We study asymmetric all-pay auctions where two privately informed agents bid for a prize. We show that capping the bids is profitable for a designer who wants to maximize the sum of bids (revenue). This finding confims the results of Che and Gale (1998) in the context of incomplete information and completes the analysis of Gavious, Moldovanu and Sela (2002) by analyzing the case of ex-ante asymmetric players.All-pay auctions

Nicolas Sahuguet

English

Research Papers in Economics

Caps in asymmetric all−pay auctions with incomplete
information 
Nicolas Sahuguet
HEC Montréal
Abstract
We study asymmetric all−pay auctions where two privately informed agents bid for a prize.
We show that capping the bids is profitable for a designer who wants to maximize the sum of
bids (revenue). This finding confims the results of Che and Gale (1998) in the context of
incomplete information and completes the analysis of Gavious, Moldovanu and Sela (2002)
by analyzing the case of ex−ante asymmetric players.
Citation: Sahuguet, Nicolas, (2006) "Caps in asymmetric all−pay auctions with incomplete information." Economics
Bulletin, Vol. 3, No. 9 pp. 1−8
Submitted: February 23, 2006.  Accepted: April 6, 2006.
URL: http://www.economicsbulletin.com/2006/volume3/EB−06C70004A.pdf1 Introduction
In many competitive situations, we observe constraints imposed on contestants. For
instance, in many professional sports, entry is regulated by imposing minimum re-
quirements. Similarly, in auctions, a reserve price or an entry fee is often imposed.
These practises have been recognized by the literature to be bene￿cial. A seller can
increase his revenue in an auction, a contest designer can raise the average e⁄ort in
a contest. The intuition behind the result is to exclude players with low valuation
to increase competition among active bidders. Another type of constraint that has
been analyzed is to place upper bounds on bids. Salary caps are common practise in
US professional sport leagues. The rationale is to level the ￿eld and give small cities
the ability to compete with larger and richer cities. This leads to a more competitive
league in which the average e⁄ort is larger. The e⁄ect of caps is also debated in the
context of political economy. Lobbying and to a lesser extent political campaigns can
be understood as contests in which the side which spends most wins. The point has
been made that a cap on expenditures can lead to a global increase in expenditures,
which in these contests would just be wasteful rent-seeking. At the center of the
argument lies the potential asymmetry between players. A cap has a positive e⁄ect if
it restores competitive balance. Weak players, believing they have a chance to com-
pete, will make more e⁄orts than if they thought the strong player had no limit in
the resources he could spend to outbid him. The importance of asymmetry between
players is thus crucial for a better understanding of the role of caps.
Che and Gale (1998) in the context of lobbying were the ￿rst to show the paradoxi-
cal e⁄ect that caps can in fact increase average expenditures. They model lobbying as
an all-pay auction with complete information but with asymmetric players. Lobbyist
value di⁄erently the political prize. They derive the equilibrium of such a game and
show that a cap can increase lobbying expenditures. Gavious, Moldovanu and Sela
(2002) analyze the role of caps in all-pay auctions with incomplete information. They
show that the result of Che and Gale does not extend to ex-post asymmetry, when
the cost of bidding is linear. If players are ex-ante symmetric (their valuation are
drawn from the same distribution function), but ex-post asymmetric (since they have
di⁄erent valuations) a cap can not increase total expenditures. They show that when
bidding costs are convex, a cap can increase revenue even with ex-ante symmetric
players.
The present note completes these two papers by analyzing the role of caps in all-
pay auctions when there is incomplete information and ex-ante asymmetries between
bidders. We show that the result of Che and Gale extends to the case of incomplete
information with ex-ante asymmetric bidders. For uniform distribution functions, we
show that there exists an appropriate choice of cap that increases total expenditures.
The most closely related papers are the ones already mentioned by Che and Gale
(1998) and Gavious, Moldovanu and Sela (2002). Another related strand of the
literature looks at the role of budget constraints in auctions. See Che and Gale
1(1996) and Fang and Parreiras (2001).
In a somewhat related work, Chakraborty (2002) considers a simple example of
a common value auction in which setting up a ceiling price in addition to a reserve
price increases the revenue.
2 The model
We consider two agents bidding for an indivisible object. Bidder i0s valuation for
the object, vi is private information to bidder i. The valuations are independently
distributed according to asymmetric distribution functions F1 and F2. We assume
that F1 and F2 are continuously di⁄erentiable. We also assume that the density
functions F
0
1 and F
0
2 are strictly positive on their respective support [0;￿] and [0;￿].
Without loss of generality, we assume that ￿ ￿ ￿.
Each bidder submits a bid bi ￿ bc, where bc is a commonly known bid cap imposed
by the auctioneer. The bidder with the highest bid wins the object. Both bidders
pay their bids. If both bidders submit the same bid, then the winner is randomly
selected (each bidder has an equal chance to win the object).
3 Equilibrium without caps
Player 1￿ s objective is to maximize v1 ￿ F2
￿
b
￿1
2 (x)
￿
￿ x over x 2 R+ while player 2￿ s
objective is to maximize v2 ￿F1
￿
b
￿1
1 (y)
￿
￿y over y 2 R+. First-order conditions are:
F
0
2
￿
b
￿1
2 (x)
￿
￿
￿
b
￿1
2
￿0 (x) =
1
b
￿1
1 (x)
; and F
0
1
￿
b
￿1
1 (y)
￿
￿
￿
b
￿1
1
￿0 (y) =
1
b
￿1
2 (y)
:
To determine the equilibrium bidding functions, we use the mapping1 h(￿) = b
￿1
2 ￿
b1 (￿), mapping player 1￿ s valuation into player 2￿ s valuation making the same bid.
The ￿rst-order conditions can be rewritten as:
￿
b
￿1
2
￿0 (b1 (v)) =
1
v ￿ F 0
2
￿
b
￿1
2 ￿ b1 (v)
￿ =
1
v ￿ F 0
2 (h(v))
;
b
0
1 (v) =
1
￿
b
￿1
1
￿0 (b1 (v))
= h(v) ￿ F
0
1 (v);
1This mapping was used in Amann-Leininger (1996) to characterise equilibrium in asymmetric
all-pay auctions. See also Parreiras and Rubinchik-Pessach (2006) for a more recent treatment of
asymmetric all-pay auctions. In particular, they deal with the issue of existence of equilibria when
the support of types are not identical, which is relevant in our set-up.
2whenever the density is positive. Finally, since h0 (v) =
￿
b
￿1
2
￿0 (b1 (v)) ￿ b0
1 (v); we
obtain the following ordinary di⁄erential equation:
h
0 (v) =
h(v) ￿ F 0
1 (v)
v ￿ F 0
2 (h(v))
:
The bidding functions bi are then obtained from b
0
1(v) = F 0
1 (v)￿h(v) (with the lowest
type making a zero bid) and b2 (w) = b1 (h￿1 (w)): In order to be able to analyze
the e⁄ect of a bid cap on the revenue, we consider some particular form of bidders￿
asymmetry.
Proposition 1 Assume that player 1￿ s valuation is distributed uniformly on [0;￿]
and that player 2￿ s valuation is distributed uniformly on [0;￿]; with ￿ > ￿. Then the
equilibrium bid functions are:
b1(v) =
￿￿
￿ + ￿
￿
￿
v
￿
￿ ￿+￿
￿
; b2 (w) =
￿￿
￿ + ￿
￿
￿w
￿
￿
￿+￿
￿
:
The revenue in the auction is:
R =
￿
￿ + ￿
￿
￿
2
￿ + 2￿
+
￿
￿ + ￿
￿2
2￿ + ￿
:
4 Price Ceilings
Suppose now that the auctioneer imposes a ceiling bc on the players￿bids: Consider an
all-pay auction with a bid cap bc ￿ b￿, with b￿ being the largest bid submitted in the
auction without cap. Then, the bid cap is not e⁄ective and the unique equilibrium is
the same as in the auction without cap. Suppose now that bc < b￿. In equilibrium,
the highest types pool their bids at the cap. Since a lower bid would yield a strictly
lower probability of winning (no chance to tie with the atom of types bidding the
cap), it must be that the next bid is strictly lower. The pattern of bidding can be
seen in ￿gure 1.
Let￿ s call ￿ ￿ (￿ ￿) the smallest valuation of player 1 (2) which bids bc. Let b￿ be the
bid of the largest valuation which does not bid bc. In equilibrium, ￿ ￿ and ￿ ￿ must be
indi⁄erent between bidding bc and bidding b￿. This gives two necessary conditions
for equilibrium:
b
c ￿ b￿ = (1 ￿ F2 (￿ ￿)) ￿
1
2
￿ ￿ ￿ (1)
b
c ￿ b￿ =
￿
1 ￿ F1
￿￿ ￿
￿￿
￿
1
2
￿ ￿ ￿ (2)
Since the local distribution of valuations does not change with the presence of price
ceilings, the O:D:E: de￿ning the h(￿) function remains unchanged. Only the bound-
ary conditions are changed. We now have h
￿￿ ￿
￿
= ￿ ￿. From this modi￿ed mapping,
3b
v
a
b*
b
c
_
a
_
b
Figure 1: Bid functions with a cap.
we can derive the bidding functions under the bid cap ￿ b1 and ￿ b2: This gives us a third
equation that characterizes the equilibrium: ￿ b1(￿ ￿) = b￿.
Proposition 2 Assume that player 1￿ s valuation is distributed uniformly on [0;￿]
and that player 2￿ s valuation is distributed uniformly on [0;￿]; with ￿ > ￿. Then the
equilibrium bid functions are :
￿ b1(v) =
￿ ￿￿ ￿
￿ + ￿
￿
￿
v
￿ ￿
￿ ￿+￿
￿
;￿ b2 (w) =
￿ ￿￿ ￿
￿ + ￿
￿
￿w
￿ ￿
￿ ￿+￿
￿
with ￿ ￿ and ￿ ￿ being de￿ned as follows.
￿ ￿ = ￿; ￿ ￿ =
￿ (￿ + ￿)
(￿ ￿ ￿)
2bc ￿ ￿
￿
:
with
￿ =
￿ + ￿
￿
b
c +
￿
2
￿
￿ + ￿
￿ ￿ ￿
￿
￿
￿ + ￿
￿
s
(bc)
2 ￿
￿￿
￿ + ￿
bc +
￿
￿￿
2(￿ ￿ ￿)
￿2
:
Proof. See Appendix.
We want to show the optimality of a price ceiling. We show that the derivative
of the revenue with respect to bc at bc =
￿￿
￿+￿ is equal to zero and that the second
derivative is positive at that point, which demonstrates that the revenue increasing
when the caps decreases and becomes binding.
4Proposition 3 The revenue in an all-pay auction with a bid cap bc <
￿￿
￿+￿ is
R
c = b
c
￿
￿ ￿ ￿ ￿
￿
+
￿ ￿ ￿ ￿
￿
￿
+
￿ ￿2￿ ￿
(￿ + ￿)(￿ + 2￿)
+
￿ ￿￿ ￿
2
(￿ + ￿)(2￿ + ￿)
There exists an auction with a binding cap bc that generates a revenue Rc strictly
larger than the revenue in the auction without cap.
Proof. See Appendix.
5 Conclusion
We have analyzed the role of caps in a model of an all-pay auction under incomplete
information with two asymmetric players. We have shown in the case of uniform
distributions that the designer of the auction can increase the revenue by imposing an
appropriate cap on the bids. A cap on bids lowers the bids of high valuation types but
increases the bids of types with a low valuation. It also increases the competitiveness
of the contest and leads to more aggressive bidding. This leads to a higher revenue.
This result has interesting implications for asymmetric auctions. Bid caps can be
an e⁄ective way of increasing competition when bidders are asymmetric. Our results
also have implications for auctions with budget constrained buyers when the seller can
provide ￿nancing. It is not obvious that providing ￿nancing to asymmetric bidders
would increase revenue. Possible extensions to other auction mechanisms and to more
general distribution functions are left for future research.
6 Appendix
Proof of proposition 1
When the distributions are uniform, F 0
1 (v) and F 0
2 (h(v)) are constant. The or-
dinary di⁄erential equation becomes
h0(v)
h(v) = 1
v ￿ ￿
￿ with unknown h, which yields
h(v) = K ￿ v
￿
￿: The constant K is easily calculated using the boundary condition
h(￿) = ￿. This yields K = ￿=￿
a=￿. The bidding functions bi are then obtained from
b
0
1(v) = 1
￿ ￿h(v) (with the lowest type making a zero bid) and b2 (w) = b1 (h￿1 (w)).We
get :
b1(v) =
￿￿
￿ + ￿
￿
￿
v
￿
￿ ￿+￿
￿
; b2 (w) =
￿￿
￿ + ￿
￿
￿w
￿
￿ ￿+￿
￿
:
The revenue generated
R =
Z ￿
0
b1(v) ￿
1
￿
dv +
Z ￿
0
b2(v) ￿
1
￿
dv =
￿￿
￿ + ￿
￿
￿
￿ + 2￿
+
￿
2￿ + ￿
￿
:
5Proof of proposition 2
The boundary condition is now h
￿￿ ￿
￿
= ￿ ￿. This yields:
K ￿ ￿ ￿
￿
￿ = ￿ ￿; h(v) = ￿ ￿ ￿
￿
v
￿ ￿
￿ ￿
￿
We get the bid functions:
￿ b1(v) =
￿ ￿￿ ￿
￿ + ￿
￿
￿
v
￿ ￿
￿ ￿+￿
￿
; ￿ b2 (w) =
￿ ￿￿ ￿
￿ + ￿
￿
￿w
￿ ￿
￿ ￿+￿
￿
.
Hence, we have the following equation de￿ning b￿ :
b￿ = ￿ b1(￿ ￿) = ￿ b2 (￿ ￿) =
￿ ￿￿ ￿
￿ + ￿
. (3)
Along with (1 ￿ 2), we have a system of 3 equations in three unknowns b￿; ￿ ￿ and ￿ ￿.
We get:
￿ ￿ = ￿; ￿ ￿ =
￿ (￿ + ￿)
(￿ ￿ ￿)
2b ￿ ￿
￿
; b￿ =
￿
￿ ￿ ￿
(2b ￿ ￿):
with ￿ is the positive root of
￿
￿
2 ￿ ￿￿
￿
Z2+
￿
2￿2bc ￿ ￿2￿ ￿ ￿￿
2 ￿ 2bc￿
2￿
Z+2￿2￿bc+
2￿￿
2bc:
￿ =
￿ + ￿
￿
b
c +
￿
2
￿
￿ + ￿
￿ ￿ ￿
￿
￿
￿ + ￿
￿
s
(bc)
2 ￿
￿￿
￿ + ￿
bc +
￿
￿￿
2(￿ ￿ ￿)
￿2
:
￿
Proof of proposition 2
R
c = b
c
￿
￿ ￿ ￿ ￿
￿
+
￿ ￿ ￿ ￿
￿
￿
+
Z ￿ ￿
0
b1 (v) ￿
1
￿
dv +
Z ￿ ￿
0
b2 (v) ￿
1
￿
dv;
= b
c
￿
￿ ￿ ￿ ￿
￿
+
￿ ￿ ￿ ￿
￿
￿
+
Z ￿ ￿
0
￿ ￿￿ ￿
￿ + ￿
￿
￿w
￿ ￿
￿ ￿+￿
￿
￿
1
￿
dv +
Z ￿ ￿
0
￿ ￿￿ ￿
￿ + ￿
￿
￿
v
￿ ￿
￿ ￿+￿
￿
￿
1
￿
dv;
= b
c
￿
￿ ￿ ￿ ￿
￿
+
￿ ￿ ￿ ￿
￿
￿
+
￿ ￿2￿ ￿
(￿ + ￿)(￿ + 2￿)
+
￿ ￿￿ ￿
2
(￿ + ￿)(2￿ + ￿)
:
￿
Proof of proposition 3
For bc =
￿￿
￿+￿, we have ￿ ￿ = ￿ = ￿ and ￿ ￿ = ￿.
6Let￿ s evaluate the following derivatives d￿ ￿
dbcjbc=
￿￿
￿+￿;
d￿ ￿
dbcjbc=
￿￿
￿+￿;
d￿
dbcjbc=
￿￿
￿+￿:
@￿
@bc =
￿ + ￿
￿
￿
￿ + ￿
￿
0
B
B
@
1
2
2bc ￿
￿￿
￿+￿ r
(bc)
2 ￿
￿￿
￿+￿bc +
￿
￿￿
2(￿￿￿)
￿2
1
C
C
A;
@￿
@bcjbc=
￿￿
￿+￿ =
￿ + ￿
￿
 
1 ￿
1
2
2
￿￿
￿+￿ ￿
￿￿
￿+￿
￿￿
2(￿￿￿)
!
=
2￿
￿
;
@￿ ￿
@bcjbc=
￿￿
￿+￿ =
2￿ (￿ + ￿)
(￿ ￿ ￿)
￿ ￿ bc ￿
d￿
dbc
￿2 =
2￿ (￿ + ￿)
(￿ ￿ ￿)
￿ ￿
￿￿
￿+￿ ￿
￿
2￿
￿
￿
￿2 = 2
￿
￿
;
@￿ ￿
@bcjbc=
￿￿
￿+￿ =
@￿
@bc =
2￿
￿
:
We now show that the derivative of the revenue with respect to bc is zero at bc =
￿￿
￿+￿.
We have:
@Rc
@bc = b
c
￿
￿@￿ ￿=@bc
￿
￿
@￿ ￿=@bc
￿
￿
+
￿
￿ ￿ ￿ ￿
￿
+
￿ ￿ ￿ ￿
￿
￿
+
￿
￿ ￿2 ￿ @￿ ￿=@bc + 2￿ ￿￿ ￿ ￿ @￿ ￿=@bc￿
(￿ + ￿)(￿ + 2￿)
+
h
￿ ￿
2 ￿ @￿ ￿=@bc + 2￿ ￿￿ ￿ ￿ @￿ ￿=@bc
i
(￿ + ￿)(2￿ + ￿)
:
@Rc
@bc j ￿￿
￿+￿ = ￿2 +
[2￿￿ + 4￿2]
(￿ + ￿)(￿ + 2￿)
+
￿
2￿￿ + 4￿
2￿
(￿ + ￿)(2￿ + ￿)
= 0
We now need to evaluate the sign of the second derivative of Rc at bc =
￿￿
￿+￿. We
have:
@2￿
@ (bc)
2 = ￿
￿ + ￿
2￿
0
B
B B B
@
2
r
bc2 ￿
￿￿
￿+￿bc +
￿
￿￿
2(￿￿￿)
￿2
￿ 1
2
(2bc￿
￿￿
￿+￿)
2
r
(bc)2￿
￿￿
￿+￿bc+(
￿￿
2(￿￿￿))
2
(bc)
2 ￿
￿￿
￿+￿bc +
￿
￿￿
2(￿￿￿)
￿2
1
C
C C C
A
@2￿
@ (bc)
2j ￿￿
￿+￿ = ￿
￿ + ￿
2￿
￿￿
(￿￿￿) ￿
(￿￿￿)￿￿
(￿+￿)2
￿
￿￿
(￿￿￿)
￿2 =
￿2(￿ + ￿)
￿
 
(￿ ￿ ￿)
￿￿
￿
(￿ ￿ ￿)
3
(￿ + ￿)
2 (￿￿)
!
= ￿
8(￿ ￿ ￿)
￿ (￿ + ￿)
;
@2￿ ￿
@ (bc)
2 =
2￿ (￿ + ￿)
(￿ ￿ ￿)
0
@￿
1
￿2
@￿
@bc ￿
￿2
￿
d￿
dbc + bc ￿
@2￿
@bc
￿
￿ 2￿
d￿
dbc
￿
bc d￿
dbc
￿
￿4
1
A;
@2￿ ￿
@ (bc)
2j ￿￿
￿+￿ =
2￿ (￿ + ￿)
(￿ ￿ ￿)
0
@￿
1
￿2
2￿
￿
￿
￿2
￿
2￿
￿ ￿ 8
(￿￿￿)
￿(￿+￿)
￿￿
￿+￿
￿
￿ 2￿2￿
￿
￿
￿￿
￿+￿
2￿
￿
￿
￿4
1
A =
8
￿
(￿ ￿ ￿)
￿ + ￿
;
7@2￿ ￿
@ (bc)
2j ￿￿
￿+￿ =
@2￿
@ (bc)
2j ￿￿
￿+￿ = ￿8
(￿ ￿ ￿)
￿ (￿ + ￿)
:
Now we have:
@2Rc
@ (bc)
2 = ￿2
￿
@￿ ￿=@bc
￿
+
@￿ ￿=@bc
￿
￿
￿ b
c
￿
@2￿ ￿=@bc2
￿
+
@2￿ ￿=@bc2
￿
￿
+
h
￿ ￿
2@2￿ ￿
@(bc)2 + 2￿ ￿ @￿ ￿
@bc
@￿ ￿
@bc + 2￿ ￿￿ ￿ @2￿ ￿
@bc2 + 2 @￿ ￿
@bc
￿
￿ ￿
@￿ ￿
@bc + ￿ ￿ @￿ ￿
@bc
￿i
(￿ + ￿)(￿ + 2￿)
+
h
￿ ￿
2 @2￿ ￿
@(bc)2 + 2￿ ￿
@￿ ￿
@bc
@￿ ￿
@bc + 2￿ ￿￿ ￿
@2￿ ￿
@bc2 + 2
@￿ ￿
@bc
￿
￿ ￿
@￿ ￿
@bc + ￿ ￿ @￿ ￿
@bc
￿i
(￿ + ￿)(2￿ + ￿)
=
(￿ ￿ ￿)
2
2￿3 + 7￿2￿ + 7￿￿
2 + 2￿
3 > 0:
Hence, the revenue function Rc is convex in the cap bc at bc =
￿￿
￿+￿. This proves that
introducing a binding cap bc <
￿￿
￿+￿ increases the revenue.￿
7 References
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8

Caps in asymmetric all-pay auctions with incomplete information

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Caps in asymmetric all-pay auctions with incomplete information

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