Second-price Auctions with Power-Related
Distributions
Luke Froeb
Owen Graduate School of Management
Vanderbilt University
Nashville, TN 37203
[email protected]
Steven Tschantz and Philip Crooke
Department of Mathematics
Vanderbilt University
February 21, 2001
Abstract
We analyze a class of parametric second-price auction models where
asymmetry is modeled by allowing bidders to take dierent numbers
of draws from the same distribution. We compute the closed-form
distribution of price and construct likelihood and method-of-moments
estimators to recover the underlying value distribution from observed
prices. We derive a Herndahl-like formula that predicts merger effects and nd that merger eects depend on the shares of the merging bidders, the variance, and the \shape" of the distribution. We
generalize the model by allowing bidders to mix over power-related
distributions. The dominant strategy equilibrium implies that an auction among bidders who mix over distributions can be expressed as
a mixture of auctions. This implies that an auction among bidders
with potentially correlated values can be expressed as a mixture over
independent power-related auctions.
Keywords: second-price, private-values auctions; merger; antitrust.
JEL Classications: D44: Auctions; L41: Horizontal Anticompetitive
Practices. C50: Econometric Modeling;
Corresponding
author.
We wish to acknowledge useful comments from Bruce Cooil.
Support for this project was provided by the Dean's fund for faculty research.
1 Introduction
A merger, modeled as a bidding coalition, will have a price eect in a secondprice auction only when the merging parties are the two best bidders. The
frequency this event, and the magnitude of the resulting price change determine expected price eects, which depend critically on the joint distribution
of bidder values. Merger evaluation is thus a problem of estimation{how
to recover the underlying joint value distribution from available data. It
is then a simple matter to compute merger eects or the magnitude of the
marginal cost reductions necessary to oset price eects (Tschantz, Crooke,
and Froeb [2000]).
This paper is motivated by the problem of recovering bidder value distributions from price data in private-values English and second-price auctions.
Interest in second-price format is rising with the popularity of Internet auctions where use of proxy bidding serves to \Vickrify" an English auction
(Lucking-Reiley [2000]), making it resemble the Vickrey [1961] second-price,
sealed-bid auction.
Athey and Haile [2000] show that the joint bidder value distribution is
nonparametrically identied in independent private values (IPV) secondprice auctions with data on either: (i) the highest or lowest values and the
identity of the bidder; or (ii) at least two values and two bidder identities.
It is possible that IPV models are also nonparametrically identied with
data on prices (second-highest values) but, even if true, there are reasons
for using parametric models. The length and quality of some data sets make
nonparametric estimation diÆcult, and a parametric approach makes it easy
to incorporate other kinds of information into the estimation, like estimates
of the variance or the price-cost margins of bidders.
In this paper, we analyze a class of parametric second-price auction
models where asymmetry among bidders is modeled by allowing bidders to
take dierent numbers of \draws" from any base distribution. We derive
a closed-form expression for the distribution of price, conditional on the
identity of the winning bidder. One of the implications of bidder asymmetry
is a moment restriction on prices and winning probabilities or \shares," i.e.,
bidders drawing from better distributions win more frequently, and at better
prices, than other bidders. This restriction leads to a method-of-moments
estimator constructed by regressing prices on shares.
The methodology is applied to the problem of predicting merger eects.
Parametric models are likely to be particularly useful for this application
because estimation must be done using available data, and within the time
2
constraints mandated by the merger statutes.1 Power-related auctions give
rise to a Herndahl-like predictor where merger eects depend on the shares
of the merging bidders, the variance, and the "shape" of the distribution,
underscoring the importance of structural estimation as an element of merger
policy.
Mixing over a base of power-related distributions allows us to generalize
the model in the same way that mixed logit models can approximate more
general random utility models (Berry, Levinsohn, and Pakes [1995]; Brownstone and Train [1999]; McFadden and Train [1999]). The dominant strategy
equilibrium of a second-price auction implies that an auction among bidders
who draw from a mixture of power-related distributions is equivalent to a
mixture of power-related auctions. Estimation and analysis is facilitated by
the closed-form expressions of power-related auctions.
2 A Second-price Auction Model
Following the notation in Tschantz, Crooke, and Froeb [2000], the value of
the i-th bidder is specied as the sum of two random variables,
= Xi + Y:
(2.1)
Xi is the private value of bidder i and Y is a common shock to all bidders'
values. The variables (X1 ; : : : ; Xn ; Y ) are mutually independent. We assume
that the common component Y is common knowledge, and simply shifts
observed bids from auction to auction. In the taxonomy of Athey and Haile
[2000], this is a conditionally independent private-values model with with
independent components (CIPV-I).
In Tschantz, Crooke, and Froeb [2000], each Xi is assumed to follow
a Gumbel distribution with dierent means, but the same variance. For
this family of distributions, they derive a closed-form expression for the
distribution of prices. We now generalize their results to the power-related
distributions considered by Waehrer and Perry [1998].
Vi
Denition 1. The random variables
and X2 , having cumulative distribution functions F1 and F2 , respectively, are power-related if there is
some positive number s, such that F2 (x) = [F1 (x)]s = F1s (x) for all x.
X1
Denition 2. A family of power-related distributions generated by
a cumulative distribution function (CDF)
fF s (x) : s 2 R+ g.
1 Section
F (x)
7A Clayton Act, 15 U.S.C. Section 18a.
3
is the set of distributions
Any distribution in a family of power-related distributions can serve as the
base distribution for the family. For example, the Gumbel distributions used
by Tschantz, Crooke, and Froeb [2000] are from the family of power-related
distributions:
Fi (x)
(x )
i
= ee
= F si (x)
(2.2)
x
where F (x) = e e and si = ei .
In a cost auction, where the low bid wins, rather than a value auction
where the high bid wins, two distributions are power-related if F2 (x) =
1 (1 [F1 (x)])s . All of the results and closed-form expressions of the
power-related value auctions extend to power-related cost auctions.
3 Order Statistics for Power-Related Distributions
In this section, the distribution of order statistics for independent draws from
families of power-related distributions are derived. The formulas for order
statistics drawn from heterogeneous distributions can be found in Section
2.8 of David [1981] but unlike David, we compute order statistics conditional on the identity of the winning bidder. By conditioning on the bidders'
identities, we are implicitly assuming that researchers have data on auction
participants, prices, and the identity of the winner.
3.1
Distribution of the Maximum Value
Families of power-related distributions are useful for modeling second-price
auctions because the family is closed under the maximum function, i:e:,
if bidders are making independent draws from distributions in the same
power-related family, then the distribution of the maximum value belongs
to the same family. This property facilitates modeling auction equilibria. In
particular, the maximum function is used to compute winning probabilities
(the probability that a bidder will have a value higher than the maximum of
rivals' values) and prices (the maximum of rivals' values), and to compute
the eects of a merger or bidding coalition (the merged rm has a value
equal to the maximum of coalition member values).
Let Xi for i = 1; 2; : : : ; n be independent random variables from a family
of power-related distributions. Suppose Xi has the CDF Fi (x) = F si (x) for
a positive constant si . We assume that F (x) is dierentiable and therefore
continuous. The results below can be extended to discrete distributions but
the analysis is complicated by the possibilities of ties.
4
Lemma 1. If P
Xmax = maxfX1 ; X2 ; : : : ; Xn g, then Xmax has CDF F smax (x)
n
where
smax
=
i=1
si .
Proof: Using independence, the CDF of the maximum is
Fmax (x)
= Prob(Xmax x) = Prob(Xi x; i = 1; 2; : : : ; n)
n
Y
=
i=1
n
Y
=
i=1
Prob(Xi x)
F si (x)
= (F (x))
Pni=1
si
= F smax (x):
If si is a positive integer, then the F si (x) can be interpreted as the distribution of the maximum of si draws. A strong or high-mean-value bidder is one
whose value is the maximum of a large number of number of draws, and this
increases the mean of the distribution. For example, taking the maximum of
si draws from a Gumbel distribution increases the mean by log(si )=, i.e.,
(t (i + logsi ))
(x) = e
:
(3.1)
Note that the maximum of power-related family of Gumbel variates has the
same variance as the distribution of each variable from which the maximum
is computed.
Waehrer and Perry [1998] derive a closed-form expression for the probability that bidder i wins the auction, i:e:, the probability that Xi is the
maximum,
F
pi
si
e
Prob(Xi = Xmax) = s si
max
:
For power-related Gumbel distributions, winning probabilities have the familiar logit form
pi
e
= Pn
i
j =1
ej
:
In what follows, we will refer to the winning probability as the \share" of
bidder i.
5
3.2
Distribution of the Price
In an open auction, the second-highest value determines the price and, on
average, dierent bidders win at dierent prices. Bidders drawing from
more favorable distributions win more frequently, and at better prices than
those drawing from less favorable distributions because they are not bidding
against themselves. In other words, when a high-mean-value bidder wins,
the losing bidders are relatively weak by virtue of the fact that the highmean-value bidder is not among them. Consequently, they are easier to
outbid, on average.
The key to computing the distribution of the second-highest value is that
the probability that Xi is the maximum value is independent of the value
of the maximum. This independence property permits us to express the
distribution of the second-highest bid as the weighted sum of the distribution
of the maximum and the distribution of the maximum of the losing bidders.
To prove this, it will suÆce to consider two bidders.
Proposition 1. Suppose X1 and X2 are distributed as F si (x) for i = 1; 2.
Then for any x, the probability
Prob(X1 = maxfX1 ; X2 g j maxfX1 ; X2 g x) =
s1
s1
+ s2
p1
(3.2)
is a constant that is independent of x. The distribution of X1 given that
X1 is the maximum of X1 and X2 is the same as the distribution of the
maximum, i.e.,
Prob(X1 x j X1 = maxfX1 ; X2 g) = F s1 +s2 (x):
(3.3)
The distribution of X2 given that X1 is the maximum of X1 and X2 is given
by
Prob(X2 x j X1 = maxfX1 ; X2 g) =
1
p1
F
s2
(x) + 1
1
p1
F s1 +s2 (x):
(3.4)
Proof: See Appendix.
We introduce some notation for the case of n bidders, drawing random
values, X1 ; X2 ; : : : ; Xn , as originally specied.
Denition 3. The symbol, Xi = maxfXj : 1 j n; j 6= ig, denotes
the maximium value among the bidders, excluding bidder i.
6
The random variable Xi has CDF F si (x), where
si
n
X
=
sj :
j =1
6
j =i
The distribution of the second-highest value given that Xi = Xmax follows
from Proposition 1,
FXi jXi <Xi (x)
=
1
pi
1
Fi (x) + (1
pi
)Fmax (x):
(3.5)
The symbol Xi jXi > Xi denotes the second-highest value given that
bidder i has the highest value, and thus wins the auction. From the distribution of the second-highest value we compute the means and variances of
the observed bids in terms of the means and variances of the power-related
value distributions. Letting (s) and 2 (s) be the mean and variance of the
distribution with CDF F s (x), we obtain
E(Xi j Xi > Xi ) =
Var(Xi j Xi > Xi ) =
1
1
pi
+
3.3
(si ) +
pi
2 (s
1
pi
1
i ) + 1
1
1
pi
1
pi
1
(smax )
pi
(3.6)
2 (smax )
((smax)
2
(si )) :
(3.7)
Joint Distribution of the Three-Highest Values
The same kinds of closed-form expressions can be derived for the joint distribution of the k-highest values. These expressions can be used to compute
likelihood functions and moments for losing, as well as winning bids. In
this section, we compute the joint distribution of the three-highest of the Xi
conditioned on the identities of the highest and second-highest bidders. It
suÆces to consider just three random variables X1 , X2 , and X3 , taking X1
to be the highest value, X2 to be the second-highest value, and X3 to be the
third-highest value, i.e., the maximum of the other Xi . In the following, let
s12 = s1 + s2 , s13 = s1 + s3 , s23 = s2 + s3 , and s123 = s1 + s2 + s3 .
Proposition 2. If Xi has CDF F si (x) for i = 1; 2; 3, then
Prob(X1 > X2 > X3 ) =
s1
s2
p2
= p1 s123 s23
p2 + p3
7
(3.8)
The joint distribution of X1 , X2 , and X3 , given X1 and X2 are the rst and
second-highest, respectively, is
Prob(X1 x1 ^ X2 x2 ^ X3 x3 j X1 > X2 > X3 )
=
s123
s1
s3
s23
for
X2
s23
s2
s2
F s1 (x1 )F s2 (x2 )F s3 (x3 )
s2 s3
F (x1 )F (x3 ) + s
s1
s23
12 s123
x2 x1.
F s12 (x2 )F s3 (x3)
s12
F
s123
(x3 )
Hence the joint distribution of X2 and
are the rst and second-highest, respectively, is
x3
X3
given
X1
and
Prob(X2 x2 ^ X3 x3 j X1 > X2 > X3 )
s123
=
s23
s1
s2
s3
s23
The distribution of
respectively, is
s2 s3
F (x3 ) + s
s23
X3
s2
F s2 (x2 )F s3 (x3 )
given
12 s123
and
X1
s12
F
X2
s123
F s12 (x2 )F s3 (x3 )
(x3 )
:
are the rst and second-highest,
Prob(X3 x j X1 > X2 > X3 ) =
s123
s1
for
x3
x2 .
s23
s2
s
1
F s3 (x)
s12
s3
s23
F (x) + s
s23
s2 s3
12 s123
F
s123
(x)
Proof: See Appendix.
Denition 4. The symbol, Xij = maxfXk : 1 k n; k 6= i; k 6= j g,
denotes the maximum of the bidders' values not including bidders i and
The random variable Xij has CDF F sij (x), where
sij
=
n
X
k=1
6
k =i;j
8
sk :
j
Let sij = si + sj . It follows from Proposition 2 that the expectation of the
third-highest value given Xi > Xj > Xij is
E(Xij jXij < Xj < Xi ) =
si
(sij ) ssiji (si) + ssijj ssmaxij (smax)
sij
si
smax
ssjj
(3.9)
:
This expression can be used to construct a method-of-moments estimator. In
the above expression, it is worth noting that the following identities relating
si , si , sij and pi , pj hold:
si
si
si
sij
sij
sij
sij
si
=
=
1
pi
1
pi
pi + pj
1
1
pi + pj
1 pi pj
=
:
1 pi
=
4 Recovering the Value Distribution from Bid Data
In this section, we consider the problem of recovering the bidders' value
distributions from observed price data. If we adopt the convention that the
distribution of the maximum value, Fmax (x), is the base distribution for
the family, then the individual distribution functions can be expressed as
pi
Fi (x) = Fmax
(x) where pi = si=smax , and Fmax (x) = [F (x)]smax = F smax (x).
The estimation problem is to recover the winning probabilities, pi , and the
base distribution, Fmax (x), from observed bid data.
Without the common unobservable shock, it would be possible to construct maximum-likelihood estimators directly from the distributions of Section 3. With the common shock, the distribution of the price paid by bidder
i is a convolution of two distributions, i.e., Bi = (Xi jXi < Xi ) + Y .
In our private-values treatment, the variable Y is a nuisance variable that
complicates recovery of the power-related distributions of interest.
In what follows, we assume the existence of data on prices and bidder
identities and their distinguishing characteristics across a sample of auctions.
9
We treat each auction as an independent event, but recognize that this
assumption may not be appropriate in the presence of collusion, as in a
bid-rotation scheme, or with bidder capacity constraints.
4.1
Data on Prices
We develop some two-step method-of-moments estimators to recover the
value distribution from data on prices. In the rst step, we estimate the
winning probabilities, and in the second, the distribution of the maximum
value.
4.1.1 Estimating the Winning Probabilities
In the rst step, the identity of winning bidders are \predicted" as a function
of observed bidder characteristics using a maximum-likelihood estimator,
analogous to a random-utility, discrete-choice model (e:g: Train [1986]). The
log-likelihood is constructed from the probability of winning across a sample
of T auctions as
L
=
T
X
t=1
log(pit ):
(4.1)
The variable i is taken as a function of t that gives the winning bidder
in the t-th auction, i:e:; pit is the probability that the t-th auction is won
by bidder i. The estimated location parameters, p^it , are the tted values
from this estimation. For the Gumbel distribution of [2000], Equation 4.1
is equivalent to a logit log-likelihood
L
=
T
X
t=1
eit
log( Pnt
k =1
ekt
)
where it is the location parameter of i-th bidder's value distribution, dened
in Equation 3.1. With data on rank-ordered bidder identities, the liklihood
would include the probabilities of lower-valued bidders, analogous to the
rank-ordered logit model of Hausman and Ruud [1987].
4.1.2 Recovering Fmax from Prices
Once we have estimated probabilities, a straightforward method-of-moments
approach is suggested by Equation 3.6. The moment restriction for the price
paid by bidder i is
10
E(bi ) = E(Xi jXi < Xi ) + E(Y )
(4.2)
1
1
(smax (1 p^i )) + 1
(smax ) + E (Y )
=
p^i
p^i
where si = smax(1 p^i ), p^i is the estimated probability that bidder i wins
the auction, and bi is the observed price, i:e:; the realized value of the random variable, Bi . A minimum distance estimator can recover the unknown
parameters of the base distribution, Fmax (x) = [F (x)]smax from the observed
prices. The expected value of the common value component, E (Y ), can also
be estimated as a function of observable auction characteristics, but unobserved variation in Y may be correlated with the auction participation
decision. If so, this would induce a spurious correlation between the characteristics of auction participants and prices.
4.2
Within-auction Estimators
Estimators based solely on prices have diÆculty distinguishing betweenauction variation in Y from within-auction variation in Xi . The problem
is that an outlying price could mean either that the variance of Xi is large
or that the variance of Y is large. To better distinguish between the two
possibilities requires data on losing bids. In these cases, within-auction estimators based on the dierences between bids permit more precise estimation
of Fmax .
4.2.1 Dierence between the Two-highest Values
In a second-price sealed-bid auction, data on the rst two values might be
observed. Lower-ranked values are probably less precise for the same reason that lower-ranked choices are less precise (Hausman and Ruud [1987]).
A method-of-moments estimator can be constructed from the dierence between the highest and second-highest bids. Note that this is also the surplus
or price-cost margin of bidder i which may also be observed.
Denition 5. The symbol, i = Xmax (Xi jXi < Xi ), denotes the
dierence between the two highest values.
Let Æi be the observed realization of i . The moment restriction is computed
from Equation 3.6 as
E(Æi ) =
1 ((s
max )
p
^i
11
(smax (1
p^i )))
(4.3)
A minimum-distance estimator can be used to recover the parameters of
the distribution Fmax . This is equivalent to a regression of the Æi on the
right-hand-side of Equation 4.3.
In the special case of a Gumbel distribution, the distribution of the
dierences has a closed form expression that can be used to construct a
maximum-likelihood estimator,
Fi (t)
=1
1
pi + pi et
(4.4)
:
4.2.2 Dierence Between the Second and Third-highest Values
In government procurement, losing oral bids are sometimes recorded (e:g:
Brannman and Froeb [2000]). Depending on the precise bidding mechanism,
the dierence between the second and third-highest bids can be taken as
the dierence between the second and third-highest values because it is a
dominant strategy for losing bidders to bid up to their values. In contrast,
the dierence between the two-highest bids is not informative about the
value distribution because the winner is trying only to outbid the secondhighest-value bidder.
We construct a method-of-moments estimator using the dierence between the second and third-highest bids.
Denition 6. The symbol, ij = (Xi jXi < Xi ) (Xij jXi
Xi ),
< Xj <
denotes the dierence between the second and third-highest values in an
auction where the two-highest bidders are i and j respectively.
Let Æij be the observed realization of ij . The moment restriction is
computed from Proposition 2 as
E(Æij ) =
pj
pj (pi + pj )
pi + pj (1
+
1
p2i
pi pj (pi
(smax (1
pi )
1)
+ p2j pj 1
pi (pi + pj )
1
!
pi
pj ))
(smax (1
pi ))
(smax ):
As above, a method-of-moments estimator of the parameters of the distribution Fmax can be constructed from this moment restriction.
12
Again, in the special case of the Gumbel distribution, the dierences
between the second- and third-highest values has a closed-form distribution
that can be used to construct a maximum likelihood estimator,
Fi;j (t)
=1
+ pij
pj + pij et
pj
+ pj + pij
pi + pj + pij et
pi
:
(4.5)
5 Predictors of Merger Eects
In this section, we apply the model to the problem of merger prediction.
To model merger eects, we assume that the value of the merged rm is
the maximum of its coalition member values. This merger characterization
has been used by the antitrust enforcement agencies to model the eects of
mergers between hospitals, mining equipment companies, defense contractors, and others (Baker [1997]). From Lemma 1, we know that the value
distribution of the merged rm is from the same power-related family as its
coalition members. This property means that post-merger expected prices
lie on the same price/share moment restriction as the pre-merger expected
prices. Since the share of the merged coalition is the sum of their pre-merger
shares, the post-merger expected price can be computed from the moment
restriction. This relationship gives rise a Herndahl-like formula.
The expected prot to bidder i, since bidder i wins only a fraction pi of
the auctions is
E(proti )
pi E(Xmax
=
pi (smax )
=
(smax )
Bi )
1 pi
(si ) +
(smax (1
1
pi )):
1
pi
(smax )
(5.1)
Hence, a bidder's prot is simply a function of pi .
Denition 7. The expected prot to a bidder with winning probability p is
h(p)
= (smax )
(smax (1
p))
The total expected prot to all bidders is
E(prot) n
X
i=1
E(proti ) =
n
X
i=1
h(pi )
(5.2)
If bidders i and j with winning probabilities pi and pj merge, then their share
after the merger is pi + pj . Because the auction is eÆcient, the increase in
13
the expected prot to the bidders is equal to the loss in revenue to the
auctioneer. This implies that the expected revenue loss of the merger is
E(prot) = h(pi + pj ) h(pi ) h(pj ):
(5.3)
The curvature in the h function determines the loss of revenue due to a
merger.
The merger eects are also related to the standard deviation of the
Xi value components. Suppose F] (x) = F ((x
b)=a) is a translated and
rescaled version of F (x). Then the family of distributions power-related
to F] (x) results from translating and rescaling the distributions powerrelated to F (x). Dening ] and ]2 from F] , we see that for each s,
] (s) = a(s) + b, ]2 (s) = a2 2 (s). Thus taking h] dened as in Denition 7, we have h] (p) = ah(p). One is tempted to argue that a larger
standard deviation in the base distribution leads to bigger merger eects,
but a larger standard deviation also changes the winning probabilities of the
merging bidders. If it makes them smaller, then a larger standard deviation
can reduce the eects of a mergers.
6 Families of Power-related Distributions
For the uniform and extreme-value families, both (s) and (s) have closedform expressions.
6.1
Uniform Power-related Distributions
Let
(x
if x 2 [a; b]
0;
otherwise
smax
and let F (x) be the base distribution for the family. If smax > 1, then
the base distribution is \pushed" towards the upper bound of the domain.
If smax < 1, then the base distribution is pushed towards the lower bound
of the domain.
The moments for the family are
F (x)
=
a
;
b a
(s)
=
2 (s)
=
a + bs
s+1
(b
(6.1)
a)2 s
(s + 1)2 (s + 2)
14
(6.2)
so that prot is given by the function
h(p)
=
p(b a)smax
:
(1 + smax)(1 + (1 p)smax)
(6.3)
DENSITY
1, SHARE= 0.5
2, SHARE= 0.3
3, SHARE= 0.2
1+2, SHARE= 0.8
6.5
7
7.5
8
8.5
9
9.5
10
VALUE
Figure 1: Uniform Family: Value Density Functions
In Figures 1, 2, and 3, the eects of a merger on a family of uniform(6; 10)
distributions with smax = 5 are graphed. There are three bidders with
winning probabilities equal to (0:2; 0:3; 0:5) and we simulate the eects of
a merger between the last two bidders. Following the merger, the winning
probabilities are (0:2; 0:8).
In Figure 1, the value probability densities of the three bidders are
graphed, along with the density of the merged rm (thick line). The nonmerging rm has a density that is graphed with a dashed line. The merged
rm has a value distribution with more mass in the higher values, which
increases the post-merger mean by about 7 percent.
In Figure 2, the densities of the winning bids are graphed. Note that the
bidders with the higher-mean values win at lower-mean prices. The merged
rm has a price distribution with more mass in the lower values. The merger
does not aect the distribution of the bids for the non-merging rms.
In Figure 3 price vs: share relationship is plotted. Note that larger rms
win at lower prices as in Equation 3.6. Because the value distribution of
the merged rm (the maximum of the two merging rms' values) belongs
15
DENSITY
1, SHARE= 0.5
2, SHARE= 0.3
3, SHARE= 0.2
1+2, SHARE= 0.8
6.5
7
7.5
8
8.5
9
9.5
10
PRICE
Figure 2: Uniform Family: Price Density Functions
PRICE
8.5
8.25
PRE MERGER
8
7.75
7.5
7.25
POST MERGER
0.2
0.4
0.6
0.8
6.75
Figure 3: Uniform Family: Price vs: Share
16
1
SHARE
to the same family, it lies on the same price/share curve. Consequently, the
eect of a merger can be plotted as a movement along this curve, from the
average pre-merger prices to the post-merger aggregate price. We see that
the price decreases from about 8:1 to 7:1, a change of about 12 percent.
6.2
Extreme-value Distributions: Types I, II, and III
In looking for families of power-related distributions, we are led to consider
those families where all distributions have the same shape i.e., those that are
linearly-scaled and translated versions of the base distribution. When limits
exist, these are also limiting distributions for the maximum value, suitably
scaled, as the number of draws approaches innity (e:g: David [1981]). The
Type I (Frechet) distribution is the limiting distribution for distributions
whose domain is bounded from below, the Type II (Weibull) for distributions whose domain is bounded from above, and the Type III (Gumbel) for
distributions whose domain is unbounded. This nomenclature diers from
that often used in economics, where the Type I distribution is referred to as
the Gumbel. The three distributions are presented below.
8 ( b x )a
>
<e x d b a ;
F (x) = e ( d ) ;
>
:exp[ e( b dx ) ];
2 [b; 1), a > 0, d > 0
x 2 ( 1; b], a > 0, d > 0
x 2 ( 1; 1), d > 0
x
To compare the three types, we reparameterize the distributions in terms of
their means, standard deviations, and \shapes" c where
8
>
< 1=a;
c = 1=a;
>
:0;
0 for Type I
c > 0 for Type II
c = 0 for Type III:
We present the reparameterized extreme-value distributions below.
8 p (2c+1)
>
>
e (
>
>
< p (2c+1)
Fc;; (x) = e (
>
>
>
>
:exp[ e( (
c<
(c+1)2 (x )+ (c+1) ) 1c
;
(c+1)2 ( x)+ (c+1) ) 1c
;
c
c
p6
x)
p
6
)
];
17
2 [ p (2c+1)(c+1)(c+1)2 ; 1)
(c+1)
]
x 2 ( 1; + p
(2c+1) (c+1)2
x 2 ( 1; 1)
x
This parameterization implies that the likelihood of the so-called \trinity" distribution (e:g:; Cooil [1995], Embrechts, Kluppelberg and Mikosch
[1999]) is a continuous function of the shape parameter c.
DENSITY
Type I
Type II
Type III
8.5
9
9.5
10
10.5
11
11.5
VALUE
Figure 4: Extreme-value Densitities
The density function F 0 (x) is plotted for the three types in Figure 4.
Each of the three base distributions has been normalized to have the same
mean and standard deviation, = 10 and = 1. Type I and Type II
distributions are plotted with shape parameters, c = 0:3 and c = 1:1,
respectively. The distinguishing feature of the Type I and Type II distributions are the upper and lower bounds, respectively, on the domain.
To compute the eects of a merger, we calculate
s
Fc;;
(t)
with
c (s )
=+
(sc
= Fc;c (s);c (s) (t)
1) (1
c)
s
(1 2c)
c
and
if c 6= 0. In the limit as c ! 0,
0 (s)
c (s)
= sc
=+
18
p
6
(6.4)
c2
(1
c)2
(6.5)
(6.6)
log(s)
(6.7)
PRICE
10
9
8
7
PRE MERGER
POST MERGER
Type I
6
Type II
5
Type III
0.2
0.4
0.6
0.8
1
SHARE
Figure 5: Extreme-value Family: Price vs: Share
and 0 (s) = . Then, with smax 1,
hc (p)
=
=
c (1)
c (1
(1 (1
p)
p)c )
(1
c)
s
c
(1 2c)
c2
(1
c)2
(6.8)
provided c 6= 0. In the limit as c ! 0, we have
h0 (p)
=
p
6
log(1
p)
(6.9)
which is the formula derived in Tschantz, Crooke and Froeb [2000] for the
logit auction model.
In Figure 5, we plot the eects of a merger using the three extremevalue curves considered above and the same three winning probabilities,
(0:2; 0:3; 0:5). Again, we consider a merger between the two-largest rms.
The pre and post-merger share/price equilibrium points are denoted with
dots for each of the rms. The merger has its biggest eect in the Type I
distribution. The eect of the merger is not only to increase the mean of
the distribution of the merged rm, but also to increase its variance. This
means that when it wins, it is likely to be further away from the secondhighest bid, increasing its prot. In contrast, the Type II distribution has a
19
very small merger eect. Here the eect of a merger is to increase the mean
and decrease the variance. It is the decrease in variance that gives it a small
merger eect. The Type III distribution, which has a constant variance, has
a merger eect that lies in between the other two types.
Note the critical role played by the curvature of the functions in Figure
5. For the Type I merger, the price eect is only 1:8 percent; for Type II 35:7
percent; and for Type III, 4:8 percent. This nding suggests estimating the
shape parameter of the trinity distribution that combines all three extreme
value distributions.
7 Mixtures of Power-related Distributions
The biggest drawback to using power-related distributions is the restrictive
way that joint bidder value distributions are modeled. We would like to
be able to accommodate more general private-value models, including correlations across bidder values. This is especially important for the merger
application where the most critical question is whether the merging rms
are \closer" to one another than they are to the other competitors.
Mixtures of power-related distributions can approximate more general
value distributions in the same way that mixtures of logit random utility
models can approximate more general random utility models (McFadden
and Train [1999]). We dene
fi (x)
=
m
X
j =1
wij gij (x)
(7.1)
to be is the distribution of bidder i's value expressed as a discrete mixture
over mi power-related distributions fgij g with mixing weights fwij g.
A joint IPV model can be approximated as the product of independent
distributions, with each bidder mixing over a set of power-related distributions. Each bidder i takes a draw from one of their mi base distributions
fgij g. Since all base distributions are power-related to one Q
another, the resulting auction is among power-related bidders. Let M = ni=1 mi denote
the number of such auctions.
20
f (x1 ; x2 ; : : : ; xn )
=
=
=
=
n
Y
fi (xi )
i=1
n mi
YX
i=1 j =1
wij gij (xi )
m1 m2
X
X
i=1 j =1
X
M
m=1
mn
X
k =1
w1i w2j : : : wnk g1i (x1 )g2j (x2 ) : : : gnk (xn )
wm fm (x1 ; x2 ; : : : ; xn )
Independence across bidder values implies that wm = w1i w2j : : : wnk or
that the weight given to the joint occurrence of a particular combination
power-related distributions is proportional to the product of their weights
in Equation 7.1. Correlation among bidder values is induced by choosing the mixing weights, fwm g, so that some combinations of power-related
distributions occur together more frequently than would be implied by independence. For example, two bidders' values are positively correlated if
they are likely to draw from their high mean-value distributions at the same
time.
The equivalence of a single auction in which bidders mix over distributions to a mixture of auctions facilitates estimation and analysis if the base
distributions are all power-related. The distribution of price is a mixture
of the closed-form price distributions from Section 3, and the eect of a
merger is a mixture of the merger eects from Section 5. The equivalence
follows from the dominant strategy equilibrium of the second-price auction.
In a rst-price sealed-bid auction, bidding functions depend on on the entire
joint value distribution, so the equivalence breaks down (e:g: Bajari [1996]).
21
8 Conclusion
In this paper we have analyzed a parametric class of second-price independent private-values auction models that have closed-form estimators and
merger predictors. We show how to generalize the class by mixing, and
conjecture that these mixed power-related auction models will prove useful
in approximating unknown joint value distributions in the same way that
mixed logit models have proven useful in approximating unknown random
utility models.
A Appendix: Proofs of Propositions
Proposition 1
x,
Suppose
X1
and
X2
are distributed as F si (x) for i = 1; 2. Then for any
Prob(X1 = maxfX1 ; X2 g j maxfX1 ; X2 g x) =
s1
s1
+ s2
p1
(A.1)
is a constant independent of x. The distribution of X1 given that X1 is the
maximum of X1 and X2 is the same as the distribution of the maximum,
i.e.,
Prob(X1 x j X1 = maxfX1 ; X2 g) = F s1 +s2 (x)
(A.2)
The distribution of X2 given that X1 is the maximum of X1 and X2 is given
by
Prob(X2 x j X1 = maxfX1 ; X2 g) =
1
p1
F
s2
(x) + 1
1
p1
F s1 +s2 (x)
(A.3)
Proof: We rst dene
Æ (s; x; x)
=
( F s(x+x)
F s (x)
F (x+t) F (x)
0;
sF s 1 (x);
F (x + x)
F (x + x)
F (x)
F (x)
6= 0
=0
We note that limx!0 Æ(s; x; x) = Æ(s; x; 0) = 0 and
F s (x + x)
F s (x)
= [sF s 1 (x) + Æ(s; x; x)][F (x + x)
22
F (x)]:
We introduce the auxiliary function
g (x) = Prob(X1 = maxfX1 ; X2 g ^ maxfX1 ; X2 g < x)
p1 Prob(maxfX1 ; X2 g < x)
where p1 = s1s+1s2 . We shall prove that g(x) 0. We note that if F (x) > 0
for x 2 ( 1; 1), then limx! 1 g(x) = 0. If there exists a such that
F () = 0, then limx! g (x) = 0.
Suppose x > 0. Then using the denition of g(x) and the expression
for F s (x + x) F s (x), we calculate the following string of equalities and
inequalities.
jg(x + x)
g (x)
j
= jProb(X2 X1 ^ x < X1 x + x)
p1 Prob(x <
max(X1 ; X2 ) x + x)j
= jProb(X2 < x < X1 x + x) + Prob(x < X2 X1 x + x)
p1 Prob(X1 < x X2 x + x) p1 Prob(X2 < x < X1 x + x)
p1 Prob(x < X1 X2 x + x) p1 Prob(x < X2 X1 x + x)j
= jProb(X2 < x < X1 x + x) + Prob(x < X1 x + x ^ x < X2 x + x)
Prob(x < X1 X2 x + x) p1 Prob(X2 < x < X1 x + x)
p1 Prob(x < X1 x + t ^ x < X2 x + x)
p1 Prob(X1 < x X2 x + x)j
jProb(X2 x < X1 x + x)
p1 (Prob(X2 x < X1 x + x) + Prob(X1 x < X2 x + x))j
+(1 + p1 )Prob(X1 2 [x; x + x] ^ X2 2 [x; x + x])
=
jF s2 (x) [F s1 (x + x)
F s1 (x)]
p1 [(F s2 (x)(F s1 (x + x)
+(1 + p1 )jF (x + x)
s1
F
s1
F s1 (x)) + F s1 (x)(F s2 (x + x)
(x)j jF (x + x)
s2
23
F
s2
(x)j
F s2 (x))]
j
=
jF s2 (x)(s1 F s1 1(x) + Æ(s1 ; x; x))(F (x + x)
p1 (F
F (x))
1 (x) + Æ (s ; x; x))(F (x + x)
1
(x)(s1 F
F (x))
s2 1
+F (x)(s2 F
(x) + Æ(s2 ; x; x))(F (x + x) F (x)))j
+(1 + p1 )js1 F s1 1 (x) + Æ(s1 ; x; x)j js2 F s2 1 (x) + Æ(s2 ; x; x)j jF (x + x) F (x)j2
s2
s1
s1
js1 p1(s1 + s2)j jF (x + x) F (x)jF s1 +s2 1(x)
+jÆ(s1 ; x; x)j jF (x + x) F (x)jF s2 (x)
+p1 (jÆ(s1 ; x; x)jF s2 (x) + jÆ(s2 ; x; x)jF s1 (x)) jF (x + x) F (x)j
+(1 + p1 )js1 F s1 1 (x) + Æ(s1 ; x; x)j js2 F s2 1 (x) + Æ(s2 ; x; x)j jF (x + x) F (x)j2
=
jÆ(s1 ; x; x)j F s2 (x) (F (x + x) F (x))
+p1 (jÆ(s1 ; x; x)jF s2 (x) + jÆ(s2 ; x; x)jF s1 (x)) (F (x + x) F (x))
+(1 + p1 )js1 F s1 1 (x) + Æ(s1 ; x; x)j js2 F s2 1 (x) + Æ(s2 ; x; x)j (F (x + x)
since s1
p1 (s1
F (x))2
+ s2 ) = 0. This shows that
jg(x + x)
g (x)
j (F (x + x)
F (x))
where is a function of a single variable such that limz!0 (z ) = 0 and
in particular, jg(x + x) g(x)j A(F (x + x) F (x)) where A is some
constant. A similar type of inequality can be found if x < 0. This implies
that for any > 0 and any x 2 R with F (x) 6= 0, there is a x > 0 such
that for 2 [x x; x + x], jg( ) g(x)j < jF ( ) F (x)j. We now show
that g(x) must be identically 0 by showing that jg(x)j < for any positive
.
If F (x) = 0, then g(x) = 0. Suppose g(x) > 0 and let be an arbitrary
positive number. Choose a number x0 , 1 < x0 x, such that 0 < g(x0 ) <
=2. Consider a sequence of points x0 < x1 < x2 < : : : < xn = x so that on
each subinterval [xi ; xi+1 ], we have
jg(xi+1 )
j 2 (F (xi+1 )
g (xi ) <
24
F (xi )) :
Consider the covering of [x0 ; x]: f[x2i x2i ; x2i + x2i ]g. We chose this
covering so that x2i+1 lies in the overlap between x2i+2 x2i+2 and x2i +
x2i . Using the bound jg(xi+1 ) g(xi )j < 2 (F (xi+1 ) F (x)), we have
jg(x)j = g(x0 ) +
X1
n
i=0
jg(x0 )j +
<
Having shown that
2
+
X1
n
X1 n
i=0
(g(xi+1)
2
i=0
g (xi ))
jg(xi+1 )
(F (xi+1 )
2 + 2 (F (x)
2 + 2 = :
j
g (xi )
F (xi )
F (x0 ))
Prob(X1 = maxfX1 ; X2 g ^ maxfX1 ; X2 g t) = p1 Prob(maxfX1 ; X2 g t);
we have
p1
= Prob(X1 = maxfX1 ; X2 g j maxfX1 ; X2 g x)
Prob(X1 = maxfX1 ; X2 g ^ maxfX1 ; X2 g x)
:
Prob(maxfX1 ; X2 g x)
This proves the rst part of the propostion.
Next consider the conditional probability:
=
Prob(X1 x j X1 = maxfX1 ; X2 g)
=
=
=
=
Prob(X1 x ^ X1 = maxfX1 ; X2 g)
Prob(X1 = maxfX1 ; X2 g)
Prob(X1 = maxfX1 ; X2 g j maxfX1 ; X2 g x) Prob(maxfX1 ; X2 g x)
Prob(X1 = maxfX1 ; X2 g)
f
p1 Prob(max X1 ; X2 )
F s1 +s2 (x):
p1
xg
25
The nal part of the propostion is proved using the following string of identities.
Prob(X2 x j X1 = maxfX1 ; X2 g)
=
Prob(X2 x
^ X2 X1 )
=
Prob(X2 x < X1 ) + Prob(X2 X1 x)
=
Prob(X2 x) Prob(X2 x
^ X1 x) + Prob(X2 X1 x)
=
Prob(X2 x) Prob(X2 x
^ X1 x) + p1Prob(X2 x ^ X1 x)
p1
p1
F
=
s2
(x) + (p1
1
=
p1
1)F
p1
F s2 (x) +
1
s1 +s2
1
p1
p1
(x)
p1
F s1 +s2 (x):
This completes the proof of the proposition.
Proposition 2 If Xi has CDF F si (x) for i = 1; 2; 3, then
Prob(X1 > X2 > X3 ) =
s1
s2
p2
= p1 s123 s23
p2 + p3
(A.4)
The joint distribution of X1 , X2 , and X3 , given X1 and X2 are the rst and
second-highest, respectively, is
Prob(X1 x1 ^ X2 x2 ^ X3 x3 j X1 > X2 > X3 )
=
s123
s1
s3
s23
for
X2
s23
s2
F s1 (x1 )F s2 (x2 )F s3 (x3 )
s2 s3
F (x1 )F (x3 ) + s
s1
s23
12 s123
F
s2
s12
s123
F s12 (x2 )F s3 (x3)
(x3 )
x2 x1. Hence the joint distribution of X2 and
are the rst and second-highest, respectively, is
x3
26
X3
given
X1
and
Prob(X2 x2 ^ X3 x3 j X1 > X2 > X3 )
s123
=
ss23 s1
2
s3
s23
The distribution of
respectively, is
s2 s3
F (x3 ) + s
s23
X3
s2
F s2 (x2 )F s3 (x3 )
12 s123
given
X1
and
s12
F
X2
s123
F s12 (x2 )F s3 (x3 )
(x3 )
:
are the rst and second-highest,
Prob(X3 x j X1 > X2 > X3 ) =
s123
s1
for
x3
s23
s2
s1
s12
F (x)
s3
s3
s23
F (x) + s
s23
s2 s3
12 s123
F
s123
(x)
x2 .
Proof: Using the previous proposition, we have the following:
Prob(X1 > X2 > X3 )
= Prob(X2 > X3 ) Prob(X2 > X3 ^ X2 X1 )
s
s2
= 2
=
=
s23
s123
s1 s2
s123 s23
s1
s2
s123
s
23
:
This proves the rst result.
Next consider the probability
Prob(X1 x1
^ X2 x2 ^ X3 x3 ^ X1 > X2 > X3 )
= Prob(X1 x1 ^ X2 minfx1 ; x2 g ^
X3 minfx1 ; x2 ; x3 g ^ X1 > X2 > X3 ):
27
We restrict ourselves to the case when x3 x2 x1 . Then
Prob(X1 x1 ^ X2 x2 ^ X3 x3 ^ X1 > X2 > X3 )
= Prob(X3 < X2 < X1 x3 ) + Prob(X3 < X2 x3 < X1 x1 )
+Prob(X3 x3 < X2 < X1 x2 )
+Prob(X3 x3 < X2 x2 < X1 x1 )
= Prob(X3 < X2 < X1 x3 ) + Prob(X3 < X2 x3 < X1 x1 )
+Prob(X3 x3 ) Prob(X2 < X1 x2 )
Prob(X2 < X1 x3 ) Prob(X2 x3 < X1 x2 )
+Prob(X3 x3 < X2 x2 < X1 x1 )
=
s1 s2
s123 s23
F s123 (x3) + ss2 F s23 (x3) [F s1 (x1)
+F (x3 )
s3
s1
s12
s
23
1
s12
F (x2 )
s12
F (x3)
s12
+F s3 (x3 )(F s2 (x2 )
=
F
s2
(x3 ) [F (x2 )
s23
s1
F s2 (x3 ))(F s1 (x1 )
F s1 (x1 )F s2 (x2 )F s3 (x3 )
s3
F s1 (x3 )]
F s1 (x1 )F s23 (x3 ) +
s2
F
s1
(x3 )]
F s1 (x2 ))
F s12 (x2 )F s3 (x3 )
s12
s2 s3
F s123 (x3 ):
s12 s123
The second formula of the propostion follows from this identity by letting
x1 ! 1. The remaining identities of the proposition follow by letting
x2 ! 1 and x3 = x. This completes the proof of the proposition.
28
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