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Reserve
Reserve prices in allpay auctions with complete information
Dipartimento di economia politica e metodi quantitativi
We introduce reserve prices in the literature concerning allpay auctions with complete information, and reconsider the case for the socalled Exclusion Principle (namely, the fact that the seller may find it in her best interest to exclude the bidders with the largest willingness to pay for the prize). We show that a version of it extends to our setting. However, we also show that the Exclusion Principle: a) does not apply if the reserve price is large enough; 2) does not extend if the seller regards bidders’ valuations as identically independently distributed according to a monotonic hazard rate. Preliminary results for the case of independent exante asymmetric bidders suggest that the case for it in settings with positive reserve prices is actually tenuous.
Keywords: allpay auctions, reserve price, economic theory of lobbying.
§ Faculty of Economics, University of Pavia, Via San Felice, 5  I27100  Pavia, ITALY; Email: paolo.bertoletti@unipv.it; Tel. +39 0382 986202; Fax +39 0382 304226. I thank Guido Ascari and Lorenzo Rampa for attracting my attention to these topics and for helpful discussions. Useful suggestions were also provided by participants in seminars at the Universities of Turin, Padova, Pavia and Salerno, and in particular by Marco Pagnozzi. Pietro Rigo’s generous firstorder advice on order statistics is gratefully acknowledged. Carolina Castagnetti kindly provided the relevant MATLAB simulations. I am also very grateful to Domenico Menicucci who pointed out a number of mistakes in a previous version: those remaining are mine.
1. Introduction
Auction models are prototypes of competitive settings, and they are used in several branches of the
economic literature. In particular, the socalled (firstprice) allpay auction is used (among others)
by Hillman and Riley (1989), Baye
et alii (1993) and Che and Gale (1998) to model the lobbying
process. This type of auction fits the lobbying game well, since a lobbyist's contribution is not
typically returned if his efforts are unsuccessful,1 and indeed this literature has elaborated a number
of interesting results. In particular, Hillman and Riley (1989) prove that, if there is some asymmetry
among bidders/lobbyists, the politically contestable rent is not totally dissipated even in the case of
a large number of potential contenders. In addition, Baye
et alii (1993) show that a seller/politician
wishing to maximize her revenue may find it in her best interest to exclude certain lobbyists from
the "finalist" short list (the socalled "Exclusion Principle"), particularly those lobbyists valuing the
political prize most (in order to raise incentives to spend for the likely losers). Che and Gale (1998)
show a somehow related result: namely, the imposition of an exogenous cap on individual lobbying
contributions may have the adverse effect of increasing total expenditure (by increasing competition
It has to be stressed that the quoted literature refers to the case of
complete information
(according to standard terminology: see e.g. MasColell
et alii, 1995: section 23, Appendix B): this
means that, at the time of bidding, any detail of the setting is common knowledge to all the bidders,
including others’ evaluations of the prize. In particular, the working of the Exclusion Principle also
requires that bidders’ evaluations are known to the seller (at the time exclusion is decided), a rather
unusual assumption in auction theory (in fact, a general rationale for using an auction mechanism is
exactly the fact that how much bidders value the prize to be allocated among them is their private
knowledge). In addition, it has been noticed by Gale and Stegeman (1994) that the Exclusion
Principle depends on the assumption that the politician must award the prize (i.e., it cannot withhold
it nor use takeitorleaveit offers).2 This assumption can be justified in the lobbying setting if the
politician is unable to refuse credibly to allocate the political rent: e.g., Baye
et alii (1993) refer to
the choice of a city to host the Olympic Games.
In this paper we discuss the case in which the seller can possibly use a different exclusion
tool, namely a reserve price (a common mechanism in auction theory),3 which does not require the
seller to know bidders’ evaluations. After characterizing the equilibrium of the allpay auction with
an exogenously given reservation price, we show that the seller would indeed prefer a strictly
1 This feature is also shared by other economic and social games, such as patent races and sports. 2 Gale and Stegeman (1994) discuss a mechanism in which the seller need not to award the prize to the highest bidder, and prove that it delivers to the seller an higher revenue. 3 In a lobbying game a positive “reserve price” could perhaps be interpreted as the politician’s ability to postpone (possibly
sine die) the final decision.
positive reserve price, which also increases the overall outcome efficiency (it might decrease the
efficacy of the lobbying process through higher rent dissipation). We show that a generalized
version of the Exclusion Principle holds: namely, that a seller lacking the fullfledged bargaining
ability to make a takeitorleaveit offer to the highestevaluation bidder (the intuitive optimal
mechanism for the seller) could still find optimal to exclude some bidders from her short list even
when using a positive reserve price. However, the case for the Exclusion principle becomes weaker
in such a setting, since it cannot apply if the reserve price is high enough.
We then discuss a setting in which, in an allpay auction with complete information among the
bidders, the seller is not fully informed while setting her reserve price and/or considering some
exclusion. Namely, we consider the case the seller ex ante regards bidders’ “adinterim” valuations
as unknown realizations of random variables.4 In such a setting Menicucci (2006) strikingly shows
that, even if the seller regards the bidders’ private valuations as identically and independently
distributed (iid), for some information structures excluding all but two bidders (randomly selected)
increases the seller’s expected revenue (yet another version of the Exclusion Principle). We
characterize the optimal reserve price in such a setting and extend the result in Bertoletti (2008):
namely, while using a positive reserve price, the seller wishes no exclusion if she regards bidders’
valuations as iid distributed according to a monotonic hazard rate (a feature of many common
distributions). Preliminary results for the case of independent but exante asymmetric valuations
seem to suggest that the case for the Exclusion Principle in settings with positive reserve prices is
2. The allpay auction with complete information and a reserve price
Consider the following setting:
n (riskneutral) agents (the “buyers”) bid for a prize (there is no
resale possibility). Bidder
i's (private) valuation of the prize is
vi (
i = 1, …,
n), and we order the
bidders in such a way that
v1 >
v2 >…>
vn1 >
vn > 0.5 The “rules” of the auction can include a
reserve (minimum) price
pr ≥ 0, i.e., a price below which the prize is not assigned. In particular, let
us indicate with
bi the bid of agent
i. In an (firstprice) allpay auction, bidder
i receives the prize if
bi > Max{
bj≠
i} and
bi ≥
pr, and in that case his payoff is
vi 
bi, whereas his payoff is 
bi if he loses
(ties are broken randomly). Assuming
pr = 0, Hillman and Riley (1989), and Baye
et alii (1993) and
(1996) show that in the unique Nash equilibrium agent 1 uses the uniform distribution
F1(
b1) =
b1/
v2
4 Actually, this is the standard assumption in a “complete information” setting: again see MasColell
et alii (1995). 5 The possibility of ties in the valuations is ignored here. This can be justified by assuming that the
vi are exante continuously independently distributed, so that case has
a priori a zero probability (in an allpay auction ties may imply the existence of multiple Nash equilibria which are not necessarily revenue equivalent: see Baye
et alii, 1996 and footnote 9 below).
on the support [0,
v2], while agent 2 uses
F2(
b2) = 1 
v2/
v1 +
b2/
v1 on the same support (note that this
amounts to the fact that agent 2 randomises between
b2 = 0 and the uniform distribution on [0,
v2]
with probabilities respectively 1 
v2/
v1 and
v2/
v1). Agents
j = 3, …,
n bid
bj = 0 with probability 1.
The prize is then given to agent 1 with probability 1 
v2/(2
v1) > ½ and to agent 2 with probability
v2/(2
v1) < ½ (note that in the latter event the result is not expost efficient, and thus it would not be
stable in the case of a resale opportunity). Agent 1 receives a (expected) payoff of
U1(
v1,
v2) =
v1 
v2, while the (expected) payoffs of the other agents are zero; i.e.,
Uj(
v1,
v2) = 0,
j = 2, …,
n. The
expected total payment to the seller is
p(
v1,
v2) =
p1(
v1,
v2) +
p2(
v1,
v2) =
v2/2 + (
v2/
v1)(
v2/2) =
v2 (1 +
v2/
v1)/2 <
v2, where
pi is the expected payment of agent
i =1,2.
The previous results show that the outcome of an allpay auction is not exante efficient, since
for the expected social welfare
W the following inequalities hold
v2 <
W(
v1,
v2) =
v1 
v2 +
p(
v1,
v2) <
v1.6 From the perspective of the economic theory of lobbying, they illustrate the possibility that,
even if the number of potential contenders is large, asymmetries among players might imply that the
political rent is not fully dissipated (see Hillman and Riley, 1989: pp. 1819). In addition, note that
∂
p/ ∂
v1 < 0 and ∂
p/ ∂
v2 > 0 (
p(⋅) can be proved to be convex): indeed, Baye
et alii (1993) show
that a politician (the seller in the auction) wishing to maximize her revenue should be willing to
select the two active lobbyists (the bidders)
i* and
i*+1 in order to maximize
p(
vi,
vi+1). This implies
that she might find it in her best interest to exclude lobbyists from 1 to
i*1 from her “finalists short
list”, if she is allowed to (there is no point in excluding bidders from
i*+2 to
n). This could be
worthwhile for her because while the expected payment from any
i* ≠ 1 in the finalist list is
necessarily less than the payment expected from 1 in the largest auction, the expected payment from
i* + 1 may rise with respect to that of 2 and more than compensate the decrease of the other
This is the Exclusion Principle, which is intuitively based on the idea to raise (overall)
incentives to spend for the active asymmetric participants by putting them on a more equal footing.
More formally, the Exclusion Principle works by raising the equilibrium probability of winning of
the least favourite contender (between the two who are active in equilibrium). From the perspective
of economic theory of lobbying, Baye
et alii (1993: p. 290) argues that the politician (the seller),
under plausible circumstances, has an adverse incentive to preclude the lobbyists who most value
the prize from participating in the lobbying game. The idea of handicapping the favourite is simple,
interesting and it has some counterpart both in the auction literature with
incomplete information7
(if agents’ valuations are not identically and independently distributed: see e.g. Myerson, 1981 and
6 For the sake of simplicity, we assume that the seller’s evaluation of the prize is zero. 7 This means that the valuation of each bidder is private information to himself at the bidding stage: see e.g. MasColell
et alii (1995: section 23) for this terminology.
Klemperer, 2004: pp. 218) and in sport (e.g., in golf competitions).8 However, note that bidder 1's
exclusion decreases the expected social welfare.
Now consider an exogenously given positive reserve price. It turns out that the bidding Nash
equilibrium is unique. In particular, if
v1 >
pr ≥
v2, the prize is efficiently allocated to bidder 1 for
his bid of
r =
pr, while the other bidders bid zero with probability 1 (if
v1 =
pr bidder 1 is indifferent
to receiving the prize and there is a continuum of Nash equilibria in which he bids
b1 = 0 with a
positive probability). Things are more interesting if
pr <
v2. The relevant results are summarized in
Proposition 1.
Consider an (firstprice) allpay auction with complete information (no resale
possibility). Suppose v2 ≥
pr ≥ 0
. Then, in the unique bidding Nash equilibrium: i) F1(
b1) =
b1/
v2
on
the support [
pr,
v2]
; ii) F2(
b2) = 1 
v2/
v1 +
b2/
v1
on the support {0 ∪ [
pr,
v2]}
; iii) Fj(0) = 1,
j = 3, …,
Proof: see Appendix 1. Note that Proposition 1 says that agent 2 randomises between
b2 = 0 and the
uniform distribution on [
pr,
v2] with probabilities of respectively 1 – (
v2 
pr)/
v1 and (
v2 
pr)/
v1, and
that agent 1 randomises between
b2 =
pr and the uniform distribution on [
pr,
v2] with probabilities of
respectively
pr/
v2 and 1 
pr/
v2. The equilibrium cumulative distribution function of agents 1 and 2
FIGURE 1: The equilibrium distribution functions for
pr <
v2
8 Sport event organizers are typically interested in some "competitive balance" among players: a famous example comes from the history of the Giro d'Italia ("Tour of Italy"), the Italian most important cycling stagerace. It is reported that at the beginning of the twentieth century cyclist Alfredo Binda was so much stronger than his possible competitors (he had already won the Giro d’Italia five times) that the organisers paid him not to participate.
Also note that, exactly as in the case without a positive reserve price, agent 1 receives an
(expected) payoff of
U1(
v1,
v2,
pr) =
v1 
v2, while the (expected) payoffs of the other agents are
zero. However, the prize is now won by agent 1 with probability 1 – (
v 2
2 
pr )/(2
v1
v2) > 1 
v2/(2
v1);
i.e., the introduction of a positive reserve price raises the probability of an expost efficient outcome
by raising the probability that the prize is allocated to agent 1. Moreover, the expected total
payment to the seller is given by (
v2 ≥
pr):
p(
v ,
v ,
p ) =
p (
v ,
v ,
p ) +
p (
v ,
v ,
p )
~ (
v1,
v2,
pr) is continuous (and differentiable) for any
v2 ≥
pr ≥ 0).9 Equation (1) shows
that, as it should be expected, the payment by agent 1 increases (on expectation), while the payment
of agent 2 decreases, with respect to the case of a null reserve price. In particular, the increase is
r /2)(
v2 –
v1 ): note that
∂ / ∂
pr > 0 and that
p~ (
v1,
v2,
pr) goes continuously from
p(
v1,
v2) to
v1 as
pr goes from 0 to
v1 (
p
~ = 0 if
v1 <
pr and
p~ =
pr if
v1 >
pr ≥
v2). This is described in
Figure 2. Note, finally, that since
W(
v1,
v2,
pr) =
v1 
v2 +
p
~ (
v1,
v2,
pr), also the expected social
welfare increases with respect to the case of a null reserve price.
FIGURE 2: The expected revenue as a function of
pr
9 For
v1 =
v2 ≥
pr > 0 there is more than a single Nash equilibrium, and equation (1) does not apply to all of them (see footnote 5).
By interpreting the reserve price
level as a measure of the seller’s bargaining power (following
the suggestion by Milgrom, 1987), we can conclude that a fullyinformed seller “strong” enough
would use
pr >
v2, and in fact
pr =
v1 (i.e., she would make a takeitorleaveit offer to agent 1),
without excluding any bidders. In such a case, there will not generally be full “rent dissipation”
(unless in the extreme cases of either
v1 =
v2 or
pr =
v1), independently from the number of
competitors. But, clearly, the Exclusion Principle does not apply, since it will always be better for
the seller to use a large enough reserve price (
pr ≥
v2) rather than to exclude through a “finalists
short list” some of the bidders who value most the prize.
However, the situation is more complex if the seller is
not “strong enough” to set a reserve
price
pr ≥
v2. Since
p
∂ / ∂
v2 > 0, it is still possible that the exclusion of some agents
is in her interest. In particular, she should choose
i,
j (>
i)
and pr in order to maximize
p
under the “bargaining constraints” she faces. Notice that, if the reserve price that the seller can
adopt does not depend on the agents she selects, she will always choose the largest possible reserve
r , and also agent
i+1 when he chooses agent
i: i.e., things are very much as in Baye
et
alii (1993), and a generalized version of the Exclusion Principle holds. But, in such a case, it cannot
be strictly better to exclude agent from 1 to
i  1 if
p +
r ≥
vi+1 (since
p
~ (
vi,
vi+1,
pr) >
pr if
vi+1 >
pr),
which implies that there will be no profitable exclusion at all if
p +
r ≥
v3. In other words, the case for
3. Exclusion for a seller facing incomplete information
The assumption that a fully informed seller can credibly exclude some bidder from her “short
list” while she is unable to ask him a price not higher than his valuation does not seem particularly
palatable as a general bargaining feature. For this reason in this section we refer to the setting
introduced by Menicucci (2006) and Bertoletti (2008), and investigate the case of a seller facing
incomplete information. In such a setting,
p
~ (
v1,
v2,
pr) is the revenue the seller expects “ad interim”
(before bidding takes place but after the definition of a possible “short list” of auction participants),
where from her point of view
v1 and
v2 are respectively the first (highest) and the second (second
highest) order statistics of
n stochastic variables (see e.g. Krishna, 2002: Appendix C). We
generalize the assumptions of Bertoletti (2008) by assuming that
v1 and
v2 are jointly distributed on
the support [
v,
v ]2,
v >
v ≥ 0, according to a continuous density function
g(
v1,
v2) which is strictly
positive for
v >
v1 >
v2 >
v.10
10 Note that, obviously,
g(⋅) depends on the joint distribution of the bidders’ valuations: see Appendix 3 for the case of independent bidders’ valuations.
It follows that the seller should set the optimal reserve price by maximizing with respect to
pr:
+ ∫ ∫
p(
v ,
v ,
p )
g(
v ,
v )
dv dv ,
where
PE(
pr) =
E{
p
~ (
v1,
v2,
pr)} (the adinterim expected value of
p~ ) is a continuous and
differentiable function,
g1(⋅) is the density function of
v1, γ(⋅) =
g1(⋅)/(
G2(⋅) 
G1(⋅)) and
G1(⋅) and
G2(⋅) are the (marginal) distributions functions of respectively
v1 and
v2. Of course,
G2(
pr) 
G1(
pr)
2 <
pr <
v1 . We call γ the
generalized hazard rate, since it is equal to the hazard rate λ(⋅)
=
h(⋅)/(1 
H(⋅)), where
h(⋅) is the density function which corresponds to distribution
H(⋅), if the
bidders’ valuations are iid according to
H. The role played by it in auction theory comes from the
fact that
E{
v1 
v2} =
E{1/γ(
v1)}, as it is easily seen (thus the expected value of the inverse of the
generalized hazard rate measures the bidders’ components of the ex ante expected social welfare).
Note that
G2 ≥
G1 and thus γ ≥ 0, and lim γ(
v) → ∞ for
v →
v . Also note that
PE(
v ) = 0, and
PE(
v)
)
g(
v ,
v )
dv dv ,
where ϕ(⋅) = (⋅) – 1/γ(⋅). We call ϕ the
generalized virtual value, because it satisfies the property
E{ϕ(
v1)} =
E{
v2} which is possessed by the “virtual value” studied by the literature on auction
with incomplete information and iid valuations (see e.g. Krishna, 2002: section 5.2). Note that lim
ϕ(
v) →
v for
v →
v . Since
g1(
v) = 0, (3) implies that
dPE(
v)/
dpr > 0 if
v > 0; moreover,
dPE(
v)/
dpr
= 0 if
v = 0 but in such a case ϕ(
v) < 0. Thus the optimal reserve price
pr* is larger than
v, smaller
than
v , and must satisfy
dPE(
pr*)/
dpr = 0.
Notice that
pr* depends on the set of the shortlisted participants to the auction through
g(⋅).
Thus, in general, the optimal reserve price should be expected to change after a change in the short
list. In particular, intuition suggests that the seller could prefer to exploit the opportunity to set an
higher reserve price rather than exclude the contender who is (expected) to be “more eager” to buy.
Unfortunately, it appears impossible to solve explicitly
dPE(
pr)/
dpr = 0 even in the simplest case in
which the seller regards bidders’ valuations (here denoted by
vj, where
j = 1, … ,
n indicates the
bidders whose valuations are not ex ante ordered) as iid according to a uniform distribution on the
support [0,1], and thus
g(
v1,
v2) = (
n2 –
n)(
v2)
n – 2 (see Appendix 3 and e.g. Krishna, 2002: p. 267).
In spite of this difficulty, some results can be gained even discharging the reserve price
optimal adjustment. Indeed, a solvable case arises if, as in Bertoletti (2008), the seller regards
bidders’ valuations as iid according to a continuous cumulative distribution function
H(⋅) on the
support [
v,
v ]. The following Proposition 2 holds.
Proposition 2.
Consider an allpay auction with complete information among bidders and any
given reserve price. Suppose that the bidders’ valuations are exante iid according to a strictly
increasing distribution H(·
) with a continuous density function and a monotonic increasing hazard
rate. In this case the seller maximizes her expected revenue by getting the largest possible number
Proof: see Appendix 3 (it extends Bertoletti, 2008 to the case of a positive reserve price).
Proposition 2 shows that no Exclusion Principle can apply if the seller regards the bidders’
valuations as iid according to a monotonic hazard rate (a technical condition, equivalent to the “log
concavity” of [1 
H(·)], satisfied by many distributions: see Bagnoli M. and Bergstrom, 2005). This
implies that
E{
v1 
v2} decreases with respect to the number of (exante identical) bidders and this,
in turn, implies that a larger
n cannot harm the seller (for a discussion see Bertoletti, 2008),
whatever is the reserve price adopted. Thus, in this sense, the result of Menicucci (2007) holds
However, we believe that a fair assessment of the Exclusion Principle in this setting should
rather refer to the case of valuations which are not identically distributed. It is intuitive and easy to
see that, if potential bidders’ valuations are independent, associated to a bidders’ group there are
distributions for the first and the second order statistics such that they firstorder stochastically
dominate the corresponding distributions associated to any (strictly smaller) bidder subgroup (see
Appendix 3 and Shaked and Shanthikumar, 1994: section 1.B.4). We speculate that, as in the case
of iid bidders’ valuations, a key question is the effect of enlarging the set of bidders on
E{
v1 
v2},11
but this appears hard to characterize in the general case, since the enlargement also affects the
generalized hazard rate γ. In particular, 1/γ is an average of the different hazard rate 1/λ
j = (1 
Hj)/
hj
whose (variable) weights are the normalized values of the socalled
reverse hazard rates σ
j =
hj/
Hj:
see Appendix 3. Clearly, even monotonicity of the generalized hazard rate would not be enough to
guarantee that
E{
v1 
v2} decreases while the set of (independent but not identical) bidders enlarges.
A simpler case arises if there are only two types of bidders, say
s and
w, with
Hs(⋅) = (
Hw(⋅)) ,
11 Note that, up to the second term of its Taylor expansion with respect to
v1,
p~ ≈
v2 – (
v1 
v2)[1 – (
pr/
v2)2]/2.
θ > 1: following Krishna (2002: section 4.3), we call them the strong (
s) and the weak (
w) bidder
types, since
Hs likelihoodratio stochastically dominates
Hw (see e.g. Krishna, 2002: Appendix B
and Shaked and Shanthikumar, 1994: section 1.C). In such a case it is easy to see that the weights in
the average that defines the generalized hazard rate are constant (with respect to its argument) and
equal respectively to θ
ns/(θ
ns +
nw) and
nw/(θ
ns +
nw) (where
ns and
nw are the numbers of the strong
and the weak bidders, with
n =
ns +
nw), and that the addition of any type of bidder generates a
distribution for
v1 such that
g1(
v1;
n + 1) (with obvious notation) likelihoodratio stochastically
dominates
g1(
v1;
n): see (A.9) in Appendix 3. Moreover, the monotonicity of λ
w then implies the
monotonicity of λ
s and is sufficient to guarantee the monotonicity of γ.
Assuming that λ
w is monotonic, a
sufficient condition for getting a decrease
E{
v1 
v2} by the
addition of one
strong bidder is:
as it can be seen by taking the difference of the expected values of the generalized hazard rate
before and after the addition, and integrating by parts. Since computation shows that
d(1/λ
w)/
dv >
d(1/λ
s)/
dv if λ
w is monotonic, it is clear that (4) is satisfied and accordingly that adding a “strong”
independent bidder to the set of the auction participants does decrease the expected difference of {
v1

v2}. One can prove that such an addition actually decreases γ, and simulations for θ,
ns and
nw
using the uniform distribution on [0,1] for
Hw indeed suggest that it also always (whatever
pr)
increases
PE, as in the iid valuations case. That is, the Exclusion principle appears not to extend to
6. Conclusions.
It is known that, without a reserve price, (firstprice) allpay auctions are inefficient trade
mechanisms under complete information among bidders. In this paper we have characterized their
(mixed strategy) Nash equilibrium under a
positive reserve price. As it is intuitive, a fully informed
seller would like to set a reserve price (if she can credibly do that), since this increases overall
efficiency and is profitable for her. Indeed, the ability to set a positive reserve price (to commit to
refuse to sell) is obviously an important part of the seller’s bargaining power. We have shown that,
even with a positive reserve price, a fullyinformed seller might find it better to exclude a bidder
who is especially eager “to buy” (an extension of the socalled Exclusion Principle). However, we
have also showed that, once the possibility that the reserve price is optimally set is taken into
account, the case for the Exclusion Principle becomes weaker.
We have also argued that an appealing model should assume that the seller is not fully informed
when setting her optimal reserve price and/or considering exclusion, and characterized her optimal
reserve price for an allpay auction with complete information of the bidders. In this setting, we
have extended a result due to Bertoletti (2008) and showed that, if the seller regards the bidders’
valuations as iid according to a monotonic hazard rate, the Exclusion principle cannot apply.
Preliminary results for the case of independent but exante asymmetric valuations (perhaps a more
apt setting) seem to restrict the case for it to situations in which reserve prices cannot be used.
Appendix 1
Proof of Proposition 1. Clearly, for each agent
i the set of (weakly) undominated strategies is given
by {0 ∪ [
pr,
vi)}. Moreover, it can be shown that in equilibrium no bidder plays
bi ∈ (
pr,
vi), and no
more than one agent bid
pr, with a positive probability. This is so because if at least two of them do
the latter, both would have an incentive to move the probability mass slightly higher, so increasing
their payoffs (the conditional probability of winning would jump, and so would the payoff). If
exactly one agent
i has a mass point at some
bi ∈ (
pr,
vi), then no other agent would place density
immediately below that bid (it would be better to move that density above the mass point). But then
agent
i would do strictly better by moving that mass down (see Che and Gale, 1998: p. 645, Lemma
1, and Hillman and Riley, 1989: pp. 2223, Proposition 1, for a formal proof). Thus, all equilibrium
cumulative distribution function
Fi(
bi) must be continuous on (
pr,
vi).
Now note that agent 1 can secure himself a payoff equal to
v1 
v2 > 0 by bidding
b1 =
v2 with
probability 1. It follows that his equilibrium strategy support cannot include
b1 ∈ {[0,
pr) ∪ (
v2,
v1)}.
Suppose that there is an agent
j ≠ 1 who gets in equilibrium a positive expected payoff. Then it must
be the case that he bids
bj >
pr with probability 1 (he cannot neither bid zero nor bid
pr with positive
probability, because otherwise he would get respectively a null and a negative payoff, while he
should be indifferent among all bids that belong to the support of his own equilibrium cumulative
distribution function
Fj(
bj)). And it must also be the case that his infimum bid does coincides with
the infimum bid of agent 1, say
b, because otherwise at least one of them would get a negative
payoff by bidding his own infimum bid. In fact, we have found a contradiction, because even by
bidding
b at least one of them must get a negative payoff (the conditional probability of winning is
zero). Thus no agent other than 1 can get a positive payoff in the equilibrium, or bid
pr with a
In addition, any agent different from 1 bidding more than
pr with a positive probability must
have an “infimum” bid (≥
pr) not smaller than
b (otherwise he would get less than zero from that
bid), and at least one must bid
b (otherwise it would pay to someone to move down some density).
Similarly, at least two agents must share the maximum bid, say
b+, larger than
pr. Let us now
suppose that two agents different from 1, say
j and
h, bid more than
pr with a positive probability. It
v Prob(
j wins b =
b) −
b =
v
for any
b >
pr belonging to the support of both
Fj(⋅) and
Fh(⋅). This implies that, for
all such a
b:
which implies that
Fh(⋅)
strictly firstorder stochastically dominates
Fj(⋅) if
j <
h. Let
k the largest
agent number among those bidding in equilibrium more than
pr with positive probability. This
implies that the maximum bid larger than
pr belongs to the support of both
F1(⋅) and
Fk(⋅). In turn,
but then by bidding
b+ with probability 1 agent 2 would get a positive payoff, unless both
k = 2 and
It follows that in equilibrium only agent 1 and 2 are active, with agent 1 using
F1(
b1) on a
support [
b,
v2], while
F2(
b2) possibly has support {0 ∪ [
b,
v2]}. Since it must be the case that for any
0
v F (
b) −
b ≥
v −
v ,
we can conclude that
b =
pr, that
F2(
b2) = 1 
v2/
v1 +
b2/
v1 has in fact the support {0 ∪ [
pr,
v2]}, and
that agent 1 uses
F1(
b1) =
b1/
v2 on the support [
pr,
v2].
Q.E.D.
Appendix 2
Proof of Proposition 2. Since the density function of the joint distribution of the first and second
order statistics (see Appendix 3 and e.g. Krishna, 2002: p. 267) of
n independent draws from
H is
n −
n)(
H (
v ))
n 2
h(
v )
h(
v )
I
(where
I(⋅)(
·) is the appropriate indicator function),
the density function of
v1 conditional on
v2 is
( −
H (
v ))(
H (
v ))
n−
on the support [
v2,
v ] (note that it does not depend on
n). Clearly,
E{
p
1,
v2,
pr)}}, with obvious notation for the previous expectations. Now consider, for
any given
pr, the function
t(⋅):
∫ ~
p(
v ,
v ,
p )
and note that it is continuous, and differentiable for any
pr ≠
v2. Clearly,
t(⋅) increases with respect
to
v2 if
pr >
v2.
Now consider the case
pr ≤
v2 and compute the derivative
h(
v )
p(
v ,
v ,
p )
h(
v )
p(
v ,
v ,
p )
∫[
v + (
v −
v )(
r − )]
h(
v )
dv −
h(
v )
v }
1 ∫(
v −
v )
h(
v )
dv
v λ(
v )
h(
v )
λ(
v ) 1−
H (
v ) 1
{
p(
v1,
v2)} is an everywhere increasing function of
v2
if the hazard rate is monotonic.
Finally, since
G2(
v2;
n + 1) firstorder stochastically dominates
G2(
v2;
n) for any number
n of
bidders with independent valuations (where
Gi(
vi;
n) is the distribution function of
vi,
i = 1,2, for
n
draws from independent random variables: see Appendix 4), any reduction in
n decreases the
expected revenue of the seller
if the hazard rate of
H(⋅) is monotonic.
Q.E.D.
Appendix 3
Consider the joint distribution of the first and second order statistics of
n independent continuous
random variables
vj,
j = 1, …,
n, whose distributions are indicated with
Hj(⋅) (
hj(⋅) is the
corresponding density function) on the common support [
v,
v ]. Clearly,
G
1(
v1) = ∏
H (
v ) , and
G(
v ,
v ) =
G (
v )[
G (
v ) =
G (
v )[∑
g (
v ) =
G (
v )
g (
v ) =
G (
v )[
g(
v ,
v ) =
G (
v )[
G (
x) −
G (
x) =
G (
x)[∑
Note that
Gi(
vi;
n) firstorder stochastically dominates
Gi(
vi;
m) for any
n >
m,
i = 1,2, and that 1/γ is
an average of the different 1/λ
j = (1 
Hj)/
hj whose weights are the values of the socalled
reverse
hazard rates weights σ
j =
hj/
Hj divided by Σσ
j. Moreover, if the random variables are iid according
to
H and
h, then 1/γ = (1 
H)/
h and
g1(
vi;
n + 1) even likelihoodratio stochastically dominates
g1(
vi;
n): see e.g. Krishna (2002: Appendix B) and Shaked and Shanthikumar (1994: section 1.C).
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