Sequential bidding in asymmetric …rst price auctions
Gal Cohensius Ella Segevyz
November 25, 2014
Abstract
We study asymmetric …rst price auctions in which bidders place their bids sequentially, one after the
other and only once. We show that with a strong bidder and a weak bidder (in terms of …rst order
stochastic dominance of their valuations’distribution functions), already with small asymmetry between
the bidders, the expected revenue in the sequential bidding …rst price auction (when the strong bidder
bids …rst) is higher than in the simultaneous bidding …rst price auction. Moreover it is higher than the
expected revenue in the second price auction. The expected payo¤ of the weak bidder is also higher in
the sequential …rst price auction. Therefore a seller interested in increasing revenue facing asymmetric
bidders may …nd it bene…cial to order them and let them bid sequentially instead of simultaneously. In
terms of e¢ ciency, both the simultaneous …rst price auction and the sequential …rst price auction cannot
guarantee full e¢ ciency (as opposed to a second price auction). The sequential bidding auction when
the stronger bidder bids …rst achieves lower e¢ ciency than the simultaneous auction. However, when
the order is reversed and the asymmetry is large enough the sequential …rst price auction achieves higher
e¢ ciency than the simultaneous auction.
Keywords: Sequential bidding, asymmetric auctions, …rst price auction, second price auction.
JEL classi…cation: D44
1
Introduction
The question of designing an auction mechanism that will ensure high revenue for the seller when bidders
have private information on their valuation of the object and are ex ante asymmetric, has long been an open
question. Even the more restricted question of …nding equilibrium behavior and then ranking di¤erent known
auctions according to their expected revenue is still unresolved in the general asymmetric case. We cannot
even determine under what conditions will a …rst price auction (FPA) yield a higher expected revenue than
Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, email:
galcohen-
[email protected]
y Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, email: [email protected]
z We thank Yoav Kerner for his help in producing …gure 7. We also thank Paul Schweinzer, Anna Rubinchik, Gabrielle Gayer
and Todd Kaplan for helpful comments.
1
a second price auction (SPA) when bidders are asymmetric.1 Partial results exist for di¤erent assumptions
on the distribution of bidders values.
Here we propose a simple modi…cation of the …rst price auction in which bidders bid sequentially and
only once. We order the bidders according to some order and bidder i observes the bids of previous bidders
1; :::; i
1 when she places her own bid. As in the standard …rst price auction, the winner is the bidder who
submitted the highest bid and she pays her bid. All other bidders have a payo¤ of zero. We break ties in
favor of later bidders. Thus bidder i wins if her bid is higher than or equal to the bids of bidders 1; ::; i
1
and strictly higher than the bids of bidders i + 1; :::; n. For the two bidders case, even with small asymmetry
between bidders, we show that by ordering the bidders such that the strong bidder bids …rst and the weak
bidder bids only after the strong bidder does2 , we increase both the weak bidder’s expected payo¤ and the
expected revenue compared to the simultaneous auction.
Asymmetry between bidders is common in many auctions. For example when one of the bidders is an
incumbent …rm while the other is an entrant …rm it is plausible to assume asymmetry in their valuations for
the object (which can be a result of di¤erences in the cost function). Another source for asymmetry may
be the seller’s preferences over the bidders. In procurement auctions for example it is common that one
seller is given an advantage over the other, re‡ecting better reliability or quality. Legal interventions that
are intended for encouraging the participation of the weaker …rm or preferred seller, such as bidding credits
and set-asides are prevalent. Our results indicate that already with a small asymmetry the weaker …rm
should be given the opportunity to bid after the stronger …rm bids. This not only improves the expected
payo¤ of the weaker …rm and therefore may increase participation of weaker …rms, but also increases the
expected revenue for the seller. In most common auctions bidders bid simultaneously or equivalently without
knowing the bids of their opponents. However, in several settings the seller may approach the bidders one
by one creating a sequential bidding game if she discloses the bids of previous bidders. Thus our motivation
for studying sequential bidding FPA does not stem from an existing practice (we are not aware of such a
practice) but from the theoretical understanding that altering the common practice to the sequential one
may improve its performance.
Remarkably, the dynamic nature of bidding in the sequential bidding auction makes the game more
tractable and as a result makes it simple to …nd the perfect Bayesian equilibrium for general distributions
and a general number of bidders. This is yet another advantage of the sequential FPA over the simultaneous
one where it is extremely di¢ cult to …nd equilibrium behavior and analyze changes in the mechanism (such
as handicaps or headstarts) on the bidders’behavior. We therefore suggest that sequential mechanisms be
used when bidders are asymmetric to allow for higher revenue, higher participation of weaker bidders and
easier prediction of expected behavior. Note moreover that even if the seller does not know the distributions
1 Examples
exist for both directions - a …rst price auction may either yield a higher or a lower expected revenue than a second
price auction but no su¢ cient conditions are yet known for either of the directions.
2 Formally we say that bidder i is stronger than bidder j if the distribution of her valuation, F , …rst order stochastically
i
dominates the distribution of bidder j’s valuation - Fj .
2
of the bidders’valuations but only that they are asymmetric such that one is strong and the other is weak
our results suggest that by ordering them such that the strong bidder bids …rst he will almost always ensure
a higher revenue than by a simultaneous auction.
We also examine the e¢ ciency of the sequential …rst price auction. E¢ ciency is de…ned as the probability
that the higher valuation bidder wins the object. When the bidders are asymmetric simultaneous …rst price
auction does not guarantee e¢ ciency. This happens because in equilibrium the weaker bidder bids more
aggressively than the stronger one (does less shading down of her valuation) and therefore may win even if
her valuation is lower than that of the stronger bidder. This is true also for the sequential …rst price auction.
In the sequential auction the second bidder has an exogenous advantage of observing the …rst bid and being
able to win by placing the same bid as the …rst bidder. Therefore, the second bidder may win although she
has a lower valuation than the …rst one. We show that when the stronger bidder bids …rst the e¢ ciency
is lower compared to the simultaneous auction. However, when the order is reversed and the asymmetry
between bidders is high enough the e¢ ciency in the sequential auction is higher than in the simultaneous
one.
There are only very few analytic solutions to the equilibrium bids in a simultaneous FPA with asymmetric
bidders and all of them are restricted to two bidders. Thus our comparison will be restricted to cases in
which we could analytically solve for the equilibrium bids in the simultaneous auction. We use results by
Maskin and Riley (2000) and by Kaplan and Zamir (2012) in two di¤erent scenarios of asymmetry between
two bidders. In the …rst scenario, stretch of probabilities, bidder 2’s valuation is uniformly distributed on
the interval [0; c] (the weak bidder) while bidder 1’s valuation is stretched to the right and is uniformly
distributed on the interval [0; c + "] (the strong bidder). In the second scenario, shift of probabilities, bidder
2’s valuation is again uniformly distributed on the interval [0; c] (the weak bidder) while bidder 1’s valuation
is shifted to the right and is uniformly distributed on the interval ["; c + "] (the strong bidder). In both cases
Maskin and Riley (2000) prove that a FPA yields a higher expected revenue than a SPA and Kaplan and
Zamir (2012) specify the inverse bid functions of the bidders. We show that when asymmetry is high enough
(" gets larger) a sequential bidding …rst price auction dominates both FPA and SPA and generates higher
expected revenue.
Finally we compare the expected payo¤ of each of the bidders in each of the auctions and show that the
ex ante expected payo¤ of the strong bidder is highest in the simultaneous FPA, second highest in the SPA
and lowest in the sequential FPA while for the weaker bidder the order is reversed. A weak bidder prefers the
sequential auction where she bids second over the simultaneous auction. Therefore a sequential …rst price
auction may serve as a tool to increase participation of weak bidders.
Much e¤ort has been put in the last two decades in …nding equilibrium behavior in asymmetric auctions
and ranking FPA and SPA in terms of revenue. Tanno (2009) proposes a necessary and su¢ cient condition
for the existence of linear bid in the equilibrium of the asymmetric …rst-price auctions with two bidders
and uniform distributions. Doni and Menicucci (2013) compare the expected revenue in FPA and SPA
3
with discrete valuations with binary support. They prove that for a large set of parameters the SPA yields
higher expected revenue than the FPA. Fibich, Gavious and Sela (2004) show that the di¤erence between
the expected revenue in FPA and SPA might be very small when all distributions of the bidders’valuations
are on the same support and are perturbations of a common distribution. Gavious and Minchuk (2014)
consider “close to uniform” distributions with identical support and show that the expected revenue in
SPA may exceed that in FPA. Hubbard, Kirkegaard and Paarsch (2013) discuss numerical approximations
to equilibrium behavior in asymmetric FPA. Kirkegaard (2009) suggests a new approach to examining
equilibrium bids in FPA. This approach leads to identifying necessary conditions for the bid functions of
the bidders to have either no crossings or a unique crossing. Kirkegaard (2011) generalizes Maskin and
Riley (2000) results and identi…es su¢ cient conditions on the distribution functions that ensure that a FPA
yields higher expected revenue than a SPA. Kirkegaard (2014) proves that FPA remains optimal (over SPA)
when the weak bidder’s distribution falls between a shift and a stretch of the strong bidder’s distribution.
Mares and Swinkels (2014) de…ne and discuss the connection between the -concavity of the underlying type
distributions and the bidders’bid functions. They are able to determine bounds on the equilibrium behavior
in asymmetric auctions.
Not much e¤ort has been devoted to examining auctions in which bidders bid one by one and only once.
Fischer et al (2014) study a very similar model to ours. They assume an exogenous probability of a leak
which is the probability that the second bidder will observe the …rst bidder’s bid. When this probability
is 1 their model coincides with ours. However they only assume symmetric bidders and therefore show
(both theoretically and in an experiment) that the sequential bidding game yields less revenue than the
simultaneous game. Roberts and Sweeting (2013) study a sequential bidding auction with an entry cost. In
their model bidders learn their private valuations only after they decide to enter and pay the entry fee and bids
can only be increased by a giving jump each round. The entry decision is therefore dependent on the entry
decision of previous bidders and the bidder’s own signal. They show that a sequential mechanism can give
both buyers and sellers signi…cantly higher payo¤s than the commonly used simultaneous bid auction. In an
earlier work, Segev and Sela (2014a) suggest a similar modi…cation for the all-pay auction and …nd the perfect
Baysian equilibrium of the sequential all-pay auction. However, they do not compare the expected revenue
to expected revenue in the simultaneous all-pay auction. Linster (1993) studied Stackelberg rent seeking
games in the form of Tullock contests where players move sequentially. He de…ned the perfect Bayesian
equilibrium and compared the results with that of a simultaneous Tullock contest. Morgan (2003) studies
sequential Tullock contests as well with incomplete information and symmetric distributions. He shows
that the sequential contest may yield higher expected revenue than the simultaneous one. Finally, Ludwig
(2006) compares sequential and simultaneous Tullock contests with complete and incomplete information
and symmetric players. She allows for correlation between the players’types and …nds that the seller prefers
sequential over simultaneous contests when he aims at maximizing the expected e¤ort sum. Moreover, the
contestants may prefer sequential contests, too.
4
2
The Model
We consider a sequential bidding …rst price auction with n
Biders bid one by one. Bidder j, 1
j
2 bidders who bid for a single indivisible object.
n observes the bids of bidders 1; 2; :::; j
1 and then places a bid bj .
The winner is the bidder who placed the highest bid. We break ties in favor of later bidders. Thus, bidder
j wins if her bid is larger than or equal to the bids of bidders 1; :::; j
1 and strictly larger than the bids of
bidders j + 1; :::; n. Bidder j’s valuation of the object is denoted by vj and is private information to bidder
j. It is common knowledge among the bidders that bidder j’s valuation vj is drawn from a continuos and
twice di¤erentiable distribution Fj with a positive density function fj on the interval [aj ; cj ] ; 0
aj < cj .
We also assume that Fj has no atom at aj .3 Our aim is to characterize the perfect Bayesian equilibrium
bid strategies of the bidders. In this section we do that for two bidders while in section 4 we discuss the
generalization for any number of bidders.
Assume n = 2. We start by analyzing the second bidder’s behavior. Bidder 2 observes the bid of bidder
1, b1 . Bidder 2 will then either bid b1 , if v2
2 bids zero: b2 (v2 ; b1 ) = 0 for all a2
a2
v2
b1 or otherwise will bid zero. Therefore, if c2
v2
c2 . If b1
b1 then bidder
a2 then bidder 2 bids b1 : b2 (v2 ; b1 ) = b1 for all
c2 . Finally, if a2 < b1 < c2 then,
8
< 0
b2 (v2 ; b1 ) =
: b
1
if
a2
v2 < b1
if
b1
v2
(1)
c2
Bidder 1 considers the probability that bidder 2 will bid zero which is her probability of winning. If she
bids less than b2 then her payo¤ is zero. Otherwise, her maximization problem is given by
max
a2 b1 minfc2 ;v1 g
fF2 (b1 ) (v1
We need to consider several cases. First, If c1
b1 )g
(2)
a2 then bidder 1 can never ensure a positive payo¤.
In the following we assume that in this case bidder 1 bids zero. Obviously she could bid any positive bid
smaller than a2 and get the same payo¤ but we want to identify the PBE with the smallest possible expected
revenue for the sequential auction and therefore assume she bids zero. Thus, whenever v1
a2 we have
b1 (v1 ) = 0.
The …rst order condition for the maximization problem is given by
f2 (b1 ) (v1
b1 )
F2 (b1 ) = 0
or equivalently we can derive the inverse bid function as
v1 = b1 +
We de…ne the function
3 An
2
(x) = x +
F2 (x)
f2 (x) .
F2 (b1 )
f2 (b1 )
We need the following condition on F2 .
atom at aj if j is the second bidder creates discontinuity in the bidders 1; :::; j
to a case where no perfect Bayesian equilibrium exists.
5
1 maximization problem and may lead
De…nition 1 A continuos and twice di¤ erentiable distribution function F with a positive density f on the
interval [a; c] is said to be sequentially regular (SeqR) if the following function
(x) = x +
F (x)
f (x)
(3)
is strictly increasing on the interval [a; c].
In the following we assume that the distribution F2 of bidder 2’s valuation is sequentially regular. This
will allow us to de…ne the bid function of bidder 1 which is the inverse function of
2
(x). Note that if a
distribution is a convex function and sequentially regular then it is also regular in terms of the Myerson’s
(1981) condition (a distribution is regular if the virtual valuation
1 F (x)
f (x)
(x) = x
is a strictly increasing
function).4 Moreover, if F (x) is concave then it is trivially sequentially regular. Finally, note that for bidder
i,
i
(ai ) = ai and
i
(ci ) = ci +
1
fi (ci ) .
Proposition 1 The following is a perfect Bayesian equilibrium (PBE) of the sequential bidding …rst price
auction with two bidders. If b1
a2 then b2 (v2 ; b1 ) = b1 for a2 v2 c2 . Otherwise
8
< 0 if a
v2 < min fb1 ; c2 g
2
b2 (v2 ; b1 ) =
: b if min fb ; c g v
c
1
If c1
If
2
1
2
2
2
a2 then b1 (v1 ) = 0. If a2 a1 and a1
2 (c2 ) then
8
1
< max 0;
if a1 v1 min f 2 (c2 ) ; c1 g
2 (v1 )
b1 (v1 ) =
:
c2
if min f 2 (c2 ) ; c1 g v1 c1
(c2 )
a1 then
b1 (v1 ) = c2
for all a1
v1
c1
Finally, if a1 < a2 then
b1 (v1 ) =
8
>
>
>
<
>
>
>
:
0
max 0;
if
1
2
(v1 )
a1
if a2
c2
if
v1
min f
2
v1 < a2
min f
(c2 ) ; c1 g
2
(c2 ) ; c1 g
v1
c1
Proof. Bidder 2 best responds to bidder 1’s behavior since he places the same bid as bidder 1 whenever
it is below his valuation. Bidder 1 solves the maximization problem (2). This together with the assumption
that if v1
a2 then bidder 1 bids zero since she cannot ensure any positive payo¤ given bidder 2’s strategy
(as explained above), gives us the described PBE.
Using proposition 1 we can express the expected revenue of the seller in this PBE. Note that with only
two bidders the expected revenue of the seller is equal to the expected bid of the …rst bidder (the second
4 Since
d
dx
if
(x) >
(x) =
f 0 (x)
(f (x))2
(x) +
1
f (x)
is strictly increasing then
d
dx
(x) =
> 0.
6
d
dx
(x)
f 0 (x)
(f (x))2
> 0 and the convexity of F (x) implies
bidder either places the same bid or bids zero. Therefore the expected revenue
8
>
>
0
if
>
>
>
R c1
>
1
>
>
max 0; 2 (v1 ) f1 (v1 ) dv1
if
>
a1
>
>
R
>
(c
)
2
2
1
<
max 0; 2 (v1 ) f1 (v1 ) dv1 + c2 (1 F1 ( 2 (c2 ))) if
a1
FPA
Rseq
=
>
>
c2
if
>
>
>
R
>
c
1
1
>
>
max 0; 2 (v1 ) f1 (v1 ) dv1
if
>
a2
>
> R (c )
>
2 2
1
:
max 0; 2 (v1 ) f1 (v1 ) dv1 + c2 (1 F1 ( 2 (c2 ))) if
a2
of the seller is given by
c1
a2
a2
a1 < c1
a2
a1 <
a2 <
2
2
(c2 )
a1 < a2
c1
a1
a2 <
2
(c2 )
(c2 )
c1
a1 < c1
2
2
(c2 )
(c2 )
c1
The following example is with uniform distributions and it applies to the …rst two scenarios we examine
below.
Example 1 Assume that v1 is uniformly distributed on [a1 ; c1 ] and v2 is uniformly distributed on [a2 ; c2 ]
then F2 is sequentially regular and
have max 0;
1
2
(v1 ) =
FPA
Rseq
=
8
>
>
>
>
>
>
>
>
>
>
>
>
<
1
2
2
(x) = 2x
a2 and
1
2
(x) =
1
2
(x + a2 ). Therefore, for any v1 we
(v1 + a2 ) and then
0
1
4
1
4(c1 a1 )
>
>
>
>
>
>
>
>
>
>
>
>
:
if
(a1 + 2a2 + c1 )
4c2 (a2 + c1
c2 )
(a1 + a2 )
c2
1
4(c1 a1 )
1
c 1 a1
(c1
2
c1
a2
if
a2
a1 < c1
2c2
a2
if
a2
a1 < 2c2
a2
c1
if a2 < 2c2
a2
a2 ) (3a2 + c1 )
if
a1 < a2
c1
a22
if
a1
(a2 + c1 ) c2
c22
a2 < 2c2
a1 < c1
2c2
a2
a2
c1
In the next section we use the above results to compare the expected revenue in the PBE of the sequential
bidding …rst price auction with the expected revenue of a simultaneous bidding …rst price and second price
auctions. We follow Maskin and Riley (2000) and examine two cases of asymmetry while using Kaplan and
Zamir (2012) to derive the explicit bidding functions of the bidders. In both cases we have a strong bidder
and a weak bidder. We say that a bidder is stronger than another bidder if the distribution from which her
valuation is drawn, …rst order stochastically dominates that of the other. When examining the sequential
bidding auction we would therefore …rst need to determine what is the optimal (in terms of the seller’s
revenue) order of the bidders.
3
Ranking the auctions
For simultaneous bidding …rst price auction there are only limited results for the Bayesian equilibrium
behavior of bidders. Kaplan and Zamir (2012) …nd the equilibrium behavior of two bidders with uniform
distributions and general supports. Therefore we restrict attention to uniform distributions in order to be
able to derive an explicit expression for the expected revenue in the Bayesian equilibrium of the simultaneous
FPA.
7
3.1
Stretch of probabilities
Assume that F1 is a uniform distribution on [0; c + "], for " > 0 while F2 is uniform on [0; c]. Bidder 1 is
therefore the strong bidder. If bidder 1 bids …rst then we have
8
< 1 (c + ") if 0 " c
4
FPA
Rseq
(1; 2) =
c"
:
if
c "
(c+")
while if bidder 1 bids second then
FPA
Rseq
(2; 1) =
1
c
4
Therefore the optimal order is when the stronger bidder bids …rst. We thus assume in the following that
we are able to determine the order of the bidders and we let the weaker bidder bid second. Then we have
8
< 1 (c + ") if 0 " c
4
FPA
Rseq
=
c"
:
if
c "
(c+")
We use Kaplan and Zamir (2012) to derive the equilibrium bid function of the simultaneous bid FPA.
The inverse bid functions, v1 (b) and v2 (b) solve the following set of di¤erential equations:
f2 (v2 (b)) 0
v (b)
F2 (v2 (b)) 2
f1 (v1 (b)) 0
v (b)
F1 (v1 (b)) 1
with the following boundary conditions: v1
c(c+")
2c+"
=
=
1
v1 (b)
1
v2 (b)
(4)
b
b
c(c+")
2c+"
= c + " and v2
= c and v1 (0) = v2 (0) = 0.
Therefore we have
2
v2 (b)
=
v1 (b)
=
2bc2 (" + c)
2
c2 (" + c) + " (" + 2c) b2
2
2bc2 (" + c)
c2 (" + c)
2
" (" + 2c) b2
Or
c (" + c)
b1 (v)
b2 (v)
=
=
c (" + c) +
q
2
c2 (" + c) + " (" + 2c) v 2
" (" + 2c) v
q
2
c (" + c) c (" + c)
c2 (" + c)
"v 2 (" + 2c)
"v (" + 2c)
Note that for every v 2 [0; c] we have b2 (v) > b1 (v). The weaker bidder bids more aggressively in the
equilibrium of the simultaneous FPA. For example, for c = 1 and " =
bid functions of the bidders.
8
1
4
the following …gure describes the
b(v)
0.5
0.4
0.3
0.2
0.1
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
v
…gure 1: Eq. bid functions, b1 (v) in black and b2 (v) in red.
In order to …nd the expected revenue of the seller in the simultaneous FPA we de…ne the random variable of
h
i
the winning bid P . This random variable is distributed on 0; c(c+")
according to the following distribution
2c+"
function
G (p) = Pr (max fb1 (v) ; b2 (v)g
p) = F1 (v1 (p)) F2 (v2 (p))
with a density function
3
8pc3 (" + c)
2
4
"2 (" + 2c) p4 + c4 (" + c)
g (p) =
4
c4 (" + c)
2
"2 (" + 2c) p4
2
Therefore we have
FPA
Rsim
=
Z
c(c+")
2c+"
pg (p) dp
0
2
c2 (" + c)
ln
=
p
q
"("+2c)
"
p
+ 2 arctan
"+2c
"+2c+ "("+2c)
p
" (" + 2c) " (" + 2c)
"+2c
+
c (" + c)
(" + 2c)
Finally, in the SPA the bidders bid their valuation and then the random variable P is distributed on [0; c]
with the following distribution function
K (p) = Pr (min fv1 ; v2 g
p) = p
(" + 2c p)
c (" + c)
and a density function
k (p) =
and therefore
R
SP A
=
Z
" + 2c 2p
c (" + c)
c
pk (p) dp =
0
9
1 3" + 2c
c
6 "+c
We are now ready to compare the auctions in terms of expected revenue. We already know from Maskin
and Riley (2000) that the expected revenue from the second price auction is always lower than the expected
FPA
revenue from the simultaneous …rst price auction. In other words RSP A (") < Rsim
("). We have the
following two propositions
Proposition 2 For every c, there exists a cut-o¤ 0 < " (c)
FPA
FPA
> Rseq
c such that for all " < " (c) ; Rsim
FPA
FPA
and for all " > " (c) ; Rsim
< Rseq
: Moreover, this cuto¤ is increasing in c.
Proof. In the appendix
Proposition 3 For every c, there exists a cut-o¤ 0 < "
and for all " > "
FPA
(c) ; RSP A < Rseq
: The cuto¤ "
FPA
(c) ; RSP A > Rseq
(c) < c such that for all " < "
(c) is increasing in c and "
(c) < " (c).
Proof. In the appendix
For example, for c = 1 we plot the expected revenue in all three auctions as a function of " in the following
…gure
R 0.8
0.7
0.6
0.5
0.4
0.3
0
1
2
3
4
5
epsilon
FPA
FPA
Figure 2: RSP A in green, Rsim
in black, Rseq
in red
We therefore conclude that when the asymmetry is high enough, " > " (c) then the sequential bidding
auction yields higher expected revenue than both simultaneous auctions. Note that " (c) < c and therefore
the asymmetry need not be so large in order to conclude that the sequential bidding auction dominates the
simultaneous auctions.
In terms of e¢ ciency, however, the sequential bidding auction always yields lower e¢ ciency than the
simultaneous auctions if the stronger bidder bids …rst. We de…ne e¢ ciency as the probability that the
winner is the bidder with the higher valuation. Recall that the simultaneous …rst price auction does not
guarantee e¢ ciency when bidders are asymmetric since the weaker bidder bids more aggressively than the
10
stronger one (less shading down bids) and then the weaker bidder may win although her valuation is lower
than the stronger bidder’s valuation. This phenomenon is also present in the sequential …rst price auction.
When the weaker bidder bids second she has an advantage over the stronger one since she gets to observe
his bid and can win by placing the same bid. This leads to cases where the weaker bidder wins although
her valuation is lower than the stronger bidder’s valuation. In the following we compare the e¢ ciency of the
mechanisms. For the second price auction we have
Ef f SP A = 1
For the simultaneous …rst price auction
FPA
Ef fsim
=
1
=
1
Pr (b2 (v2 ) > b1 (v1 ) and v1 > v2 )
!
cv2 ("+c)
Z c Z p
1
1 2"2 + 4c2 + 5"c
1
(c2 ("+c)2 "("+2c)v22 )
dv1
dv2 =
c1 + "
c
2 (" + c) (" + 2c)
v2
0
For the sequential FPA, when the stronger bidder bids …rst, we have, for "
FPA
Ef fseq
([0; c + "] ; [0; c])
=
1
Pr (b1 (v1 ) < v2 and v2 < v1 )
!
Z c Z v1
Z c+"
1
1
dv2
dv1 +
1
c
c+"
0
c
2 v1
1
Z
c
1
2 v1
1
dv2
c
!
1
dv1
(c + ")
!
1
dv1
c+"
!
1 "2 + 3c2 + 2"c
4
c (" + c)
=
while for "
c
c
FPA
Ef fseq
([0; c + "] ; [0; c])
=
1
1
=
Pr (b1 (v1 ) < v2 and v2 < v1 )
!
Z c Z v1
Z 2c
1
1
dv2
dv1 +
1
c
c+"
0
c
2 v1
1 2" + c
2 "+c
Z
c
1
2 v1
1
dv2
c
!
Therefore we have for every c and every "
FPA
FPA
Ef fseq
([0; c + "] ; [0; c]) < Ef fsim
< Ef f SP A = 1
However, if we order the bidders such that the weaker bidder bids …rst than we can increase e¢ ciency.
In this case we have
8
< 0
b2 (v2 ; b1 ) =
: b
1
and
b1 (v1 ) =
if
if
0
b1
1
v1 for all 0
2
v2 < b1
v2 < c + "
v1
c
and
FPA
Ef fseq
([0; c] ; [0; c + "])
=
1
=
1
Pr (b1 (v1 ) < v2 and v2 < v1 )
!
!
Z c Z v1
1
1
1 3c + 4"
dv2
dv1 =
1
c+"
c
4 c+"
0
2 v1
11
Then
FPA
FPA
Ef fseq
([0; c + "] ; [0; c]) < Ef fseq
([0; c] ; [0; c + "])
and for " < 2c we have
FPA
FPA
Ef fseq
([0; c] ; [0; c + "]) < Ef fsim
while for " > 2c we have
FPA
FPA
Ef fseq
([0; c] ; [0; c + "]) > Ef fsim
For example, for c = 1 the following …gure describes the e¢ ciency as a function of ". In the …gure Ef f SP A
FPA
FPA
FPA
appears in yellow, Ef fsim
in black, Ef fseq
([0; c + "] ; [0; c]) in red and Ef fseq
([0; c] ; [0; c + "]) in light
green
Eff
1.0
0.9
0.8
0.7
0
1
2
3
4
5
6
7
8
epsilon
Figure 3
When the asymmetry is very large the e¢ ciency in all auctions approaches 1. We have
FPA
FPA
FPA
lim Ef fseq
([0; c] ; [0; c + "]) = lim Ef fseq
([0; c + "] ; [0; c]) = lim Ef fseq
=1
"!1
"!1
"!1
Finally we discuss the di¤erences between the auctions in the eyes of the bidders. For each of the auctions
we calculate the expected revenue of the strong bidder and the weak bidder. For the simultaneous SPA we
have
A
uSP
1
(v)
A
uSP
(v)
2
=
=
8
<
Rv
(v
0
v2 ) 1c dv2 =
(v
0
: R c (v
0
Z v
v2 )
v1 )
1
c dv2
=v
1 v2
2 c
1
2c
2
1
1 v
dv1 =
c+"
2c+"
12
if
if c
0
v
v
c
c+"
and ex ante we have
U1SP A
=
Z
Z c+"
1 v2
1
dv +
v
2 c c+"
c
1 v2
1
1 c2
dv =
2c+" c
6c+"
c
0
U2SP A
=
Z
c
0
For the simultaneous FPA we have
cv(c+")
Z p
(c2 (c+")2 +v2 "(2c+"))
simF P A
u1
(v) =
(v
1
b1 (v)) dv2
c
0
=
PA
usimF
(v)
2
=
=
1
1 3c" + 3"2 + c2
dv =
c+"
6
c+"
1
c
2
q
2
2
(c + ") " (" + 2c) v 2 + c2 (" + c)
c (" + c) c2 (" + c) + " (" + 2c) v 2
r
2
c2 (c + ") + v 2 " (2c + ")
" (" + 2c)
Z p
cv(c+")
1
dv1
c
+
"
0
q
2
2
c "v 2 (" + 2c) c2 (" + c) + c (" + c) c2 (" + c)
"v 2 (" + 2c)
r
2
" (" + 2c)
c2 (c + ")
v 2 " (2c + ")
(c2 (c+")2
v 2 "(2c+"))
(v
b2 (v))
and ex ante we have
U1simF P A
=
=
U2simF P A
=
=
1
q
2
2
2
2 (" + c)2 + " (" + 2c) v 2
c
(c
+
")
"
("
+
2c)
v
+
c
("
+
c)
c
("
+
c)
C 1
B
C
B
r
A c + " dv
@
0
2
2
2
" (" + 2c)
c (c + ") + v " (2c + ")
!
!
!
p
2
c + " + " (2c + ")
(c + ")
c+"
c2
p
3 ln
2 arccosh
(c ")
2" (2c + ")
c
c
" (2c + ")
1
0
q
Z c c "v 2 (" + 2c) c2 (" + c)2 + c (" + c) c2 (" + c)2 "v 2 (" + 2c)
B
C1
C dv
B
r
Ac
@
0
2
2
2
" (" + 2c)
c (c + ")
v " (2c + ")
!!
p
2
" (2c + ")
c2 (c + ")
1 c2 (c + 2")
1
c
p
+
arcsin
2 arcsin
c+"
(c + ")
2 " (2c + ")
2" (2c + ") " (2c + ") 2
Z
c+"
0
Finally, for the sequential FPA we have, when 0
FPA
useq
(v)
1
FPA
useq
2
and when c
" then
(v)
=
=
v2
4c
8
<
R 2v
0
: R c+" v
0
FPA
useq
(v)
1
FPA
useq
(v)
2
v
1
2 v1
=
1
2 v1
"
1
c+" dv1
1
c+" dv1
8
<
v2
4c
: v
=v
if
=
v2
c+"
1
4
v
c+"
13
if 0
(c + ") if
0
c if 2c
2
=
c
v
v
2c
c+"
1
2
v
(c + ")
1
2
(c + ")
v
c
Note that although the second bidder is the weaker one she enjoys a higher expected revenue for every
type since she has the bene…t of bidding second. The ex ante
8
< 1 (c+")2 if
seq F P A
12
c
=
U1
: 1 3"2 +c2 if
6 c+"
8
< 1 7c2 4c"+"2
seq F P A
24
c
=
U2
1 c2
:
3 c+"
expected payo¤ is given by
0
"
c
if
c
"
0
if
"
c
c
"
We prove the following proposition
Proposition 4 For every c and " we have
U1seq F P A < U1simF P A < U1SP A
and
U2SP A < U2simF P A < U2seq F P A
Proof. In the appendix
The above proposition establishes the fact that the weaker bidder prefers the sequential bidding auction
over the two simultaneous bidding auctions while the stronger one does not. Therefore letting the weaker
bidder bid after the stronger bidder bids, both increases her ex-ante expected payo¤ and the expected revenue
of the seller (when the asymmetry is large enough).
For example, when c = 1 the next two …gures present the ex ante expected payo¤ of the strong bidder,
bidder 1 (…gure 4) and the weak bidder, bidder 2 (…gure 5) in all three auctions
U_1
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.5
1.0
1.5
2.0
2.5
Figure 4: U1SP A in green, U1simF P A in black, U1seq F P A in red
14
3.0
epsilon
U_2
0.25
0.20
0.15
0.10
0.05
0.0
0.5
1.0
1.5
2.0
2.5
3.0
epsilon
Figure 5: U2SP A in green, U2simF P A in black, U2seq F P A in red
3.2
Shift of probabilities
Assume that F1 is a uniform distribution on ["; c + "] while F2 is uniform on [0; c]. Bidder 1 is therefore the
strong bidder. If bidder 1 bids …rst then we have
8
R c+" 1 1
1
>
>
>
2 v c dv = 4 (c + 2")
"
< R
R
2c 1
c+" 1
FPA
1
1 4c
Rseq
(1; 2) =
2 v c dv + 2c c c dv = 4 " c
"
>
>
>
:
c
"
if
0
"
c
if
c
"
2c
if
2c
"
while if bidder 1 bids second then
FPA
Rseq
(2; 1) =
8 R
< c
1
" 2
(v + ") 1c dv =
:
1
4c
(c
") (c + 3") if
0
0
"
if
c
c
"
Therefore the optimal order is again for the stronger bidder to bid …rst. We thus assume in the following
that we are able to determine the order of the bidders and we let the weaker bidder bid second. Then we
have
FPA
Rseq
8
R c+" 1 1
1
>
>
>
2 v c dv = 4 (c + 2")
"
< R
R c+" 1
2c 1
1
1 4c
=
2 v c dv + 2c c c dv = 4 " c
"
>
>
>
:
c
"
if
0
"
c
if
c
"
2c
if
2c
"
We again use Kaplan and Zamir (2012) to derive the equilibrium bid function of the simultaneous bid
FPA. We …rst note that if "
2c the stronger bidder will ensure winning by bidding c therefore the following
is an equilibrium of the simultaneous FPA:
b1 (v1 )
= c
b2 (v2 )
= v2
15
and then
FPA
Rsim
=
Z
c+"
"
When 0
"
1
c dv1 = c
c
2c the inverse bid functions, v1 (b) and v2 (b) solve the same set of di¤erential equations
1 4c" "2 +4c2
8
c
(4) as before, with the following boundary conditions: v1
and v1
"
2
"
2
= " and v2
v2 (b)
= c + " and v2
1 4c" "2 +4c2
8
c
=c
= 2" .5 Therefore we have
"2 (2c
= "+
2 (2b
") (2c + ") e
"
2b "
")
(2c
4"c
")(2c+")
2b (2c
")
2
v1 (b)
" (2c + ")
=
2 (2b
Again, for every b 2
h
" 1 4c" "2 +4c2
2; 8
c
i
") (2c
4"c
")(2c+")
") e (2c
"
2b "
+ 2 ("
b) (2c + ")
we have v2 (b) < v1 (b). The weaker bidder bids more aggressively in
the equilibrium of the simultaneous FPA. We again de…ne the random variable of the winning bid P . This
h
i
2
2
random variable is distributed on 2" ; 18 4c" "c +4c according to the following distribution function
G (p) = Pr (max fb1 (v) ; b2 (v)g
p) = F1 (v1 (p)) F2 (v2 (p))
with a density function
g (p) =
(2c
") (2c + ") "4 e (2c
2c2
4"c
")(2c+")
0
e 2p " B
B
B
(2p ") @ +
"
Therefore we have
FPA
Rsim
=
Z
1
1
1 4c"
8
2p(2c ")e (2c
4"c
")(2c+")
"
+(2p ")(2c+")e 2p
2
"
1
( 2p+")(2c ")e (2c
4"c
")(2c+")
"
+2(p ")(2c+")e 2p
2
"
C
C
C
A
"2 +4c2
c
pg (p) dp
"
2
Unfortunately we could not derive here the analytic expression of the revenue. However, we could
numerically evaluate the integral for any set of parameters (c and ").
Finally, in the SPA the bidders bid their valuation and then the random variable P is distributed on [0; c]
with the following distribution function for 0
K (p) = Pr (min fv1 ; v2 g
Then
k (p) =
and
RSP A =
Z
0
5 The
"
p
dp +
c
Z
"
c
8
<
:
2c
"
c
p) =
8
<
:
1
c
2c 2p+"
c2
2p + "
c2
p
c
2cp c"+p" p2
c2
if
0
p
"
if
"
p
c
pdp =
1 "3
6
if 0
p
"
if
p
c
3c"2 + 3c2 " + 2c3
c2
equilibrium is such that the weaker bidder bids her true valuation up to
1
b2 (v) = v2 (b).
16
"
"
2
and from there on bids according to
For c
", we have for 0
p
c
K (p) = Pr (min fv1 ; v2 g
p) =
p
c
Then
k (p) =
and
R
SP A
=
Z
c
0
1
c
1
p
dp = c
c
2
We are now ready to compare the auctions in terms of expected revenue. We already know from Maskin
and Riley (2000) that the expected revenue from the simultaneous second price auction is always lower than
FPA
the expected revenue from the simultaneous …rst price auction. In other words RSP A (") < Rsim
("). We
have the following proposition.
Proposition 5 For every c, there exists a cut-o¤ 0 < "
and for all " > "
FPA
: The cuto¤ "
(c) ; RSP A < Rseq
FPA
(c) ; RSP A > Rseq
(c) < c such that for all " < "
(c) is increasing in c.
Proof. In the appendix
FPA
and RSP A
For example, for c = 1, …gure 6 presents Rseq
R
1.0
0.8
0.6
0.4
0
1
2
3
4
5
epsilon
FPA
Figure 6: Rseq
in red and RSP A in black
Since we do not have the explicit expression for the revenue in the simultaneous FPA we can not prove
a similar proposition to Proposition 2. However, for c = 1 we are able to plot the revenue as a function of
". Figure 7 demonstrates that indeed for some 1
Note that for "
FPA
2 we have RF P A < Rseq
"
2c we have
FPA
FPA
Rseq
= Rsim
=c
17
Figure 7:
4
in blue and
in purple.
More than two bidders
We now present the PBE of the sequential bidding auction for more than two bidders. We focus on the case
of stretch of probabilities in order to demonstrate the features of the equilibrium. Assume therefore that Fi
is a uniform distribution on [0; ci ]. Moreover, order the players by their strength, c1
c2
by analyzing bidder n’s behavior. Bidder n observes all previous bids. Denote by
j
highest bid among the …rst j
between any bid below
Bidder n
1 bids, then bidder n will either bid
n,
if vn
n
:::
= max1
cn . We start
i j 1 bi
the
or otherwise is indi¤erent
n.
We assume that she bids zero in this case. Therefore we have
8
< 0 if 0 v < min f ; c g
n
n n
bn (vn ; b1 ; :::; bn 1 ) =
:
if min f n ; cn g vn cn
n
1 considers both the probability that bidder n will bid zero and the bids of bidders 1; :::; n
(5)
1.
Her maximization problem is given by
max
max fFn (b) (vn
0 b cn
s.t. b
or zero if Fn
n 1
vn
1
n 1
<0
18
1
b)g ; vn
n 1
1
cn
(6)
Then, if 0
bn
1
while if cn
cn then
n 1
(vn
1 ; b1 ; :::; bn 2 )
cn
n 1
bn
1 (vn
1
8
>
>
>
>
>
>
<
=
0
1 ; b1 ; :::; bn
1
2 vn 1
if
cn
if
2) =
8
>
>
>
<
cn
1
2 vn 1
and so on. If cj
then it is
8
<
max
max
: j b cn
1
j
1
8
< nY1
:
then bj (vj ) = 0.
vn
0
if
n 1
if
min 2
1g
vn
vn
n 1
1
min f2cn ; cn
vn
1
1
cn
n 1 ; cn 1
vn
1
b2 (v2 ; b1 ) =
while if c
b1
c+"
b2 (v2 ; b1 ) =
and
b1 (v1 ) =
8
>
>
>
<
>
>
>
:
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
8
>
>
>
<
>
>
>
:
9
=
b) ; :::;
max
fFj+1 (b) (vj
;
cj+2 b cj+1
0
if
0
b1
if
1
2 v2
min fb1 ; c + "g
if
min f2b1 ; c + "g
c
if
0
if
0
b1
if
1
2 v2
min fb1 ; c + "g
if
2
3 v1
c
if
1
2 v1
if
v2
min fb1 ; c + "g
v2
v1
3
2 c; c
+ 2"
v2
min
v1
min f2c; c + 2"g
19
min f2c; c + "g
v2
c+"
min fb1 ; c + "g
min f2b1 ; c + "g
0
min
v2
min f2b1 ; c + "g
v2
min f2c; c + "g
if
cn
1
i=j+1
j) " for 1 j 3 we have
8
< 0 if 0 v < min f ; cg
3
3
b3 (v3 ; b1 ; b2 ) =
:
v3 c
3 if min f 3 ; cg
b1 < c then
1
n 1 ; cn 1
For example,for n = 3, if cj = c + (3
If 0
1g
n 1
min 2
1
n 1 ; cn 1
cn is given by
j
8
9
< nY1
=
Fi (b) (vj b) ; :::;
max
fFj+1 (b) (vj
;
cj+2 b cj+1
1 :
Fi (b) (vj
i=j+1
n 1 ; cn 1
if
1
min 2
1
min f2cn ; cn
n 1
>
>
>
:
i=j+1
j
min 2
vn
vn
n 1
0
The maximization problem of bidder j if 0
8
8
9
n
<
< Y
=
max
max
Fi (b) (vj b) ; max
: j b cn :
; cn b cn
If cn
0
if
n 1
>
>
>
>
>
>
:
then
if
min f2b1 ; c + "g
v2
3
2 c; c
c+"
+ 2"
min f2c; c + 2"g
v1
c + 2"
9
=
b)g
;
9
=
b)g
;
Note that when there are more than two bidders the expected revenue is no longer equal to the expected
bid of the …rst bidder. We have
Z
Z c1 Z c2 Z
FPA
:::
Rseq
=
(max fb1 (v1 ) ; :::; bn
1
For the example above with three bidders we have, for 0
FPA
Rseq
Z
=
Z
3
3
4 c+ 4 "
0
+
Z
3
3
4 c+ 4 "
and for any " we have
FPA
Rseq
=
4.1
4
3 v1
2
v1
3
0
c+2"
1
dvn
cn 1
1
0
0
0
cn
Z
c+"
2
v1
3
0
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
Z
1
dv2 +
c+"
1
dv2
c+"
(vn
1 )g)
"
1
4c
c+"
4
3 v1
1 34c"+" +5c
16
c+2"
2
1 18c"+17"2 +9c2
16
c+2"
3
2
if
0
if
1
4c
!
1
dv1
c1
1
dv1
c + 2"
3
1 23" +51c" +51c "+7c
24
(c+")(c+2")
1
4c
"
1
2c
"
1
2c
if
2
1
dv2
c+"
1
dv2
c2
1
dv1
c + 2"
1 70c"+67"2 +19c2
48
c+2"
2
1
v2
2
1 :::
if
"
c
c
"
Monotonicity in the number of bidders
We emphasize that determining the optimal order is crucial in the sequential model. As in the sequential
all-pay auction in Segev and Sela (2014a) there exists no monotonicity of the revenue in the number of
bidders. For example, with two symmetric bidders such that F1 and F2 are uniform on [0; c] we have
Z c
1
1
1
FPA
Rseq ([0; c] ; [0; c]) =
v1
dv1 = c
2
c
4
0
If we now add a strong bidder with F3 uniform on ["; c + "] such that 0
we have
b1 (v1 ) =
and, for 0
v1
"
8
>
<
>
:
1
3
v1 + " +
b2 (v2 ) =
for "
v1
(3c ")(c+")
4c
b2 (v2 ) =
and for
8
>
>
>
<
>
>
>
:
0
(3c ")(c+")
4c
:
1
3
v1 + " +
v1
p
2
") + v1 "
b2 (v2 ) =
0
if 0
: b (v ) if
1
1
1
3
"
if
v1
c
"
v2
"
(v2 + ") if
"
v2
c
1
3
v2
v1 + " +
"v1
2v1
"+2
v2
1
3
c
8
<
v1
0
"2 + v12
1
3
(v2 + ") if
(v1
p
v2
20
p
"2 + v12
1
3
"2 + v12
v1 + " +
p
v1 + " + "2 + v12
c and place her last then
if 0
if
0
if
0
0
1
2
if
b1 (v1 )
1
2
8
<
q
"
p
2v1
"v1
"2 + v12
"v1
"v1
p
" + 2 "2 + v12
v2
c
"v1
v2
c
"v1
Therefore
FPA
Rseq
([0; c] ; [0; c] ; ["; c + "])
=
1
48
3
2
2
q
(" 2c)
(c
12c
3
42" + 5c" + 22c " + 11c
c2
0
2
2
") + c" +
"
ln @
8c
1
2"
c
+
q
(c
2
") + c"
3
2c
1
A
FPA
FPA
For example, for c = 1 we plot in …gure 8 both Rseq
([0; c] ; [0; c]) in red and Rseq
([0; c] ; [0; c] ; ["; c + "])
FPA
in black as a function of " and when " is large enough (" > 0:847) then we have Rseq
([0; c] ; [0; c] ; ["; c + "]) <
FPA
Rseq
([0; c] ; [0; c]) i.e., adding a third bidder decreased the expected revenue.
R
0.5
0.4
0.3
0.2
0.1
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
epsilon
Figure 8
However, if we place the stronger bidder …rst instead of last then we have
FPA
Rseq
(["; c + "] ; [0; c] ; [0; c])
8 R3 R4
Rc
R c+" 2
2
1
1
1
1
4c
3 v1
>
>
3 v1 c dv2 + 43 v1 2 v2 c dv2 c dv1 + 43 c
3 v1
"
0
>
>
>
3
2
2
3
>
1
64" +144c" +180c "+171c
>
>
= 432
>
c2
< R3 R4
R
R 3c
c
v
c
1
2
1
1
4
3
=
v1 c dv2 + 4 v1 2 v2 1c dv2 1c dv1 + 32c 23 v1
3
"
0
3
>
4
>
R c+" 1
>
>
1
64"3 +324c2 "+135c3
>
+
c
dv
=
3
1
>
c
432
c2
c
>
>
R 32 c 2 2 1
R c+" 1
>
:
1 12c" 4"2 +3c2
3 v1 c dv1 + 3 c c c dv1 = 12
c
"
2
1
c dv1
1
c dv1
if
0
if
1
2c
if
3
4c
1
2c
"
"
"
3
4c
c
FPA
which is indeed always higher than Rseq
([0; c] ; [0; c]).
5
Concluding remarks
We analyze a sequential bidding …rst price auction and demonstrate that it may yield a higher expected
revenue than both the simultaneous bidding …rst price auction and the second price auction when the stronger
21
bidder bids …rst. We therefore argue that when there is reason to believe that bidders are asymmetric the
seller may want to consider ordering them and let them bid one by one. By doing that she may not only
increase the expected payo¤ of the weak bidder but also her expected revenue. If bidders are asymmetric
enough we show that the sequential FPA may also increase e¢ ciency compared to the simultaneous FPA.
This happens however when the stronger bidder bids second.
The lower e¢ ciency achieved in the sequential bidding auction when the stronger bidder bids …rst is
mainly due to the preferred position of the second bidder (when she is the weaker bidder) which allows her
to win the auction easily by bidding the same bid as the …rst bidder. We leave for future research examining
mechanisms that could potentially increase the e¢ ciency (when the revenue is already higher than in the
simultaneous FPA) by reducing the second bidder’s advantage. One such mechanism may be giving a head
start to the …rst bidder. In Segev and Sela (2014b) we discuss head starts in the sequential all-pay auction
in which the second bidder can win either by bidding strictly more than the …rst bidder by some constant
(an additive head start) or by bidding the …rst bidder’s bid multiplied by some constant larger than one
(multiplicative head start) and show that head starts can increase both e¢ ciency and expected revenue.
Other such mechanisms may be a reserve price, a minimal bid or a winning bid mechanisms.
Other important questions such as determining the optimal order of bidders for a general number of
bidders and general distributions are also still open and will have to be addressed in continuation research.
References
[1] Doni N., and D. Menicucci (2013), "Revenue comparison in asymmetric auctions with discrete valuations", The B.E. Journal of Theoretical Economics, 13(1), pp. 429–461.
[2] Fibich G., A. Gavious and A. Sela (2004), "Revenue equivalence in asymmetric auctions", Journal of
Economic Theory, 115, pp. 309–321.
[3] Fischer S., W. Guth, T. Kaplan and R. Zultan (2014), "Auctions and Leaks: A Theoretical and Experimental Investigation", mimeo, Munich Personal RePEc Archive (MPRA).
[4] Gavious A., and Y. Minchuk (2014), "Ranking asymmetric auctions", International Journal of Game
Theory, 43, pp. 369–393.
[5] Hubbard T., R. Kirkegaard and H. Paarsch (2013), "Using Economic Theory to Guide Numerical Analysis: Solving for Equilibria in Models of Asymmetric First-Price Auctions", Computational Economics,
42, pp. 241–266.
[6] Kaplan T., and S. Zamir (2012), "Asymmetric …rst-price auctions with uniform distributions: analytic
solutions to the general case", Economic Theory, 50, pp. 269–302.
22
[7] Kirkegaard R., (2009), "Asymmetric …rst price auctions", Journal of Economic Theory, 144, pp. 1617–
1635.
[8] Kirkegaard R., (2011), "Ranking Asymmetric Auctions using the Dispersive Order", mimeo, University
of Guelph.
[9] Kirkegaard R., (2014), "Ranking asymmetric auctions: Filling the gap between a distributional shift
and stretch", Games and Economic Behavior, 85, pp. 60-69.
[10] Linster B., (1993), "Stackelberg rent-seeking", Public Choice, 77, pp. 307-321.
[11] Ludwig S., (2006), "Private Information in Contests - Sequential versus Simultaneous Contests", mimeo,
University of Bonn.
[12] Maskin E., and J. Riley (2000),"Asymmetric Auctions",The Review of Economic Studies, 67(3), pp.
413-438.
[13] Mares V., and J. Swinkels (2014), "On the analysis of asymmetric …rst price auctions", Journal of
Economic Theory, 152, pp. 1-40.
[14] Morgan J., (2003), "Sequential contests", Public Choice, 116, pp. 1–18.
[15] Myerson R., (1981), "Optimal Auction Design", Mathematics of Operations Research, 6, pp. 58-73.
[16] Roberts J., and A. Sweeting (2013), "When Should Sellers Use Auctions?", American Economic Review,
103(5), pp. 1830–1861.
[17] Segev E., and A. Sela (2014a), "Multi-Stage Sequential All-Pay Auction", European Economic Review,
70, pp. 371-382.
[18] Segev E., and A. Sela (2014b), "Sequential All-Pay Auctions with Head Starts", Forthcoming in Social
Choice and Welfare.
[19] Tanno T., (2009), "Linear Bid in Asymmetric First-Price Auctions", mimeo.
6
Appendix
Proof of Proposition 2
23
For " = 0; Rsim = 31 c > Rseq = 41 c. De…ne the di¤erence between the expected revenue in both auctions
as
g (")
FPA
FPA
= Rsim
Rseq
8
c2 ("+c)2
>
>
< "("+2c)p"("+2c)
=
>
c2 ("+c)2
>
p
:
("+2c)"
ln
"("+2c)
p
p
q
"("+2c)
"
1 (2c ")" "("+2c)
p
+ 2 arctan
+
"+2c
4
c2 ("+c)
"+2c+ "("+2c)
p
p
q
"+2c
"("+2c)
c" "("+2c)
"
p
+ 2 arctan
ln
"+2c +
("+c)3
"+2c
"+2c+
We …rst show that g (") < 0 for every "
" + 2c
f (") = ln
" + 2c +
p
p
" (" + 2c)
+ 2 arctan
" (" + 2c)
therefore we show that f (") < 0 for every "
f (") < 0 ,
3
(" + c)
c, g (") =
r
"
" + 2c
c2 ("+c)2
("+2c)"
+
c"
p
"("+2c)
0
if
"("+2c)
c. Note that when "
p
c" " (" + 2c)
if
"
c
c
"
f (") for
p
" (" + 2c)
(" + c)
3
c. Now
" + 2c +
< ln
" + 2c
p
p
" (" + 2c)
2 arctan
" (" + 2c)
r
"
" + 2c
p
"("+2c)
p
p "
p
q
(("+c) "2 +2c" c("+2c) "+2c
)
"
The r.h.s is an increasing function of " since
ln
2 arctan
=
3
2
2
"+2c
" +3" c+2"c
"+2c
"("+2c)
q
q
p
p
2
"
"
and (" + c) "2 + 2c" c (" + 2c) "+2c
> 0 , (" + c) "2 + 2c" > c (" + 2c) "+2c
, (" + c) "2 + 2c"
p
p
q
"+2c+ "("+2c)
(3+ 3)
2
"
2
2
p
2 arctan
j"=c = ln 3 p3
c (" + 2c) " = " (" + 2c) > 0. Moreover ln
"+2c
(
)
"+2c
"("+2c)
p
q
q
"+2c+ "("+2c)
1
"
p
2 arctan
2 arctan
= 0:269 76. Therefore ln
0:269 76 for every "
c.
3
"+2c
"("+2c)
"+2c
p
p
c("2 +"c 3c2 )
"("+2c)
c" "("+2c)
d
The l.h.s is …rst increasing in " and then decreasing, since d"
=
and
3
4
"+2c
("+c)
("+c)
p
"2 + "c 3c2 > 0 , " > 12 c 13 1 = 1: 302 8c. Therefore, the maximum of the l.h.s is achieved when
pp
p
p
p
p
2( 13 1) 2 13+10
c" "("+2c)
c" "("+2c)
1
p
p
j"= 1 c( 13 1) =
= 0:221 3. Thus
" = 2 c 13 1 and then
3
("+c)3
("+c)3
2
(1+ 13)
p
pp
p
p
q
q
"+2c+ "("+2c)
2( 13 1) 2 13+10
(3+ 3)
1
"
p
p
2 arctan
2 arctan
< ln 3 p3
ln
3
3
"+2c . We now show
1+
13
(
)
"+2c
"("+2c)
(
)
that g (") is decreasing in " on [0; c]. We have
0
1
p
"2 ("+2c)("3 +4c3 +4"c2 +4"2 c)
2
2
2
2
p
4c (" + c) " + 3c + 2"c " + 2c"
d
4c2 ("+c)("2 +3c2 +2"c) "2 +2c1 "
@
A<0,
q
g (") =
p
3
2
3
"
d"
4" (" + 2c)
p" +2c" + 2 arctan
+ ln "+2c
2
"+2c
"+2c+ " +2c"
!
p
r
2
3
3
2
2
" (" + 2c) " + 4c + 4"c + 4" c
" + 2c + "2 + 2c"
"
p
p
> ln
2 arctan
2
2
2
2
2
" + 2c
4c (" + c) (" + 3c + 2"c) " + 2c"
" + 2c
" + 2c"
d
d"
"+2c+
Both the r.h.s and the l.h.s of this inequality are increasing functions of " on [0; c] since
d
d"
"2 ("+2c)("3 +4c3 +4"c2 +4"2 c)
p
4c2 ("+c)("2 +3c2 +2"c) "2 +2c"
=
q
p
(2"7 +36c7 +188"2 c5 +215"3 c4 +150"4 c3 +63"5 c2 +104"c6 +16"6 c)
"+2c+p"2 +2c"
d
"
p
> 0 and d"
ln "+2c
2 arctan "+2c
=
3
2 +2c"
2 2
2
2
2
2
"
c ( " +2"c) ("+c) (" +3c +2"c)
q
p
"2 ("+2c)("3 +4c3 +4"c2 +4"2 c)
"+2c+p"2 +2c"
"
p"
> 0. Moreover, lim"!0 4c2 ("+c)("2 +3c2 +2"c)p"2 +2c" = 0 and lim"!0 ln "+2c
2 arctan "+2c
=
("+c) "2 +2c"
"2 +2c"
1 2
4"
(" + 2c)
0.
24
Finally, the di¤erence between the l.h.s and the r.h.s is increasing in " on [0; c] since
!!
!
p
r
"2 (" + 2c) "3 + 4c3 + 4"c2 + 4"2 c
d
" + 2c + "2 + 2c"
"
p
p
ln
2 arctan
d" 4c2 (" + c) ("2 + 3c2 + 2"c) "2 + 2c"
" + 2c
" + 2c
"2 + 2c"
=
"3 (" + 2c) 2"6 + 20c6 + 159"2 c4 + 130"3 c3 + 59"4 c2 + 100"c5 + 16"5 c
> 0:
p
3
2
2
4 "2 + 2"c c2 (" + c) ("2 + 3c2 + 2"c)
Moreover, since
p
g (0) = lim"!0
"("+2c)
"("+2c)
0
and
g (c) =
4c
p
3 3
o¤ 0 < " (c)
p
p
q
("+2c "2 +2c")
"
ln "+2c+p"2 +2c" + 2 arctan
"+2c +
)
(
c2 ("+c)2
1 (2c ")" "("+2c)
4
c2 ("+c)
=
1
12 c
>
p
q
p
(3 p3)
3
1
+
2
arctan
= 0:04099 5c < 0 we conclude that there exists a cut3 + 8
(3+ 3)
FPA
FPA
FPA
FPA
c such that for all " < " (c) ; Rsim
> Rseq
and for all " > " (c) ; Rsim
< Rseq
. It
ln
remains to show that " (c) is increasing in c. The cuto¤ " (c) is de…ned by the following equation
p
p
r
" (" + 2c)
" + 2c
1 (2c ") " " (" + 2c)
"
=0
g (") = 0 , ln
+ 2 arctan
+
p
" + 2c
4
c2 (" + c)
" + 2c + " (" + 2c)
p
("+2c "("+2c)) +2 arctan p " + 1 (2c
p
( "+2c ) 4
"("+2c))
"+2c+
(
p
Then d"dc(c) =
("+2c "("+2c)) +2 arctan p " + 1 (2c
d
ln
p
( "+2c ) 4
d"
("+2c+ "("+2c))
Proof of Proposition 3
d
dc
We have
SP A
Rsim
FPA
Rseq
=
Therefore it is easy to check that "
FPA
Rseq
(" ) =
p
")" "("+2c)
c2 ("+c)
ln
p 2
c(1+ 3)
r
p
p
(1+2 3) 1+23 3
959 7c > 0 therefore "
8
<
:
1 3"+2c
6 c "+c
1 3"+2c
6 c "+c
1
4
p
")" "("+2c)
c2 ("+c)
(c + ") =
"c
"+c
!
!
=
1 3"2 c2
12 "+c
1 3" 2c
6 c "+c
=
"
c
> 0.
if
0
"
if
c
c
"
p1 c < c. Moreover " (c) is increasing in c. Finally, RF P A ("
sim
3
p p !
p p
p
p
q
1+2 3
1+2 3
2
3
1
1+2
3
)p
1 (
1p
+ 12
=
+ 2 arctan
p p
p
1+2
3
3
1+
)
(
1+2 3+ 1+2 3
(c) =
ln
(c) < " (c).
)
0:01
Proof of Proposition 4
We …rst need to prove that
U1seq F P A < U1simF P A < U1SP A
where
U1seq F P A
=
U1simF P A
=
U1SP A
=
8
<
:
2
1 (c+")
12
c
if 0
1 3"2 +c2
6 c+"
if
c
2
(c + ")
2" (2c + ")
p
"
c
"
c2
" (2c + ")
3 ln
c+"+
p
" (2c + ")
c
!
2 arccosh
c+"
c
!
!
(c
")
+2 arccosh
c+"
c
1 3c" + 3"2 + c2
6
c+"
Now U1simF P A < U1SP A , g (") =
3
2
2
3
1 (3" +7c" +8c "+3c )
3
c(c+")3
0. Now
25
p
"(2c+")
3 ln
p
c+"+
"(2c+")
c
>
dg(")
d"
=
p
3
2
2
3
1 (3" +7c" +8c "+3c )
3
c(c+")3
d
d"
"(2c+")
3 ln
p
c+"+
"(2c+")
c
+ 2 arccosh
c+"
c
0 and for " = 0; g (0) = 0. Therefore we conclude that indeed g (") > 0 for every ".
p
c+"+ "(2c+")
Moreover, for 0
"
c, U1seq F P A < U1simF P A , f (") = 3 ln
c
p
2
2
1 ( 4c"+" +6c ) "(2c+")
> 0. Now
6
c3
p
p
2
2
"(2c+")
c+"+
df (")
d
1 ( 4c"+" +6c ) "(2c+")
c+"
=
=
3
ln
2
arccosh
3
d"
d"
c
c
6
c
for " = 0; f (0) = 0: Therefore we conclude that indeed f (") > 0 for every 0 " c.
p
c+"+ "(2c+")
For c ", U1seq F P A < U1simF P A , h (") = 3 ln
2 arccosh c+"
c
c
0. Now
dh(")
d"
=
d
d"
3 ln
p
c+"+
for " = c; h (c) = 3 ln 2 +
h (") > 0 for every c
"(2c+")
c
p
3
2 arccosh
2 arccosh (2)
c+"
c
p
3
2
p
2
2
1 (8c"+" +3c ) "(2c+")
3
(c+")3
=
"2 ("3 +4c"2 +5c2 "+c3 )
=
p
c(c+")4
c+"
c
2 arccosh
"(c+")(2c ")
2c3
p
"(2c+")
> 0 and
"(2c+")
p
2
2
1 (8c"+" +3c ) "(2c+")
3
(c+")3
"2 (3c+")2
(c+")4
p
"(2c+")
>
> 0 and
= 0:450 93 > 0. Therefore we conclude that indeed
".
Next we wish to prove that
U2SP A < U2simF P A < U2seq F P A
where
U2SP A
=
1 c2
6c+"
2
U2simF P A
=
U2seq F P A
=
c2 (c + ")
1
p
arcsin
2" (2c + ") " (2c + ") 2
8
< 1 7c2 4c"+"2 if 0 " c
24
c
1 c2
:
if
c "
3 c+"
c
c+"
2 arcsin
p
" (2c + ")
(c + ")
!!
First we show that U2SP A < U2simF P A . We have U2SP A < U2simF P A , g (") = 21
p
p
"(2c+")
(7c"+5"2 +3c2 ) "(2c+")
2 arcsin (c+")
> 0. Now
+ 13
3
(c+")
p
p
2
2
"(2c+")
dg(")
d
1
1 (7c"+5" +3c ) "(2c+")
c
=
=
arcsin
2
arcsin
+
3
d"
d"
2
c+"
(c+")
3
(c+")
+
1 c2 (c + 2")
2 " (2c + ")
arcsin
2 2
cp
"
(c+")4 "(2c+")
and for " = 0; g (0) = 0: Therefore we conclude that indeed g (") > 0 for every ".
p
3
2
2
3
1 (" 3c" +2c " 12c ) "(2c+")
Moreover, for 0 " c we have U2simF P A < U2seq F P A , f (") = 12
3
c (c+")
p
"(2c+")
c
arcsin c+" + 2 arcsin (c+")
> 0. Now
df (")
d"
=
for every 0
"("3 (c+")+3c2 (c ")(2c+"))
p
4c3 (c+")2 2c"+"2
"
=
c.
3
2c
p"
(c+")4 "(2c+")
> 0
1
2
+
> 0 and for " = 0; f (0) = 0. Therefore we conclude that indeed f (") > 0
Finally, for c " we have U2simF P A < U2seq F P A , h (") =
p
"(2c+")
2 arcsin (c+")
> 0. Now
dh(")
d"
c
c+"
> 0 and for " = c; h (c) =
p
3
2
Therefore we conclude that indeed h (") > 0 for every c
Proof of Proposition 5
26
1
2
".
p
2
2
1 (5c"+4" +3c ) "(2c+")
3
(c+")3
+ arcsin
1
2
+ 2 arcsin
1
2
p
3
2
+arcsin
c
c+"
+
= 0:181 17 > 0.
>
We have
SP A
Rsim
FPA
Rseq
SP A
FPA
For " = 0 we have Rsim
Rseq
=
and the solution is:
0
"
B
(c) = B
@1
Moreover
d
d"
1
1
r
2 3 q 7
16
SP A
Rsim
1
12 c.
>
>
>
:
1 2"3 6c"2 +c3
12
c2
2
2
1 (2c (2c ") )
4
c
if
0
"
c
if
c
"
2c
1
2c
if
When 0
"
0
+
3
4
1
1 p B
+ i 3B
@ rq
2
3
7
FPA
Rseq
=
FPA
Rseq
and for all " > "
=
8
>
>
>
<
16
d
d"
1 2"3 6c"2 +c3
12
c2
3
4
=
sr
1 2c "
2 " c2
SP A
FPA
(c) ; Rsim
< Rseq
: Finally when c
FPA
< 0.
Rseq
27
"
SP A
FPA
c, we have Rsim
Rseq
= 0 , 2"3 6c"2 +c3 = 0,
3
+
2c
1
7
3C
+ C
16 4 A
1
2
sr
3
1
7
3C
+ C c = 0:442 13c
16 4 A
< 0. Therefore, for all " < "
"
2c and when 2c
SP A
(c) ; Rsim
>
SP A
" we have Rsim