fu
G( ˆ f)(1-H(s)) . (B9)
max 0, ˆf-b(s) η(s|s)-ˆ
fG( ˆ f)(1-H(s))ζ(s)
Given F, define the map Λas follows: Λ(b,ζ)=( ˇ b,
ˇ b(s)=a(ˆs)+ ˆss max{v(x,x)-b(x),0}G(F(x|b,ζ))h(x) ζ(x) dx (B10)
To develop an intuition for F, consider Figure B1, which sketches the value outputted by
Ffor a fixed s, b, and ζ. Ideally, Fwould like to output a solution to max
. When there are two solutions, it selects the smaller one (Figures B1b and B1c).
(There can be at most two solutions.) When the desired solution does not exist, Fminimizes
the distance between the two sides of the equation (Figure B1a). (This minimum exists and
will be unique.) Clearly, Fis continuous in s, b, and ζ.
ˇ ζ)where
ˇ ζ(s)=H(ˆs)+ ˆss G(F(x|b,ζ))h(x)dx (B11)
43
L
LR
F
LR
fb
R
Min Distance
w η Ff b
w ηFf b
(a) No
Solution
(b) Two
Solutions
Figure B1: Definition of F. Notation: L(f)=max 0,
wη
(c) Solution at w
f-b(s)
η(s|s)-f
, R(f)=
G(f)(1-H(s))
ζ(s)
.
Λis continuous, Λ(B×Z)⊂B×Z, and by Schauder’s fixed point theorem, Λhas a fixed point.
Lemma B12. Let (b,ζ) be a fixed point of Λ and let F(s) = F(s|b,ζ). Then, (1) b(s) satisfies
the integral equation b(s)=a(ˆs)+s ˆs[v(x,x)-b(x)]G(F(x|b,ζ))h(x) ζ(x)dx, (2) b(s) is strictly increasing if
and only if F(s)>w , and (3) b(s)=w.
(1) Suppose there is an interval [ˇs,ˇs] ⊂ [ˆs,1] such that for all x∈ [ˇs,ˇs'], v(x,x)b(x)
=0. Then b(x) =b(ˇs) for all x∈ [ˇs,ˇs']. Since b(s) is continuous and b(ˆs)=a(ˆs)< v(ˆs,ˆs),
without loss of generality, we may assume that at ˇs, v(ˇs,ˇs) = b(ˇs). However, for all x>
ˇs, v(x,x) >v(ˇs,ˇs). Hence, x∈ (ˇs,ˇs) =⇒ v(x,x) >b(ˇs) = b(x), which is a contradiction. (2)
By the preceding part, v(s,s)-b(s)>0a.e. Thus, the conclusion follows. (3) Suppose b(s)>w
for some s. Since b(s)is nondecreasing there exists ˇs>ˆssuch that for all s>ˇs, b(s) >w .
But from the definition of F, if b(s) = w, F(s) = w; thus, the integrand above is zero for all
x> ˇs. Coupled with b(·)’s continuity, however, this is a contradiction.
Lemma B13. Let (b,ζ)be a fixed point of Λand let F(s)=F(s|b,ζ). Then for all s=ˆs,
F(s)-b(s)
= G(F(s))(1-H(s))ζ(s) . (B12)
η(s|s)-F(s)
'Proof.
Proof. If b(s) = w , then (B12) holds since b(s) = F(s) = w ensures equality. Suppose b(s)
< w. We first establish that (B12) holds at s = ˆs. When s = ˆs, (B12) becomes
44
F(ˆs)-a(ˆs)
η(ˆs|ˆs)-F(ˆs)
d df
=
d
d
s
=
g
(
f
)
(
η
f
)
H
G(F(ˆs))(1-H(ˆs))
H(ˆs)
g(f) -
=
g
(
f
)
1
H
. From Lemma B11, if F(ˆs)=µ(ˆs), then the preceding equation is
satisfied. Thus, F(ˆs) = µ(ˆs). Suppose ∃f < µ(ˆs) such that. Rearranging terms and
suppressing the dependence on ˆs,
-a=(η-f)G(f)(1-H)/H-f (B13)
Differentiating the right-hand side of (B13) with respect to fgives
(η-f)G(f)(1-H)/H-f
1-H -G(f) 1-HH -1
H
=g(f) 1-H
η-fG(f)
H g(f)(1-H)
H (η(ˆs|ˆs)-d(f,ˆs|ˆs)).
When f<µ(ˆs), η(ˆs|ˆs)-d(f,ˆs|ˆs)>0. So the right-hand side of (B13) is strictly increasing as
f↑µ(ˆs). Indeed, the above derivative is zero only at f=ψ(ˆs)=µ(ˆs). Therefore, there is no
f<µ(ˆs)that satisfies (B13). Thus, F(ˆs)=µ(ˆs).
As F(s), b(s), and ζ(s)are continuous, if (B12) fails to be satisfied, it must fail on some
interval, say (˜s,˜s''). For all such s∈(˜s,˜s),F(s)-b(s) η(s|s)-F(s)=G(F(s))(1-H(s)) ζ(s). Equivalently,
(F(s)-b(s))ζ(s)=(η(s|s)-F(s))G(F(s))(1-H(s)). (B14)
'As the inequality in (B14) becomes strict for s∈(˜s,˜s), at s=˜s, the derivative of the
lefthand side of (B14) with respect to s(holding F(˜s)>w fixed) must exceed the derivative
of the right-hand side of (B14) with respect to s. Intuitively, the right-hand side escapes
downward from the left-hand side locally at ˜s. Therefore, the following must be true,
[(F(˜s)-b(s))ζ(s)]|s=˜s = dds [(η(s|s)-F(˜s))G(F(˜s))(1-H(s))]|
.s=˜
s
(˜s) =
Some algebra and substituting b'
∂
∂s[v(˜s,˜s)-b(˜s)]G(F(˜s))h(˜s)and
ζ(˜s)v(s,y) s=˜s
ζ ''
45
(˜s) = G(F(˜s))h(˜s) gives
0=G(F(˜s))1 ˜s∂ ∂sv(s,y) s=˜sh(y)dy. However,>0. Since ˜s=1and F(˜s)>w , this is a
contradiction.
Lemma B14. Let (b,ζ)be a fixed point of Λand let F(s)=F(s|b,ζ). There exists ˆs',
b(s)=F(s)=w .∈(ˆs,1] such that for all s=ˆs
Proof. As (B12) holds, at s=1, we have F(1)=b(1). Moreover, as b(s)= w =F(s) we
must have b(1)=F(1)=w. The lemma’s conclusion follows since whenever b(s)=w, then
F(s)=w as well, and b(s)is nondecreasing.
Lemma B15. Let (b,ζ)be a fixed point of Λand let F(s)=F(s|b,ζ). Then,
(1-H(s))[g(F(s))(η(s|s)-F(s))-G(F(s))]=ζ(s).
Proof. Suppressing sinthenotation, F(s)satisfies F-b=(η-F)G(F)(1-H)/ζ. ForF>w , the
right-hand side of this expression is concave in F, therefore at the minimal solution to this
equation, the slope of the right-hand side must exceed the slope of the left-hand side.
Differentiation gives [g(F)(η-F)-G(F)](1-H)/ζ=1, as desired.
Lemma B16. Let (b,ζ)be a fixed point of Λ. F(s)=F(s|b,ζ)is non-increasing on [ˆs,1]. Proof.
Noting Lemma B13, Ui(F(s)|s,F(s))=Ui(b(s)|s,F(s))for all s>ˆs. Moreover, since band ζare
differentiable a.e., F(s)is differentiable a.e. Suppressing a bidder’s budget-type in the
following notation, differentiating the preceding expression with respect to sgives
∂Ui(F(s)|s) dF(s) + ∂Ui(F(s)|s) = ∂Ui(b(s)|s) db(s + ∂Ui(b(s)|s)
(B15
∂F(s)
ds
∂s
∂b(s)
) ds ∂s
)
=
d
=
(1-H(s))(g(F(s))(η(s|s)-F(s))-G(F(s)))-ζ(s)
F'= 0ˆsd
dF(s)
dsv(s,y)h(y)
dsv(s,y)h(y)dy+ +G(F(s)) s1d dsˆssd dsv(s,y)h(y)G(F(y))dy
0
dy+ ˆss
ˆ
ds
d dsv(s,y)h(y)G(F(y))dy (B19)
s
i
d
b
(
s
)
d
s
i
i
(s) (B16) ∂U(F(s)|s) ∂s
v(s,y)h(y)dy (B17) ∂U(b(s)|s) ∂b(s)=0 (B18) ∂U(b(s)|s) ∂s
w
here
∂Ui(F(s)|s)
∂F(s)
(B18)iszero by theenvelope theorem. The firstterms of (B16),
collectively
+G(F(s)) 1 s d ds
*
∂Ui(F(s)|s) dF(s) ds
∂F(s)
1
*
∂Ui(F(s)|s)
∂F(s)
'
*,ζ
*,ζ*
46
,
isnonnegative by Lemma B15. Simplifying (B15),v(s,y)h(y)dy=0. When F(s)>w , the second
term on the left-hand side is positive. So, F(s)=0. Choose a fixed point of Λ, (b), and define
b(s) = b(s) and f(s) = F(s|b). Therefore, the strategy ß(s,w)exists with the stated properties.
Lemma B17. Suppose w is intermediate. Then ß(s,w)is a symmetric equilibrium.
Proof. We divide the argument into cases. As usual, a bidder will not have a profitable
deviation to a bid outside the range of ß(s,w).
1. Consider a bidder with a value-signal of s<ˆs. This bidder’s expected payoff when
following the strategy ß(s,w)is Ui(a(s)|s,w)= s 0(v(s,y)-a(s))h(y)dy.(a) By the usual argument
confirming an equilibrium in a first-price auction, this type of bidder will not have a
profitable deviation to any bid a(x), x∈[0,ˆs].
(b) Suppose this bidder bids '(x), x∈[ˆs,ˆs]. The expected payoff from this bid is
b1
Ui(b1(x)|s,w)= 0ˆs(v(s,y)-b
1(x))h(y)dy+ ˆsx(v(s,y)-b1(x))G(f(y))h(y)dy.
ˆsx G(f(y))h(y)dy .
' 1(x)]G(f(x))h(x)-b(x) H(ˆs)+
1Hence,
d dxUi(b1(x)|s,w)=[v(s,x)-b
d dxThus, s→Ui(b1 d dxUi(b1(x)|s,w) = 0 and x<s =⇒d dxUis=x=0. Therefore, s<x =⇒i(b1(x)|s,w) = 0.
Ui(
Hence, when s = ˆs = x, U(x)|s,w) = Ui(b1(b1(ˆs)|s,w) = U(a(ˆs)|s,w) = Ui'(a(s)|s,w).
Therefore, this is not a profitable deviation. (c) Suppose this bidder bids f(x),
x∈[ˆs,ˆsi]. The expected payoff from this bid is
(v(s,y)-f(x))h(y)dy.
x ˆs(v(s,y)-f(x))h(y)dy+
(v(s,y)-f(x))G(f(y))h(y)dy
U
i
(
+G(f(x)) x1f
(
x
)
|
s
,
w
)
=
ˆs
0
Consider the difference
Ui(f(x)|s,w)-U= 0= 0ˆs[bˆs1[b1i(b1(x)|s,w)(x)-f(x)]h(y)dy+ ˆs(x)-f(x)]h(y)dy+
ˆsxx[b1[b1(x)-f(x)]h(y)G(f(y))dy+ x(x)-f(x)]h(y)G(f(y))dy+ x11[v(s,y)-f(x)]h(y)dy[v(x,y)-f(x)]h(y)dy
=Ui(f(x)|x,w)-Ui(b1(x)|x,w)=0
By case 1b, Ui(f(x)|s,w)=Ui(b1(x)|s,w)=Ui(a(s)|s,w).
47
(d) Suppose this bidder bids ˆ b(x), x∈[ˆs,1]. Analogously to the proof of Theorem
4and noting case 1c, Ui( ˆ b(x)|s,w)=Ui( ˆ b(ˆs)|s,w)=Ui(f(ˆs)|s,w)=Ui(a(s)|s,w).
2. Consider a bidder with a value-signal s=ˆsand a budget w=µ(ˆs). Thus, ß(s,w)= min{ˆ
b(s),w}. Hence, ∃sw∈[ˆs,s]such that ß(s,w)= ˆ b(sw).
(a) Following Theorem 3, we can rule out all deviations to bids above µ(ˆs). (b) Suppose
this bidder bids a(x), x∈[0,ˆs]. Analogously to the proof of Theorem 4,x=s =⇒
Ui(a(x)|s,w)=Ui(a(ˆs)|s,w). Noting that ˆ b(ˆs)=f(ˆs)=µ(ˆs),
ˆs
[ ˆ b(ˆs)-a(ˆs)]h(y)dy-G(f(ˆs))
Ui(a(ˆs)|s,w)-Ui( ˆ b(ˆs)|s,w)=
(a(ˆs)|ˆs,w)-Uii=U= 0ˆs[ ˆ
Thus, Ui(a(x)|s,w)=Ui
b(ˆs)-a(ˆs)]h(y)dy-G(f(ˆs))( ˆ
b(ˆs)|ˆs,w)=0.
(a(ˆs)|s,w)=Ui( ˆ b(ˆs)|s,w)=Ui( ˆ b(sw
)|s,w). (c) Suppose this bidder bids f(x), x=s. A calculation gives
d Ui(f(x)|s,w)=f'(x) g(f(x)) x1[v(s,y)-f(x)]h(y)dy-G(f(x))(1-H(x))
dx
dx 1
dx
0
ˆ
b
b
v(x,y)h(y)dy=0
=-G(f(x))
=f'(x)
H(ˆs)+ H(ˆs)+ ˆsx
G(f(y))h(y)dy
g(f(x)) x1 [v(x,y)-f(x)]h(y)dy-G(f(x))(1-H(x))
x G(f(y))h(y)dy
ˆs
(f(x)|s,w)= U
Therefore, Ui
(f(ˆs)|s,w)= Ui( ˆ b(ˆs)|s,w)= Ui( ˆ b(s)|s,w). If instead x > s, by the reason
part 1c, Ui(f(x)|s,w) = Uw(x)|s,w) = Ui(b1(s)|s,w)=Ui(f(s)|s,w)=Ui( ˆ b(s)|s,w
Suppose this bidder bids b1w'(x), x∈ [ˆs,ˆsi(b1']. Suppose first that s= ˆs. B
reasoning in case 1b, Ui(b1(x)|s,w) = Ui(b1(s)|s,w). However, Ui(b1(s)|s,w
48
i ')|s,w).
If s>ˆs, then Ui(b1(x)|s,w)=
Ui(b1)|s,w).
i
i(f(ˆs
(ˆs'
'
1(s).
i(a(ˆs)|s,w)=Ui(b1
i(f(x)|s,w)=Ui(b1
(a) Bids of f(x),
x∈[ˆs,sw
i
i(f(sw
i(f(sw)|s,w)=Ui
''
i(b1(x)|s,w)=Ui(b1
49
1
)|s,w)= U')|s,w)=Ui( ˆ b(sw
U
3.
Consider
a bidder with a value-signal s= ˆsand a budget w= f(s). In this cas
i(f(s)|s,w)=Ui( ˆ
ß(s,w)=b
b(sw
(b1(s)|s,w)=U
(a) Standard arguments show that U
i(b1
(ˆs)|s,w)=U
i(b1(s)|s,w).
(b) Regarding
bids f(x), x ∈
[sw
(x), x∈[ˆs,ˆs
w
i(f(s)|s,w)=U
(b1(x)|s,w)for all x∈[ˆs,ˆs
i )=w.
w
].
(b) Since s=ˆs, by a standard argument we can see that for all x=ˆs, U(a(x)|s,w)=
U(s)|s,w). (c) Deviations to a bid above f(s) are not feasible. Thus, suppose this
bidder
bids f(x) for some x = s. Replicating the argument in case 1c shows that
U(x)|s,w)=U
4. Consider a bidder with a value-signal s= ˆsand a budget f(s)=w=µ(ˆs). In this case,
ß(s,w)=wand there exists some s=ssuch that f(s
), or ˆ b(s), x=ˆs, are not feasible
deviations.
,s], then the reasoning of case 2c confirms that
U(f(x)|s,w). (c)
Regardingbidsf(x),x=s,thenthereasoningofcase1cshowsthatU(f(x)|s,w)=
U(s)|s,w)=U)|s,w). (d) Bids of b], and a(y), y=ˆs, can be ruled out by reasoning
similar to the arguments already presented above.
The discussion above has studied all cases; thus, ß(s,w)is a symmetric equilibrium.
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Salant, David J. 1997. Up in the Air: GTE’s Experience in the MTA Auction for Personal
Communication Services Licenses. Journal of Economics & Management Strategy, 6(3),
549–572.
Simon, Leo K., & Zame, William R. 1990. Discontinuous Games and Endogenous Sharing
Rules. Econometrica, 58(4), 861–872.
Topkis, Donald M. 1998. Supermodularity and Complementarity. Princeton, NJ: Princeton
University Press.
Vickrey, William. 1961. Counterspeculation, Auctions, and Competitive Sealed Tenders.
Journal of Finance, 16(1), 8–37.
Zheng, Charles Z.2001. High Bids and Broke Winners. Journal of Economic Theory, 100(1),
129–171.
53
X Online Appendix
X.1 Example 1
In this section we prove that the strategy proposed in Example 1 is an equilibrium. The proof
is in a more general environment rendering the construction more transparent.
Suppose bidders have independent private values which are distributed according to the
c.d.f. H:[0,¯s]→[0,1]. H(s)admits a density, h(s), which is strictly positive for s∈(0,¯s). With
probability pa bidder has abudget w which we assume satisfies 0<w<¯s-¯s 0H(z)dz. With
probability 1-pa bidder has a budget of ¯w= ¯s. (Example 1 does not meet this requirement
but the argument is unchanged since a bidder with a high budget bids less than 3/4 in
equilibrium anyway.) Ties are resolved with a fair coin flip.
We begin with two preliminary lemmas which define ˆsand ˆs'1 H(˜s). Let ˜sbe the solution to w
=˜s-˜s 0H(z)dz.Lemma X1. There exists a unique ˆs∈(w ,˜s)such that Proof. Let
t(s)=[p+(1-p)H(s)](s-w )s 0ˆs 0H(z)dz=[p+(1-p)H(ˆs)](ˆs-w ).H(z)dz. It suffices to show that
t(s)crosses zero exactly one time. When s=w , t(w)<0. If instead s=˜s,
t(˜s)=[p+(1-p)H(˜s)](˜sw)-H(˜s)(˜s-w)>0. Thus, there exists ˆs∈(w,˜s)such that t(ˆs)=0.To
show that ˆsis unique, note that t'(s)=p(1-H(s))+(1-p)(s-w )h(s)>0when w =s<¯s. Thus, t(s)is
strictly increasing and therefore ˆsis unique.
Lemma X2. Let ˆsbe as defined in Lemma X1. There exists a unique ˆs'0ˆs>ˆssuch that
H(z)dz+ ˆs'ˆs[(1-p)H(ˆs)+pH(z)]dz= (1-p)H(ˆs)+p1+H(ˆs') 2 (ˆs'-w ).
Proof. Let
t(s)= χ(s)= 0ˆsH(z)dz+ ˆss[(1-p)H(ˆs)+pH(z)]dz,
and(1-p)H(ˆs)+p 1+H(s)2 (s-w ).
X1
t(ˆs)= 0
At s=ˆs,
Instead, if s=¯s,
1+H(s) 2
ˆs
H(z)dz
=[p+(1-p)H(ˆs)](ˆs-w )>
(1-p)H(ˆs)+p 1+H(ˆs)2
(ˆs-w )=χ(ˆs).
[(1-p)H(ˆs)+pH(z)]dz
<[p+(1-p)H(ˆs)](ˆs-w )+[p+(1-p)H(ˆs)](¯s-ˆs)
t(¯s)=[
=[p+(1-p)H(ˆs)](¯s-w)=χ(¯s).
p+(1p)H(ˆs
)](ˆs-w
)+ ˆs¯s
'
'
)=χ(ˆs'
'∈(ˆs,¯s)such that t(ˆsis
'(s) <χ
unique, note that t
'
p 2h(s)s. Since p1+H(s) 2
'
>
p
H
(
s
)
,
'
t
Thus, there exists ). To show that ˆs(s) = (1-p)H(ˆs)+pH(s) but χ+(s) = (1p)H(ˆs)+p'(s) and s→t(s)-χ(s)
ˆs'
strictly decreasing. Thus, ˆsis unique. Theorem X1. Let ˆsand ˆsbe as defined in
Lemmas X1 and X2. The strategy profile where all bidders follow the strategy
1
H(s)
s 0 s ˆs[p+(1-p)H(z)]dz p+(1-p)H(s)s
ˆs[pH(z)+(1-p)H(ˆs)]dz
pH(s)+(1-p)H(ˆs)
'
i
f
ˆs 0
X2
ˆs 0
s
∈
(
ˆ
s
,
ˆ
s
'
i
,
¯
s
]
,
( w
X =
1) w
s0
H(z)dz+ H(z)dz if s∈[0,ˆs],w∈{w ,¯w} s-H(z)dz+ if s∈(ˆs,¯s],w=
¯w s-],w=w w if s∈(ˆs
ß
(
s
s
is a symmetric
Bayesian-Nash
equilibrium. Proof. To simplify notation, let¯
,
w
ß(s)=ß(s,¯w)and
ß (s)=ß(s,w). We will also hold fixedthe strategy of player j=iat the
)
= strategy ßand suppress it from our notation. If any type has a profitable
prescribed
deviation from ß(s,w), it will be to different feasible bid in
the range of ß; thus, we need only verify that such deviations are not worthwhile. In tota
there are five cases to consider.
1. Consider bidder iof type s∈ [0,ˆs] with budget ¯w. If this bidder follows the prescribed
strategy, her expected payoff is U( ¯ ß(s)|s,¯w)=H(z)dz. There are 4 kinds
of alternative bids to consider.
(a) A deviation to a bid ¯ ß(x), x∈ [0,ˆs], is unprofitable. The standard argument'], will
give this bidder a payoff ofconfirming an equilibrium in a symmetric, first-price
sealed-bid auction applies. (b) A deviation to a bid ß (x), x∈(ˆs,ˆs
Ui(ß (x)|s,¯w)=[(1-p)H(ˆs)+pH(x)](s-x)+ 0ˆsH(z)dz+ ˆsx[(1-p)H(ˆs)+pH(z)]dz.
The net gain from this alternative bid is
Ui(ß (x)|s,¯w)-Ui( ¯ ß(s)|s,¯w)=[(1-p)H(ˆs)+pH(x)](s-x)+ = sˆssˆsH(z)dz+
H(z)dz+[(1-p)H(ˆs)+pH(x)](s-ˆs)+pH(x)(ˆs-x)+ ˆsxpH(z)dzˆsx[(1-p)H(ˆs)+pH(z)]dz
=H(ˆs)(ˆs-s)+[(1-p)H(ˆs)+pH(ˆs)](s-ˆs)+pH(x)(ˆs-x)+pH(x)(x-ˆs)=0
Thus, Ui(ß (x)|s,¯w)=Ui( ¯ ß(s)|s,¯w).
(c) Since the bid w is placed with strictly positive probability by bidder j=iwhen she is
following the prescribed strategy, bidder iwill never wish to deviate to this bid since a
bid slightly higher will guarantee a strictly higher payoff. A slightly higher bid is
feasible for bidder isince she has a budget of ¯w.
(d) Finally, consider a deviation to a bid ¯ ß(x), x∈(ˆs,¯s]. This bid will give bidder ian
expected payoff of
0 ˆ H(z)dz+ ˆsx[p+(1-p)H(z)]dz.
s
Ui( ¯ ß(x)|s,w)=[p+(1-p)H(x)](s-x)+
X3
The net gain from this alternative bid is
Ui( ¯ ß(x)|s,¯w)-U= sˆsi( ¯
ß(s)|s,¯w)H(z)dz+[p+(1-p)H(x)](s-ˆs)+[p+(1-p)H(x)](ˆs-x)+
ˆsx[p+(1-p)H(z)]dz=H(ˆs)(ˆs-s)+[p+(1-p)H(x)](s-ˆs)+ ˆsx(1-p)[H(z)-H(x)]dz=0
The final inequality follows from x=ˆs=s. Thus, Ui( ¯ ß(x)|s,¯w)=Ui( ¯ ß(s)|s,¯w).
2. Consider bidder iof type s∈ (ˆs,¯s] with budget w= ¯w. If this bidder follows the
prescribed strategy, her expected payoff is Ui( ¯ ß(s)|s,¯w) =ˆs 0H(z)dz+ s ˆs[p+(1p)H(z)]dz.
There are four kinds of alternative bids she may consider.
(a) A deviation to a bid ¯ ß(x), x∈[0,ˆs], will give this bidder a payoff of
Ui( ¯ ß(x)|s,¯w)=H(x)(s-x)+
0 x
H(z)dz.
The net gain from this alternative bid is
Ui(
¯
ß(
x)|
s,¯
w)
-Ui
(¯
ß(
s)|
s,¯
w)
=H
(x)
(sx)
+
0=
H(
x)(
ˆsx)
+
=
xˆsˆ
sxx
H(
z)dz0ˆs
H(
z)
dzˆ
sH
(z)
dz
+H
(x)
(sˆs)
sˆs[
p+
(1p)
H(
z)]
dz
s[p
+(
1p)
H(
z)]
dz
[H(x)-H(z)]dz+H(x)(s-ˆs)
-[p+(1-p)H(ˆs)](s-ˆs)=0
'Therefore,
this deviation is
not profitable. (b) A
deviation to a bid ß (x),
x∈[ˆs,ˆs], will give this
bidder a payoff of
Ui(ß
(x)|s,¯w)=[(1-p)H(ˆs)+pH(x)](s-x)+
0ˆsH(z)dz+ ˆsx[(1-p)H(ˆs)+pH(z)]dz.
X4
The net gain from this alternative bid is
Ui(ß (x)|s,¯w)-Ui( ¯ ß(s)|s,¯w)=[(1-p)H(ˆs)+pH(x)](s-x)+
ˆss[p+(1-p)H(z)]dzˆs=[(1-p)H(ˆs)+pH(x)](ˆs-x)+
ˆsxx[(1-p)H(ˆs)+pH(z)]dz[(1-p)H(ˆs)+pH(z)]dzs
ˆ [p+(1-p)H(z)]dz
s
ˆ
s
p[H(z)-H(x)]dz
=
+[(1-p)H(ˆs)+pH(x)](s-ˆs)x
+[(1-p)H(ˆs)+pH(x)](s-ˆs)-[p+(1-p)H(ˆs)](s-ˆs)
=ˆsxp[H(z)-H(x)]dz+p(s-ˆs)(H(x)-1)=0
Therefore, this deviation is not profitable. (c) As in the previous case, a bidder
can strictly improve upon the bid w by bidding
omore. (d) A deviation ¯ ß(x), x∈(ˆs,¯s], will give this bidder a payoff
to a bid
of
0 ˆ H(z)dz+ ˆsx[p+(1-p)H(z)]dz.
Ui ( ¯ ß(s)|s,¯w)=[p+(1-p)H(x)](s-x)+
s
(1-p)[H(z)-H(x)]dz=0
Therefore, this deviation is not profitable.
3. Consider a bidder of type s∈ [0,ˆs] with budget w= w . By identical arguments
to those presented above fora high-budgetbidder with avalue-signal less than ˆs,
this type of bidder will not have a profitable deviation to any bid, except possibly
¯w= ß (x),
X5
x
The net gain from this alternative bid is =[p+(1-p)H(x)](s-x)+ = sxs
[p+(1-p)H(z)]dz
Ui( ¯ ß(x)|s,¯w)-Ui( ¯ ß(s)|s,¯w)
x>ˆs'. Noting the uniform tie breaking rule, this deviation will yield a payoff of
1+H(ˆs') (s-w ).
Ui(w|s,w)= (1-p)H(ˆs)+p
2
The net gain from this alternative bid is
'
[(1-p)H(ˆs)+pH(z)]dz=0
ˆs
(1-p)H(ˆs)+p 1+H(ˆs) 2
[(1-p)H(ˆs)+pH(z)]dz
'(1-p)H(ˆs)+p
1+H(ˆs) 2
+
(1-p)H(ˆs)+p 1+H(ˆs) 2
Ui(w |s,w)-U= = =
i(ß(s)|s,w)
'(1-p)H(ˆs)+p
(s-ˆs' (s-ˆs')+
1+H(ˆs')
-w )0
(1-p)H(ˆs)+p)+ (s-ˆs)+'
2
(ˆs'
(ˆs-ˆsssˆsˆsH(z)dz+ H(z)dz)+
'
1+H(ˆs) 2
'ˆsˆsˆs'
Therefore, this deviation is not profitable.
Consider a bidder of type s∈ (ˆs,ˆs] with budget w= w . If this bidder follows the
prescribed strategy, her expected payoff is Ui(ß (s)|s,w)= ˆs 0H(z)dz+ s
ˆs[pH(z)+(1p)H(ˆs)]dz.
(a) A deviation to a bid ß (x), x∈[0,ˆs], will give this bidder a payoff of
Ui(ß (x)|s,w)=H(x)(s-x)+ 0x
H(z)dz
'4.
The net gain from this alternative bid is
U
i
(
ß
(
x
)
|
s
,
X6
[
(
1
p
)
H
(
ˆ
s
)
+
p
H
(
z
)
]
d
z
s
[
(
1
p
)
H
(
ˆ
s
)
+
p
H
(
z
)
]
d
z
=0+H(x)(s-ˆs)-[(1-p)H(ˆs)
+pH(ˆs)](s-ˆs)
=[H(x)-H(ˆs)](s-ˆs)=0
Therefore, this deviation is not profitable.
'(b)
A deviation to a bid ß(x), x∈[ˆs,ˆs], will give this bidder a payoff of
Ui(ß (x)|s,w)=[(1-p)H(ˆs)+pH(x)](s-x)+ 0
ˆ H(z)dz+
s
[(1-p)H(ˆs)+pH(z)]dz
The net gain from this alternative bid is
Ui(ß (x)|s,w)-Ui(ß(s)|s,w) =[(1-p)H(ˆs)+pH(x)](s-x)+
x [(1-p)H(ˆs)+pH(z)]dz.
ˆs
sx
= sx
p[H(z)-H(x)]dz=0
Therefore, this deviation is not profitable. (c) A deviation to a bid of w can be
ruled out by an argument fully analogous to that
presented in case 3 above.
Consider a bidder of type s∈(ˆs,¯s]with budget w=w . If this bidder follows the
prescribed strategy, her expected payoff is Ui'1+H(ˆs) 2
(a) A deviation to a bid ß (x), x∈[0,ˆs], will give this bidder a payoff of
H(z)dz.
Ui(ß (x)|s,w)=H(x)(s-x)+ 0x
'5.
X7
w).
[(1-p)H(ˆs)+pH(z)]dz
The net gain from this alternative bid is
1+H(ˆs')
Ui(ß (x)|s,w)-Ui(ß(s)|s,w)=H(x)(s-x)+ =H(x)(s-x)+ 2
00xxH(z)dz (1-p)H(ˆs)+pH(z)dz'(1-p)H(ˆs)+p
1+H(ˆs=H(x)(ˆs-x)+ ˆsx0) 2H(z)dz'ˆsˆsH(z)dz (s-ˆs')ˆsˆs'
'+H(x)(ˆs -ˆs)ˆs
[(1-p)H(ˆs)+pH(z)]dz
+H(x)(s-ˆs ' ) (1-p)H(ˆs)+p
1+H(ˆs') (s-ˆs')=0
2
(s-w )
'Therefore,
this deviation is not profitable. (b) A deviation to the bid ß (x), x∈[ˆs,ˆs],
can be ruled out with reasoning similar to the preceding case.
The above discussion has exhausted all possible cases; therefore, (X1) is a symmetric
equilibrium strategy profile.
Remark X1. This examplecanbegeneralized tothecaseofN>2bidders. The equilibrium’s
qualitative features remain unchanged.
X.2 Example 2
The reasoning below echos the proof of Theorem X1. We specialize to the parameters of
interest to simplify the overall argument.
X8
16
s
11-3)s224(s+ v
11-3)2
s2
2s2+13 v11-42
4(s+1)
3+24( v
''if
s∈(ˆs ,ˆs'
]
''if
s∈[0,ˆs]
if s∈[0,ˆs]
+33(19 v 11-63)
2s
3
Theorem X2. The following is a symmetric equilibrium in Example 2. ß(s,1/4)=
if s∈(ˆs,1]
if s∈(ˆs
ß(s,3/4)=
1
2
',1]
(ˆs-1/4)
2
ˆs'
+24( v11-3)s2
wh
er
e
ˆs
=v
where q(s)=
Pro
of.
As
in
the
pro
of
of
The
ore
m
X1
ther
e
'
'
=
1
1
3
,
11-3), and ˆs+ˆs 2solves the
equation (ˆs-q(ˆs))= 1 2'· ˆs+1 2+
4
v 12 ˆs
3(
ˆ
s
16s
3
+33(19 v11-63) 24(s+v
11-3)
. (Approximately,
ˆs'
˜0.298.)
14
'
1
2
are
mul
tipl
e
cas
es.
Cle
arly
,
we
nee
d
onl
y
con
sid
er
pos
sibl
e
dev
iati
ons
into
the
ran
ge
of
ß(s,
w).
To
sim
plify
not
atio
n,
we
will
writ
e
¯
ß
(
s
)
=
ß
(
s
,
3
/
4
)
a
n
d
ß
(
s
)
=
ß
(
s
,
1
/
4
)
.
1.
C
o
n
si
d
er
a
bi
d
d
er
wi
th
a
b
u
d
g
et
of
w
=
3/
4
a
n
d
a
v
al
u
esi
g
n
al
s
∈
[0
,ˆ
s]
.
If
th
is
bi
d
d
er
pl
a
c
e
s
a
bi
d
¯
ß(
x)
,
x
∈
[0
,ˆ
s]
,
s
h
e
wi
ll
d
ef
e
at
al
l
hi
g
hb
u
d
g
et
o
p
p
o
n
e
nt
s
wi
th
a
v
al
u
esi
g
n
al
le
ss
th
a
n
x
a
n
d
al
l
lo
w
-b
u
d
g
et
o
p
p
o
n
e
nt
s
wi
th
a
v
al
u
esi
g
n
al
le
ss
th
a
n
3
x/
4.
T
h
u
s,
th
e
pr
o
b
a
bi
lit
y
th
at
s
h
e
wi
n
s
th
e
a
u
cti
o
n
is
7
x/
8.
T
h
e
e
x
p
e
ct
e
d
p
a
y
o
ff
fr
o
m
th
is
bi
d
is
th
er
ef
or
e
Ui( ¯
ß(x)|s,¯w)=
M
a
xi
m
izi
n
g
th
e
a
b
o
v
e
e
x
pr
e
ss
7x8
s-
x .
2
io
n
wi
th
re
s
p
e
ct
to
x
=ˆ
s
gi
v
e
s
a
s
ol
ut
io
n
of
x
=
s.
T
h
u
s,
¯
ß(
s)
is
th
e
o
pt
i
m
al
bi
d
w
h
e
n
c
o
n
st
ra
in
e
d
to
a
bi
d
in
th
e
ra
n
g
e
[0
,ˆ
s/
2]
.
S
u
p
p
o
s
e
in
st
e
a
d
th
is
bi
d
d
er
bi
d
s
¯
ß(
x)
a
n
d
x
∈
(ˆ
s,
1]
.
T
hi
s
bi
d
wi
ll
d
ef
e
at
al
l
lo
w
-b
u
d
g
et
o
p
p
o
n
e
nt
s
si
n
c
e
it
is
a
b
o
v
e
1/
4
a
n
d
it
wi
ll
d
ef
e
at
al
l
hi
g
hb
u
d
g
et
o
p
p
o
n
e
nt
s
wi
th
a
v
al
u
esi
g
n
al
le
ss
th
a
n
x.
T
h
er
ef
or
e,
th
e
e
x
p
e
ct
e
d
p
a
y
o
ff
fr
o
m
th
is
bi
d
is
Ui( ¯
ß(x)|s,¯w)=
X
9
1+x2
(s¯ ß(x))
It is easy to calculate that
definition of ˆs, we see
that
1
1+ˆs 2 s-
d Ui(
dx
¯ ß(x)|s,¯w) = (s-x)/2 = 0. However, given our
4
2
= ( v 11-2)(4s-1)7s8
=16 ∀s∈[0,ˆs]
Since Ui( ¯
ß(s)|s,¯w)=
Ui( ¯ ß(x)|s,¯w)=
lim+x→ˆs=
'
Ui( ¯
ß(x)|s,¯w)
'').
'),
For x∈[ˆs
7s2
16
= 7s∆(x,s)=Ui216 -
''
i
(s-ß (x)).
,ˆs, this proposed deviation is not profitable. The remaining
candidate deviation is to a bid ß (x), x∈[ˆs
(¯
(ß
ß(s)|s,¯w)-U (x)|s,¯w)
x+ˆs2
' ,ˆs'
'
It is sufficient to establish that ∆(x,s) = 0 for all x∈
''
[ˆs''(noting that ß (s)is continuous at ˆs''
= 78 (s-ˆs)=0.
∂∆(ˆs'',s)
∂s
''Hence, for all s=ˆs, ∆(ˆs
8(v 11-3-3s)+32x49(19 v
11-63)(x+ v
=0
= 148
11-3)
2
8(v 11-3-3ˆs)+32ˆs''
=
(ˆs + v 11-3)2
''
148
) and s= ˆs. From the definition of ˆsand ˆs), ∆(ˆs,ˆs)=0. Next, note that
,ˆ
s
let
,s)=0. Thus, it is sufficient to confirm that
49(19v 11-63)
∂∆(x,s)
∂x
∂∆(x,s) ∂x
=0for all s∈[0,ˆs]. A direct computation gives
where we used the fact that 19 v11-63>0. Therefore, a bidder of type s<ˆshas no
incentive to deviate into this range of bids.
2. Consider a bidder with a budget of w = 3/4 and a value-signal s∈ (ˆs,1]. By an
argument fully analogous to the preceding case, all deviations to bids¯ ß(x), x∈[0,1]can
be ruled out. We therefore consider a possible deviation to a bid ß (x) where
X10
1+s
2
'
'
¯
ß(s)),ˆs'). Let
(s-ß (x)).
( ¯ ß(s)|s,¯w)-U (s-i(ß
(x)|s,¯w) x+ˆs2
x
∈
∆(x,s)=Ui
[
=
ˆ
s
'',ˆs)=0. Note that for any x∈[ˆs
'' ,ˆ
s'
∂∆(x,s) = 4- v 11+s-x2 =
4v11+ˆs-x 2
>0.
∂s
]. As before, ∆(ˆs'',ˆs'],
Again, it is sufficient to verify that ∆(x,s)=0for s∈(ˆs,1]and x∈[ˆs
∂∆(x,ˆs) =0. A direct calculation gives
Thus, it is sufficient to confirm that ∂x
''=
2ˆs3
11 +1=0.
3
∂∆(x,ˆs) = 2x3 - 49(19 v11-63) 48(x+v +1
∂x
11-3)49(19 v211-63)
48(ˆs''+ v11-3)v211 3v
Thus, a bidder with a value-signal of s>ˆshas no profitable
deviation from ¯ ß(s).
''3. Consider a bidder with a budget of w= 1/4 and a value-signal s∈
[0,ˆs'']. If this bidder places a bid of ß (x), x∈[0,ˆs], she will defeat all
low-budget opponents with a value-signal less than xand all
high-budget opponents with a value-signal less than 4x/3. Therefore,
the expected payoff from this bid is
1/2
Ui (ß (x)|s,w)= 7x6 s2x 3
.
''Maximizing the above expression with respect to x= ˆs
Ui(ß(x)|s,w)= x+ˆs 2 (s-ß (x))
X11
1
2
.
gives a solution of x= s. Thus, ß(s)is the
the range [0,'',ˆs'3]. If instead this bidder plac
payoff would be
''2ˆs
=0.
Differentiating this expression with respect to xgives
2
49(19 v
11-63)(x+ v
11-3)
d dxUi(ß (x)|s,w)= v8(3s+ v6(s-x) 11-3)-16xTherefore, since ß is continuous at ˆs'', Ui(ß(x)|s,w)=Ui (ß(ˆ '' )|s,w)=U(ß(s)|s,w). Finally, since ˆs''=3 4( vi11-3)˜0.23, a
s
negative payoff. Therefore, this type of bidder does not
(ß (s)|s,w)= 8s-49( 19 v 11-63) ( s+ = v 11-33 2)2(ˆs+
,ˆ
v+8( v 11-3)8ˆs''11-3) = (-49(19 v 11-63)(ˆs''+ v 11-3)v
s'
11-3) 382v 11-1262+8( v 11-3)˜0.563
]. By an argument fully analogous to th
''4. Consider a bidder with a budget of w = 1/4 and a value-signal s ∈ (ˆs
(x), x∈ [0,ˆs] can be ruled out. We there
the bid of w = 1/4. Define ∆(s) = U(w|s,
''
(s) = 0; equivalently, that
i(ß(s)|s,w)-Ui) = 0. Therefore, it is sufficient to
, ∆(ˆs verify that ∆
= 11-5 4v 4ˆs-1= 11-29 16
˜0.901and
ˆs
'
d dsU
v
1
1
3
U
i
i(
ß(s)|s, w )=11-5 4v
4s-1
3 s+
2
5. Considerabidderwithabudgetofw=1/4andavalue-signals∈(ˆs,1]. Byarguments
analogous to those presented in the previous cases, it is simple to verify that ß(ˆs ')is
the utility-maximizing, feasible bid other than w . Thus, we verify that U(w|s,w)= Ui(ß
(ˆs'')|s,w). By definition of ˆs, Ui'(w|ˆs,w)=Ui(ß(ˆs'')|ˆsi,w). Thus, it is sufficient to
X12
'
verify that for all s>ˆs',
d dsUi
d
d
s
U
i
=
(
ˆ
s
11+
'
'
+
2
v
d
d
s
(
ß
(
w
(
ˆ
s
|
s
,
w
)
Ui (ß (ˆs')|s,w)=
)
|
s
,
w
)
Ui(w |s,w). Consider
therefore,
( +2 v11-5)
ˆ 8(3s+ v2 '
s v24s-6(ˆs
- '11-3)-16ˆs+
v11-3)
'
=
'
11-5) 2 v
11)s-86 v8 v
8(3s+ v 11-3)-16ˆs-
49(19 v
11-63)(ˆs''+ v
11-3)2
=
(
24s-6(ˆs+ v11-3)
3
4s-1 This final expression is decreasing
in
sand at s=1we can
3
3
conclude that
+
6
8
v
50
23
d (ß (ˆs
'i
U
i
which is the desired d
(
s
result.
w
d ds
The
pre
ced
ing
disc
ussi
on
has
exh
aus
ted
all
pos
d
d
s
|
s
,
w
)
U
)|s,w) = 1 1501-54v 11˜1.51=1
2
4
49(19 v
11-63)(ˆs'+ v
11-3)2
sibl
e
cas
es;
ther
efor
e,
ß(s,
w)is
a
sy
mm
etri
c
equ
ilibri
um.
X
.3
R
el
a
xi
n
g
A
s
s
u
m
pt
io
n
A
3
(
1
b
)
As
not
ed
in
foot
not
e 8,
in
the
cas
e of
two
bid
der
s
with
qua
silin
ear
pref
ere
nce
s
Ass
um
ptio
n
A-3
(1b)
is
not
nec
ess
ary.
In
this
sec
tion
we
ela
bor
ate
on
this
rem
ark.
˜
A,
d
e
fi
n
e
d
b
el
o
w,
m
a
y
n
ot
b
e
a
s
u
bl
at
tic
e.
20
T
h
e
o
nl
yr
ol
e
of
A
ss
u
m
pt
io
n
s
A3(
1
a)
a
n
d
A3(
1
b)
isi
nt
h
e
v
er
ifi
c
at
io
n
of
a
si
n
gl
e
cr
o
ss
in
g
c
o
n
di
ti
o
n.
T
h
er
ef
or
e,
it
s
u
ffi
c
e
s
to
s
h
o
w
th
at
L
e
m
m
a
A
3
o
bt
ai
n
s
wi
th
o
ut
A
ss
u
m
pt
io
n
A
3(
1
b)
.
L
e
m
m
a
X
4
d
e
m
o
n
st
ra
te
s
th
is
c
o
n
cl
u
si
o
n.
G
e
n
er
al
ly,
th
is
ar
g
u
m
e
nt
d
o
e
s
n
ot
a
p
pl
y
if
N
=
3
si
n
c
e
th
e
s
et
Lemma X3. Let s' i =si and suppose =sj =¯sj, then
s' j
|s' i
Proof. We note that for all sj, ¯ H(sj|si)= ¯ H(sjj'
i=siand zj=tj
20
;
thu
s,
the
upp
er
bou
nd
foll
ows
.
Sin
ce
h(·)
is
logsup
erm
odu
)
¯ H(sj|s'
i)
¯H(sj|si)
=
¯ H(s' j|s'
i) ¯H(s'
j|si)
=1.
|si)h(tj |s' i)=
lar,
for
s,
the
foll
owi
ng
ine
qua
lity
hol
ds:
h(z
T
h
i
s
d
i
s
c
u
s
s
i
o
n
c
o
r
r
e
c
t
s
a
n
d
c
l
a
r
i
fi
e
s
a
r
e
s
u
l
X
1
3
t
o
r
i
g
i
n
a
l
l
y
i
n
c
l
u
d
e
d
i
n
t
h
e
w
o
r
k
i
n
g
p
a
p
e
r
K
o
t
o
w
s
k
i
(
2
0
1
0
)
.
i)dzj
sj
|s' i)h(tj|si). Therefore,
j |s' i)=h(zj
|si)h(tj
j=s' j
h(tj|s' i
h
(
z
'j
t
j
j
j=s
) =⇒=zh(zj|stj=sj)dt= sj=zj=sj
h(zj
¯ H(s' ¯ H(s|sj ' i)= ¯ H(s' ' i|s' i)¯
) ¯ j|s' i)= ¯ H(s' H(sj|s) ¯
j|s
H(sj j
H(sj|si)
(si,s-i,wi,b
j
j
) ¯
H(sj|si
H(sj
H(sj
)=v
' i'i i)
-i
= ¯ H(s|s|si)
H(s
(wi)=wi
i)gj(wj)dwjdsj.
i
,
u
2
0= ˜ A
i,
˜A
=⇒ ¯ H(s' j|si)¯ H(sj|si
|s' i)h(tj |si
h(z |s' i)dz
i)
=⇒=⇒ ¯|s) ¯|s|s' j' i) ¯' j
j
'i
i(si,s
)+wi -b
i
Lemma X4. Suppose
¯ui
')
i
i
,b
)h(sj|s
θi
and w
i
, and N=2.
Then Lemma A3 continues to hold without Assumption A-3(1b).
Proof. We first note that the =ordering reduces to the coordinate-wise partial
order of R. We specialize our notation to the case of two bidders. Noting the proof
of Lemma A3, it is sufficient to show that for s=s=w
ui(s' i,sj,w' i,bi)h(s |s' (wj)dwjdsj =
ui(si,sj ,wi i
i)gj
is independent of , the random variables (Si,Sj,W
Sj
h(
j
i
i
×Sj ×W j
and wj
where the ˜
A={(sSince W jj,wj
follows: (si,sj,wj) ˜ =(s, (Si' i,S,sj' j,w,W j' j)if and =s' i, sj =s' j
only if si
j jas
j
)
< '
(sj,wj)=bi}and Pr[ ˜ A]>0.
j
:
ß i
j
b
)are affiliated. Redefine the ordering of S=w' j. Again by the independence of
W)are affiliated when “affiliationness” is defined with the˜ = order.
S×S×Wremains a lattice under this alternative ordering.
j ×W
under the ˜ = ordering. To see this, suppose=
and
w
j
The set ˜ A is a sublattice
w' j. Then, since ßj
of S) ∈ ˜ Abut s= s' jj
(sj,wj),(s' j,w' j is nondecreasing in the usual coordinate-wise partial order,
=ßj(s' j ,w' j)=ßj (sj,w' j)=ßj (sj˜ ∨s',w
j j˜ ∨w' j)=ßj (sj,wj)=bi
(sj,wj)=bi.
(s
j˜ ∧s' j
b' i
,wj˜ ∧w' j)=ßj
=ßj(s' j ,w' j)=ßj (s' j,wj)=ßj
'i
, and b
X14
Thus, (sj ˜ ∧s' j˜ ∧w' j),(sj˜ ∨s' j,wj˜ )∈ ˜ A. We can now establish the
j,w
following:
∨w' j
ui(s' i,sj,w' i,bi)h(sj |s' (wj)dw jdsj
i)gj
j ' i)
|s' i)gj(wj)dwjdsj
j|s
¯
˜A
=
˜ A= u˜ i(s' i,sj,w' i,bi) h(sj|s' i) H(sj
A
¯H(s
|s' i)gj(wj)dwjdsj
)
¯H(s|s
ui(si,sj,wi,bi) h(s
|s
H(sj
j i ) ¯i
=Pr[ ˜
=si
,wi,bi ) ¯H(Sj|s' i) ˜
=si
A|Si
¯H(Sj|si) A,Si
i
ii
¯H(Sj|s' i)
i)| ˜
E ¯|s)
˜
A,S
i
=si
j
]E ]E uu]E
,wi,buiiii(s(s(siii
,S,S,S)h(sjjjj|si
=
=Pr[ ˜
si
A|Si
=si
=Pr[ ˜
A|S= ˜ ,sj
Auii(si
,wi,
b
,w ,b
)| ˜
A,S
=
si
H(
Sj
A,S
i
i =si
dsj
)g (wj)dwj
The first inequality follows from Assumption A-3(1a). The second inequality
follows from the fact that ui(si,·,wi,bi):Sj×W j→ R is nondecreasing,¯ H(·|s' i)
¯H(·|si):Sj×W ¯ H(sj|s' i→ R is nonincreasing (Lemma X3), and Milgrom & Weber
(1982, Theorem 23). The final inequality follows from) ¯H(sj|si)being positive and
bounded above by 1.j
X.4 Multiple Critical Points
Multiple critical points can arise if ψ(s) is not monotone. Our conclusions carry
over to this case as well. The analysis’ essence is to identify a family of
connected, upward sloping solutions from (4) and to link appropriate solutions at
critical points to define the bidding strategy. Here we sketch how these
definitions may be carried out.
Definition of ¯ b(s)(Low w ). Figure X1 presents a typical case of three critical
points and
w=0. The central critical point is an unstable node while the others are saddle
points. The thick arrows indicate the motion of (4) in the various regions.
Confining attention to the regions between ν(s) and ψ(s), we are able to link
adjacent nodes constructing a path for
¯ b(s)which satisfies the desired
conditions.
Definition of ˆ b(s) (Intermediate w ). The preceding procedure can be
generalized to
define ˆ b(s)whenthere aremultiple critical pointsandw isintermediate. Inthiscase
however,
X15
b
b
ν
¯b(s)
ψ
0
s
Figure X1: Multiple critical points and the
definition of ¯ b(s).
explicitconsiderationneedstobegiventoµ(s). Consider, forexample,
thesituationinFigure X2 with three critical points. Suppose the equilibrium
strategy is discontinuous at ˆs. µ(s) is indicated by the thick dotted curve.
Since solutions to (4) radiate from the critical point at s* 2, the initial
segment of ˆ b(s)approaches this critical point from below. Beyond s * 2, ˆ
b(s)
equals ¯ b(s)which ensures continuous passage through the final
critical point at s* 3.
X.5 Comparing First- and Second-Price Auctions
In this section we prove the following standard result.
LemmaX5. Supposew =0andletß1(s,w)andß2(s,w)besymmetric
equilibriumstrategies for the first-price and second-price auctions
respectively. Then, ß1(s,w)=ß(s,w)and the inequality is strict if ß1(s,w)<w.2
Proof. If there are no critical points, the result is immediate as
ß1(s,w)=v(s,s)=ß(s,w). Suppose instead that there is a critical point
(s*,b*2)and let ¯ b(s)be the solution (3) passingthrough (s* ,b*). It is sufficient to show that ¯ b(s)<b2(s). From their definitions, it is easy
X16
b
ψ
ν
s
¯
b(s)
s
*
2
s
ˆs a
ˆ b(s)
s* 3
*
1
w
s0
Figure X2: Construction of ˆ b(s) (orange) when w is intermediate
and there are multiple
critical points. µ(s)is the dotted curve. For s>s* 2, ˆ b(s)= ¯
b(s)
.
. As both functions are continuous, theremustex
Notingthatd(b,s|s)>b, b' 2(ˇs)< ¯ b'(ˇs). Therefore
when s<s*. This contradicts b2(s*)>v(s*,s*). Thus
t
o see that for all s∈[s*,1], bSuppose instead that
b22(s)>v(s,s)= ¯ b(s).(s)= ¯ b(s)for some 0<s<s*
G0(w)= w2 , G1(w)= w 2, G
X17
'
(w)= w(4-w)14 .
X.6 Change in the Budget Distribution
In this section we detail our comparative static c
constraints. Our argument is a simple example.
with three alternative distributions of budgets on
ψ1(s)= 9s likelihood ratio dominates both G1 and G
k
9s5
4
3' .
The critical p
2
implies a distinct ψk(s):
ψ0(s)= 3s2
-7s2
v9s4
1
0(s* 0).
+3s ) occurs at
G the intersection of ν(s)and
each distribution G
+3s
-2s+1) ψ' (s)= 9-6s-3s12-36s3+48s2-84s+57 6-6s
1
1
1
Table 1 summarizes
approximate
values for the critical point in each case. The critica
s
8
(points are strictly ordered: s* 1<s* 0<s* 1'. In a neighborhood of s* 0, unconstrained bidd
s
bid less when budgets are distributed according to Gversus G 0: ¯ b1(s* 0)< ¯ b0(s* 0). T
converse is true when budgets follow G' : ¯ b1' (s1* 0)> ¯ b
Table 1: Critical Points for Different Budget Distributions
b
+2s-1 4s-43+3s+3-4 v2
0
Distribution s* k
*k
1
0.0864G
0.1728 G00.1056
0.2111 G' 0.1264 0.25281
X18