First-Price Auctions with Budget Constraints∗
Maciej H. Kotowski†
June 28, 2016
Abstract
Consider a first-price, sealed-bid auction where participants have interdependent
valuations and private budget constraints; that is, bidders have private multidimensional types. Budget constraints affect equilibrium bidding in three distinct ways. The
direct effect caps bids as participants hit their spending limit. The strategic effect
encourages more aggressive bidding by participants with large budgets. And the countervailing incentive effect depresses bids as agents’ preferences respond to their wealth.
We characterize the interaction among these forces and we document their economic
implications. Symmetric and monotone equilibria may feature discontinuous bidding
strategies that stratify competition along the budget dimension. The strategic and
incentive implications of budget constraints confound the interpretation of bidding behavior in auctions.
Keywords: First-Price Auctions, Budget Constraints
JEL: D44
∗
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. Part of this research was
completed with the support of a SSHRC Doctoral Fellowship, which I gratefully acknowledge. This paper
supersedes and corrects results originally communicated in Kotowski (2010) and included in Kotowski (2011).
It is a substantial revision of both preceding studies.
†
John F. Kennedy School of Government, Harvard University, 79 JFK Street, Cambridge MA 02138.
E-mail: <[email protected]>.
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.
Once bidders have multidimensional private information composed of a value signal,
which informs their preferences, and a budget, which defines their feasible strategy set, even
simple auctions feature nontrivial equilibrium outcomes. For concreteness, consider a firstprice, sealed-bid auction for one item. Bidders simultaneously submit their best offer to the
seller, the highest bidder wins, and the winner pays his own bid for the item. In this common
setting, private budgets affect bidding in three distinct ways. A direct effect caps bids. A
strategic effect amplifies and stratifies competition. And a countervailing incentive effect
skews interests and preferences. The cumulative implications are substantial. For example,
in a symmetric, pure-strategy equilibrium, high-budget bidders may adopt a bidding strategy
with jump discontinuities to strategically outbid others. Monotone equilibria, as commonly
defined, may not exist due to countervailing incentives.1 A low-budget bidder may bid more
than a similar high-budget rival if his preference for risk responds to money holdings. Thus,
the interpretation of observed bidding data is vulnerable to multiple atypical confounds. In
summary, many classic intuitions demand significant qualification once private budgets enter
into bidders’ strategic calculus.
First-price auctions with private budget constraints have been studied before, notably by
Che and Gale (1998). Despite their importance, many first-order economic implications of
private budgets, particularly the strategic and the incentive effects, have remained elusive.
By relaxing key restrictions, our analysis isolates these effects in a general environment
and we offer a new comprehensive assessment of auction performance, equilibrium behavior,
and agents’ welfare. As budget constraints are a common feature of everyday economic
interaction, neglecting their implications may result in empirical misidentification and may
lead to misguided normative conclusions. Budgets carry significantly deeper consequences
than simply biasing bids downward. We explain the resulting complications and, where
possible, suggest approaches to account for the enriched pattern of equilibrium behavior.
We introduce and summarize our argument with two simple examples. The first example
extends Vickrey’s (1961) classic model.2 As explained below, its key features generalize.
1
In a (nondecreasing) monotone equilibrium agents with a higher type take a higher action. In an auction,
it usually means that a bidder with a greater valuation and with a larger budget bids more.
2
Milgrom (2004, p. 132) presents a variant of Example 1 with a fixed and known budget constraint.
2
Example 1. There are two, ex ante identical risk-neutral bidders participating in a firstprice, sealed-bid auction. Each bidder i ∈ {1, 2} observes an independent, uniformlydistributed signal si ∈ [0, 1] regarding his value for the item. Signals are private, but their
distribution is common knowledge. If a bidder with value signal si wins the auction with the
bid bi , his payoff is si − bi ; otherwise, it is zero. A fair coin flip resolves ties. In this auction’s
unique Bayesian-Nash equilibrium, both bidders adopt the strategy α(si ) = si /2.
Suppose, additionally, that each bidder has a budget wi , which is independent of his value
signal. Budgets are private, but their distribution is common knowledge. With probability
1/2, a bidder’s budget is 1/4; with probability 1/2, it is 3/4. A budget is a hard constraint
on bids. Thus, a type-(si , wi ) bidder cannot bid above wi . Clearly, si /2 cannot be an
equilibrium strategy since a low-budget bidder cannot afford to bid above 1/4 as required
when si > 1/2.
A tempting conjecture is that
min{si /2, wi }
(1)
is the auction’s new symmetric equilibrium strategy. In (1), a bidder follows the original
equilibrium strategy α(si ) = si /2 until he exhausts his budget. However, this arrangement
also cannot be an equilibrium. If bidder j follows (1), then a high-budget bidder i of type
si = 1/2 − ǫ has a profitable deviation. According to (1), he should bid 1/4 − ǫ/2. But,
if he bids ǫ more—1/4 + ǫ/2—he is assured of defeating any budget-constrained opponent.
His probability of winning jumps from approximately 1/2 to over 3/4, which more than
compensates for the higher bid’s infinitesimal added cost.
Reflecting on the preceding reasoning, we are led to an equilibrium strategy featuring a
jump discontinuity. A sketch of the new symmetric equilibrium, β(si , wi ), is presented in
Figure 1 where s̃ = 1/3 and s̃′ = 11/27.3 More formally,
β(si , wi ) =



si /2




 5+9s2i
18+18si
1+9s2i



 6+18si


1/4
if si ∈ [0, s̃]
if si ∈ (s̃, 1] and wi = 3/4
if si ∈ (s̃, s̃′ ] and wi = 1/4
.
(2)
if si ∈ (s̃′ , 1] and wi = 1/4
Both direct and strategic effects are evident in Figure 1. The direct effect is obvious.
A high-valuation (si > s̃′ ), low-budget bidder hits his spending limit in equilibrium. The
3
In Appendix A we verify that β(si , wi ) is an equilibrium in a more general version of the example.
Though we omit the argument, it is the unique equilibrium in this example.
3
bi
β(si , 34 )
1
4
β(si , 14 )
si
0
s̃′
s̃
1
Figure 1: The equilibrium strategy in Example 1.
strategic effect appears in two forms. First, a high-budget bidder’s equilibrium strategy
features a prominent jump discontinuity at s̃ due to an important strategic tradeoff. A
high-budget bidder has an option of bidding above 1/4, thus outbidding with certainty any
low-budget competitor. This lets him dramatically increase the probability of winning the
auction. Due to the added cost, exercising this option is worthwhile only when his value is
sufficiently high, i.e. si > s̃. Private budget constraints therefore encourage more aggressive
bidding among privileged participants than would occur otherwise. The intuition for the
discontinuity at s̃′ is similar, but it involves a tie among low-budget bidders at 1/4.
Second, while some bids may jump up discontinuously, on the margin strategies become
less brazen. In Figure 1, the slope of β(·, wi ) declines moving past s̃. The intuition here hinges
on the endogenous stratification of competition along the budget dimension. When placing
a bid above 1/4, a bidder need only worry about competing bids from another high-budget
competitor. The probability of that case, however, is only 1/2. Similarly, in equilibrium, a
low-budget bidder of type si ∈ (s̃, s̃′ ] need only worry about competing bids from low-budget
adversaries. In each case, the decline in effective competition reduces the incentive to bid
aggressively. This competitive stratification further compromises the auction’s efficiency
and revenue potential. Inefficient allocations may occur even if all agents bid less than their
budget and thus are not observed to be financially constrained.
Although the preceding conclusions 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,
4
and the associated discontinuities and stratification, can arise even when budgets assume
a continuum of values and are continuously distributed. While the construction is more
elaborate, the intuition is unchanged. We detail this argument in Section 2 and we offer a
generalizable methodology to analyze these and more complex cases.
Though direct and strategic effects influence strategies overtly, private budgets also impact equilibrium bidding in more subtle ways. An incentive effect appears if we eschew
the confines of quasilinear preferences. For example, an agent’s attitude toward risk often
varies with his wealth, which in our model is a part of his private type. The equilibrium
implications may be counterintuitive. If a bidder’s wealth increases and his aversion to risk
decreases, which is behaviorally plausible, then he will want to bid less.4 Such behavior
conflicts with the usual understanding of a monotone, nondecreasing strategy, which is at
the core of the overwhelming majority of auction studies.
Example 2. Modifying Example 1, now suppose that a winning bidder of type (si , wi )
1
receives a payoff of (si − bi )wi+ 4 . Payoffs following a loss are zero.5 Though stylized for
illustration, these payoffs capture the idea that a bidder with a large budget is less riskaverse. A high budget (wi = 3/4) implies risk-neutrality while a low budget (wi = 1/4)
implies strict risk-aversion. The following strategy, sketched in Figure 2, defines the new
symmetric equilibrium:
β(si , 1/4) =



2s /3

 i
√
√
16s3i +24( 11−3)s2i +33(19 11−63)
√
24(si + 11−3)2



1/4

s /2
i
β(si , 3/4) = 2s2 +13√11−42
 i
4(si +1)
where s̃ =
√
if si ∈ [0, s̃′′ ]
if si ∈ (s̃′′ , s̃′ ]
if si ∈ (s̃′ , 1]
if si ∈ [0, s̃]
if si ∈ (s̃, 1]
√
11 − 3, s̃′′ = 3( 11 − 3)/4, and s̃′ ≈ 0.298 (see Appendix A).
For low values of si , the budget’s incentive effect is dominant and a low-budget agent bids
aggressively. As si increases, the strategic effect takes over and bids jump up. As a result,
equilibrium strategies fail to have a discernible ordering confounding analysis and inference.
Interestingly, however, some systematic patterns remain. For instance, an increase in wi
alone may decrease an agent’s equilibrium bid. But if wi and si increase together, a higher
4
5
Intuitively, the insurance provided by a higher bid is less valuable (Krishna, 2002, p. 38).
Cox et al. (1988) consider a setting with a similar utility function, though without budget constraints.
5
bi
β(si , 43 )
1
4
β(si , 41 )
si
0
s̃′′
s̃′ s̃
1
Figure 2: Equilibrium strategy in Example 2.
equilibrium bid will eventually be optimal. As shown by Reny (2011), such regularities can
be used to redefine a monotone strategy in a context-appropriate manner thereby assuring
the existence of a well-behaved equilibrium, suitable for analysis and interpretation. As
explained in Section 3, such an analysis applies to our environment as well, thus taming the
incentive effect’s countervailing instincts.
This paper is organized as follows. In the next section we briefly review the literature
on auctions with budget constraints. Thereafter, we flesh out the phenomena identified in
Examples 1 and 2, confirming their generality. In Section 2 we study a setting analogous to
Milgrom and Weber’s (1982) general symmetric model with continuously distributed types,
interdependent valuations, and affiliated signals. Despite the “smooth” setting, we construct
a discontinuous symmetric equilibrium in the first-price auction. Even without the incentive
effect, the resulting empirical confounds are substantial. For example, a uniform distribution
of signals and budgets can yield a bimodal equilibrium bid distribution (Example 3). An
endogenous segregation among high- and low-budget bidders causes this outcome. Quantitively, the strategic effect may be substantial, boosting the bids of high-budget bidders
by over 50 percent. While the construction is complex, our analysis allows for a simple
qualitative description of equilibrium bidding and facilitates numerical approximation, when
desired. Building upon the observations of Example 2, in Section 3 we address the existence
of a monotone equilibrium once private budget constraints and their incentive implications
are accounted for. This analysis extends our model beyond the quasilinear case and allows
6
for asymmetries among bidders. Section 4 concludes. An online appendix collects proofs
and supporting arguments.
1
Related Literature
While budget constraints have long been recognized as a feature of many auctions,6 interest
in their implications has grown recently due to their appearance in many internet-motivated
applications (Ashlagi et al., 2010; Balseiro et al., 2015) and their salience in spectrum auctions (Cramton, 1995; Salant, 1997; Bulow et al., 2009). Few studies, however, treat both
valuations and budgets as private information. In fact, most examine budget constraints
in multi-unit or sequential settings (Pitchik and Schotter, 1988; Benoît and Krishna, 2001;
Pitchik, 2009; Brusco and Lopomo, 2008, 2009). Given such markets’ extant complexity,
simplifications of the information structure are a practical necessity.
We contend that complex or multi-unit environments, like those above, are not necessary
to appreciate the strategic implications posed by budgets—classic, static, single-item settings
are sufficient, although (surprisingly) research here has been more limited. Che and Gale
(1996a,b) examine first-price, second-price, and all-pay auctions in a common-value setting
where only budgets are private information. In an important paper, Che and Gale (1998)
discuss first-price auctions allowing for private values and private budget constraints. As suggested above, their model is a special case of the analysis pursued herein and, consequently,
we identify a richer collection of implications than they propose. Fang and Parreiras (2002,
2003) study second-price auction with private budget constraints allowing for affiliated and
interdependent valuations. In a similar setting, Kotowski and Li (2014a,b) analyze all-pay
auctions. Our analysis in Section 2 complements these studies due to the parallel environment. Section 3, which focuses on the incentive effect, considers a more general asymmetric
setting than the aforementioned studies.
The optimal auction design problem following Myerson (1981) incorporating budget constraints is particularly challenging and has been considered under various guises (Laffont and
Robert, 1996; Monteiro and Page, 1998; Che and Gale, 1999, 2000; Maskin, 2000; Malakhov
and Vohra, 2008; Richter, 2011; Pai and Vohra, 2014; Kojima, 2014; Baisa, 2015). Auction
design accounting for budget limits has also spurred a growing interest computer scientists.7
Beyond discussing reserve prices in Section 2, we do not tackle this problem directly. Instead,
6
7
Rothkopf (1977) and Palfrey (1980) are early discussions of multi-unit auctions with budgets.
Borgs et al. (2005) and Dobzinski et al. (2012) are representative contributions.
7
we study a particular, canonical auction format. Che and Gale (2006) develop techniques
that help characterize auction revenue when bidders have multidimensional types. Their
analysis also considers the case of private budget constraints.
Whereas most models assume “hard” budget constraints, some papers explore ex post
default or asymmetries in bid financing ability (Zheng, 2001; Jaramillo, 2004; Rhodes-Kropf
and Viswanathan, 2005). These more subtle treatments of bid financing may mask some of
the strategic effects of budget constraints alone; thus, we treat budgets as a rigid bound for
bids. Cho et al. (2002) suggest that overcoming budget constraints can encourage collusion
or promote bidding consortia. We assume non-collusive behavior. Burkett (2016) argues that
budget constraints are a tool that resolves a principal-agent problem when auction bidding is
delegated to a third party. We agree with that motivation in some applications, but budgets
in our setting are exogenous primitives, as they would be in the absence of delegation. We
remain agnostic concerning their origins.
2
The Strategic Effect and Competitive Stratification
Examples 1 and 2 both illustrate the strategic effect associated with private budget constraints. Its most vivid consequence is the symmetric equilibrium strategy’s jump discontinuity and the resulting segregation among high- and low-budget bidders. It is perhaps
surprising that this phenomenon is more general. In contrast to the examples, casual intuition suggests that when agents’ types are continuously distributed such discontinuities
should be “smoothed out” since the equilibrium bid distribution can neither feature an atom
nor can it have a gap in its support.8 This intuition is correct in some special cases, but
does not apply generally.
To analyze a general case, we build on Milgrom and Weber’s (1982) canonical model
of a first-price auction with interdependent valuations. For simplicity, assume there are
two ex ante symmetric bidders.9 As above, let si ∈ [0, 1] be the value signal observed by
8
Abstracting from winner’s curse effects, the standard reasoning is as follows. If the equilibrium bid
distribution has an atom at b̂, then a strictly positive measure of bidder-types with budgets w > b̂ must
bid b̂. Each of these types can afford to increase his bid by ǫ. The infinitesimally greater bid leads to a
discrete increase in the probability of winning and, thus, a strictly higher payoff. This contradicts b̂ being
an equilibrium bid. Likewise, if there is a gap in the equilibrium bid distribution, any agent bidding slightly
above the gap can increase his payoff by lowering his bid. The decrease in bid is an order of magnitude
larger than the decrease in the probability of winning, making it worthwhile.
9
Accommodating N bidders with interdependent values requires an additional combinatorial accounting
to compute expected payoffs. This extension does not substantively change the conclusions of our analysis.
See Kotowski and Li (2014a).
8
bidder i ∈ {1, 2}. It defines bidder i’s information about the item for sale. If si and sj
are the bidders’ value signals, v(si , sj ) is bidder i’s valuation for the item. The function
v(·, ·) is continuously differentiable, strictly increasing in the first argument, nondecreasing
in the second, v(0, 0) = 0, and v(1, 1) = v̄ < ∞. To minimize confounds, assume that
bidders’ value signals are independent and identically distributed. (We relax independence
below.) The probability density function (p.d.f.) of bidder i’s value signal is h(si ). It is
continuously differentiable and strictly positive. As usual, H(si ) is the associated cumulative
distribution function (c.d.f.). Though a bidder’s realized value signal is private information,
its distribution is common knowledge. If bidder i wins the auction with a bid of bi , his
utility is v(si , sj ) − bi . Otherwise, it is zero. Ties are resolved with a uniform randomization.
When budget constraints are absent, Milgrom and Weber (1982) show that there exists
an equilibrium where all bidders adopt a common bidding strategy, α(si ), that solves the
differential equation
α′ (si ) = (v(si , si ) − α(si ))
h(si )
,
H(si )
α(0) = 0.
(3)
To introduce budget constraints, we pursue the simplest generalization beyond the introductory examples. Budgets are independently and identically distributed according to the
c.d.f. G(wi ).10 This distribution is twice continuously differentiable and has support [w, w̄].
We assume that w = 0, but we will vary this parameter below, notably in Example 3. To
limit the number of cases, we assume that (i) v̄ < w̄ and (ii) the p.d.f. g(wi ) := G′ (wi ), is
strictly positive on [w, w̄] and bounded, 0 < g < g(wi ) < ḡ. We relax (i) below. Condition
(ii) is technical and economically innocuous. As standard, Bayesian-Nash equilibrium is our
solution concept.11 In this section, we focus on symmetric equilibria where all agents adopt
the same strategy and we henceforth suppress bidder subscripts when possible.
To derive an equilibrium strategy, we conjecture that the auction has a symmetric equilibrium where each bidder’s strategy is
β(s, w) = min{b̄(s), w}
(4)
10
Assuming independence between value signals and budgets is natural in many applications. For example,
a bidder’s assessment of a painting’s authenticity (“si ”) should not be impacted by his private budget (“wi ”).
11
A strategy profile is a Bayesian-Nash equilibrium if for each agent there does not exist an alternative
feasible strategy that yields a higher expected payoff given the strategy followed by the other agent(s). An
equilibrium strategy is a best response for almost every type of bidder.
9
and b̄ : [0, 1] → R is differentiable and strictly increasing.12 The strategy β(s, w) implies
Leontief “isobid” curves in (s, w)-space with b̄(s) defining the locus of kink-points. If the
opposing bidder is following (4), bidder i’s expected payoff when he bids b̄(x), x ∈ [0, 1], is
U(b̄(x)|s, w) =
Z
x
0
Z
1
(v(s, y) − b̄(x))h(y)g(z) dzdy
0
+
Z
1
x
Z
0
b̄(x)
(v(s, y) − b̄(x))h(y)g(z) dzdy.
(5)
The first term is the payoff gain from defeating an opponent with a value signal less than x
who is bidding less than b̄(x). The second term is the payoff gain from defeating an opponent
with a value signal greater than x who has a budget less than b̄(x). Computing the necessary
first-order condition, ∂U (b̄(x)|s,w)
= 0, and defining the terms
∂x
x=s
δ(b, x) := b +
γ(b) :=
G(b)
g(b)
+
H(x)
,
g(b)(1−H(x))
g(b)
,
1−G(b)
λ(y) :=
R1
v(s,y)h(y)
dy,
x 1−H(x)
h(y)
,
1−H(y)
η(x, s) :=
gives the following characterization of b̄(s):
1 − G(b̄(s))
h(s)
b̄(s) − v(s, s)
·
·
1 − H(s) η(s, s) − δ(b̄(s), s)
g(b̄(s))
b̄(s) − v(s, s)
λ(s)
=
·
.
γ(b̄(s)) η(s, s) − δ(b̄(s), s)
b̄′ (s) =
(6)
(7)
A natural step at this point is to try to solve (7) subject to the boundary condition
b̄(0) = 0. This is not an appropriate strategy for two reasons. First, (7) may not have
a strictly increasing solution defined for all s as assumed. And second, b̄(0) = 0 may not
be the best bid for an agent with a value signal of s = 0. A bidder observing this signal
may wish to bid more if values are interdependent. A positive bid may defeat a budgetconstrained opponent who holds favorable information concerning the item’s value. Thus,
budget constraints attenuate the winner’s curse and, counterintuitively, they may inflate
bids, ceteris paribus. This phenomenon is distinct from, but complementary to, the strategic
effect. We will revisit its implications in Section 3.
To resolve the above complications, we first determine the qualitative behavior of solu12
Since Che and Gale (1998), (4) has been the usual functional form considered in the literature on auctions
with budget constraints.
10
tions to (7). Unlike Fang and Parreiras (2002), we will not analyze (7) directly.13 Instead,
we will use phase-plane analysis to examine the closely-related plane-autonomous system
ṡ(s, b) = γ(b) (η(s, s) − δ(b, s))
ḃ(s, b) = λ(s) (b − v(s, s))
(8)
where (s, b) ∈ [0, 1] × [w, w̄].14,15 The direction of motion as a function of time in (8) is
immaterial for our purposes. Instead, we are interested in the solution paths, which are
(locally) integral curves of (7) since, following Birkhoff and Rota (1978, Chapter 5), we
observe that
ḃ(s, b)
db . ds
db
=
=
= b̄′ (s).
ds
dt dt
ṡ(s, b)
In this complementary space, our goal is to identify solutions to (8) whose paths can be
represented by a function of s meeting the desiderata prescribed to b̄(s) above. Interpreting
the problem as proposed has two advantages. First, it allows for the use of elementary
techniques from phase-plane analysis to discern the behavior of candidate solutions. Second,
it placates the recurrent singularities encountered in a direct analysis of (7). Hence, the
presence of an admissible b̄(s) can be quickly ascertained.
To describe a plane-autonomous system, we are initially interested in two related objects,
the nullclines and the critical points. The nullclines of (8) are loci in (s, b)-space where
solution paths are horizontal or vertical. We can express these sets as correspondences:
ν(s) := {b | b = v(s, s)}
and
ψ(s) := {b | η(s, s) = δ(b, s)}.
Along ν(s), the solution paths of (8) parallel the s-axis. Along ψ(s), they parallel the b-axis.
This set corresponds to the solutions of η(s, s) = δ(b, s). For each s, this equation may
have zero, one, or many solutions.16 As we explain below, the nature and number of these
solutions will be tied to the features of the auction’s equilibrium. Critical points occur where
nullclines meet. Thus, (s∗ , b∗ ) is a critical point if ṡ(s∗ , b∗ ) = ḃ(s∗ , b∗ ) = 0.
Going forward, we consider two representative cases to simplify exposition and to better
identify the source of the strategic effect. In the first case, we impose a (strong) regularity
13
Solving (7) in closed form is generally not possible.
See Strogatz (1994) for an introduction to dynamical systems and phase-plane analysis.
15
ṡ and ḃ are not defined at the boundary where b = w̄ and s = 1, respectively. This technical fact is not
a concern and we suppress it for expositional clarity. We can perform all required analysis on the restricted
domain [0, 1 − ǫ] × [w, w̄ − ǫ]. The identified solution can be extended to the boundary using standard results
in the theory of ordinary differential equations.
16
Generally, ψ(0) 6= ∅ and ψ(s) = ∅ when s is sufficiently large (Lemma B.2).
14
11
condition that subdues the strategic effect. Consequently, the direct effect dominates and
we can define a suitable function b̄(s) by “pasting-together” particular solutions of system
(8). Analysis of this simplified situation, which enjoys a symmetric equilibrium in continuous
strategies, is primarily pedagogical. It provides a straightforward environment where we can
introduce notation, explain methodology, and relate our results to those of Che and Gale
(1998), which is a special case. Thereafter, we relax the imposed regularity condition to
construct a discontinuous equilibrium analogous to Example 1 above. This analysis builds
upon the simpler case, but requires an additional argument to define a bidding strategy with
an appropriate discontinuity.
2.1
Case 1: A Continuous Equilibrium
The first case depends on a simplifying assumption that we will subsequently relax.
Assumption 1. There exists a function ψ̄(s) such that (i) b < ψ̄(s) =⇒ η(s, s)−δ(b, s) > 0
and (ii) b > ψ̄(s) =⇒ η(s, s) − δ(b, s) < 0.
We postpone the economic interpretation of Assumption 1 to Remark 1 below. Paraphrased,
it says b 7→ η(s, s) − δ(b, s) crosses zero at most one time and from above. The model of
Che and Gale (1998) satisfies this condition. A sufficient condition ensuring Assumption 1
is satisfied is for the budget c.d.f. G : [0, w̄] → [0, 1] to be concave.
When Assumption 1 is satisfied, analysis of (8) is routine. Figure 3 illustrates a representative case and we explain it in detail.
1. The curve ν(s) runs from the origin to the figure’s upper-right corner. Along this
curve, solutions of (8) are parallel to the horizontal axis.
2. When ψ̄(s) > 0, ψ̄(s) = ψ(s) and ψ̄(s) is continuous. Along this curve solutions of (8)
are parallel to the vertical axis.
3. The functions ψ̄(s) and ν(s) intersect at least once (Lemma B.5). To illustrate a more
general analysis, Figure 3 presents a case with three intersections, at points A, B, and
C. These are the critical points. Generically, there is an odd number of critical points.
4. The ratio ḃ/ṡ is positive only at points in-between ν(s) and ψ̄(s) (Lemma B.6). These
regions, labelled R1 through R4 , are shaded in Figure 3. In these regions, the solution
paths of (8) are upward sloping as functions of s.
12
b
ν
b
b
R4
C
R3
b
B
R2
b
R1
0
ψ̄
A
s∗A
s∗B
s∗C
1
s
Figure 3: A phase-portrait of the (ṡ, ḃ) system when Case 1 applies.
5. Each critical point is either a saddle point or a node (Lemma B.7). When there is one
critical point, it is a saddle point. Otherwise, critical points alternate between saddles
and nodes. In Figure 3, A and C are saddle points, while B is an node.
We can use the preceding facts to define b̄ : [0, 1] → R+ . The function b̄(s) must be
increasing, continuous, and its derivative must satisfy (7). If such a function exists, it must
begin in region R1 , pass through the critical point A and continue on in region R2 . In those
regions ḃ/ṡ > 0 and, therefore, b̄′ (s) > 0. By similar reasoning, it must continue to R3 and
R4 , necessarily passing through points B and C en route. The only paths satisfying these
requirements correspond to the stable manifolds of (8) approaching points A and C. These
are illustrated as thick, black curves in Figure 3. Define the set b as the closure of the union
of these paths and express it as a function of s: b̄(s) := {b | (s, b) ∈ b}. The function b̄(s)
is continuous, strictly increasing, and an integral curve of (7). The value b̄(0) is defined
13
b
w
w̄
b̄
s
w̄
1
s
0
1
0
(a) Level sets of β(s, w).
(b) Illustration of β(s, w).
Figure 4: The continuous strategy β(s, w) in Case 1.
implicitly by the requirement that A, B, and C are waypoints as the function traverses its
domain. With an admissible b̄(s) defined, we can verify the following claim.
Theorem 1. There exists an equilibrium where each bidder adopts the bidding strategy
β(s, w) = min{b̄(s), w}. The function b̄ : [0, 1] → R is strictly increasing, continuous, and
for a.e. s, b̄′ (s) is given by (7). b̄(s) is identified as in the preceding discussion.
Figure 4 illustrates a representative strategy implied by Theorem 1 and we provide a
computed example in the following subsection.17 The preceding analysis can be adapted
and extended as needed. If w̄ < v̄ then it is sufficient to define b̄(s) only on a restricted
domain, say [0, s̄], such that b̄(s̄) = w̄. If there are fewer than three critical points, the
construction of b̄(s) is appropriately abbreviated (see Example 3 below). The argument’s
repetition accommodates more critical points, if necessary.
Value signal independence can also be relaxed if we further assume affiliation (Milgrom
and Weber, 1982). Though notationally more burdensome, and thus omitted from our initial
exposition, the analysis is essentially unchanged. In this case, let h(si , sj ) be the strictly
17
Closed-form expressions for b̄(s) are rarely available. To numerically approximate b̄(s), the following
procedure may be used. First, identify the saddle point(s) in (8) through which b̄(s) must pass. The stable
manifolds from system (8) approach these points with a slope equal to that of the negative eigenvector from
the Jacobian matrix of (8) evaluated at that point. A linearized solution can be defined locally around each
saddle point. This solution can be extended to the domain’s remainder with standard numerical methods.
14
positive, bounded, permutation symmetric, and log-supermodular joint density of bidders’
value signals.18 Define h(si |sj ) and H(si |sj ) as the conditional p.d.f. and c.d.f., respectively.
R1
H(x|s)
h(x|s)
+ g(b)(1−H(x|s))
, η(x, s) := x v(s,y)h(y|s)
dy, and λ(x|s) := 1−H(x|s)
, then
If δ(b, x|s) := b + G(b)
g(b)
1−H(x|s)
Theorem 1 generalizes as follows.19
Theorem 1′ . Suppose s 7→ v(s, x)λ(x|s) is nondecreasing for each x. There exists a symmetric equilibrium where each bidder adopts the bidding strategy β(s, w) = min{b̄(s), w}.
The function b̄ : [0, 1] → R is strictly increasing, continuous, and for a.e. s,
b̄′ (s) =
λ(s|s)
b̄(s) − v(s, s)
·
.
γ(b̄(s)) η(s, s) − δ(b̄(s), s|s)
b̄(s) is identified as in the preceding discussion.
Assuming that v(·, x)λ(x|·) is nondecreasing is a very common restriction in the auction
literature (Krishna and Morgan, 1997; Lizzeri and Persico, 2000; Fang and Parreiras, 2002).
Intuitively, it limits the “strength” of affiliation. It is implied by independence, but it does not
preclude strict affiliation. For example, it holds when v(si , sj ) = (si + sj )/2 and h(si , sj ) =
4(1 + si sj )/5.
2.2
Case 2: A Discontinuous Equilibrium
When Assumption 1 is not satisfied, it may not be possible to define b̄(s) for all s, as above.
For instance, the relevant solutions of system (8) may fail to meet if if ψ(s) is multivalued.
Figure 5 illustrates this possibility. Again, there are three critical points but the increasing
portions of the relevant solutions to (8) passing through A and C—now labeled b̄0 and b̄1 ,
respectively—do not pass through B. In this case, a jump discontinuity is required to define
the equilibrium bidding strategy.
For simplicity we focus on a case paralleling Figure 5 and we again assume that there
are three critical points. Define b̄0 : [0, s̄] → R and b̄1 : [s, 1] → R as in Figure 5 and further
assume that s∗A ≤ s < s̄.20 Given this setup, we replace Assumption 1 with the following
alternative.
Assumption 2. For each s ∈ [s, s̄], there exist values ψ(s) ≤ ψ̄(s), both greater than b̄0 (s),
such that
If h(si , sj ) is permutation symmetric, then h(s, s′ ) = h(s′ , s). Log-supermodularity implies affiliation.
Both conditions follow from Milgrom and Weber (1982) and are standard in the literature.
19
In Assumption 1, δ(b, s|s) replaces δ(b, s).
20
The requirement that s∗A ≤ s can be relaxed without substantively changing our basic argument.
18
15
b
ν
b̄1
ψ̄
R2
R3
b
C
µ
b
B
ψ
R4
b̄0
b
R1
0
A
s∗A s
s∗B
s∗C
s̄
1
s
Figure 5: A phase-portrait of the (ṡ, ḃ) system when Case 2 applies.
(i) b̄0 (s) < b < ψ(s) =⇒ η(s, s) − δ(b, s) < 0;
(ii) ψ(s) < b < ψ̄(s) =⇒ η(s, s) − δ(b, s) > 0; and,
(iii) b > ψ̄(s) =⇒ η(s, s) − δ(b, s) < 0.
In Figure 5, ψ(s) and ψ̄(s) are the dashed portions of the enclosure of regions R2 and R4 .
Remark 1 (Interpretation of Assumptions 1 and 2). It is instructive to interpret Assumption 2 and to contrast it with Assumption 1. Paraphrased, Assumption 2 says that for
certain values of s and over a particular domain the function b 7→ η(s, s) − δ(b, s) crosses
zero two times—first from below and then from above.21 When it crosses zero from below,
21
Note that η(s, s) − δ(b, s) < 0 for b sufficiently close to w̄. Thus, if b 7→ η(s, s) − δ(b, s) crosses zero
multiple times, the final crossing must always be from above even in the absence of Assumptions 1 or 2.
16
H(s)
+ g(b)(1−H(s))
is decreasing in b, which is possible only if g(b) is increasδ(b, s) = b + G(b)
g(b)
ing sufficiently rapidly. This means that the budget distribution’s density becomes more
concentrated in the neighborhood of ψ(s). This concentrated mass increases the return to
a higher bid and, intuitively, should encourage a bidder with a large enough budget to increase his bid substantially. Example 1, where the budget distribution featured an atom,
involved a particularly extreme instance of this phenomenon. In light of Assumption 2, we
can interpret Assumption 1 as the absence of (relatively) large changes or concentrations in
the budget density. This is consistent with the observation that Assumption 1 is satisfied
when the budget c.d.f. G(·) is concave, as noted above. In that case, its density is always
non-increasing.
To define an equilibrium strategy, we first pin-down a value signal, s̃, at which a bidder
with a high budget discontinuously changes his bid. To define this value, we conjecture
that the candidate low bid will be b̄0 (s̃) and the candidate high bid will be µ(s̃) where
µ(s) := min{b̄1 (s), ψ̄(s)}. This function, which is defined on [s, s̄] and labeled in Figure 5,
enjoys the following important property. Passing through each point (s, µ(s)) is a solution
of (8) that can be continuously extended as an increasing function (of s) to the boundary.
This solution either coincides with b̄1 (s) or lies below it. Several of these candidate solutions
are included in Figure 5 for illustration. Intuitively, µ(s) defines a set of candidate “landing
points” to which a jump bid may aspire.
By usual reasoning, bids of b̄0 (s̃) and µ(s̃) must yield the same expected payoff in equilibrium. We conjecture that the expected payoff of a bidder who bids b̂ ∈ {b̄0 (s̃), µ(s̃)}
is
Z
Z
s
Ū(b̂|s, w) :=
0
1
(v(s, y) − b̂)h(y)dy + G(b̂)
s
(v(s, y) − b̂)h(y)dy.
(9)
As confirmed by Lemma B.1, there exists an s̃ ∈ [s, s̄] such that Ū (b̄0 (s̃)|s̃, w) = Ū (µ(s̃)|s̃, w).22
Having defined b̄0 (s), s̃, and µ(s), we can introduce the strategy β(s, w). It is composed
of three functions, b̄(s), b̃(s), and φ̃(s), which we explain below. Together, these functions
22
If there are multiple such values, let s̃ be the largest one. We can repeat the presented argument with
any s̃ ∈ [s, s̄] satisfying Ū (b̄0 (s̃)|s̃, w) = Ū (µ(s̃)|s̃, w). Thus, we cannot rule out equilibrium multiplicity.
17
w
b
w̄
b̄
µ(s̃)
φ̃
b̃
s
s
0
s̃′
s̃
w̄
1
s̃′
1
s̃
µ(s̃)
0
(a) Level sets of β(s, w). The function is discontinuous along the thick, dashed curve.
(b) Illustration of β(s, w).
Figure 6: The discontinuous strategy β(s, w) in case 2. (Case where b̃(s̃′ ) = φ̃(s̃′ ) illustrated.)
define β(s, w) as follows:



min{b̄(s), w} if s < s̃




min{b̃(s), w} if s ∈ [s̃, s̃′ ], w < φ̃(s)
β(s, w) =
.

′

min{
b̄(s),
w}
if
s
∈
[s̃,
s̃
],
w
≥
φ̃(s)




min{b̄(s), w} if s > s̃′
(10)
We explain the above terms using Figure 6 as a reference.
1. The function b̄ : [0, 1] → R defines the bid of a unconstrained bidder. For s < s̃,
b̄(s) coincides with b̄0 (s), defined above. For s ≥ s̃, b̄(s) coincides with the increasing
solution to system (8) passing through (s̃, µ(s̃)). It either coincides with b̄1 (s) or lies
below it, depending on the particular case. Thus, b̄(s) has a discontinuity at s̃ and for
a.e. s, b̄′ (s) is given by (7).
2. The function b̃ : [s̃, s̃′ ] → R describes the bids placed by a bidder who lacks the funds to
match the jump-bid of high-budget agents. It accounts for the decreased competition
at low bid levels given the discontinuous strategy of a relatively high-budget opponent.
18
Echoing (6), b̃(s) solves
b̃′ (s) =
h(s)
b̃(s) − v(s, s)
G(φ̃(s)) − G(b̃(s))
·
·
1 − H(s) η(s, s) − δφ̃ (b̃(s), s)
g(b̃(s))
(11)
subject to b̃(s̃) = b̄0 (s̃) where
Hφ̃ (s) := H(s̃) +
Z
s
s̃
G(φ̃(y))h(y)dy and δφ̃ (b, s) := b +
Hφ̃ (s)
G(b)
+
.
g(b)
g(b)(1 − H(s))
3. Expression (11) captures the equilibrium’s competitive stratification through the first
and final terms. These depend on the function φ̃ : [s̃, s̃′ ] → R, which is continuous,
decreasing, and φ̃(s̃) = µ(s̃) = lims→s̃+ b̄(s). Intuitively, this function is an endogenous
boundary between high- and low-budget types. A bidder with value signal s ∈ [s̃, s̃′ ]
is indifferent between the bids b̃(s) and φ̃(s), but he would like to bid above φ̃(s) if he
has the resources to do so. A bidder with budget w < φ̃(s), bids min{b̃(s), w} and a
bidder with budget w ≥ φ̃(s) bids min{b̄(s), w} when following β(s, w). More formally,
if
Ũϕ (b̂|s, w) :=
Z
Z
s̃
s
(v(s, y) − b̂)h(y)dy +
(v(s, y) − b̂)G(ϕ(y))h(y)dy
0
s̃
Z 1
+ G(b̂)
(v(s, y) − b̂)h(y)dy,
(12)
s
then φ̃(s) and b̃(s) are such that Ũφ̃ (φ̃(s)|s, w) = Ũφ̃ (b̃(s)|s, w) for all s ∈ [s̃, s̃′ ]. The
value s̃′ is defined in Appendix C and is primarily technical.
The functions b̃(s) and φ̃(s) are the above definition’s new and key components. To
provide intuition, consider the bidding problem of an agent with value signal s = s̃ + ǫ and
budget w = µ(s̃) − ǫ. This bidder is tempted to increase his bid discontinuously. A high
bid wins the auction with a relatively high probability and an amelioration of the winner’s
curse, mentioned above, reinforces the expected payoff benefit. However, this bidder lacks
the resources to bid µ(s̃). The most he can bid is µ(s̃) − ǫ. To determine whether he
should increase his bid to this marginally lower value, this bidder does not compare bidding
w = µ(s̃) − ǫ versus b̄0 (s̃ + ǫ). Instead, he compares bidding w versus b̃(s̃ + ǫ), which accounts
for the change in the marginal benefit of a low bid. Since an opponent with a comparable
value signal but a higher budget is bidding above µ(s̃) given β, the lower-budget bidder faces
19
less competition on the margin if he bids low. The same tradeoff appears in Example 1 for
a low-budget bidder with value signal of s ∈ [s̃, s̃′ ]. For a type-s bidder, only bids exceeding
φ̃(s) deliver a higher expected payoff than b̃(s). As s increases, the gain from a win increases
and bidders with ever smaller budgets find it worthwhile to increase their bid discontinuously
to their maximum feasible value. The resulting jumps decrease in magnitude and bidders
smoothly segregate on either side of the φ̃(s) boundary.
Theorem 2. When Assumption 2 holds, the strategy β(s, w) defined in (10) constitutes a
symmetric equilibrium.
An important technical subtly is that b̃(s) and φ̃(s) are jointly determined. The rate
at which competition changes—captured by φ̃(s)—determines the low bid level, b̃(s). This
affects the incentives to “jump bid,” which feeds back into φ̃, and so on. Therefore, proving
Theorem 2 requires verifying that b̃ and φ̃ exist with the prescribed properties. This technical
detour relies on a fixed-point argument. Once defined, verification of the equilibrium proceeds
like in the proof of Theorem 1. Necessarily, many additional cases need to be considered to
accommodate the strategy’s piecewise definition.
Example 3. An example illustrates both continuous and discontinuous equilibria. Suppose
value signals are uniformly distributed on the interval [0, 1] and v(si , sj ) = si + sj . In the
absence of budget constraints, α(s) = s is this auction’s symmetric equilibrium strategy.
Now suppose budgets are distributed uniformly on [0, 2]. In this case,
b̄′ (s) =
2(2 − b̄(s))(2s − b̄(s))
(1 + s)(3s − 1) − 4(s − 1)b̄(s)
and there is one critical point at s∗ ≈ 0.105. The eigenvalues of the Jacobian of (ṡ, ḃ) at the
critical point are −6.62 and 4.83. Therefore, the critical point is a saddle point and the stable
manifolds approach it at a slope of 0.85. Figure 7a plots b̄(s), ψ̄(s), and ν(s). Representative
solutions from the associated plane-autonomous system and α(s) are included for context.
For small s, unconstrained bidders bid more than in the absence of budget constraints, i.e.
b̄(s) > α(s), highlighting the attenuation of the winner’s curse discussed above.
Suppose instead that budgets are distributed uniformly on the interval [0.2, 2]. While the
w > 0 case was not discussed above, the preceding analysis readily carries over. The density
of budget constraints increases rapidly at w = 0.2—g(w) jumps from zero to 5/9—and we
can identify an equilibrium in discontinuous strategies with the same reasoning as above.23
23
The definition of ψ̄(s) is unchanged, ψ(s) = w = 0.2 and b̄0 (s) = α(s).
20
Figure 7b illustrates the functions b̄(s), ψ̄(s), φ̃(s), and b̃(s). Representative solutions from
the associated plane-autonomous system and α(s) are included for context.
Like in Example 1, b̄(s) and α(s) coincide when s is small, but b̄(s) jumps up at s =
s̃ ≈ 0.181. Bidders with that value signal (and with a sufficient budget) increase their
bid from 0.181 to 0.286—an increase of 58 percent. Therefore, the strategic effect can be
quantitatively large. The strategic effect is also a serious confound for the interpretation
of bidding data. Figure 7c plots a histogram approximating the resulting equilibrium bid
distribution. Despite the uniform type-space, the equilibrium bid distribution is bimodal. If
budget constraints were absent, this distribution would be uniform.
Remark 2 (Further Generalizations). For expositional ease, we have restricted attention to
instances of our environment satisfying Assumptions 1 or 2. Of course, more general cases
can and do arise. Our analysis readily extends to accommodate more unusual circumstances
as well. For example, the argument may be repeated to define multiple discontinuities, if
necessary. Whether an expanded analysis is necessary depend directly on the characteristics
of ψ(s) and indirectly on the budget distribution G(·). The equilibrium strategy’s basic
rationale and supporting intuition remain unchanged.
2.3
Discussion and Applications
To add context to the preceding analysis, we briefly relate our model to that of the secondprice auction. We also comment on reserve prices.
The Second-Price Auction
Fang and Parreiras (2002) study the second-price auction with private budget constraints.
They identify a symmetric equilibrium of the form βII (s, w) = min b̄II (s), w . In our
notation, b̄II (s) solves the boundary-value problem
b̄′II (s) =
b̄II (s) − v(s, s)
λ(s)
·
,
γ(b̄II (s)) η(s, s) − b̄II (s)
b̄II (1) = v̄.
We can relate this equilibrium to our analysis in three ways. First, βII (s, w) is continuous
when w = 0.24 Thus, the strategic implications of budgets in the second-price auction are
less pronounced than in the first-price auction. To emphasize this contrast, recall Example
24
Fang and Parreiras (2002) show that b̄II (s) may be discontinuous when w > 0. The second-price auction
can have a discontinuous equilibrium even in the absence of budget constraints (Krishna, 2002, p. 116).
21
(a) Equilibrium strategy when budgets are distributed uniformly on [0, 2].
(b) Equilibrium strategy when budgets are distributed uniformly on [0.2, 2].
(c) Distribution of equilibrium bids when budgets are distributed uniformly on [0.2, 2]. Histogram of one million simulated bids (rescaled to unit area).
Figure 7: Equilibrium bidding in Example 3.
22
1. In that environment, min{s, w} is a dominant strategy in the second-price auction. As
shown in (2), the first-price auction’s equilibrium is far more intricate.
Second, bidding is more subdued in the first-price auction than in the second-price auction. If βI (s, w) and βII (s, w) are symmetric equilibria in the first- and second-price auctions,
respectively, then βI (s, w) ≤ βII (s, w) (Lemma D.1). A comparison of bids across auction
formats offers an avenue to test for the presence of private budget constraints in applications.
Specifically, when βI (s, w) = βII (s, w) we can exploit the invariance of bids across formats to
infer whether a participant is constrained by a budget. Though unusual as a source of variation, the case suggests that budget constraints, as distinct from valuations, can be identified
from bidding behavior alone.25 Without novel forms of variation, it is otherwise difficult to
draw an inference about the model’s two unobserved primitives, s and w.
Finally, first- and second-price auctions differ in their comparative statics. For example,
consider the budget distributions G0 (w) and G1 (w) where G0 (w) likelihood ratio dominates
G1 (w). Intuitively, financial constraints are more severe when budgets are distributed according to G1 (w) rather than G0 (w). In the second-price auction, changing the budget
distribution from G0 (w) to G1 (w) increases the bids of unconstrained bidders (Fang and
Parreiras, 2002). In the first-price auction, in contrast, the change has ambiguous effects.
An unconstrained bidder may bid more or less (see Appendix D). The associated intuition
centers on two competing effects. When values are interdependent, budget constraints encourage higher bids by attenuating the winner’s curse. The countervailing force concerns
the marginal impact of higher bids on the probability of winning. There is less incentive
to bid aggressively because competition is less intense at higher bid levels. Both forces are
magnified when the budget distribution is skewed toward lower values. The final alignment
of incentives may favor one or the other.
Reserve Prices
As in the case of the second-price auction, there does not exist a straightforward expression
for expected revenues in the first-price auction with budget constraints. However, equilibrium
strategies hint at some revenue implications, particularly when reserve prices are considered.
In the presence of budget constraints, reserve prices screen bidders on both the value and the
budget dimension. Expected revenues may decline if bidders are disproportionately screened
along the latter. For example, if a reserve price of r < b̄(0) is set, all bidders for whom w < r
25
For example, bidders could submit format-contingent bids. We are unaware of any field or laboratory
experiments investigating this possibility.
23
are excluded from the auction while the bids of agents for whom w ≥ r are unaffected. As
a result, expected revenue falls.
Given its multi-channeled impact, setting the revenue-maximizing reserve price when
bidders are possibly budget constrained is a delicate exercise. In the introductory model
of Example 1 the revenue-maximizing reserve price varies discontinuously with the model’s
parameters. If high budgets are likely, a reserve price of 1/2 is optimal. If bidders are
budget-constrained with high probability, the optimal reserve price is strictly lower.
3
The Incentive Effect and Monotone Equilibrium
The symmetric, quasilinear model is sufficient to isolate the strategic implications of budget
constraints. As illustrated by Example 2, an agent’s budget or wealth will impact his bid in
more complex ways in more general circumstances. While such cases suggest that systematic
conclusions concerning equilibrium bidding may be elusive, this is not the case. A “monotone”
equilibrium with economically-relevant properties exists quite generally, though monotonicity
needs appropriate qualification. This conclusion is not immediate.26 Reny and Zamir (2004)
show that a monotone equilibrium may not exist in a first-price auction when signals are
multidimensional, as they are here. In some auctions, budget constraints may preclude the
existence of an equilibrium altogether (Ghosh, 2014).
Generalizing our model, suppose that there are N bidders, each with a private type
θi = (si , wi ) ∈ [si , s̄i ] × [wi , w̄i ]. Let s = (s1 , . . . , sN ) be the vector of all bidders’ value
signals and let s−i = (s1 , . . . , si−1 , si+1 , . . . , sN ) be the vector of value signals with the signal
of agent i removed. We define w and w−i analogously. Suppose each bidder has a set of valid
bids Bi ⊂ R. The set Bi may incorporate a reserve price ri ≥ 0.27 It also includes a minimal
“null” or “non-competitive” bid ℓ < ri guaranteeing a loss in the auction. The (measurable)
function βi : [si , s̄i ] × [wi , w̄i ] → Bi is an admissible (pure) strategy if it respects the agent’s
budget constraint.
We assume that payoffs satisfy the following conditions whenever ri ≤ bi ≤ wi :
(i) If bidder i wins the auction, his payoff is ūi (si , s−i , wi , bi ). ūi is bounded, nondecreasing
in (si , s−i , wi ), strictly decreasing in bi , and differentiable. Moreover, ∂ ūi /∂si ≥ κi > 0.
26
See de Castro and Karney (2012) for a survey concerning equilibrium existence in auction games.
Like Athey (2001), we allow for bidder-specific reserve prices. Kotowski (2015) studies equilibrium
bidding in first-price auctions with heterogenous reserve prices.
27
24
(ii) If bidder i does not win the auction, his payoff is ui (wi ). ui is bounded, nondecreasing,
and differentiable.
(iii) If bi ≤ 0, then ūi (si , s−i , wi , bi ) ≥ ui (wi ); however, there exists B̄ such that for all
bi > B̄, ūi (si , s−i , wi , bi ) < ui (wi ).
(iv) For all bi ≥ b′i ≥ ri , ūi (si , s−i , wi , bi ) − ūi (si , s−i , wi , b′i ) is nondecreasing in (si , s−i , wi ).
(v) inf
∂ ūi
∂wi
du
≤ sup dwii < ∞.
Conditions (i)–(v) adapt assumptions from Maskin and Riley (2000), Reny and Zamir (2004),
or McAdams (2007), among others, to our setting. Points (i) and (ii) are standard. The
lowerbound on ∂ ūi /∂si in (i) is similar to Athey’s (2001) Assumption A2(v). Point (iii)
says that the item is desirable, but has finite value. Point (iv) is an often-encountered
increasing-differences condition. The final condition is primarily technical. Its role will be
apparent shortly. To simplify notation, we define ui (si , s−i , wi , bi ) := ūi (si , s−i , wi , bi )−ui (wi ).
We can nest the model from Section 2 in this new notation by defining ūi (si , s−i , wi , bi ) =
v(si , s−i ) + wi − bi and u(wi ) = wi .
As highlighted by Example 2, when bidders’ preferences are sufficiently general, equilib-
rium behavior may not conform with traditional definitions of a monotone strategy. However,
the definition of a monotone, nondecreasing strategy may be weakened while retaining its key
economic implications. Reny (2011) pioneered this approach in games of incomplete information and, interestingly, we can draw on his analysis of a multi-unit auction with risk-averse
bidders to study our single-item setting. The analogy between our model of a single-item
auction with private budgets and a multi-unit auction is the following. An auction with
private budget constraints can be interpreted as an unusual multi-item auction: the highest
bidder wins the item for sale, as usual, and all bidders “win” a private cash endowment equal
to their private budget irrespective of their admissible bid.
Leveraging the above analogy, we follow Reny (2011) by first defining an alternative
ordering of the type space. If
du
ξi :=
sup dwii − inf
inf
∂ ūi
∂si
∂ ūi
∂wi
,
then (si , wi ) ≥θi (s′i , wi′ ) ⇐⇒ wi ≥ wi′ and si − s′i ≥ ξi (wi − wi′ ). Figure 8 sketches the
greater- and less-than sets for type θ̂i = (ŝi , ŵi ). In the quasilinear case, ξi = 0 and ≥θi
reduces to the standard coordinate-wise partial order of R2 . With ≥θi given, we call βi
monotone nondecreasing if θi ≥θi θi′ =⇒ βi (θi ) ≥ βi (θi′ ).
25
w̄i
Slope =
θ̂i
ŵi
b
1
ξi
θi′ ≥θi θ̂i
θi′ ≤θi θ̂i
wi
si
ŝi
s̄i
Figure 8: The ≥θi ordering.
Second, concerning the information structure, we maintain an environment paralleling Theorem 1′ . The joint density of agents’ types is given by f (θ) := h(s)g(w) and is
strictly positive and bounded. Value signals are affiliated, though not too strongly. Thus,
R
R
if H(s−i |si ) := t−i ≤s−i h(t−i |si )dt−i and H̄(s−i |si ) := t−i ≥s−i h(t−i |si )dt−i , then we define
h(s−i |si )
and σ(s−i |si ) :=
λ(s−i |si ) := H̄(s
−i |si )
we assume that for a.e. s−i ,
h(s−i |si )
.
H(s−i |si )
If (si , wi ) ≥θi (s′i , wi′ ) and ri ≤ bi ≤ wi′ , then
(i) ui (s′i , s−i , wi′ , bi ) ≥ 0 =⇒ ui (si , s−i , wi , bi )λ(s−i |si ) ≥ ui (s′i , s−i , wi′ , bi )λ(s−i |s′i ), and
(ii) ui (si , s−i , wi , bi ) < 0 =⇒ ui (si , s−i , wi , bi )σ(s−i |si ) ≥ ui (s′i , s−i , wi′ , bi )σ(s−i |s′i ).
Conditions like (i) and (ii) are not new (cf. Theorem 1′ and the associated discussion). They
are satisfied when value signals are independent, but they can also hold when correlation
is strict.28 Intuitively, they ensure that the direct influence of si on payoffs dominates its
informational content concerning s−i .29 As above, budgets are mutually independent and
Q
independent of value signals, i.e. g(w) = i gi (wi ).30
Theorem 3. There exists an equilibrium in monotone, nondecreasing strategies if:
(i) Bidders’ sets of valid bids are finite and Bi ∩ Bj = {ℓ} for all i 6= j.
28
For example, consider N = 2, ri = 0, ui (si , s−i , wi , bi ) = (si + sj )/2 − bi , and h(si , sj ) = 4(1 + si sj )/5.
Continuity implies that it will also hold in nearby economies where the ≥θi ordering has bite.
29
Formally, restrictions on the degree of affiliation facilitate the verification of a single crossing condition
when the conditioning event is not a sublattice. Such restrictions are unnecessary in the first-price auction
without budget constraints due to the one dimensional
type space (Reny and Zamir, 2004).
Q
30
We may abuse notation by writing g(w−i ) ≡ j6=i gj (wj ).
26
(ii) Bidders’ have private values,31 Bi = {ℓ} ∪ [ri , ∞), and ties among high bidders are
resolved with a uniform randomization.
Theorem 3’s proof rests on verifying 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 a multi-unit auction with risk-averse bidders. The key
lemmas proceed similarly and quickly lead to part (i), which is the theorem’s economicallysalient conclusion. In part (i), agents’ competitive bids (those other than ℓ) are restricted
to finite and disjoint sets. These sets can be arbitrarily fine and, we believe, constitute
an economically innocuous modeling device. Focusing on disjoint bid sets is common in
the literature, particularly when types are multidimensional, and avoids the complications
that arise from tied bids.32 Part (ii) shows that ties among high bidders pose no problem
when values are private. It generalizes Che and Gale’s (1998) private-values model to an
asymmetric setting.
Recently, Gentry et al. (2015) have also adapted Reny’s (2011) argument and methodology to confirm the existence of a monotone equilibrium in a first-price auction where agents’
degree of risk aversion and wealth is private information. In contrast to our model, their
analysis focuses on bidders exhibiting either constant relative or absolute risk aversion. Their
analysis is also restricted to the private-values case and they employ a different partial order
of the type space than we propose. Importantly, wealth is not synonymous with a hard
budget constraint in their model. Rather, it is only a variable affecting an agent’s attitude
toward risk. This different interpretation has economic consequences. To illustrate, recall
Example 2 from the introduction. Gentry et al. (2015) show that bids decline in wealth,
ceteris paribus. Such a decline indeed occurs in Example 2, but only for a subset of bidder
types. The presence of the strategic effect ensures that a high-budget bidder with a high
value signal outbids a low-budget competitor with the same value signal.
It is interesting to note that the complications associated with extending part (ii) to the
interdependent-values case stem from economic, and not purely technical, considerations.
They are related to the way budget constraints ameliorate the winner’s curse and a bidder’s
desire to break ties in his favor. It is instructive to relate this point to prior studies of
equilibrium existence in first-price auctions. After establishing the existence of a monotone
equilibrium in a finite-action setting, Athey (2001) uses a limiting argument to study the
continuum-action version of her model. Her Assumption A3 is crucial for that extension.
That is, ui (si , s−i , wi , bi ) = ui (si , s′−i , wi , bi ) for all s−i and s′−i . With private values Theorem 3(ii) also
extends to the case where si and wi are correlated.
32
McAdams (2006) adopts a similar approach to study monotone equilibria in multi-unit auctions.
31
27
w̄j
Ã
b′i
bi
A−
wj
sj
A+
s∗
s̄j
Figure 9: The danger of defeating an opposing bidder with a “higher” type.
Roughly, that assumption says that whenever bidder i’s opponents employ nondecreasing
strategies, bidder i’s expected payoff increases when he defeats opponents with higher types
holding his bid fixed. Figure 9 illustrates a possible failure of this assumption in a twobidder instance of our model. Given bidder j’s monotone strategy, when bidder i bids bi ,
he defeats bidder j when θj ∈ A− ∪ A+ . When bidder i bids b′i > bi , he additionally wins
when θj ∈ Ã. Now suppose that when sj < s∗ , ui (si , sj , wi , bi ) < 0 and when sj > s∗ ,
ui (si , sj , wi , bi ) > 0. As informally suggested by the figure, E[ui (si , Sj , wi , bi )|A− ∪ A+ ] >
E[ui (si , Sj , wi , bi )|A− ∪ A+ ∪ Ã] since ui (si , sj , wi , bi ) < 0 on most of Ã. Thus, even when bi
is fixed, defeating an opponent who is bidding slightly more need not be worthwhile. Reny
and Zamir (2004) generalize Athey’s (2001) analysis. In passing to a continuum action set
in their model, they show that a bidder benefits from bidding infinitesimally more to “win a
tie.” For reasons similar to those above, this common intuition may fail in our setting. By
winning a tie, a bidder may defeat an opponent who holds relatively unfavorable information.
The concerns raised by ties are not unique to our environment. When bidders have multidimensional private information, the tie breaking procedure matters and standard (uniform)
tie breaking may be inadequate (Araujo et al., 2008). Though pursuing these considerations
is beyond the scope of this study, several approaches can be invoked to ensure existence
despite the associated complications. One method relies on “endogenous” tie breaking rules
(Simon and Zame, 1990; Jackson et al., 2002; Jackson, 2009). In a working paper version
of this study, we adopted a special endogenous tie breaking rule proposed by Araujo and de
Castro (2009) as an example. Many alternative specifications are possible too.
28
4
Concluding Remarks
Private budget constraints are a key feature of many economic environments and, as argued
above, carry first-order implications within the first-price auction. Equilibrium analysis
demands a marked amendment of existing intuition. Private budgets have direct, strategic,
and incentive effects. Each carries qualitatively meaningful and quantitatively significant
implications that should be accounted for in auction bidding, analysis, and design.
Our results suggest many possibilities for further research. For instance, the degree to
which post auction resale ameliorates or exacerbates the inefficiencies introduced by budgets
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 largely open questions. We have briefly touched upon these
issues, though much work remains. Untangling the more subtle relationship between bids,
valuations, and budgets stands as a new, and in some cases continuing, challenge for theoretical and empirical researchers alike.
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33
All appendices are intended for online publication only.
A
Examples 1 and 2
A.1
Example 1
In this section we prove in a more general environment that the strategy proposed in Example
1 is an equilibrium. There are two, risk-neutral agents bidding for a single item in a first-price,
sealed-bid auction. Ties are resolved with a fair coin flip. Each bidder has a two-dimensional
type, composed of a value signal (si ) and budget (wi ). If a bidder of type (si , wi ) wins the
auction his payoff is si −bi ; else, it is zero. As the model is symmetric, we henceforth suppress
subscripts. Value signals are independent and identically distributed according to the c.d.f.
H : [0, s̄] → [0, 1], with strictly positive density h(s). With probability p ∈ (0, 1) a bidder’s
R s̄
R s̄
budget is w, 0 < w < s̄ − 0 H(z)dz. With probability 1 − p, it is w̄ > s̄ − 0 H(z)dz. It is
R s′
1
easy to verify that there exists a unique s′ ∈ (w, s̄) such that w = s′ − H(s
′ ) 0 H(z)dz. The
following lemmas define the values s̃ and s̃′ . These constants define the points of discontinuity
in the bidding strategy that we introduce below.
R s̃
Lemma A.1. There exists a unique s̃ ∈ (w, s′ ) such that 0 H(z)dz = [p+(1−p)H(s̃)](s̃−w).
Rs
Proof. Let τ (s) := [p + (1 − p)H(s)](s − w) − 0 H(z)dz. It suffices to show that τ (s) crosses
zero exactly once. When s = w, τ (w) < 0. If instead s = s′ , τ (s′ ) = [p + (1 − p)H(s′ )](s′ −
w) − H(s′ )(s′ − w) > 0. Thus, there exists s̃ ∈ (w, s′ ) such that τ (s̃) = 0. Moreover,
τ ′ (s) = p(1 − H(s)) + (1 − p)(s − w)h(s) > 0 for all w ≤ s < s̄. Thus, τ (s) is strictly
increasing and s̃ is unique.
Lemma A.2. Define s̃ as in Lemma A.1. There exists a unique s̃′ ∈ (s̃, s̄) such that
Z
s̃
H(z)dz +
0
Z
s̃
s̃′
1 + H(s̃′ )
[(1 − p)H(s̃) + pH(z)]dz = (1 − p)H(s̃) + p
(s̃′ − w). (A.1)
2
Proof. Let
Z
s̃
Z
s
[(1 − p)H(s̃) + pH(z)]dz, and
1 + H(s)
(s − w).
χ(s) := (1 − p)H(s̃) + p
2
τ (s) :=
0
H(z)dz +
s̃
34
At s = s̃,
τ (s̃) =
Z
s̃
H(z)dz
0
= [p + (1 − p)H(s̃)](s̃ − w)
1 + H(s̃)
(s̃ − w) = χ(s̃).
> (1 − p)H(s̃) + p
2
Instead, if s = s̄,
τ (s̄) = [p + (1 − p)H(s̃)](s̃ − w) +
Z
s̃
s̄
[(1 − p)H(s̃) + pH(z)]dz
< [p + (1 − p)H(s̃)](s̃ − w) + [p + (1 − p)H(s̃)](s̄ − s̃)
= [p + (1 − p)H(s̃)](s̄ − w) = χ(s̄).
Thus, there exists s̃′ ∈ (s̃, s̄) such that τ (s̃′ ) = χ(s̃′ ). To verify uniqueness, note that
τ ′ (s) = (1 − p)H(s̃) + pH(s)
and
χ′ (s) = (1 − p)H(s̃) + p
1 + H(s) p
+ h(s)s.
2
2
Hence, τ ′ (s) < χ′ (s) and s 7→ τ (s) − χ(s) is strictly decreasing. Thus, s̃′ is unique.
Theorem A.1. Define s̃ and s̃′ as above. The strategy profile where all bidders follow the
strategy
β(s, w) =



s−




s −


s−




w
R
s
1
H(z)dz
H(s) 0
Rs
R s̃
0 H(z)dz+ s̃ [p+(1−p)H(z)]dz
p+(1−p)H(s)
Rs
R s̃
0 H(z)dz+ s̃ [pH(z)+(1−p)H(s̃)]dz
pH(s)+(1−p)H(s̃)
if s ∈ [0, s̃], w ∈ {w, w̄}
if s ∈ (s̃, s̄], w = w̄
if s ∈ (s̃, s̃′ ], w = w
(A.2)
if s ∈ (s̃′ , s̄], w = w
is a symmetric Bayesian-Nash equilibrium.
Proof. The following proof closely mirrors the proof of the analogous claim in the absence
of budget constraints (Krishna, 2002, p. 17–18). However, there are many more cases due
to the piecewise definition of β(s, w). To simplify notation, define β̄(s) := β(s, w̄) and
β(s) := β(s, w). Let U(b|s, w) be the expected utility of bidder i of type (s, w) when he bids
b and the other bidder follows the prescribed strategy, β.
35
1. Consider bidder i of type s ∈ [0, s̃] with budget w̄. If this bidder follows β, U(β̄(s)|s, w̄) =
Rs
H(z)dz. There are four kinds of alternative bids to consider.
0
(a) A deviation to β̄(x), x ∈ [0, s̃], is unprofitable. The standard argument confirming
an equilibrium in a symmetric, first-price sealed-bid auction applies. For example,
see Krishna (2002, p. 17–18).
(b) A deviation to β(x), x ∈ (s̃, s̃′ ], yields a payoff of U(β(x)|s, w̄) = [(1 − p)H(s̃) +
R s̃
Rx
pH(x)](s − x) + 0 H(z)dz + s̃ [(1 − p)H(s̃) + pH(z)]dz. The net gain is
U(β(x)|s, w̄) − U(β̄(s)|s, w̄)
Z
s̃
= [(1 − p)H(s̃) + pH(x)](s − x) +
H(z)dz +
s
Z s̃
=
H(z)dz + [(1 − p)H(s̃) + pH(x)](s − s̃)
s
Z x
+ pH(x)(s̃ − x) +
pH(z)dz
Z
x
s̃
[(1 − p)H(s̃) + pH(z)]dz
s̃
≤ H(s̃)(s̃ − s) + [(1 − p)H(s̃) + pH(s̃)](s − s̃) + pH(x)(s̃ − x) + pH(x)(x − s̃) = 0
Thus, U(β(x)|s, w̄) ≤ U(β̄(s)|s, w̄).
(c) Given β, the bid w is placed with strictly positive probability by bidder j. A bid
of w + ǫ gives a bidder i a strictly higher payoff.
(d) Consider a deviation to a bid β̄(x), x ∈ (s̃, s̄]. Bidder i’s expected payoff is
R s̃
Rx
U(β̄(x)|s, w) = [p + (1 − p)H(x)](s − x) + 0 H(z)dz + s̃ [p + (1 − p)H(z)]dz. The
net gain is
U(β̄(x)|s, w̄) − U(β̄(s)|s, w̄)
Z s̃
=
H(z)dz + [p + (1 − p)H(x)](s − s̃)
s
Z x
+ [p + (1 − p)H(x)](s̃ − x) +
[p + (1 − p)H(z)]dz
s̃
Z x
≤ H(s̃)(s̃ − s) + [p + (1 − p)H(x)](s − s̃) +
(1 − p)[H(z) − H(x)]dz ≤ 0.
s̃
The final inequality follows from x ≥ s̃ ≥ s. Thus, U(β̄(x)|s, w̄) ≤ U(β̄(s)|s, w̄).
2. Consider bidder i of type s ∈ (s̃, s̄] with budget w = w̄. If this bidder follows the
36
R s̃
Rs
prescribed strategy, U(β̄(s)|s, w̄) = 0 H(z)dz + s̃ [p + (1 − p)H(z)]dz. There are four
kinds of alternative bids to consider.
(a) A deviation to β̄(x), x ∈ [0, s̃], yields a payoff of U(β̄(x)|s, w̄) = H(x)(s − x) +
Rx
H(z)dz. The net gain is
0
U(β̄(x)|s, w̄) − U(β̄(s)|s, w̄)
Z x
Z s̃
Z s
= H(x)(s − x) +
H(z)dz −
H(z)dz −
[p + (1 − p)H(z)]dz
0
0
s̃
Z x
Z s
= H(x)(s̃ − x) +
H(z)dz + H(x)(s − s̃) −
[p + (1 − p)H(z)]dz
s̃
s̃
Z s̃
≤
[H(x) − H(z)]dz + H(x)(s − s̃) − [p + (1 − p)H(s̃)](s − s̃) ≤ 0.
x
Therefore, this deviation is not profitable.
(b) A deviation to β(x), x ∈ [s̃, s̃′ ], yields a payoff of U(β(x)|s, w̄) = [(1 − p)H(s̃) +
R s̃
Rx
pH(x)](s − x) + 0 H(z)dz + s̃ [(1 − p)H(s̃) + pH(z)]dz. The net gain is
U(β(x)|s, w̄) − U(β̄(s)|s, w̄)
= [(1 − p)H(s̃) + pH(x)](s − x) +
Z s
−
[p + (1 − p)H(z)]dz
s̃
= [(1 − p)H(s̃) + pH(x)](s̃ − x) +
≤
Z
=
Z
Z
x
Z
x
[(1 − p)H(s̃) + pH(z)]dz
s̃
[(1 − p)H(s̃) + pH(z)]dz
Z s
+ [(1 − p)H(s̃) + pH(x)](s − s̃) −
[p + (1 − p)H(z)]dz
s̃
s̃
x
p[H(z) − H(x)]dz
s̃
+ [(1 − p)H(s̃) + pH(x)](s − s̃) − [p + (1 − p)H(s̃)](s − s̃)
s̃
x
p[H(z) − H(x)]dz + p(s − s̃)(H(x) − 1) ≤ 0.
Therefore, this deviation is not profitable.
(c) As above, a bidder can strictly improve upon the bid w by bidding ǫ more.
37
(d) A deviation to β̄(x), x ∈ (s̃, s̄], yields a payoff of
U(β̄(x)|s, w̄) = [p + (1 − p)H(x)](s − x) +
Z
s̃
H(z)dz +
0
Z
s̃
x
[p + (1 − p)H(z)]dz.
The net gain is
U(β̄(x)|s, w̄) − U(β̄(s)|s, w̄)
Z
x
= [p + (1 − p)H(x)](s − x) +
[p + (1 − p)H(z)]dz
s
Z x
=
(1 − p)[H(z) − H(x)]dz ≤ 0.
s
Therefore, this deviation is not profitable.
3. Consider bidder i of type s ∈ [0, s̃] with budget w = w. By identical arguments to
those presented above, this bidder will not have a profitable deviation to any bid,
except possibly w = β(x), x > s̃′ . This deviation yields a payoff of U(w|s, w) =
i
h
′)
(s − w). The net gain is
(1 − p)H(s̃) + p 1+H(s̃
2
U(w|s, w) − U(β(s)|s, w)
1 + H(s̃′)
= (1 − p)H(s̃) + p
(s − s̃′ )
2
Z s
1 + H(s̃′)
′
(s̃ − w) −
H(z)dz
+ (1 − p)H(s̃) + p
2
0
Z s̃
Z s̃′
1 + H(s̃′)
′
(s − s̃ ) +
= (1 − p)H(s̃) + p
H(z)dz +
[(1 − p)H(s̃) + pH(z)]dz
2
s
s̃
Z s̃
1 + H(s̃′)
(s − s̃) +
H(z)dz
= (1 − p)H(s̃) + p
2
s
Z s̃′
1 + H(s̃′)
′
+ (1 − p)H(s̃) + p
(s̃ − s̃ ) +
[(1 − p)H(s̃) + pH(z)]dz ≤ 0.
2
s̃
Therefore, this deviation is not profitable.
4. Consider bidder i of type s ∈ (s̃, s̃′ ] with budget w = w. If this bidder follows the
R s̃
Rs
prescribed strategy, U(β(s)|s, w) = 0 H(z)dz + s̃ [pH(z) + (1 − p)H(s̃)]dz.
(a) A deviation to β(x), x ∈ [0, s̃], yields a payoff of U(β(x)|s, w) = H(x)(s − x) +
38
Rx
0
H(z)dz. The net gain is
U(β(x)|s, w) − U(β(s)|s, w)
Z x
Z s̃
Z s
= H(x)(s − x) +
H(z)dz −
H(z)dz −
[(1 − p)H(s̃) + pH(z)]dz
0
Z0 x
Zs̃ s
= H(x)(s̃ − x) +
H(z)dz + H(x)(s − s̃) −
[(1 − p)H(s̃) + pH(z)]dz
s̃
s̃
≤ 0 + H(x)(s − s̃) − [(1 − p)H(s̃) + pH(s̃)](s − s̃)
= [H(x) − H(s̃)](s − s̃) ≤ 0.
Therefore, this deviation is not profitable.
(b) A deviation to β(x), x ∈ [s̃, s̃′ ], yields a payoff of U(β(x)|s, w) = [(1 − p)H(s̃) +
R s̃
Rx
pH(x)](s − x) + 0 H(z)dz + s̃ [(1 − p)H(s̃) + pH(z)]dz. The net gain is
U(β(x)|s, w) − U(β(s)|s, w)
= [(1 − p)H(s̃) + pH(x)](s − x) +
Z x
=
p[H(z) − H(x)]dz ≤ 0.
Z
x
s
[(1 − p)H(s̃) + pH(z)]dz
s
Therefore, this deviation is not profitable.
(c) A deviation to w can be ruled out by an argument similar to that from case 3
above.
5. Consider bidder i of type s ∈ (s̃′ , s̄]h with budget w = w. iIf this bidder follows the
′)
prescribed strategy, U(β(s)|s, w) = (1 − p)H(s̃) + p 1+H(s̃
(s − w).
2
(a) A deviation to β(x), x ∈ [0, s̃], yields a payoff of
U(β(x)|s, w) = H(x)(s − x) +
39
Z
0
x
H(z)dz.
The net gain is
U(β(x)|s, w) − U(β(s)|s, w)
Z x
1 + H(s̃′ )
= H(x)(s − x) +
(s − w)
H(z)dz − (1 − p)H(s̃) + p
2
0
Z x
Z s̃
Z s̃′
= H(x)(s − x) +
[(1 − p)H(s̃) + pH(z)]dz
H(z)dz −
H(z)dz −
s̃
0
0
1 + H(s̃′ )
(s − s̃′ )
− (1 − p)H(s̃) + p
2
Z x
= H(x)(s̃ − x) +
H(z)dz
s̃
Z s̃′
′
[(1 − p)H(s̃) + pH(z)]dz
+ H(x)(s̃ − s̃) −
s̃
1 + H(s̃′ )
′
+ H(x)(s − s̃ ) − (1 − p)H(s̃) + p
(s − s̃′ ) ≤ 0.
2
Therefore, this deviation is not profitable.
(b) A deviation to β(x), x ∈ [s̃, s̃′ ], can be ruled out with reasoning similar to the
preceding case.
Therefore, (A.2) is a symmetric equilibrium strategy.
A.2
Example 2
Theorem A.2. The following is a symmetric equilibrium in Example 2.
β(s, 3/4) =
β(s, 1/4) =

s
2
if s ∈ [0, s̃]
√
2s2 +13


2s



3
11−42
4(s+1)
if s ∈ (s̃, 1]
√
√
16s3 +24( 11−3)s2 +33(19 11−63)
√
24(s+ 11−3)2



1
4
where s̃ =
√
if s ∈ [0, s̃′′ ]
if s ∈ (s̃′′ , s̃′]
if s ∈ (s̃′ , 1]
√
11 − 3 ≈ 0.317, s̃′′ = 43 ( 11 − 3) ≈ 0.237, and s̃′ solves the equation
s̃′ + s̃
2
1
2
(s̃ − q(s̃)) =
1
1 s̃′ + 1 1
·
+ s̃ (s̃ − 1/4) 2
2
2
2
40
where q(s) =
√
√
16s3 +24( 11−3)s2 +33(19 11−63)
√
.
2
24(s+ 11−3)
(s̃′ ≈ 0.298.)
Proof. We mimic the reasoning outlined in the proof of Theorem A.1. Again, there are
multiple cases. To simplify notation, let β̄(s) := β(s, 3/4) and β(s) := β(s, 1/4).
1. Consider bidder i with budget w = 3/4 and value signal s ∈ [0, s̃]. The bid β̄(x), x ∈
[0, s̃], defeats a high-budget opponent with a value signal less than x and a low-budget
opponent with a value signal less than 3x/4. The expected payoff is U(β̄(x)|s, w̄) =
(7x/8)(s−x/2). Maximizing this expression with respect to x, subject to the constraint
x ≤ s̃, gives a solution of x = s. Thus, β̄(s) is the optimal bid when constrained to a
bid in the range [0, s̃/2].
Instead, suppose i bids β̄(x), x ∈ (s̃, 1]. This bid defeats a low-budget opponent
and a high-budget opponent with a value signal less than x. The expected payoff is
d
(s− β̄(x)). Noting that dx
U(β̄(x)|s, w̄) = (s−x)/2 ≤ 0 and given
U(β̄(x)|s, w̄) = 1+x
2
1+s̃
1
+ U(β̄(x)|s, w̄) =
the
definition
of
s̃,
we
see
that
U(
β̄(x)|s,
w̄)
≤
lim
s
−
=
x→s̃
2
4
√
( 11−2)(4s−1)
7s2
≤ 16 = U(β̄(s)|s, w̄). Hence, this proposed deviation is not profitable.
8
The remaining candidate deviation is to the bid β(x), x ∈ [s̃′′ , s̃′ ). For x ∈ [s̃′′ , s̃′ ), let
7s2
∆(x, s) := U(β̄(s)|s, w̄) − U(β(x)|s, w̄) =
−
16
x + s̃
2
(s − β(x)).
It is sufficient to establish that ∆(x, s) ≥ 0 for all x ∈ [s̃′′ , s̃′ ) when s ≤ s̃. From the
definition of s̃ and s̃′′ (noting that β(s) is continuous at s̃′′ ), ∆(s̃′′ , s̃) = 0. Next, note
that ∂∆(s̃′′ , s)/∂s = 7(s − s̃)/8 ≤ 0. Hence, for all s ≤ s̃, ∆(s̃′′ , s) ≥ 0. Thus, it is
≥ 0 for all s ∈ [0, s̃]. A direct computation gives
sufficient to confirm that ∂∆(x,s)
∂x
1
∂∆(x, s)
=
∂x
48
≥
1
48
!
√
49(19 11 − 63)
√
8( 11 − 3 − 3s) + 32x −
(x + 11 − 3)2
!
√
√
49(19
11
−
63)
√
8( 11 − 3 − 3s̃) + 32s̃′′ −
= 0.
(s̃′′ + 11 − 3)2
√
Therefore, a bidder of type s ≤ s̃ has no incentive to deviate into this range of bids.
2. Consider bidder i with budget w = 3/4 and value signal s ∈ (s̃, 1]. By an argument
analogous to the preceding case, all deviations to bids β̄(x), x ∈ [0, 1] can be ruled out.
41
We therefore consider a possible deviation to the bid β(x), x ∈ [s̃′′ , s̃′ ). Let
∆(x, s) := U(β̄(s)|s, w̄) − U(β(x)|s, w̄) =
1+s
2
(s − β̄(s)) −
x + s̃
2
(s − β(x)).
It is sufficient to verify that ∆(x, s) ≥ 0 for s ∈ (s̃, 1] and x ∈ [s̃′′ , s̃′ ]. As before,
∆(s̃′′ , s̃) = 0. And for any x ∈ [s̃′′ , s̃′ ],
4−
∂∆(x, s)
=
∂s
√
11 + s − x
4−
≥
2
Thus, it is sufficient to confirm that
∂∆(x,s̃)
∂x
√
11 + s̃ − x
≥ 0.
2
≥ 0. A direct calculation gives
√
√
11
2x
49(19 11 − 63)
∂∆(x, s̃)
√
−
=
−
+1
2
∂x
3
3
48(x + 11 − 3)
√
√
11
2s̃′′
49(19 11 − 63)
√
≥
−
+ 1 = 0.
−
3
3
48(s̃′′ + 11 − 3)2
Thus, a bidder with a value signal of s > s̃ has no profitable deviation from β̄(s).
3. Consider bidder i with budget w = 1/4 and value signal s ∈ [0, s̃′′ ]. The bid β(x),
x ∈ [0, s̃′′ ], defeats a low-budget opponent with a value signal less than x and a highbudget opponent with a value signal less than 4x/3. Therefore, the expected payoff
from this bid is
1/2
7x
2x
U(β(x)|s, w) =
s−
.
6
3
Maximizing the above expression with respect to x, subject to the constraint x ≤ s̃′′ ,
gives a solution of x = s. Thus, β(s) is the optimal bid when considering bids in the
′′
range [0, 2s̃3 ].
If instead bidder i bids β(x), x ∈ [s̃′′ , s̃′ ], his expected payoff is
U(β(x)|s, w) =
1
x + s̃
(s − β(x)) 2 .
2
Substituting and differentiating with respect to x gives
√
6(s − x)
d
U(β(x)|s, w) = q
√
dx
8(3s + 11 − 3) − 16x −
42
√
49(19 11−63)
√
(x+ 11−3)2
≤ 0.
Since β(s) is continuous at s̃′′ , U(β(x)|s, w) ≤ U(β(s̃′′ )|s, w) ≤ U(β(s)|s, w).
√
Finally, since s̃′′ = 43 ( 11 − 3) ≈ 0.23, the bid w = 1/4 would yield a negative payoff.
Therefore, this bidder does not have a profitable deviation.
4. Consider bidder i with budget w = 1/4 and value signal s ∈ (s̃′′ , s̃′ ]. By an argument
analogous to the preceding case, all deviations to bids β(x), x ∈ [0, s̃′ ], by i can be ruled
out. We therefore consider only the bid of w = 1/4. Define ∆(s) := U(β(s)|s, w) −
U(w|s, w). From the definition of s̃′ , ∆(s̃′ ) = 0. Therefore, it is sufficient to verify
d
d
U(β(s)|s, w) ≤ ds
U(w|s, w). This fact is true since
that ∆′ (s) ≤ 0; equivalently, that ds
√
√
√
′′ +2 11−5
s̃′ +2
11
d
11−5
11−29
s̃
U(w|s, w) = 4√4s−1 ≥ 4√4s̃−1 = √ √
≈ 0.901 and
ds
16
d
U(β(s)|s, w) = r
ds
8s −
4( 11−3)−1
q
3
2
s+
√
11 − 3
√
49(19 11−63)
√
2
(s+ 11−3)
q
√
√
+ 8( 11 − 3)
3
(s̃ + 11 − 3)
2
√
√
49(19 11−63)
√
+
8(
11
′′
2
(s̃ + 11−3)
≤q
8s̃′′ −
√
( 11 − 3)2
=p √
≈ 0.563.
38 11 − 126
− 3)
5. Consider bidder i with budget w = 1/4 and value signal s ∈ (s̃′ , 1]. By arguments
analogous to those presented in the previous cases, we can verify that β(s̃′ ) is the
utility-maximizing, feasible bid other than w.
To verify that U(w|s, w) ≥ U(β(s̃′ )|s, w), recall that at s̃′ , U(w|s̃′ , w) ≥ U(β(s̃′ )|s̃′ , w).
Thus, it is sufficient to verify that for all s > s̃′ ,
d
U(w|s, w)
ds
d
U(β(s̃′ )|s, w)
ds
√
d
U(β(s̃′ )|s, w)
ds
≤
d
U(w|s, w).
ds
q
√
√
11−63)
√
(s̃ + 2 11 − 5) 8(3s + 11 − 3) − 16s̃′ − 49(19
′
(s̃ + 11−3)2
√
=
√
2 24s − 6(s̃′ + 11 − 3)
q
√
√
√
11−63)
′′
√
(s̃ + 2 11 − 5) 8(3s + 11 − 3) − 16s̃ − 49(19
′′
(s̃ + 11−3)2
√
≥
√
2 24s − 6(s̃ + 11 − 3)
q
√
√
(333 + 68 11)s − 86 11 + 502
3
√
=
.
8 4s − 1
′
43
But,
This final expression is decreasing in s. Evaluating at s = 1 shows that
d
U(w|s, w)
ds
d
U(β(s̃′ )|s, w)
ds
1
≥
24
q
√
1501 − 54 11 ≈ 1.51 ≥ 1,
which is the desired result.
As we have exhausted all possibilities, β(s, w) is a symmetric equilibrium.
B
Proof of Theorems 1 and 1′
We prove Theorem 1′ , which implies Theorem 1. We define the following:
H(x|s)
G(b)
+
,
δ(b, x|s) := b +
g(b)
g(b)(1 − H(x|s))
g(b)
γ(b) :=
,
1 − G(b)
Z
1
v(s, y)h(y|s)
dy,
1 − H(x|s)
x
.
h(y|s)
λ(y|s) :=
1 − H(y|s)
η(x, s) :=
Now, (7) becomes
b̄′ (s) =
λ(s|s)(b̄(s) − v(s, s))
.
γ(b̄(s))(η(s, s) − δ(b̄(s), s|s))
(B.1)
And (8) reads as
ṡ = γ(b) (η(s, s) − δ(b, s|s))
ḃ = λ(s|s) (b − v(s, s)) .
(B.2)
Assumption 1 generalizes in the obvious manner.
Assumption 1′ . There exists a function ψ̄(s) such that (i) b < ψ̄(s) =⇒ η(s, s)−δ(b, s|s) >
0 and (ii) b > ψ̄(s) =⇒ η(s, s) − δ(b, s|s) < 0.
Henceforth, we assume Assumption 1′ is satisfied. We first consider several preliminary
lemmas concerning system (B.2).
Lemma B.1. Fix b. The function δ(b, x|s) is increasing in x and decreasing in s. η(x|s) is
increasing in both arguments.
Proof. δ(b, x|s) is increasing in x as H(x|s) is increasing in x. From affiliation, s′ > s =⇒
H(y|s′ )
H(y|s)
′
H(y|s′) ≤ H(y|s) and thus, 1−H(y|s
′ ) ≤ 1−H(y|s) . Hence, δ(b, x|s ) ≤ δ(b, x|s). The claims
concerning η follow from its definition and affiliation.
44
Lemma B.2. There exists an s′ < 1 such that ψ(s) = ∅ for all s > s′ .
s→1
H(s|s)
H(s|1)
Proof. Since g(b) < ḡ, δ(b, s|s) > ḡ(1−H(s|s))
≥ ḡ(1−H(s|1))
−−→ ∞. Hence, ∃s′ < 1 such that
∀s > s′ , δ(b, s|s) > v̄ ≥ η(s|s). Thus, ψ(s) = ∅ when s > s′ .
Lemma B.3. There exists a constant ǫ > 0 such that ψ̄(s) < v̄ − ǫ.
Proof. For all s, δ(v̄, s|s) ≥ v̄ + G(v̄)/g(v̄) > v̄. Thus, there exists ǫ > 0 such that for all
b > v̄ − ǫ, δ(b, s|s) > v̄ ≥ η(s|s). Hence, if δ(b, s|s) = η(s, s), then b < v̄ − ǫ.
Lemma B.4. The function ψ̄(s) is continuous.
Proof. Given Assumption 1′ , if ψ(s) 6= ∅, then ψ̄(s) = ψ(s). Otherwise, ψ̄(s) = 0. From
inspection, we known that ψ̄(s) is continuous when it equals ψ(s). Suppose that at some ŝ,
ψ̄(ŝ) = 0 but ψ̄(s) is not continuous at ŝ. Thus, there exists a sequence sk → ŝ such that for
all k, ψ̄(sk ) > ǫ > 0. Since ψ̄(sk ) = ψ(sk ) and ǫ ≤ ψ(sk ) ≤ v̄, without loss of generality we
can assume that ψ(sk ) converges to a limit and, thus, limsk →ŝ ψ(sk ) = ψ(ŝ) ≥ ǫ. But this
implies ψ̄(ŝ) = ψ(ŝ) ≥ ǫ, which is a contradiction.
Lemma B.5. There exists at least one critical point. Generically, there is an odd number.
Proof. It is easy to see that ψ(0) 6= ∅. There are two possibilities. If ψ(0) = 0, then a
critical point occurs at the origin. If instead ψ(0) > 0, then to verify that a critical point
exists it is sufficient to show that s 7→ ψ̄(s) − ν(s) crosses zero. Thus, it is sufficient to show
that lims→1 ψ̄(s) − ν(s) < 0. Thus, it is sufficient to show that lims→1 ψ̄(s) < v̄. By Lemma
B.2, ψ̄(s) = 0 for all s sufficiently close to 1. Hence, lims→1 ψ̄(s) < v̄, as required. Since
ψ̄(0) − ν(0) ≥ 0 and lims→1 ψ̄(s) − ν(s) < 0, s 7→ ψ̄(s) − ν(s) will (generically) cross zero an
odd number of times.
Lemma B.6.
(i) If b > ν(s), then ḃ(s, b) > 0. If b < ν(s), then ḃ(s, b) < 0.
(ii) If b > ψ̄(s), then ṡ(s, b) < 0. If b < ψ̄(s), then ṡ(s, b) > 0.
Proof. Part (i) is trivial. Part (ii) is an immediate consequence of Assumption 1′ and the
definitions of ψ̄(s) and ψ(s).
Lemma B.7. Consider the system (ṡ, ḃ) defined in (8). If (s∗ , b∗ ) is a critical point, then it
is (generically) either a node or a saddle point.
45
Proof. It is sufficient to confirm that the eigenvalues of the Jacobian matrix evaluated at the
critical point (s∗ , b∗ ),
!
∂ ṡ
∂ ṡ
∂s
∂b
,
J=
∂ ḃ
∂ ḃ
∗ ∗
∂s
∂b
are real-valued. The eigenvalues will be real if
is non-negative. Thus,
(s ,b )
∂ ṡ
∂s
−
∂ ḃ
∂b
2
∂ ḃ
when evaluated at (s∗ , b∗ )
+ 4 ∂∂bṡ ∂s
!
∂ ṡ ∂δ
∗
′ ∗
∗ ∗
∗ ∗ ∗
∗ ∂δ +γ(b
)
−
=
γ
(b
)
(η(s
,
s
)
−
δ(b
,
s
|s
))
=
−γ(b
)
.
{z
}
|
∂b (s∗ ,b∗ )
∂b (s∗ ,b∗ )
∂b (s∗ ,b∗ )
0
We know that γ(b∗ ) 6= 0 and (s∗ , b∗ ), δ(b∗ , s∗ |s∗ ) − η(s∗ , s∗ ) = 0. Assumption 1′ implies that
∗ |s∗ ) = ∂δ(b,s
> 0. Hence, ∂∂bṡ (s∗ ,b∗ ) < 0. Also,
(generically) ∂δ
∂b (s∗ ,b∗ )
∂b
b=b∗
∂ ḃ ∂s (s∗ ,b∗ )
∂λ dv(s, s) ∗
∗ ∗
∗ ∗
(b − v(s , s )) +λ(s |s ) −
< 0.
=
{z
}
∂s (s∗ ,b∗ ) |
ds s=s∗
0
∂ ḃ
Therefore, 4 ∂∂bṡ ∂s
> 0 as needed.
Proof of Theorem 1′ . Suppose the other bidder is following the prescribed strategy β. The
expected utility of bidder i of type (s, w) who bids b̄(x) ≤ w is U(b̄(x)|s, w) as defined in (5).
Differentiating with respect to x and collecting terms,
∂U(b̄(x)|s, w)
λ(x|s)
′
v(s, x) − b̄(x) + b̄ (x) η(x, s) − δ(b̄(x), x|s) .
= g(b̄(x))(1 − H(x|s))
∂x
γ(b̄(x))
Suppose x > s. Therefore, b̄(x) > b̄(s) = β(s, w). By affiliation, η(x, x) ≥ η(x, s),
δ(b̄(x), x|x) ≤ δ(b̄(x), x|s), and λ(x|x) ≤ λ(x|s). Moreover, v(s, x)λ(x|s) ≤ v(x, x)λ(x|x).
Thus,
∂U(b̄(x)|x, w)
∂U(b̄(x)|s, w)
≤
= 0.
∂x
∂x
Hence, bidder i can improve his payoff by bidding less than b̄(x). A parallel argument shows
that when x < s, a bidder of type (s, w) can increase his payoff by bidding more than b̄(x).
Together, the two cases imply that no bidder has an incentive to deviate from β(s, w) to
some other bid in the range of b̄(·). (This also confirms that a bidder who is constrained,
β(s, w) = w, cannot benefit from lowering his bid.)
Suppose instead a bidder of type (s, w) places a bid b̂ outside the range of b̄(s). Bids
46
exceeding b̄(1) are clearly dominated. Bids of b̂ < b̄(0) give an expected payoff of
U(b̂|s, w) =
Z bZ
0
1
0
v(s, y) − b̂ h(y|s)g(w)dydw = G(b̂)(η(0, s) − b̂).
∂U (b|s,w) ∂b
b=b̂
It is sufficient to show that
(B.3)
> 0 for all b̂ < b̄(0). Differentiating (B.3),
∂U(b|s, w) = g(b̂)(η(0, s) − b̂) − G(b̂) ≥ g(b̂)(η(0, 0) − b̂) − G(b̂).
∂b
b=b̂
If b̄(0) > 0, then ψ̄(0) > b̄(0). Thus, for all b̂ < ψ(0), η(0, 0) > δ(b̂, 0|0) =⇒ η(0, 0) >
b̂ + G(b̂)/g(b̂) =⇒ g(b̂)(η(0, b̂) − b̂) − G(b̂) > 0, which implies the desired conclusion.
Thus, a bidder of type (s, w) has no profitable deviation from β(s, w) = min{b̄(s), w}
when the other bidder adopts the same strategy. Therefore, β(s, w) is a symmetric equilibrium strategy.
C
Proof of Theorem 2
The proof of Theorem 2 has three parts. First, Lemma C.1 defines s̃. Second, Section C.1
defines b̃(s), φ̃(s), and s̃′ . Finally, Section C.2 verifies that the introduced strategy is an
equilibrium.
Lemma C.1. Define b̄0 (·), s, s̄, and µ(·) as in Section 2.2. Let
Ū(b̂|s, w) :=
Z
0
s
(v(s, y) − b̂)h(y)dy + G(b̂)
Z
s
1
(v(s, y) − b̂)h(y)dy.
(C.1)
There exists s̃ ∈ [s, s̄] such that Ū(b̄0 (s̃)|s̃, w) = Ū (µ(s̃)|s̃, w).
Proof. By continuity, it is sufficient to show that (i) Ū(b̄0 (s)|s, w) ≥ Ū (µ(s)|s, w) and (ii)
Ū (b̄0 (s̄)|s̄, w) ≤ Ū(µ(s̄)|s̄, w). After collecting terms,
dŪ(b|s, w)
= g(b)(1 − H(s))(η(s, s) − δ(b, s)).
db
We rely upon Assumption 2 to sign the preceding expression. (i) If s = s, then η(s, s) −
δ(b, s) < 0 for all b ∈ (b̄0 (s), µ(s)). Hence, b 7→ Ū (b|s, w) is decreasing on this interval. Thus,
U(b̄0 (s)|s, w) ≥ Ū(µ(s)|s, w). (ii) If s = s̄, then η(s̄, s̄) − δ(b, s̄) > 0 for all b ∈ (b̄0 (s̄), µ(s̄)).
Consequently, b 7→ Ū (b|s̄, w) is increasing and Ū(b̄0 (s̄)|s̄, w) ≤ Ū(µ(s̄)|s̄, w).
47
Henceforth, define s̃ as the largest value satisfying Lemma C.1. With s̃ fixed, we record
several auxiliary functions that will be used below.
Z
Z
s̃
s
(v(s, y) − b̂)h(y)dy +
(v(s, y) − b̂)G(ϕ(y))h(y)dy
0
s̃
Z 1
+ G(b̂)
(v(s, y) − b̂)h(y)dy,
s
Z s
Hϕ (s) := H(s̃) +
G(ϕ(y))h(y)dy,
Ũϕ (b̂|s, w) :=
(C.2)
(C.3)
s̃
δϕ (b, s) := b +
Hϕ (s)
G(b)
+
.
g(b)
g(b)(1 − H(s))
(C.4)
Definitions of b̃(s), φ̃(s), and s̃′
C.1
To define b̃(s), φ̃(s), and s̃′ , we proceed in four steps:
1. We define two compact subsets of the space of continuous function, B and F , and we
characterize some key properties of b(·) ∈ B and F (·) ∈ F (Lemmas C.2 and C.3).
2. For an ǫ > 0, we define b̃ and F̃ on [s̃, s̃ + ǫ] as a fixed point of a particular operator,
Λ : B × F → B × F , that we introduce below. Using Definition C.1, b̃ and F̃ we then
define φ̃.
3. In Remark C.3 and Lemmas C.4–C.7 we characterize several key properties of b̃ and
φ̃. In particular, we show that Ũφ̃ (b̃(s)|s, w) = Ũφ̃ (φ̃(s)|s, w) for all s ∈ [s̃, s̃ + ǫ] where
Ũ(·) (·|s, w) was defined above in (C.2).
4. We repeat the preceding construction on an incrementally larger domain until such
extensions are no longer possible. We define [s̃, s̃′ ] as the maximal domain accommodating the preceding construction.
C.1.1
Definitions of B and F
We begin by defining B and F . The set B depends on a constant K defined in next lemma.
Lemma C.2. Let ϕ : [s̃, 1] → [b̄0 (s̃), µ(s̃)] be a continuous function such that ϕ(s̃) = µ(s̃).
Let
h(s)
b − v(s, s)
G(ϕ(s)) − G(b)
·
·
.
(C.5)
ξϕ (s, b) :=
g(b)
1 − H(s) η(s, s) − δϕ (b, s)
48
There exists an ǫ > 0 and a K > 0 such that for all (s, b) ∈ [s̃, s̃ + ǫ] × [b̄0 (s̃), b̄0 (s̃) + ǫ],
0 ≤ ξϕ (s, b) ≤ K < ∞.
Proof. At s̃, δϕ (b, s̃) = δ(b, s̃). Thus, η(s̃, s̃) − δϕ (b̄0 (s̃), s̃) = η(s̃, s̃) − δ(b̄0 (s̃), s̃) < 0. By
continuity, there exists a neighborhood of (s̃, b̄0 (s̃)) such that when (s, b) belongs to that
neighborhood, η(s, s) − δϕ (b, s) ≤ η(s, s) − δ(b, s) < 0 for all ϕ satisfying the conditions
above. Similarly, since b̄0 (s̃) − v(s̃, s̃) < 0 for a neighborhood sufficiently close to (s̃, b̄0 (s̃)),
b − v(s, s) < 0 for all (s, b) in that neighborhood. Therefore, for some ǫ > 0 sufficiently small
b−v(s,s)
is bounded. Thus, again when
if (s, b) ∈ [s̃, s̃ + ǫ] × [b̄0 (s̃), b̄0 (s̃) + ǫ], then 0 < η(s,s)−δ
ϕ (b,s)
(s, b) ∈ [s̃, s̃ + ǫ] × [b̄0 (s̃), b̄0 (s̃) + ǫ],
"
1
0 ≤ ξϕ (s, b) ≤ sup
b∈[0,w̄] g(b)
#"
#
h(s)
b − v(s, s)
sup
<K<∞
s∈[s̃,s̃+ǫ] 1 − H(s) η(s, s) − δϕ (b, s)
for some K > 0.
Choose ǫ > 0 sufficiently small such that Lemma C.2 holds. To simplify notation, let
s̃ǫ := s̃ + ǫ.
Define B and F as follows:
• B is the set of all continuous, nondecreasing, functions with domain [s̃, s̃ǫ ] such that
b(s̃) = b̄0 (s̃) and ∀ s, s′ ∈ [s̃, s̃ǫ ] and b(·) ∈ B, |b(s) − b(s′ )| ≤ K|s − s′ |.
• F is the set of all continuous, nondecreasing functions with domain [s̃, s̃ǫ ] such that
F (s̃) = H(s̃) and ∀ s, s′ ∈ [s̃, s̃ǫ ] and F (·) ∈ F , |F (s) − F (s′ )| ≤ sups∈[0,1] h(s) |s − s′ |.
For any b(·) ∈ B and F (·) ∈ F , let
h
i
∆(b,F ) (φ, s) := (G(φ)(1 − H(s))(η(s, s) − φ) − φF (s)
h
i
− G(b(s))(1 − H(s))(η(s, s) − b(s)) − b(s)F (s) .
Lemma C.3. Fix b(·) ∈ B and F (·) ∈ F .
d∆(b,F ) (φ,s̃) < 0.
1.
dφ
(C.6)
φ=b(s̃)
2. For s sufficiently close to s̃, φ 7→ ∆(b,F ) (φ, s) attains one local minimum and one local
maximum on the interval (b(s), w̄).
49
ψ̄(s̃)
φ
b
φ
b
∆(φ, s̃)
b(s̃) ψ(s̃)
b(s̃)
µ(s̃) ψ̄(s̃)
(a) Case where µ(s̃) < ψ̄(s̃).
ψ(s̃)
µ(s̃)
∆(φ, s̃)
(b) Case where µ(s̃) = ψ̄(s̃).
Figure C.1: The function ∆(φ, s̃) from the proof of Lemma C.3.
d∆(b,F ) (φ,s̃) dφ
φ=µ(s̃)
3.
≥ 0.
Proof. In this proof, let ∆(φ, s) := ∆(b,F ) (φ, s). Then,
d∆(φ, s)
G(φ)
F (s)
.
= g(φ)(1 − H(s)) η(s, s) − φ −
−
dφ
g(φ)
g(φ)(1 − H(s))
(C.7)
d∆(φ,s̃)
dφ
= g(φ)(1 − H(s̃)) (η(s̃, s̃) − δ(φ, s̃)). Since
< 0.
b(s̃) = b̄0 (s̃), Assumption 2 implies that η(s̃, s̃)−δ(b(s̃), s̃) < 0. Thus, d∆(φ,s̃)
dφ 1. F (s̃) = H(s̃) and (C.7) simplifies to
φ=b(s̃)
2. Noting the previous part, by continuity of (C.7), ∆(φ, s) must be decreasing when
φ ≈ b(s). By Assumption 2, φ 7→ η(s̃, s̃)−δ(φ, s̃) crosses zero two times, first from below
and then from above. Thus, ∆(φ, s̃) attains a local minimum and a local maximum
at these crossings, respectively. By continuity, the preceding holds for all s sufficiently
close to s̃.
3. Recall that µ(s) := min{b̄1 (s), ψ̄(s)}. If µ(s̃) = b̄1 (s̃) < ψ̄(s̃), then η(s̃, s̃) − δ(µ(s̃), s̃) >
0. Else, if µ(s̃) = ψ̄(s̃) then η(s̃, s̃) − δ(µ(s̃),
s̃) = 0, which follows from the definition
d∆(φ,s̃) of ψ̄(s̃). Each case implies that dφ ≥ 0. Figure C.1 presents both cases.
φ=µ(s̃)
C.1.2
Definition of φ̃
To define φ̃, we first introduce the continuous function Φ. Intuitively, it is defined as follows.
Given s, b(·), and F (·), it identifies the smallest zero of ∆(b,F ) (·, s) strictly greater than b(s).
50
Φ
b
Φ
φ
φ
b
b
∆(b,F ) (φ, s)
b(s)
b(s)
∆(b,F ) (φ, s)
(a) Case 1: Φ identifies the smallest zero greater (b) Case 2: Φ identifies the local maximum zero
than b(s).
greater than b(s).
Figure C.2: Definition of Φ.
If such a value does not exist, i.e. ∆(b,F ) (φ, s) < 0 for all φ > b(s), then Φ picks out an
approximate zero by identifying the smallest value greater than b(s) at which ∆(b,F ) (·, s) has
a local maximum. Figure C.2 illustrates this definition in each case.
Definition C.1. Define Φ : [s̃, s̃ǫ ] × B × F → [b̄0 (s̃), w̄] as follows:
1. If ∆(b,F ) (φ, s) ≥ 0 for some φ > b(s), then Φ(s, b, F ) equals the minimal value of φ
greater than b(s) such that ∆(b,F ) (φ, s) = 0.
2. If ∆(b,F ) (φ, s) < 0 for all φ > b(s) and ∆(b,F ) (φ, s) attains a local maximum on the
interval (b(s), w̄), then Φ(s, b, F ) is the least value of φ greater than b(s) at which
∆(b,F ) (φ, s) attains a local maximum.
3. If for some s′ ≤ s, neither case 1 nor case 2 applies, then Φ(s, b, F ) = Φ(s′′ , b, F ) where
s′′ < s′ is the largest value such that case 1 and/or case 2 applies for all ŝ ∈ [s̃, s′′ ].
Remark C.1. From the proof of Lemma C.3, if b(·) ∈ B and F (·) ∈ F , then Φ(s̃, b, F ) = µ(s̃)
and Φ(s, b, F ) > b(s) for all s sufficiently close to s̃.
Define the mapping Λ : B×F → B×F as follows. If (b, F ) ∈ B×F , then Λ(b, F ) = (b̌, F̌ )
where
Z s
b̌(s) = b̄0 (s̃) +
ξΦ(y,b,F ) (y, b(y))dy, and
Zs̃ s
F̌ (s) = H(s̃) +
G(Φ(y, b, F ))h(y)dy.
s̃
51
Λ is a continuous self-map defined on a compact, convex space. By Schauder’s theorem,
there exists a fixed point, (b̃, F̃ ) = Λ(b̃, F̃ ). Given such a fixed point (b̃, F̃ ), we define
φ̃(s) := Φ(s, b̃, F̃ ).
Remark C.2. We record the following two observations that follow from substitutions of
already defined terms.
Rs
1. F̃ (s) = H(s̃) + s̃ G(φ̃(y))h(y)dy = Hφ̃ (s).
Rs
2. b̃(s) = b̄0 (s̃) + s̃ ξφ̃ (y, b̃(y)) dy where
ξφ̃ (s, b̃(s)) = b̃′ (s) =
C.1.3
h(s)
b̃(s) − v(s, s)
G(φ̃(s)) − G(b̃(s))
·
.
·
1 − H(s) η(s, s) − δφ̃ (b̃(s), s)
g(b̃(s))
(C.8)
Properties of b̃ and φ̃
To make notation more compact, henceforth we define
˜
∆(φ,
s) := ∆(b̃,F̃ ) (φ, s).
˜
For each s, let ψ̄φ̃ (s) be the least local maximum of φ 7→ ∆(φ,
s) greater than b̃(s). Likewise,
define ψ φ̃ (s) as the least local minimum greater than b̃(s). These values exist for s sufficiently
close to s̃ and, without loss of generality, we can assume ǫ is sufficiently small such that they
are defined for all [s̃, s̃ǫ ].
Remark C.3. From the definitions above, we conclude the following:
1. b̃(s) ≤ ψ φ̃ (s) ≤ φ̃(s) ≤ ψ̄φ̃ (s).
2. At s = s̃, ψ φ̃ (s̃) = ψ(s̃) and ψ̄φ̃ (s̃) = ψ̄(s̃).
˜
3. ∆(φ,
s) is continuous, differentiable, and
˜
d∆(φ,
s)
= g(φ)(1 − H(s)) η(s, s) − δφ̃ (φ, s)
dφ
where δφ̃ (φ, s) = φ +
G(φ)
g(φ)
+
(C.9)
F̃ (s)
.
g(φ)(1−H(s))
Above, we have relied on the substitution F̃ (s) = Hφ̃ (s). As ψ̄φ̃ (s) and ψ φ̃ (s) define
a local maximum and minimum, respectively, we infer that η(s, s) − δφ̃ (ψ̄φ̃ (s), s) = 0
and η(s, s) − δφ̃ (ψ φ̃ (s), s) = 0. Furthermore, φ < ψ φ̃ (s) =⇒ η(s, s) − δφ̃ (φ, s) < 0 and
ψ φ̃ (s) < φ < ψ̄φ̃ (s) =⇒ η(s, s) − δφ̃ (φ, s) > 0.
52
After making the appropriate substitutions in (C.2) and collecting terms, we see that
Ũφ̃ (b̂|s, w) =
Z
s̃
v(s, y)h(y)dy +
Z
s
v(s, y)G(φ̃(y))h(y)dy
+ G(b̂)(1 − H(s))η(s, s) − b̂ Hφ̃ (s) + G(b̂)(1 − H(s)) .
0
s̃
˜ φ̃(s), s) = U (φ̃(s)|s, w) − U (b̃(s)|s, w).
Since Hφ̃ (s) = F̃ (s), we can write ∆(
φ̃
φ̃
˜ φ̃(s), s) = 0.
Lemma C.4. For all s ∈ [s̃, s̃ǫ ], ∆(
˜ φ̃(s̃), s̃) = 0. Suppose there exists ŝ ≥ s̃ such that ∆(
˜ φ̃(ŝ), ŝ) = 0
Proof. Note that ∆(
˜ φ̃(ŝ + ǫ′ ), ŝ + ǫ′ ) < 0. (By definition, ∆(
˜ φ̃(s), s) cannot be strictly
and for all ǫ′ > 0, ∆(
positive.) Hence, for ǫ′ > 0 arbitrarily small φ̃(s) = ψ̄φ̃ (s) for all ŝ < s < ŝ + ǫ′ . (This
is because Φ equals a local maximum per definition, C.1.) Differentiating with respect to
s,
˜ φ̃(s),s)
d∆(
ds
=
dŨφ̃ (φ̃(s)|s,w)
ds
−
dŨφ̃ (b̃(s)|s,w)
.
ds
After substituting b̃′ (s) with (C.8), φ̃(s) = ψ̄φ̃ (s),
η(s, s) − δφ̃ (φ̃(s), s) = η(s, s) − δφ̃ (ψ̄φ̃ (s), s) = 0, and canceling terms,
˜ φ̃(s), s)
d∆(
= G(φ̃(s))
ds
Z
1
s
∂v(s, y)
h(y)dy − G(b̃(s))
∂s
Z
1
s
∂v(s, y)
h(y)dy.
∂s
˜
˜ φ̃(ŝ), ŝ) = 0, this implies
> 0 for ŝ < s < ŝ + ǫ′ . Since ∆(
Since φ̃(s) > b̃(s), d∆(φ̃(s),s)
ds
˜ φ̃(s), s) > 0—a contradiction.
∆(
Lemma C.5. For all s ∈ [s̃, s̃ǫ ], η(s, s) − δφ̃ (φ̃(s), s) ≥ 0.
˜
Proof. The function φ 7→ ∆(φ,
s) is nondecreasing at φ = φ̃(s). From (C.9), this implies
η(s, s) − δφ̃ (φ̃(s), s) ≥ 0.
Lemma C.6. φ̃(s) is decreasing.
˜ φ̃(ŝ), ŝ) = 0. Holding φ̃(ŝ) fixed,
Proof. Let (ŝ, ŝ′ ) ⊂ [s̃, s̃ǫ ]. We know that ∆(
˜ φ̃(ŝ), s)
∂ ∆(
= G(φ̃(ŝ))
∂s
Z
s
1
∂v(s, y)
h(y)dy − G(b̃(s))
∂s
Z
s
1
∂v(s, y)
h(y)dy > 0.
∂s
˜ φ̃(ŝ), ŝ′) > 0, and by definition of φ̃(s), it must be that φ̃(ŝ′ ) < φ̃(ŝ).
This implies that ∆(
C.1.4
Extension of the Domain to [s̃, s̃′ ]
Above, we defined b̃, F̃ and φ̃ on the interval [s̃, s̃ǫ ]. If b̃(s̃ǫ ) < ψ(s̃ǫ ) then η(s̃ǫ , s̃ǫ ) −
δφ̃(s̃ǫ ) (b̃(s̃ǫ ), s̃ǫ ) < 0. Hence, ξφ̃ (s̃ǫ , b̃(s̃ǫ )) < ∞. Thus, the preceding argument can be repeated
53
with s̃ǫ replacing s̃, b̃(s̃ǫ ) replacing b̄0 (s̃), and φ̃(s̃ǫ ) replacing µ(s̃) throughout. As a result,
the definition of b̃, F̃ and φ̃ can be extended to the interval [s̃, s̃ǫ + ǫ′ ] for some ǫ′ > 0.
Repeating this reasoning, let [s̃, s̃′ ) be the maximal interval on which b̃, F̃ and φ̃ can be
defined as above. Extend this domain to [s̃, s̃′ ] by taking the appropriate limits of b̃ and F̃
from the left. Both are monotone functions; hence, the limits exist.
Lemma C.7. For all b̃(s̃′ ) ≤ b̂ ≤ φ̃(s̃′ ), Ũφ̃ (b̃(s̃′ )|s̃′ , w) = Ũφ̃ (φ̃(s̃′ )|s̃′, w) = Ũφ̃ (b̂|s̃′, w).
Proof. The first equality follows by continuity. To verify the second equality, there are two
˜ b̂, s̃′ ) > 0 and ∆(
˜ b̂, s) > 0 for
cases. First, if Ũφ̃ (b̂|s̃′ , w) > Ũφ̃ (φ̃(s̃′ )|s̃′ , w) for some b̂, then ∆(
˜ s) ≤ 0 for all b ∈ [b̃(s), φ̃(s)], which is a contradiction.
all s sufficiently close to s̃′ . But, ∆(b,
˜ b̂, s̃′ ) < 0. Then φ 7→ ∆(φ,
˜
If instead Ũ (b̂|s̃′ , w) < Ũ (φ̃(s̃′ )|s̃′ , w) for some b̂, then ∆(
s̃′ )
φ̃
φ̃
has a local minimum on (b̃(s̃′ ), φ̃(s′ )). Hence b̃(s̃′ ) < ψ φ̃ (s̃′ ). But this implies that b̃, F̃ , and
φ̃ can be defined on a larger domain, which is also a contradiction.
C.2
Equilibrium Verification
To verify that β is a symmetric equilibrium strategy, we show that no bidder has a profitable
deviation to an alternative feasible bid in the range of β, given that the other bidder is
following β. For each bid outside of the range of β, there is a bid in the range of β that
yields a higher payoff. We can classify each bid b̂ in the range of β into the following classes:
1. b̂ < b̄(0).
2. b̂ = b̄(x) for some x ∈ [0, s̃].
3. b̂ = b̃(x) for some x ∈ [s̃, s̃′ ]. (Recall that b̄(s̃) = b̃(s̃).)
4. b̂ ∈ (b̃(s̃′ ), φ̃(s̃′ )).
5. b̂ = φ̃(x) for some x ∈ [s̃, s̃′ ].
6. b̂ = b̄(x) for some x ∈ (s̃, 1]. (Recall that limx→s̃+ b̄(x) = φ̃(s̃) = µ(s̃).)
We organize this argument on a case-by-case basis depending on the bidder’s type, (s, w).
Remark C.4. When a type-(s, w) agent bids b̂ < b̄(0), his expected payoff is
U(b̂|s, w) = G(b̂)
Z
1
0
(v(s, y) − b̂)h(y)dy.
54
(C.10)
The same argument as in the proof of Theorem 1′ establishes that (C.10) is increasing in
b̂ when b̂ ≤ b̄(0). Therefore, the bid β(s, w) = w is optimal for a bidder with a budget
w < b̄(0). And, for a bidder with a budget w ≥ b̄(0), the bid b̄(0) is superior to any b̂ < b̄(0).
Below we do not consider deviations into this range again and we assume that w ≥ b̄(0).
C.2.1
s ≤ s̃ and β(s, w) < w
Consider a bidder of type (s, w) where s ≤ s̃ and β(s, w) < w. In this case β(s, w) = b̄(s).
1. If this bidder places a bid b̂ = b̄(x), x ∈ [0, s̃], his expected payoff is
U(b̄(x)|s, w) =
Z
x
0
(v(s, y) − b̄(x))h(y)dy + G(b̄(x))
Z
x
1
(v(s, y) − b̄(x))h(y)dy. (C.11)
λ(s)(b̄(s)−v(s,s))
, the same argument as in the proof of Theorem 1′ conAs b̄′ (s) = γ(b̄(s))(η(s,s)−δ(
b̄(s),s))
firms that U(b̄(s)|s, w) ≥ U(b̄(x)|s, w) for all x ∈ [0, s̃].
2. If this bidder places a bid of b̂ = b̃(x), x ∈ (s̃, s̃′ ], his expected payoff is
U(b̃(x)|s, w) =
Z
Z
s̃
x
(v(s, y) − b̃(x))h(y)dy +
(v(s, y) − b̃(x))G(φ̃(y))h(y)dy
0
s̃
Z 1
+ G(b̃(x))
(v(s, y) − b̃(x))h(y)dy.
(C.12)
x
Differentiating with respect to x and collecting terms gives
h
dU(b̃(x)|s, w)
′
= b̃ (x) H(s̃) +
dx
Z
x
G(φ̃(y))h(y)dy + G(b̃(x))(1 − H(x))
i
− g(b̃(x))(1 − H(x))(η(x, s) − b̃(x))
+ G(φ̃(x)) − G(b̃(x)) v(s, x) − b̃(x) h(x)
= G(φ̃(x)) − G(b̃(x)) h(x)
s̃
η(x, s) − δ (b̃(x), x)
φ̃
v(s, x) − b̃(x) − v(x, x) − b̃(x)
×
η(x, x) − δφ̃ (b̃(x), x)
≤ G(φ̃(x)) − G(b̃(x)) h(x) (v(s, x) − v(x, x))
!
≤ 0.
The first inequality is because η(x, s) − δφ̃ (b̃(x), x) ≤ η(x, x) − δφ̃ (b̃(x), x) ≤ 0. The
55
second inequality is because v(s, x) − v(x, x) < 0. Thus, U(b̃(x)|s, w) ≤ U(b̃(s̃)|s, w) =
U(b̄(s̃)|s, w) ≤ U(b̄(s)|s, w).
3. If this bidder places a bid of b̂ ∈ (b̃(s̃′ ), φ̃(s̃′ )), his expected payoff is
U(b̂|s, w) =
Z
Z
s̃
s̃′
(v(s, y) − b̂)h(y)dy +
(v(s, y) − b̂)G(φ̃(y))h(y)dy
s̃
Z 1
+ G(b̂)
(v(s, y) − b̂)h(y)dy.
0
(C.13)
s̃′
Differentiating with the respect to b̂ shows that
dU(b̂|s, w)
db̂
Z
s̃′
G(φ̃(y))h(y)dy
= −H(s̃) −
s̃
Z 1
+ g(b̂)
(v(s, y) − b̂)h(y)dy + G(b̂)(1 − H(s̃′ ))
s̃
Z s̃′
G(φ̃(y))h(y)dy
≤ −H(s̃) −
s̃
Z 1
+ g(b̂)
(v(s̃′ , y) − b̂)h(y)dy + G(b̂)(1 − H(s̃′ ))
s̃
=
′
dU(b̂|s̃ , w)
db̂
= 0.
The final equality follows from Lemma C.7. Therefore, U(b̂|s, w) ≤ U(b̃(s̃′ )|s, w). And
by the preceding case, U(b̃(s̃′ )|s, w) ≤ U(b̄(s)|s, w); hence, U(b̂|s, w) ≤ U(b̄(s)|s, w).
4. If this bidder places a bid of b̂ = φ̃(x), x ∈ [s̃, s̃′], his expected payoff is
U(φ̃(x)|s, w) =
Z
Z
s̃
x
(v(s, y) − φ̃(x))h(y)dy +
(v(s, y) − φ̃(x))G(φ̃(y))h(y)dy
0
s̃
Z 1
+ G(φ̃(x))
(v(s, y) − φ̃(x))h(y)dy.
(C.14)
x
56
Recalling that U(b̃(x)|s, w) is given by (C.11), we see that
Z
U(φ̃(x)|s, w) − U(b̃(x)|s, w) = −(φ̃(x) − b̃(x)) H(s̃) +
x
G(φ̃(y))h(y)dy
s̃
− φ̃(x)G(φ̃(x))(1 − H(x)) + b̃(x)G(b̃(x))(1 − H(x))
Z 1
+ G(φ̃(x)) − G(b̃(x))
v(s, y)h(y)dy
x
≤ U(φ̃(x)|x, w) − U(b̃(x)|x, w) = 0.
(C.15)
The inequality is because v(s, y) ≤ v(x, y). The final equality follows from the defi-
nition of φ̃(x). For a type s ∈ [s̃, s̃′ ] agent, U(φ̃(s)|s, w, ) = U(b̃(s)|s, w). Therefore,
U(φ̃(x)|s, w) ≤ U(b̃(x)|s, w) ≤ U(b̄(s)|s, w).
5. If this agent bids b̂ = b̄(x), x > s̃′ , his expected payoff is given by (C.11). The same
≤ 0. Recalling
argument as in the proof of Theorem 1′ establishes that dU (b̄(x)|s,w)
dx
that limx→s̃+ U(b̄(x)|s, w) = U(φ̃(s̃)|s, w), we see that U(b̄(x)|s, w) ≤ U(φ̃(s̃)|s, w) ≤
U(b̃(s̃)|s, w) ≤ U(b̄(s)|s, w).
The preceding five cases exhaust all possibilities. In each case, the bid b̄(s) = β(s, w) yields
a higher expected payoff than the proposed alternative.
C.2.2
s ≤ s̃ and β(s, w) = w
Consider a bidder with a value signal s ≤ s̃ for whom β(s, w) = w. There exists ŝ ≤ s̃
such that β(s, w) = w = b̄(ŝ). By the same reasoning as in the proof of Theorem 1′ ,
U(b̄(x)|s, w) ≤ U(b̄(ŝ)|s, w) for all x ≤ ŝ. Deviations to bids greater than b̄(ŝ) are not
feasible.
C.2.3
s̃ < s ≤ s̃′ and b̃(s) < w < φ̃(s)
Consider a bidder of type s ∈ (s̃, s̃′ ] for whom β(s, w) = b̃(s) < w < φ̃(s).
1. If this agent bids b̂ = b̃(x), x ∈ [s̃, s̃′ ], his expected payoff is given by (C.12). Hence,
dU(b̃(x)|s, w) = G(φ̃(x)) − G(b̃(x)) h(x)
dx
!
η(x, s) − δ (b̃(x), x)
φ̃
.
×
v(s, x) − b̃(x) − v(x, x) − b̃(x)
η(x, x) − δφ̃ (b̃(x), x)
57
If x > s, then η(x, s) − δφ̃ (b̃(x), x) ≤ η(x, x) − δφ̃ (b̃(x), x) ≤ 0 and v(s, x) − v(x, x) ≤ 0.
Hence,
dU (b̃(x)|s,w)
dx
≤ 0. Thus, U(b̃(s)|s, w) ≥ U(b̃(x)|s, w).
If instead x < s, then
η(x, s) − δφ̃ (b̃(x), x)
dU(b̃(x)|s, w) ≥ G(φ̃(x)) − G(b̃(x)) h(x) v(x, x) − b̃(x)
1−
dx
η(x, x) − δφ̃ (b̃(x), x)
!
η(x, x) − η(x, s)
= G(φ̃(x)) − G(b̃(x)) h(x) v(x, x) − b̃(x)
η(x, x) − δφ̃ (b̃(x), x)
!
≥ 0.
The above expression is non-negative since the numerator and the denominator in the
final term are negative. Hence, U(b̃(x)|s, w) ≤ U(b̃(s)|s, w).
2. If this agent bids b̂ = b̄(x), x ≤ s̃, then his expected payoff is given by (C.11). The
same reasoning as in the proof of Theorem 1′ lets us conclude that U(b̄(x)|s, w) ≤
U(b̄(s̃)|s, w) and, thus, U(b̄(x)|s, w) ≤ U(b̃(s)|s, w).
3. If this agent bids b̂ ∈ (b̃(s̃′ ), φ̃(s̃′ )), then his expected payoff is given by (C.13). As
s ≤ s̃′ , the same reasoning as in case C.2.1(3) above shows that this bid cannot lead
to a greater payoff.
4. If this agent bids b̂ = φ̃(x), x ∈ (s, s̃′ ], then his expected payoff is given by (C.14).
(C.15) implies that U(φ̃(x)|s, w) ≤ U(b̃(x)|s, w) and, from above, we know that U(b̃(x)|s, w) ≤
U(b̃(s)|s, w).
The preceding cases exhaust all feasible deviations for a bidder of this type. In each case,
the bid β(s, w) = b̃(s) resulted in a greater expected payoff.
C.2.4
s > s̃ and β(s, w) = w ≤ b̃(s̃′ )
Consider a bidder with value signal s > s̃ for whom β(s, w) = w ≤ b̃(s̃′ ). In this case,
there exists ŝ < s such that b̃(ŝ) = β(s, w) = w. As in the preceding case, U(b̄(x)|s, w) ≤
U(b̃(s̃)|s, w) ≤ U(b̃(ŝ)|s, w) for all x ≤ s̃. Similarly, U(b̃(x)|s, w) ≤ U(b̃(ŝ)|s, w) for all
x ∈ [s̃, ŝ]. Therefore, all feasible alternative bids yield a lower payoff.
58
C.2.5
s̃′ ≤ s and β(s, w) = w ∈ (b̃(s̃′ ), φ̃(s̃′ ))
1. If this agent bids b̂ ∈ [b̃(s̃′ ), w], his expected payoff is given by (C.13). Hence,
dU(b̂|s, w)
db̂
Z
s̃′
= −H(s̃) −
G(φ̃(y))h(y)dy
s̃
Z 1
+ g(b̂)
(v(s, y) − b̂)h(y)dy + G(b̂)(1 − H(s̃′ ))
s̃
Z s̃′
≥ −H(s̃) −
G(φ̃(y))h(y)dy
s̃
Z 1
+ g(b̂)
(v(s̃′ , y) − b̂)h(y)dy + G(b̂)(1 − H(s̃′ ))
s̃
=
′
dU(b̂|s̃ , w)
db̂
= 0.
Therefore, all bids b̂ ∈ [b̃(s̃′ ), w] are inferior to a bid of w. Bids above w are not feasible.
2. The same arguments as in cases C.2.3(1) and C.2.3(2) above show that a deviation to
a bid of b̂ ≤ b̃(s̃′ ) is not profitable.
C.2.6
s̃ ≤ s and φ̃(s) ≤ w ≤ φ̃(s̃)
In this case, β(s, w) = w and there exists some ŝ ∈ [s̃, s] such that β(s, w) = φ̃(ŝ).
1. If this bidder places the bid b̂ = φ̃(x), x ∈ [ŝ, s], his expected payoff is given by (C.14).
Differentiating with respect to x and collecting terms,
dU(φ̃(x)|s, w)
= φ̃′ (x)g(φ̃(x))(1 − H(x))(η(x, s) − δφ̃ (φ̃(x), x)).
dx
Since s ≥ x, η(x, s) − δφ̃ (φ̃(x), x) ≥ η(x, x) − δφ̃ (φ̃(x), x) ≥ 0. The second inequality
follows from Lemma C.5. Since φ̃(x) is decreasing, U(φ̃(x)|s, w) is decreasing in x.
Hence, U(φ̃(ŝ)|s, w) ≥ U(φ̃(x)|s, w) for all x ∈ [ŝ, s].
2. If this agent bids b̂ = b̃(x), x ∈ [s̃, s̃′ ], his expected payoff is given by (C.12). Hence,
U(φ̃(s)|s, w) = U(b̃(s)|s, w) ≥ U(b̃(x)|s, w) for all x ∈ [s̃, s̃′ ]. The equality follows from
the definition of φ̃(s) while the inequality follows from case C.2.3 above. Together with
the previous case, this implies U(b̃(x)|s, w) ≤ U(φ̃(ŝ)|s, w) for all x ∈ [s̃, s̃′ ].
59
3. If this bidder places the bid b̂ = b̄(x), x ≤ s̃ ≤ s, his expected payoff is given by (C.11).
From case C.2.3 above, we know that U(b̄(x)|s, w) ≤ U(b̃(s̃)|s, w). Thus, noting the
preceding case, U(b̄(x)|s, w) ≤ U(φ̃(ŝ)|s, w) for all x ≤ s̃.
4. If this bidder places the bid b̂ = φ̃(x), x ∈ [s, s̃′ ], his expected payoff is given by (C.14).
From (C.15), we know that U(φ̃(x)|s, w) ≤ U(b̃(x)|s, w) when x ≥ s. From above,
however, we know that U(b̃(x)|s, w) ≤ U(φ̃(ŝ)|s, w).
5. If this bidder places the bid b̂ ∈ (b̃(s̃′ ), φ̃(s̃′ )), his expected payoff is given by (C.13).
As s ≤ s̃′ , the reasoning from case C.2.1(3) shows that U(b̃(s)|s, w) ≥ U(b̂|s, w).
Combined with the preceding analysis, U(b̂|s, w) ≤ U(φ̃(ŝ)|s, w).
C.2.7
s̃′ ≤ s and φ̃(s̃′ ) ≤ w ≤ φ̃(s̃)
The same argument as case C.2.6 applies, except when considering the bid b̂ ∈ (b̃(s̃′ ), φ̃(s̃′ )).
In that case, the argument from C.2.5(1) applies.
C.2.8
s̃ < s and φ̃(s̃) < β(s, w) < w
Consider a bidder of type s̃ < s for whom φ̃(s̃) ≤ β(s, w) < w.
1. If this agent bids b̂ = b̄(x), x ∈ (s̃, 1], his expected payoff is given by expression
(C.11). The same argument as in the proof of Theorem 1′ confirms that U(b̄(s)|s, w) ≥
U(b̄(x)|s, w) for all x ∈ (s̃, 1]. By continuity, U(b̄(s)|s, w) ≥ U(φ̃(s̃)|s, w) as well.
2. The same argument as in case C.2.6(1) shows that for all x ∈ [s̃, s], U(φ̃(s̃)|s, w) ≥
U(φ̃(x)|s, w). Hence, U(b̄(s)|s, w) ≥ U(φ̃(x)|s, w) for all x ∈ [s̃, s]. More generally, a
repetition of the arguments from cases C.2.6 and C.2.7 show that U(b̄(s)|s, w) exceeds
U(b̂|s, w) for all b̂ ≤ φ̃(s).
C.2.9
s̃ < s and φ̃(s̃) < β(s, w) = w
Consider a bidder of type s̃ < s for whom φ̃(s̃) ≤ β(s, w) = w. In this case, there exists
some ŝ ∈ (s̃, s) such that β(s, w) = b̄(ŝ). As in the preceding case, for all x ∈ (s̃, ŝ],
U(b̄(x)|s, w) ≤ U(b̄(ŝ)|s, w). A repetition of the above arguments further shows that this
bidder also does not have a profitable deviation to any bid b̂ ≤ φ̃(s̃).
60
The preceding cases have shown that no type of bidder (s, w) has a profitable, feasible
deviation from β(s, w) given that the other bidder also adopts this strategy. Thus, β(s, w)
defines a symmetric equilibrium.
D
Comparison of First- and Second-Price Auctions
Lemma D.1. Let βI (s, w) and βII (s, w) be symmetric equilibrium strategies for the firstprice and second-price auctions, respectively. βI (s, w) ≤ βII (s, w).
Proof. Let βk (s, w) = min{b̄k (s), w}. It is sufficient to show that b̄I (s) ≤ b̄II (s). First,
observe that for all s, b̄II (s) ≥ v(s, s). Thus, suppose for some š, b̄I (š) > b̄II (š). If b̄I (š) >
v(š, š), then there exists a critical point at some s∗ > š. At s∗ , b̄II (s∗ ) ≥ v(s∗ , s∗ ) = b̄I (s∗ ).
Noting that δ(b, s) > b, if b̄II (s) = b̄I (s), then b̄′II (ŝ) < b̄′I (ŝ). Therefore, b̄II (·) always crosses
b̄I (·) from above, which implies a contradiction.
D.1
Change in the Budget Distribution
Here we detail the comparative static conclusion regarding financial constraints. Consider
Example 3 but with three alternative budget distributions on [0, 2]: G0 (w) = w/2, G1 (w) =
p
w/2, and G1′ (w) = w(4 − w)/4. Each case satisfies Assumption 1. G0 likelihood ratio
dominates G1 and G1′ . Each Gk implies a different “ ψ̄(s)” function, which we label ψ̄k (s):
3s2 + 2s − 1
4s − 4
√
3
9s − 7s2 + 3s + 3 − 4 9s5 − 11s4 + 3s3 + 3s2
ψ̄1 (s) =
18 (s2 − 2s + 1)
√
9 − 6s − 3s2 − 9s4 − 36s3 + 48s2 − 84s + 57
ψ̄1′ (s) =
6 − 6s
ψ̄0 (s) =
Table D.1 identifies the only critical point (s∗k , b∗k ) in each case. These are strictly ordered:
s∗1 < s∗0 < s∗1′ . By continuity, in a neighborhood of s∗0 , unconstrained bidders bid less
when budgets are distributed according to G1 versus G0 . They bid more when budgets are
distributed according to G1′ .
61
Table D.1: Critical Points for Different Budget Distributions
Distribution
G1
G0
G1′
E
s∗k
b∗k
0.0864 0.1728
0.1056 0.2111
0.1264 0.2528
Proof of Theorem 3
The proof of Theorem 3 follows a standard argument, building on those developed by Athey
(2001) and Reny and Zamir (2004). We rely on Reny (2011) to confirm the existence of
a monotone equilibrium (in the qualified sense of term) in a finite-action setting. This
argument is similar to his study of a multi-unit auction, as explained in the text.
Define ϕi (bi , b−i ) as the probability that bidder i wins the auction with bid bi given that
the profile of others’ bids is b−i . If bi is the unique high bid, ϕi (bi , b−i ) = 1. If bi is strictly
less than the highest bid or bi = ℓ, ϕi (bi , b−i ) = 0. In the absence of ties, when others’
strategies are β−i , bidder i’s expected payoff at the bid bi given θi = (si , wi ) is
Ui (bi , β−i |θi ) = ui (wi ) +
Z
Θ−i
ϕi (bi , β−i (θ−i ))ui (s, wi , bi )f (θ−i |θi )dθ−i .
Bidder i’s (ex-ante) payoff when following the strategy βi is Ui (βi , β−i ) = E[Ui (βi (θi ), β−i |θi )].
∗
∗
The strategy profile β ∗ = (βi∗ , β−i
) is a Nash equilibrium if for every i, Ui (βi∗ , β−i
) ≥
∗
Ui (βi , β−i
) for all admissible strategies βi . When convenient, we use capital letters—Si ,
S−i , Wi , W−i , etc.—to refer to signals or types as random variables.
We start with two preliminary lemmas. The first shows that ui (si , s−i , wi , bi ) is ≥θi monotone in (si , wi ). The second verifies a single crossing condition.
Lemma E.1. Let ri ≤ bi ≤ wi′ . If (si , wi ) ≥θi (s′i , wi′ ), then ui (si , s−i , wi , bi ) ≥ ui (s′i , s−i , wi′ , bi ).
Proof. Recall that (si , wi ) ≥θi (s′i , wi′ ) ⇐⇒ wi ≥ wi′ and si − s′i ≥ ξi (wi − wi′ ) where
du
ξi =
sup dwii − inf
inf
62
∂ ūi
∂si
∂ ūi
∂wi
.
Thus, we can establish the following inequalities:
ūi (si , s−i , wi , bi ) −
ūi (s′i , s−i , wi′ , bi )
∂ ūi
∂ ūi
′
(si − si ) + inf
(wi − wi′ )
≥ inf
∂si
∂wi
∂ ūi
∂ ūi
′
≥ inf
ξi (wi − wi ) + inf
(wi − wi′ )
∂si
∂wi
du
≥ sup i (wi − wi′ )
dwi
≥ ui (wi ) − ui (wi′ ).
Rearranging the final inequality gives the desired result.
Lemma E.2. Let β−i be a profile of nondecreasing strategies for bidders j 6= i. Let θi ≥θi θi′
and suppose the bids bi > b′i tie with probability zero with the bids of the other agents.
1. Ui (bi , β−i |θi′ ) − Ui (b′i , β−i |θi′ ) ≥ 0 =⇒ Ui (bi , β−i |θi ) − Ui (b′i , β−i |θi ) ≥ 0.
2. Ui (bi , β−i |θi ) − Ui (b′i , β−i |θi ) ≤ 0 =⇒ Ui (bi , β−i |θi′ ) − Ui (b′i , β−i |θi′ ) ≤ 0.
Proof. We prove part 1. Part 2 follows similarly. It is sufficient to show that Ui (bi , β−i |θi′ ) −
Ui (b′i , β−i |θi′ ) ≤ Ui (bi , β−i |θi ) − Ui (b′i , β−i |θi ). First, define the following subsets of Θ−i :
A′ = {(s−i , w−i ) | max βj (sj , wj ) ≤ b′i }
j6=i
A = {(s−i , w−i ) | max βj (sj , wj ) ≤ bi }
j6=i
à = A \ A′
Ã+ = {(s−i , w−i ) ∈ Ã | ui (si , s−i , wi , bi ) ≥ 0}
Ã− = {(s−i , w−i ) ∈ Ã | ui (si , s−i , wi , bi ) < 0}.
63
Then,
0 ≤ Ui (bi , β−i |θi′ ) − Ui (b′i , β−i |θi′ )
Z
=
ui (s′i , s−i , wi′ , bi ) − ui (s′i , s−i , wi′ , b′i ) h(s−i |s′i )g(w−i )dw−i ds−i
′
{z
}
|A
(1)
+
Z
ui (s′i , s−i , wi′ , bi )h(s−i |s′i )g(w−i )dw−i ds−i
−
| Ã
{z
}
(2)
+
Z
|
Ã+
ui (s′i , s−i , wi′ , bi )h(s−i |s′i )g(w−i )dw−i ds−i .
{z
}
(3)
ui (si , s−i , wi , bi )−ui (si , s−i , wi , b′i ) is nondecreasing in s−i and negative for all (s−i , w−i ) ∈ A′ .
Since βj is nondecreasing in sj , (s−i , w−i ) ∈ A′ =⇒ (s′−i , w−i ) ∈ A′ ∀ s′−i ≤ s−i . Thus, if
1A′ (s−i , w−i ) is the indicator function for the set A′ , the function
χ(s−i , w−i |θi ) := ui (si , s−i , wi , bi ) − ui (si , s−i , wi , b′i ) 1A′ (s−i , w−i )
is nondecreasing in s−i when w−i is fixed. Hence,
Z
ui (s′i , s−i , wi′ , bi ) − ui (s′i , s−i , wi′ , b′i ) h(s−i |s′i )g(w−i )dw−i ds−i
A′
i
h
′
′
= E χ(S−i , W−i |θi ) | θi
h
i
= E E [χ(S−i , W−i |θi′ )|W−i = w−i ] | θi′
h
i
≤ E E [χ(S−i , W−i |θi′ )|W−i = w−i ] | θi
h
i
≤ E E [χ(S−i , W−i |θi )|W−i = w−i ] | θi
Z
=
ui (si , s−i , wi , bi ) − ui (si , s−i , wi , b′i ) h(s−i |si )g(w−i )dw−i ds−i
A′
64
Next, consider term (2). First,
Z
Ã−
ui (s′i , s−i , wi′ , bi )h(s−i |s′i )g(w−i )dw−i ds−i
Z
h(s−i |s′i )
H(s−i |s′i )g(w−i )dw−i ds−i
′
H(s
|s
)
−
−i i
ZÃ
h(s−i |si )
H(s−i |s′i )g(w−i )dw−i ds−i
≤
ui (si , s−i , wi , bi )
H(s
|s
)
−
−i i
ZÃ
≤
ui (si , s−i , wi , bi )h(s−i |si )g(w−i )dw−i ds−i .
=
ui (s′i , s−i , wi′ , bi )
Ã−
The second inequality follows from u(si , s−i , wi , bi ) ≤ 0 and
Similar reasoning applies to term (3):
Z
Ã+
=
Z
Z
H(s−i |s′i )
H(s−i |si )
≥ 1.
ui (s′i , s−i , wi′ , bi )h(s−i |s′i )g(w−i )dw−i ds−i
Ã+
max{ui (s′i , s−i , wi′ , bi ), 0}h(s−i |s′i )g(w−i )dw−i ds−i
h(s−i |s′i )
H̄(s−i |s′i )g(w−i )dw−i ds−i
′
H̄(s
|s
)
+
−i
i
ZÃ
h(s−i |si )
≤
max{ui (si , s−i , wi , bi ), 0}
H̄(s−i |s′i )g(w−i )dw−i ds−i
+
H̄(s
|s
)
−i i
ZÃ
≤
max{ui (si , s−i , wi , bi ), 0}h(s−i |si )g(w−i )dw−i ds−i
Ã+
Z
≤
ui (si , s−i , wi , bi )h(s−i |si )g(w−i )dw−i ds−i .
=
max{ui (s′i , s−i , wi′ , bi ), 0}
Ã+
The second inequality follows since
H̄(s−i |s′i )
H̄(s−i |si )
≤ 1. The third is because ui (si , s−i , wi , bi ) ≥ 0.
Together, the preceding cases imply that 0 ≤ Ui (bi , β−i |θi′ ) − Ui (b′i , β−i |θi′ ) ≤ Ui (bi , β−i |θi ) −
Ui (b′i , β−i |θi ).
Next, consider the following finite-action variant of our model.
1. Restrict each agent’s set of admissible bids to a finite set Bi ⊂ {ℓ} ∪ [ri , ∞) where
ri ≥ 0 and ℓ < ri . Assume that ∀i 6= j, Bi ∩ Bj = {ℓ}. Therefore, relevant ties among
bidders are not possible.
2. Modify each bidder’s utility function so that he incurs a penalty if he bids more than
this private budget irrespective of the auction’s outcome.33 Suppose that this penalty
33
Che and Gale (1998) suggest using a small penalty to ensure adherence to the budget constraint. As a
65
is sufficiently large to ensure that all bids exceeding wi are strictly dominated by the
bid ℓ for a type-(si , wi ) bidder.
3. For a small ǫ̃ > 0, restrict each agent with a value signal of si ≤ si + ǫ̃ to bid ℓ. This
technical restriction is used by Athey (2001) and Reny and Zamir (2004) and its role
here is the same. Among other considerations, it ensures that all bids bi > ℓ can win
the auction with strictly positive probability. Thus, for example, all bids above B̄ are
strictly dominated by the bid ℓ and will not be employed in equilibrium.
In this modified game, let ρi (θi |β−i ) be the set of bidder i’s best-response bids when his type
is θi and others follow the strategy profile β−i . Since Bi is finite, ρi (θi |β−i ) is not empty. It
is also bounded, max ρi (θi |β−i ) ≤ max{B̄, wi }.
Lemma E.3. Consider the modified auction game described above. Fix a nondecreasing
strategy profile β−i . ρi (θi |β−i ) is nondecreasing in the strong set order.
Proof. Let θi ≥θi θi′ and let bi ∈ ρi (θi |β−i ) and b′i ∈ ρi (θi′ |β−i ). If bi ≥ b′i for all such bids, we
are done. Otherwise, suppose b′i > bi . Since wi ≥ wi′ ≥ b′i > bi , both bids are feasible when
a bidder has type θi or θi′ . By Lemma E.2,
0 ≤ Ui (b′i , β−i |θi′ ) − Ui (bi , β−i |θi′ ) ≤ Ui (b′i , β−i |θi ) − Ui (bi , β−i |θi ) =⇒ b′i ∈ ρi (θi |β−i ),
and
0 ≤ Ui (bi , β−i |θi ) − Ui (b′i , β−i |θi ) ≤ Ui (bi , β−i |θi′ ) − Ui (b′i , β−i |θi′ ) =⇒ bi ∈ ρi (θi′ |β−i ).
Hence, ρi (θi |β−i ) is nondecreasing.
Remark E.1. 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).
Lemma E.4. There exists an equilibrium in nondecreasing strategies in the modified auction
game described above.
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.
modeling device it ensures that each agent’s strategy set is (formally) independent of his type.
66
Lemma E.5 (Theorem 3(i)). Consider the modified auction, where ǫ̃ = 0. There exists an
equilibrium in nondecreasing strategies.
Proof. Consider a sequence of auctions indexed by k where the measure of agents constrained
to bid ℓ becomes arbitrarily small, i.e. ǫ̃k → 0. Let β k be the corresponding profile of
equilibrium strategies. As each βik is monotone there exists a strategy βi such that βik (θi ) →
βi (θi ) for all i and almost every θi . For each i, βi is monotone in the ≥θi -order. We will
verify that β is an equilibrium when ǫ̃ = 0.
Suppose the contrary and that bidder i has a profitable deviation from βi to β̂i when
β−i is the strategy profile of the other bidders. Thus, Ui (β̂i , β−i ) − Ui (βi , β−i ) = ∆ > 0.
Since βik (θi ) → βi (θi ) point-wise, this convergence is uniform except possibly on a set of
arbitrarily small measure (Egorov’s Theorem). Choose ε > 0 sufficiently small such that
2ε sup ui (s, wi , b) < ∆/3. Let Θε be the set of measure (at most) ε where convergence need
not be uniform. Next, condsider
k
Ui (βik (θi ), β−i
|θi ) − Ui (βi (θi ), β−i |θi )
Z
k
=
ϕi (βik (θi ), β−i
(θ−i ))ui (s, wi , βik (θi ))f (θ−i |θi )dθ−i
Θ−i \Θε−i
−
+
Z
Z
Θ−i \Θε−i
Θε−i
−
<
Z
Z
Θε−i
Θ−i \Θε−i
ϕi (βi (θi ), β−i (θ−i ))ui (s, wi , βi (θi ))f (θ−i |θi )dθ−i
k
ϕi (βik (θi ), β−i
(θ−i ))ui (s, wi , βik (θi ))f (θ−i |θi )dθ−i
ϕi (βi (θi ), β−i (θ−i ))ui (s, wi , βi (θi ))f (θ−i |θi )dθ−i
"
k
ϕi (βik (θi ), β−i
(θ−i ))ui (s, wi , βik (θi ))
−ϕi (βi (θi ), β−i (θ−i ))ui (s, wi , βi (θi ))
#
f (θ−i |θi )dθ−i + ∆/3
On the set where convergence is uniform, Θ−i \ Θε−i , for k sufficiently large β k = β since the
k
range of β k is a finite set. Thus, for k sufficiently large, Ui (βik (θi ), β−i
|θi )−Ui (βi (θi ), β−i |θi ) <
k
0 + ∆/3. By a similar argument, Ui (β̂i (θi ), β−i |θi ) − Ui (β̂i (θi ), β−i |θi ) < ∆/3 for k sufficiently
large. But, this implies that for k sufficiently large,
i i h
h
k
k
k
k
) − Ui (βi , β−i )
) + Ui (βik , β−i
) − Ui (βik , β−i
) + Ui (β̂i , β−i
∆ = Ui (β̂i , β−i ) − Ui (β̂i , β−i
k
k
< ∆/3 + Ui (β̂i , β−i
) − Ui (βik , β−i
) +∆/3 ≤ 2∆/3,
{z
}
|
≤0
67
which is a contradiction. (The middle term is non-positive since βik is a best response to
k
β−i
.)
To prove Theorem 3(ii) we consider a sequence of discretized auctions with increasingly
finer sets of admissible bids. Athey (2001) and Reny and Zamir (2004), among others, also
consider such sequences. Specifically suppose that in auction k, bidder i’s set of feasible bids
is Bik = {ℓ} ∪ [ri , B̄] ∩ Pik where Pik = {−εi + m/2k | m = 1, 2, . . .}. {εi } are small fixed
constants ensuring that Pik ∩ Pjk = ∅ for all i, j, k. Contemporaneously, take ǫ̃k → 0 as well.
Let β k be a corresponding sequence of equilibria.
Lemma E.6 (Theorem 3(ii)). Suppose bidders have private values, Bi = {ℓ} ∪ [ri , ∞),
and ties among high bidders are resolved with a uniform randomization. There exists an
equilibrium in nondecreasing strategies.
Proof. This is a consequence of better-reply security (Reny, 1999); see especially Reny (2011,
Corollary 5.2), whom we follow, and also Jackson and Swinkels (2005). Briefly, for k very
large β k is an ǫ-equilibrium of the first-price auction where all bidders can place bids from
Bi . Given any monotone strategy profile β−i , bidder i can approximate the payoff he would
receive from the (feasible) strategy βi : Θ → Bi . Specifically, suppose bidder i bids
β̃ik (θi )
=

ℓ
max b ∈ Bk | b < min{β (θ ) +
i i
i
1
, wi }
2k
w i < ri +
1
k
w i ≥ ri +
1
k
.
As k → ∞, the (ex-ante) loss to bidder i from using β̃ik versus βi becomes arbitrarily small
and this conclusion holds versus all strategies that the other bidders may follow.
68