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DISSOLVING A PARTNERSHIP EFFICIENTLY
Peter Cramton, Robert Gibbons,
and Paul Klemperer*
November 1985
406
Number
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
DISSOLVING A PARTNERSHIP EFFICIENTLY
By
Peter Cramton, Robert Gibbons,
and Paul Klemperer*
November 1985
Number ^06
.
*Yale University, Massachusetts Institute of Technology, and St. Catherine's
College, Oxford, respectively. This project was begun at Stanford University
and financially supported by the Sloan Foundation, the National Science
Foundation, and the Center for Economic Policy Research. We are grateful to
David Gold for posing the problem and encouraging its .analysis and to Roger
Guesnerie, Christopher Harris, David Kreps, Robert Wilson, and two referees
for helpful suggestions.
,
DEG3
1988
Dissolving a Partnership Efficiently
By
Peter Cramton, Robert Gibbons, and Paul Klemperer*
November 1985
Abstract
Several partners jointly
a valuation for the
own an
asset that
asset; the valuations arc
a common probability distribution.
We
may
known
be traded
among them. Each partner has
privately and
characterize the set of
drawn independently from
all
incentive-compatible and
interim-individually-rational trading mechanisms, and give a simple necessary and sufficient
condition for such mechanisms to dissolve the partnership ex post efficiently.
is
constructed that achieves such dissolution whenever
it is
A bidding
game
possible. Despite incomplete in-
formation about the valuation of the asset, a partnership can be dissolved ex post
efficiently
provided no single partner owns too large a share; this contrasts with Myerson and Satterthwaite's result that ex post efficiency cannot be achieved
when the
asset
is
owned by a
singlf
party.
"Yale University, Massachusetts Institute of Technology, and St.
Catherine's College,
Oxford, respectively. This project was begun at Stanford University and financially supported
by the Sloan Foundation, the National Science Foundation, and the Center for Economic
Policy Research.
analysis,
We
are grateful to David Gold for posing the problem and encouraging its
and to Roger Guesnerie, Christopher Harris, David Kreps, Robert Wilson, and two
referees for helpful suggestions.
Introduction
1.
When
a partnership
When
municipalities jointly need a hazardous-waste
site
is
to be dissolved,
and how much should
it
or children divide an estate,
who should buy
out his associates and at what price?
dump, which town should provide the
be compensated by the others?
who should keep
When husband and
wife divorce,
how much should
the family house or farm, and
the others be paid?
We
consider partnerships
in
which each player
is
i
endowed with a share
be traded, and specific capital or other transaction costs
on the market and
efficiently
[3]
We
split the proceeds.
make
it inefficient
of
r,-
a good to
to sell the good
look for procedures that allocate the good ex post
while satisfying interim individual rationality. Unlike Myerson and Satterthwaite
—who show that no procedure can
with uncertainty
(r\
=
1
and
r?
=
yield both properties in two-player bargaining
0)
— we show
games
that the distributed ownership found
in
a partnership often makes the two compatible. For the case of n players whose valuations
are independently drawn from an arbitrary distribution,
is
we
derive a simple condition that
necessary and sufficient for efficient, individually-rational dissolution, and
simple bidding game that will accomplish such dissolution whenever
Bayesian Nash equilibrium
The
it
application that inspired our analysis was the Federal
proposed to make the
can be achieved as a
some extensive-form game.
in
Communication Commissions
allocation of licenses for cellular-telephone franchises. After closing the
FCC
we introduce a
final allocation
list
of applicants, the
using a simple lottery. Prior to the lottery each
applicant has an equal chance of winning, and so can be thought of as owning a l/n share in
the license. Our analysis suggests that the applicants would do better to form a cartel (that
would win the lottery with certainty) and then allocate the franchise to one
via our bidding game; this
to
resell.
A
is
more
similar example
is
efficient
than the
lottery, even
if
the winner
airlines shares in the slot equal to their weights in the
two-member
permitted
proposed
lottery,
is
and
to assign the
let
them play
-
game we propose.
As another
is
number
the Federal Aviation Administration's proposal to allocate
landing slots at busy airports by lottery. Again, a more efficient approach
the "bidding
of their
application of the theory, consider the following buy-out provision of
many
partnerships: one side submits a "bu3 -out" offer, and the other side then has
:
1
the choice of either buying or selling at these terms. Since this scheme does not guarantee
ex-post efficiency (the
will
player will not,
first
be inefficiently allocated
valuation),
it
general, submit his valuation, so the object
in
the other's valuation
if
is
between the
first
player's bid and
too can be improved upon by our bidding game.
The bidding game
that achieves
efficient, individually-rational dissolution differs
typical auction in that every player pays or receives a
players' bids. Moreover, the set of bidders
who pay
sum
of
money that
a positive
amount
is
a function of
typically
is
to the winning bidder, but includes the second and other high bidders as well.
—such as
first-
and second-price
rational dissolution. This part of our analysis
He
in
the
Since such
which
in
efficient, individually-
was inspired by Samuelson
a similar problem and solves a two-player example
distribution on [0,1].
— can achieve
all
not limited
bidding arrangements are not frequently observed, we determine the circumstances
more common auctions
from a
[4],
who
describes
which types are drawn from a uniform
finds that split-the-difference bidding (the average of first-
and
second-price) yields an efficient allocation, but he ignores the individual rationality constraint
that players should prefer participation
in
the
game
to retaining their current share.
(He
imposes instead the weaker constraint that players prefer participation to being dispossessed
of their current share.)
We
consider n players and arbitrary distributions and
and other similar auctions ma}' accomplish
only
when
efficient,
partners' shares are very close to equal.
Coase Theorem under incomplete information: instead
sion that efficiency
exactly
this
individually-rational dissolution, but
Samuelson also provides an interesting interpretation
efficiency
show that
may be
when
is
of his
work as an exploration
of the
of the complete-information conclu-
always achieved and that property rights are immaterial, he shows that
lost
and property
efficiency can
rights
may
matter. In these terms, our analysis shows
be achieved and how property rights matter.
»
The
tion
rest of the
game
paper
is
organized as follows. In Sections 2 and 3,
we analyze a
to determine the set of partnerships that can be dissolved efficiently.
introduces a bidding
game that accomplishes
and Section 5 characterizes the
Section 6 shows that
efficient dissolution
set of partnerships for
commonly observed auctions
which
whenever
it
revela-
Section 4
is
possible,
efficient dissolution is possible.
are efficient in
some circumstances.
}
The Revelation Game
2.
n players indexed by
Our model has
good to be traded
Each
are
€
(r,
player's valuation
€
N=
r<
=
and X3"=i
[0, 1]
is
«
known
1)
{ 1,
.
.
n
,
}.
Player
owns a share
i
an(^ ^ias a valuation for the entire
privately, but
drawn independently from a distribution
.
it is
F
common knowledge
with support
r,
of the
good of
t/,-.
that the valuations
and positive continuous
[r, v]
density /.
We
valuations v
t(v)
=
game
consider the direct revelation
=
{i\(v),
to player
t.
.
{v\,.
.
.
tn
,v n
(v) },
=
r
=
where
{n,.
for all v
=
and then receive an allocation s(v)
}
s; is
(Since this allocation
ownership rights,
and ^ii(v)
,
..
which players simultaneously report their
in
..
€
trader-specific,
is
We
,r n }.)
[v,v]
the ownership share and
n
The
.
may depend on
it
money
the net
is
f,
(«),..., s n (t')}
{ «i
outcome functions
transfer
the vector of initial
require that these allocations balance:
pair of
and
{s,t)
is
Cs
i'(
v)
=
1
referred to as a
trading mechanism.
A
in
player with valuation
money and
u,-,
the asset. Also,
share
to be v,r;, while afterwards
player
i
it is
t>_,-..
when he announces
S,(tv)
is
player
+
is
£,'.
Let -i
= K\i
v;r;
+ m,,
which
is
linear
endo%ved with enough money,
is
utility,
mechanism
and
let
only net transfers
(s.t) can be taken
£_,{•} be the expectation
and money transfer
define the expected share
for
by
v,
=
t?;s,-
£-i{si{v)
}
and
=
Ti (17)
£_, { U(v)
,
is
Ui{vi)
(s.t)
has utility
Because of the linear
feasible.
Then we can
so the player's expected payoff
The mechanism
m,-
before participating in the trading
i's utility
operator with respect to
and money
we assume that each
say v, that any required transfer
matter, so player
r,-,
= VjSj[vi) + Ti[vi).
incentive compatible
if all
types of
players
all
want to report
their private information truthfully:
tf.-fc) >"t-vS,(u)
By the
Revelation Principle (Myerson
+ Ti{u)
[2],
Vi
among
attention to incentive-compatible mechanisms.
3
€
TV, tv,
others),
we
«
-
€
[r,
lose
The mechanism
t>].
no generality by restricting
(s,t)
is
interim individually
rational
if all
types of
all
players are better off participating in the
their expected payoff) than holding their initial
>
Ui(vi)
The
following
Vi
r,f,-
£
lemmas develop a necessary and
incentive compatible
and individually
mechanism
(in
terms of
endowments:
A'
and
v,-
E
[r, v].
sufficient condition for
a mechanism to be
rational. Since the proofs are either simple or standard,
they are relegated to the Appendix.
Lemma
i
£ N,
The trading mechanism
1.
Si is increasing
{s,t} is incentive compatible if
and only
if for
every
and
V
T (v;)-T (v )= [
i
i
'udSi(u)
i
(IC)
i
€
for all Vi,v*
Lemma
\v,v).
1 follows
implies that
U,-
from the fact that
convex and increasing
is
in v;,
continuity of V, implies that the net utility
Lemma
at
v-
=
with derivative
—
C/,(t',)
S,-
has a
r,t>,-
Linearity
asset.
almost everywhere. The
minimum
over
v,-
€
\v,v\.
2 identifies this worst-off type, and this allows us to restate individual rationality as
a single condition
Lemma
money and the
utility is linear in
2.
in
Lemma
3.
Given an incentive-compatible mechanism (s.l), traderi's net
i[inf (V.*)
+ sup(V*)] €
=
V*
In the simplest case,
{Vi
£,•
\
is
worst-off type satisfies Si(v*)
r,-;
equal to his initial ownership right
be neither a buyer nor a
<
T{
Vw <
Vi\
S;(w)
continuous and has
=
seller of
minimized
where
[v, v],
Si(ll)
utility: is
that
r,-.
is.
r;
>
Ti
Ww>
Vi}.
range, so the valuation of the
in its
the worst-off type expects to receive a share
Intuitively, the worst-off
type expects on average to
the asset, and therefore he has no incentive to overstate or
understate his valuation. Hence, he does not need to be compensated in order to induce him
to report his valuation truthfully, which
This
is
why he
is
the worst- off type of trader.
an interesting generalization of a similar result
exchange
for bilateral
seller (v) are
longer clear
is
worst
who
is
(r
off;
=
In
{0,1}).
[3]
and who
is
Myerson and Satterthwaite
[3]
the lowest-type buyer (v) and the highest-type
here the worst-off type typically
selling
in
buying.
-1
is
between v and
v, since it is
no
(
Lemma
3.
for all
E
i
An
incentive-compatible mechanism (s,t)
individually rational
is
if
and only
N
r.(O>0,
where
is defined in
v*
i
Lemmas
4.
2.
and individually
rational, stated in
For any share function s such that
transfer function
only
Lemma
(ir)
1-3 lead to a necessary and sufficient condition for a trading mechanism to be
incentive compatible
LEMMA
t
such that (s,t)
is
Lemma
4 below.
S, is increasing for all
E N,
i
there exists a
and individually rational
incentive compatible
if
and
if
n
m
rv
V
[
i>v
Vi
J2 [ \l-F{u))udSi(v)- f
where
v.*
I
is
defined in
The "only
if"
Lemma
>
"if"
part of the
—
lemma
*
is
F(u)udSi(u)
>0,
(/)
2
lemma follows
part of the
balance conditions J2 s ( v )
The
if
anc^ T2
f
«"
v)
directly
=
from the previous lemmas and the budget
which even feasible mechanism must
-
0,
proven by constructing a transfer function
compatible and individually rational provided the inequality
following intuition precise: there exists a transfer rule
t
(7) holds.
t
that
is
satisfy.
incentive
The proof makes the
that entices the worst-off type of each
trader to participate in the mechanism, because (J) guarantees that the expected gains from
trade are sufficient to bribe every trader to
3.
Ex
The
trading mechanism
of the
tell
the truth.
Post Efficiency
(s,
t ) is
mechanism {s(v),f(v)}
is
ex post efficient
if
for each vector of valuations v the
outcome
Pareto-undominated by any alternative allocation (ignoring
incentive constraints). Thus, ex post efficiency requires that the asset go to the trader with
the highest valuation.
post
efficient
partnership (r,F) can be dissolved efficiently
trading mechanism (s,t) that
Such a mechanism
we
A
will
is
For economy of expression,,
mechanisms that are incentive compatible and
individually rational as efficient trading mechanisms.
efficiently will
there exists an ex
incentive compatible and individually rational.
be said to dissolve .the partnership.
will henceforth refer to ex post-efficient
if
A
be referred to as a dissolvable partnership.
5
partnership that can be dissolved
We
now prepared
are
to answer the central question of this paper:
At
can be dissolved efficiently?
partnerships
to
some
and
is
empty; that
first
glance, one might think that the set of dissolvable
This
F
from
A
1.
can be dissolved efficiently
v*
=
-P
an
efficient
partnership with ownership rights
-1 ^,"" 1
Proof: Ex post
and G{v t )
)
if
and only
efficiency requires that the
1
among them.
F{u)udG{u)
Si{vi)
v' satisfies
(D).
I
4..
An
,
J
Efficient
In this section,
(t;*)
n~ 1
=
Bidding
we
trading mechanism.
to dissolve
because
it
(D)
0,
good go to the trader who values
it
the most:
< max r,-
i.',-
=
if Vj
maxT'y.
=
the expected share function S,
Pr{tV > max t;,,-}
r,-,
so v'
=
= F -1 (r " -1
i
Fit;)"-
).
it.
1
=
is
given
G(v,).
game that
serves the
same purpose
Using terminology analogous to that introduced
is
Lemma
4 yields
Game
introduce a bidding
This
in
bj'
Substituting into (J) of
dissolvable partnership, one could use an efficient trading
game
>
Since ties occur with zero probability, they will be ignored
By independence,
Thus,
and valuations independently drawn
two or more traders have the highest valuation, then the shares can be
(In the event that
follows.)
trading mechanism.
= Ffa) n-\
if
what
following theorem gives a necessary
if
!0
split arbitrarily
r
\l- F{u))udG{v)-
/
where
The
not the case.
is
sufficient condition for the existence of
THEOREM
partnerships
the incomplete information about valuations necessarily leads
is,
inefficiency in trade.
What
a useful
mechanism
as an efficient
in Section 3, given
a
or an efficient bidding
complement to the revelation-game analysis
of Section 2,
uses strategy spaces familiar in practice, namely bids rather than valuations. In a
general bidding game, the
n players submit sealed
6
bids, the
good
is
transferred to the highest
bidder, and each bidder
pays a total price Pi{b\
In the efficient bidding
game
that can precede the bidding. Note that in the efficient bidding
game
t
analyzed below, the total price
P,
p, (b 1
and a side-payment
c,
.
,
sum
the
is
. . ,
the winning bidder pays a positive price
bidders.
As
bj
n
p,-,
—
)
'*—*
1
bj
may
as usual, but so
the second and other high
a standard auction, a higher bid buys the player a larger probability of winning.
in
Here, however, making a higher bid
price of losing tickets
THEOREM
of a price
=
6„)
b„).
A
2.
is
bidding
buying more lottery tickets
is like
in that
the purchase
not refunded.
game
with prices
p,(6i,...,fc n )
= 6,-
-— J^6y,
(P)
preceded by side-payments
rv'
c,-(r 1
is
,...,r„)=
an e&cient bidding game:
PROOF:
We solve for a strictly
use the strategy
fc(-),
then
t's
UfaA) =
all
=
J"
b(vj)
but the highest of the
71
|
—
[ti
-
+
b;
and "(ry
u)
1
condition
is
is
is
b(u).)
positive (negative) for
satisfied.
We
|
-b(u)
n
-
u)
=
1
The
6,-
fc,-
(C)
-6(t/)]
-
the
v,-
77
["'
1
others
is
dG(u)
F(vj)/F(u). (Since types are independent,
same expected
i
value, conditional on
therefore solves
than (greater than)
6(u,-),
the second-order
are interested in the symmetric solution, which satisfies
6(v,-)=
—
1
best response for
less
If
with valuation
+
J!
other bids generate the
the value of the highest bid: this
Since dUi/dbi
udG(u),
'
expected utility from bidding
/
dF(vj
rv'
increasing symmetric Bayesian equilibrium.
Jfc-i(6,)
where 5(u)
"
any dissolvable partnership.
dissolves
it
j
'udG[v)--J2
/
vdG{v)
+ b{v).
Since
6'
>
the good.
0, this
equilibrium
The constant
game
the rules of the
6(r)
is
ex post
is
arbitrary,
efficient:
the trader with the highest valuation receives
equals the lowest amount, presumably zero, that
(it
when
allow a player with valuation v to bid), and disappears
p,
and
b
are composed:
Pi [b( Vl ),...,b(v n )]
Some simple
= - f' udG(u)+ -L-J2
(
u )-
algebra verifies that (T) and (C) define the transfer rule used
Lemma
of the proof of
4, so individual rationality
Since the side-payments depend on
F, but not on v
is
udG
I"'
=
r
=
is
guaranteed.
{n,...,r n
}
that follows, since the prices
p,-
who
in
the "If" part
|
(through
t>*
=
{v\,
{vi,...,u„}, they can precede the bidding procedure.
to compensate large shareholders,
P)
.
.
.,
t* })
Their purpose
are effectively dispossessed in the bidding
are independent of
r
and so treat
all
and
game
shareholders alike.
Accordingly, the side-payments are zero for the equal-shares partnership (jf,...,^).
5.
Characterization Results
We now offer four propositions that characterize the set of partnerships which
efficiently.
First,
The proofs
can be dissolved
are not of interest in themselves, and so are given in the Appendix.
we formalize the
idea that
it
is
large shareholders that
make
interim individual ratio-
nality difficult to achieve: for any distribution F, the equal-shares partnership
but the partnership
PROPOSITION
in
The
1.
is
dissolvable
set of partnerships that can be dissolved efficiently is a
nonempty,
which one player owns the entire asset
convex, symmetric subset of the n
shares partnership (£,
PROPOSITION
2.
A
.
.
.
,
is
not.
— 1-dimensional simplex and
is
=
,r„
centered around the equal-
±).
one-owner partnership
{rj
l,r2
=
=
0} cannot be dissolved
i
efficiently.
Proposition 2 generalizes to
seller relationship
rationality.
many .buyers Myerson and
Satterthwaite's
[3]
result that,
a buyer-
cannot simultaneously satisfy ex post efficiency and interim individual
This speaks to the time-honored tradition of solving complex allocation problems
by resorting to
lotteries:
even if the winner
is
8
allowed to resell the object, such a scheme
is
.
inefficient
because the one-owner partnership that results from the lottery cannot be dissolved
efficiently.
These propositions are derived by making the appropriate substitutions
orem
1.
Note that each partner's ownership share
As an example,
if
into (D) of
The-
enters the inequality through v'
r,
the traders' valuations are drawn from a uniform distribution on
[0, 1],
then (D) simplifies to
z_
Thus,
if
for a
(D')
is
uniform distribution,
n
+
1,
(D
1
)
1
the extremities of the simplex cannot be dissolved
from
be dissolved
efficiently
if
i
and only
determines a convex, symmetric subset of the
simplex, shown as the unshaded region in Figure
number
y
1
a partnership r can
By Proposition
satisfied.
-
'
for the case
efficiently.
n
=
3.
Only partnerships
in
For the uniform case, as the
of partners (n) grows, the percentage of partnerships that are dissolvable increases
58%
to
93%
largest possible
to
99%
owner
in
as n increases
from 2 to 4 to
6.
Also, the percentage share of the
a dissolvable partnership increases from
79%
to
82%
increases from 2 to 20 to 200.
FIGURE
1.
Dissolvable partnerships with
?!
=
3 and F(v)
=
(0.1.0)
(.3.. 7.0)
(.7.
.3.0)
(1,0,0)
.7.0..3)
.3.0. .7)
u.
to
88%
as n
By
amount
contrast with Proposition 2, however, partnerships with an arbitrarily small
may
of distributed ownership
(Note that the proof employs a very special
be dissolvable.
distribution.)
Proposition
some
for
3.
Any
partnership not owned by a single player can be dissolved efficiently
distributions F.
Finally,
if
any partnership
number
replicated a sufficient
is
of times then
can be
it
dis-
solved efficiently. Consider the partnership
R(n
=
777)
|
{r,-
=
l/m,
i
=
1,
.
.,
.
m;
r
-
;
= 0,
j
= m+
1,
.
.
.,nm}.
m times — the
of m goods, so
This partnership results from replicating the n-player, one-owner partnership
m
partners
who own
own
positive shares each
1/771 of
endowment
the total
partnerships continue to be represented by points in a simplex.
any other n-player partnership can be represented
Proposition
4.
such that for
all
Given
m
>
F
in
replication of
a similar way.
and an n-player partnership {n
M the m-fold
The m-fold
,
r„}, there exists a finite
M
replication of the n-player partnership can be dissolved
efficiently.
We
think of this result as complementing the core-convergence theorems for exchange econ-
omies: replicating the economy sufficiently often reduces the effect of the incomplete infor-
mation to
An
zero.
interesting special case
is
R{2
|
For
771).
777
=
1,
Myerson and Satterthwaite
(and our Proposition 2 above) proves that the partnership cannot be dissolved
For larger values of
and Satterthwaite
here each bidder
771,
[l],
R
is
much
like
the double auction studied by Wilson
may demand up
to
m
units of the good.
Satterthwaite show that ex post efficiency
on each side of the market.
is
efficiently.
and Gresik
[5]
When
is
(Think of each
l/m
achieved for the
When
approached
in the limit;
each bidder wants
same example when there
m
more
specifically, for
if
there are
units, on the other hand,
are as few as
each side of the market. (To see this, check that [D') holds for R{2
10
share of
each bidder wants one unit, Gresik and
the uniform case, they find that 99.31 percent of the gains from trade are realized
ex post efficiency
s result
although there each bidder wants only one unit of the good, whereas
the partnership as one unit of the good.)
six traders
:
|
two traders on
2) for this
example.)
.
Simple Trading Rules
6.
The
efficient
Although the
same
bidding
game proposed
efficient
trading mechanism implicit
we
effect,
prefer the bidding
mentioned above,
more important,
shifted from the
of
it
in
Theorem
in
game
for
2 dissolves
in
Lemma
any dissolvable partnership.
4 and
game
mechanism designer
1
achieves the
two (somewhat imprecise) reasons.
And
uses strategy spaces familiar in practice.
the bidding
Theorem
First, as
second, and probably
a great deal of the computational burden has been
to the players: the designer
makes a simple calculation
side-payments and prices, using (C) and (P), while the players do most of the work
their calculation of the optimal bidding strategy b(v,).
In the trading
in
mechanism, on the
other hand, the players simply report their valuations while the designer shoulders
all
of the
computational burden.
In this spirit,
we
find
it
disappointing that the side-payments given by (C) in
depend on the distribution F,
for
it
than do the players. Ideally we would
seems plausible that the designer
like a
of J", so that the designer could always
will
recommend
it
may depend on F and game
We
less well
vithout assessing the distribution of
analyzes the composition of the players' strategies and the designer's
players that
know F
2
bidding game that can be described independently
the partners' private valuations. Unfortunately, the Revelation Principle
no guidance as to how to decompose this
Theorem
map from
is
game
no help here:
rules,
it
but gives
types to outcomes into strategies for the
rules for the designer that are
independent of
F
have not addressed the issue of finding the optimal bidding game or trading mech-
anism that
is
independent of F,
in part
because of the difficulty
(Asking only for independence of F, for instance,
is
in
formalizing this notion.
not enough, because the designer can
ask the players to report F, and then implement the side-payments given by (C)
ports agree, and forbid trade
if
the reports do not agree.) Instead,
approaches to bidding games that are independent of F.
in the
we
offer
if
the
re-
two speculative
hope that these ideas
will
be
expanded upon.
First,
it
may
be possible for the designer to use the players
1
bids to estimate F, and
then to use this estimate to construct side-payments that relax the individual rationality
1
constraints, like those in (C). This process seems both complex and delicate.
11
Second, some bidding games that are independent of
In particular, the
the set of dissolvable partnerships.
equal-shares partnership for any distribution F.
when compared
other virtues
and familiar. Second,
saves the cost of
all
it is
can dissolve a limited subset of
game we
discuss below dissolves the
game
in
Theorem
and the good
is
a
u
k
+
1-price auction" in
For k
k
=
=
1/2
this
it is
the
b t are
is like
first-
all
=
l[kb e
simple
=
+ (l-k)b f
important.
When
it
relies less
),
1 it is like
... ,b„},
and k E
a second-price auction,
the split-the-difference scheme described by Samuelson
is
third,
each of the others
and second-highest bids from {61,62,
a first-price auction, for k
the bidders. This
it is
required to pay.
is
[0,1].
and
for
[4].
Mote that the revenue from bidding (namely, the highest bidder's bid)
among
First,
three
which the players submit sealed bids
who pays
transferred to the highest bidder,
p{b l ,b 2 ,...,b u )
where bj and
And
losing bids by submitting only one nonzero bid.)
we consider
2.
game has
vulnerable to collusion. (In the efficient bidding game, a cartel
heavily on the risk neutrality of the bidders, since only the winner
Specifically,
bidding
In addition, this
to the efficient bidding
less
F
is
divided equally
the original ownership shares are unequal,
the individual rationality constraints could be more easily met by paying losing bidders in
proportion to their shares, but then partners owning different shares would have different
equilibrium bidding strategies so the partner with the highest valuation might not win, violating ex post efficiency. 2
We
begin by calculating an equilibrium bidding strategy for this auction. Calculating the
interim expected utility associated with this equilibrium then determines the set of partnerships that can be dissolved efficiently using the k
is
analogous to that introduced
PROPOSITION
5.
A
k
+
in
Section
1-price auction
-f-
1-price auction.
(Again, the terminology
3.)
has a symmetric equilibrium bidding strategy given
by
b
Proposition
auction
is
6.
The
^=
V
'
\F(r,)
-
•
A-]"
set of partnerships that can be dissolved efficiently using a k
a nonempty, convex, symmetric subset of the simplex and
equal-shares partnership (^
,
i).
12
is
+
1-price
centered around the
Thus, an equal-shares partnership can always be dissolved
auction. Such a simple auction only works, however,
when
equal, since the auction ignores the ownership rights
efficiently
by any k
+
1-price
1
partners' shares are approximately
and this makes large shareholders
r
unwilling to participate. For the uniform case, the A-f 1-price auction dissolves a partnership
<
(maxr,)--'
which for n
=
=
n
and n
if
and only
if
no partner's share exceeds 57.7%, 5.5%, and 0.5%, respectively.
if
the uniform distribution
is
jj^y,
The
2.)
intuition
is
efficient
=
200
satisfied
is
(A special property
of
bidding game, which are given after
that, because the auction treats all players as
share l/n, large shareholders will participate only
the cost of being, in
20,
that these results are independent of k. Contrast these results,
however, with the corresponding results for the
Proposition
2,
effect, dispossessed.
As n
if
if
they owned
the expected gain from trade exceeds
increases, the expected gain
from trade of the
worst-off type decreases: a player with high share and high valuation becomes almost certain
to be just outbid by players with slightly- higher valuations.
if
n<Kp
.
and
PROPOSITION
N
some
there will exist
in a
€
[0, 1]
the limit, only shareholders with p<l/n are willing to participate.
in
7.
As
—
n
oc, the only partnership that can be dissolved efncientiy
That
is.
by a t +
1-
letting p n be the largest
n-player partnership thai any partner can have such that the partnership can be
—
dissolved efncientiy. jj^
7.
given a share p
such that a partner with share p will be willing to participate only
p
price auction is the equal-shares partnership (£,...,£).
share
Thus
1
as n
— oc.
Conclusion
A simple extension
of
Myerson and Satterthwaite
[3]
shows that with incomplete information
no mechanism can guarantee that an object to be traded
who
values
that
if
it
most,
if
the ownership
possible.
Further,
is
it is
initially
among a
possible,
it
owned by
in
is
be allocated to the person
a single party.
In contrast,
we show
partnership, ex-post efficient allocation
is
often
can be achieved by a simple bidding game. In a
of partnerships, our observation that the
be dissolved efncientiy
way
is
distributed
when
more general model
influencing the
the object
will
range of partnerships that can
centered around equal shares suggests that this might be a factor
which partnerships are formed.
13
Footnotes
1.
More
precisely, estimating
an example, suppose that
F
F
seems complex and using the estimate seems
known
is
to be approximately uniform on
delicate.
with v €
[0,T>]
[11,
As
V].
Consider playing the bidding game of Theorem 2 without the side-payments. Then for every
pair of players {i,j}, use the remaining
?i
—
F, as follows.
2 players' bids to estimate
use the symmetric equilibrium bidding strategy identified in the text to
distribution of valuations
F
into a distribution of bids.
distributions described above in order to
And
second, vary
maximize some goodness-of-fit
the two distributions of bids, one observed and the other calculated.
calculate the side-payment cy(ri, ... r„) given in (C) and let
,
this construction, the
i
map
F
an arbitrary
over the set of
imposed on
criterion
Now
First,
use this estimate to
pay j the amount cy/(n—
payments received by any one player depend only on the other
1). In
players'
bids, so the equilibrium strategies are unaffected. Therefore, the size of the subset of the set
of dissolvable partnerships that can be dissolved in this
way depends only on how
elaborately constructed side-payments mimic those defined by (C)
2.
We
could for even- partner j divide j's bid
among the
when F
known.
is
other (n
well these
—
1)
players in
proportion to the other's relative shares, since then incentives for the bidders are unaffected
by their relative shares. This
is
an example of the type of auction discussed
in
the previous
footnote and would typically perform better than the auction considered here, at some cost
in
terms of greater complexity.
14
.
Appendix
This Appendix supplies the proofs of
Lemmas
LEMMA
(s,t) is incentive compatible if
T
i
€A
,
1.
The trading mechanism
S{ is increasing
and Propositions
1-4
1-7.
and only
if for
every
and
Titf) -
r
=
Ti{vi)
u dSi(u)
(IC)
i
for all Vi v*
,
€E
PROOF: Only
T,-(u), or
[v, v]
If (s,t) is
If.
=
incentive compatible, then Ui(v,)
+ T (vi) >
{
r,S,(u)
+
equivalently
>
Ui[vi)
Ui(u)
+
( Vi
-
u)Si{u),
implying that U; has a supporting hyperplane at v with slope
and has derivative ^Ef
=
S,
almost everywhere. Also,
U
i
(We use the
=
(v i )-U,(v:)
S,
£,-(«)
>
0.
Thus,
U,- is
convex
must be increasing, and
1^' S,iv)du.
Stiekjes integral throughout this paper, so that any discontinuities in the ex-
By
pected share function are accounted for in the integral.)
P
=
Si(v) dv
ViSiivi)
-
i.;S,-(t.;j
-
1
which together with the definition
Adding the
integration by parts,
jT' udS,(u).
1
If.
ViSj(v;)
of
17,
yields [IC).
identity
Vi\Si{vi)
-
Si[v*)]
=
dSi{u)
/
vi
to [IC) results in
Vi[Si{vi)
-
5,-(t>?)]
+ r.lv.) -
Ti{vfi
P
=
*
where the inequality follows because the integrand
Si is increasing.
is
(tV
-
«) dS,-(u)
nounegative for
Rearranging the terms on the lefthand side yields
v,s,{vi)
which
is
'»
incentive compatibility.
+ Ti{vj) >
|
15
v,Si{v*)
+ r,(i.;),
>
0,
all 7,7,11
€
[v,w], since
LEMMA
at v*
=
Given an incentive-compatible mechanism (s,t), trader
2.
i[inf(V;*)
+ sup(r')]
V =
*
{v,
|
t
PROOF: The net
by
Lemma
1.
Therefore, trader
S;(u)
>
or v'
=
r,-
and
or S,(u)
decreasing
<
r,
v, respectively.
>
such that 5,(u)
r,-
3.
for all
€
i
An
€
[
t'*
=
r,-
V
r,
v
<
net utility
bound
w,-
V
is E/,-(t',-)
w >
—
v,}.
r U]
,
,
is
convex
But
r,.
-^ =
5,
minimum
then the
in f,
and
left
almost everywhere,
First,
S,-
suppose that
occurs at the boundaries
=
i'*
(1)
suppose
jumps past
r,-,
=
5,-
r,-,
is
continuous and strictly increasing,
which minimizes trader
then the
v,-
at which
S,-
i's
net utility;
jumps minimizes net
over an interval, then each type in the interval
incentive-compatible mechanism {s.t)
is
v
w
three cases deal with the case where there exists u and
r,-:
is
is
equally worse
B
individually rational if and only
if
N
We
Lemma
>
(in)
o.
2.
need only check individual rationality at the valuation
v'
Lemma
2.
or Tjv'-rTj(v')
>
denned
r,-v'
.
in
I
Lemma
4.
For any share function
transfer function
only
which
r,-v,,
Thus, the individual-rationality constraint becomes v' Si(v*)-rT,-(v*) >
r,-v*.
minimized
arbitrarily select any valuation in the interval to be the worst-off type.
v" is defined in
PROOF:
r,
r,
Four cases need to be considered.
r,(v,')
where
is
minimized at the point where the
is
such that S,(v*)
S,-
S,(w) >
v,;
with valuation
u,-.
and S;(w) <
utility; (3) finally, if S,(u)
Lemma
in
The next
not continuous and
o5 and we can
i's
for all u
then there exists a unique
(2) if Si is
i
with respect to
{/,-
T, is
<
Sj(u)
net utility
where
[r.r],
utility to trader
right derivatives of
increasing,
e
i 's
t
such that (s,t)
s
is
such that
S,-
is
increasing for
all
i
€ N,
there exists a
incentive compatible and individually rational
if
and
if
V
y2\[
where
v' is defined in
[l-F{u)}udSi{u)- r* F{v)udS;{u)]>0,
Lemma
2.
16
(I)
PROOF: Only
Lemma
Suppose
if.
(s,i)
incentive compatible and individually rational.
is
Then from
1,
=
Tifa)
Titf)
- J\dSi(u).
t
Integrating over all types in \v,v] yields
V
£i {Ti (v )}
t
=
T,(v:)-
[
=
Ti(v-)-
udSi(u)dF( Vi
f'
—
*
)
1
V
r
I
dF{vi)udSi(u)+
J u=v' J t,=u
=
- J\l -
Tiiv-)
F(v)} v dS,(v)
+
12?=i
t>i v )
=
line follows
v,
F(u)u dSi{u),
J"'
—
I
where the second
fu= v JI"=r dF(vi)udS,(u)
•'
from changing the order of integration. Budget balance requires
we have
for all v, so
£>{r,te)ww {I>(»)}=.o.
1=1
Therefore,
summing
over
all
L r'( r r =
.
From Lemma
if.
is
traders yields
V
S
[ \l-F{u))«dS,iu)- f'
must be nonnegative
3. T,-(v*)
The proof
)
t=i
for all
TV,-
t>( v )
=
=
Lemma
c,-
-
=
c;
-
individual rationality
if
and only
= -X>,(tv)+
" ,=1
which results
in 2V(t£)
=
>'
=
0-
is
Then,
—
0-
J S.
± E"=i
u
dSJ (")i
after changing the order of integration.
incentive compatible.
F(u)]udS,-(tt),
Finally, by
Lemma
so
we can choose
l
udSiiu)-——J2 r[l-F(u)]udSj(u),
r,-(t';)>0.
3,
we have
A little algebra shows the hypothesis of Lemma 4
^"_i T,(r*)>0.
[''
/
+ -i- ?['[!-
if T,-(t:*)>0.
to be equivalent to the condition
c,
c
5_
dS,- («) -r
udSi(v)
f"
guarantees that (s,f)
1
^>'( t; *)
rvj
i
u
/
Q implies J2?=i
Ti{vi)
so
which implies ^"=1
by construction. Let
U (v)
where 32?= i
i,
F(u)udSitv)}.
"
|
17
j*i
J s.
'
PROPOSITION
1.
Tie
be dissolved efficiently
set of partnerships that can
convex, symmetric subset of the n - l-dimensional simplex and
shares partnership (^
Proof: Use (D) to
is
is
a nonempty,
centered around the equal-
£).
,
define
<j>
—
£"
:
>
Z by
V
= 21
^(r)
f
Convexity follows from concavity of
dd
-t.--dF(r-)"-
__
d 2 d>
(£ n_1
drf
—
TT
where the equality
l
dvt
.
—
1
—r <
1
r,
0.
dr,-
>
because at v'(±)
=
we have
f tv/(r,)F(tv) n -
nF"~ —
t?
—
dt,
1
and
0,
n
dr;
d-d>
=
1
-v;
dvj_ _
follows from relabeling the partners. Finally, </>(£,...,£)
)
=
1
72—1
dTidrj
F
[ vJiv^Fivi)"-
which we have because
4>,
dri
Symmetry
-
2
A' r./(v.)F(t:,)"- dr,
(n
+"
2.
n
dv,
dtV
-
A
for all
t;,-
£
B" 1
f
-
dvi
«.-/(«,-)^(tv)
dv,
:
:
/
arises after integrating each
— l)F r'<l
PROPOSITION
F(vi)
/
2
=
{rj
0,
term by parts, and the inequality holds since
\v',v}. so the first
one-owner partnership
>
term dominates the third.
I.t?
=
rn
=
I
0} cannot be dissolved
efficiently.
PROOF: For the n-player partnership
{rj
=
l.r 2
=
r„
=
0}. integrating by parts in
(D) yields
F"~ 1 dv+
which
fails for all finite n.
Proposition
for
some
3.
Any
/
F"dv >
0,
!
partnership not owned by a single player can be dissolved efficiently
distributions F.
PROOF: The proof
where a E
(0. i)
is
by construction, using the distribution F(u)
and u(v)
= 1+
{[T
-
l){v
-
v)/{v
18
-
=
v)} (so u{v)
{1
=
— \u{v)]~°}/{l — T~°}
1
and u{v)
=
T). Take
an arbitrary partnership {r
-2
for .p"
Fn~
and
l
<
r,|
n
u
*w-E=i
nOf"
in
T
=
cf
— T
1
°
1
Using a binomial expansion
.
in
(D) yields
l-2o
i/
-(n-2) 1
2q
1-2q
v 1-cr
a
—
terms
let
l-o
- Q
1
i=i
t,
-(n-1)
- Q
T
in
+•
- 2q
1
show that the above terms
to
and
and performing the integration indicated
(
It suffices
V:}
1
r-
11=1
ignored are of lower order, and so are insignificant, and
all
<
1.
limits are finite because u(v*(r,-)) approaches a finite limit, Vr;
have
r,-
terms
<
in
T
=
since u(v'{\))
1,
T, which does not stay
approach oo, replace
T
by
—
(1
1
a)
-1 /
and
T—
factor of
oo, d
(1
—
—
d).
!
1
- 2a
d(l-a)J
Since q
•
€
(0, i), 2
1
for d
Proposition
4.
such that for
all
£
vt3§--. 1)-
Given
m
—
—
<
^
(It is
crucial that
>
F
1
- 2d
d(l-a)
This yields
collect terms.
(n-l)(l-dy-i
+
rf(l-2a)
'
So 6{r) >
>
it
has an extra
—
1
provided
0,
for sufficiently large T.
and an n-player partnership
M the m-fold
we
To show that the
oo.)
the hrst term goes to oo as d
0. so
the
the terms in the lower
Therefore, the second term above can be ignored, since
1.
1
which holds
T
finite as
1
=r((i-flAs
+oo with 7\ because
the braces tend to
in
i
{rj,. ..,r„
E
tiere exists a finite
},
M
replication of the n-player partnership can be dissolved
efficiently.
PROOF: By the concavity
of
established in Proposition
<j>
holds for the n-player partnership
{n
=
= l/m =
=
j
which
is {r,-
:
and the partnership
1,
,
m;
of interest
tj
is {r,
=
=
l.rj
r„
= m + 1,
71/.V
=
j
. .
.
1
r."
nl
,
=
rj
A (A'-
r'
1
—l
j
=
.
Then
{N/n)
we have
d?/
T
,/ v -
f*'-
show that the
mn = N
=
!
1
to
J
dV
-L,
19
f
(
1)
NFN -
i
result
0}. the 772-fold replication of
mji}. Let
N/n:
1
After integrating by parts in (D) and collecting terms,
**(*)- V(JV-1)
1. it suffices
- {n -
A'(.V-l)
df n
)
du
+
m=
N/n
1,..., TV}.
=
where
t>*
=
v'{n/N)
pair of terms
= ^{(n/N)
(k -
:
Consider two arbitrary values k\ <
N—
J =
oo, v*
Jll
—
F N du. Then /£
tfr K du
(v'
-
(fci
- v)A^ +
.
and
7
A we have
7
let
<
ki
/£ F*"
<
A- 2
r
/ -r
„
>Ai
J
(A-j
(tv-^lAT
+ (* -
r-Sr
- n)At
t>*
=A]
—
e
>
0.
and
let A'i
= 1 — ie,
and
so the last ratio above approaches
fixed
ti,
as required.
Proposition
5.
Let /
.
for
=
t
= Jk F
2
1,2.
As
du and
J[K\. So
f^ du
A*o
1
= 1—
i£.
— ic and
Then
rj
A- 2 )
A'f
k2
(A-l-l)(AV^2)
Fix
v*
(/:,-)
J/K2 <
du <
1
= F
A",
A^A"^', we have
I+J
Ji:
if
n
(li,* ),
J/A'j and
first
:
1
1
(*,
>K
+J> J>
du
du
1
over [r*,U], the
positive for sufficiently large A'
>
F*-
- 1)F A'<1
(A'
J^F K du + J^F K du
_ ^Af- + /;= f»-i du + ;;•
I+J
^
/; f^-i du
Since I
<
1
F*'" du
is
f»
from
A- 2
so for sufficiently large
v,
pair
—
['•
jv
1
1) j;;:
lim
A-oo
A'F^" -
Since
].
The second
strictly positive.
is
—
A
+ K-*2)"
'
(A~i/A* 2 )
—
A
•
made
can therefore be
while v'
to exceed
—
1
A2
—
—
v
— A2
,
(l/n) for
I
AH
l-price auction has a
symmetric equilibrium bidding strategy given
by
v
f
b{Vi)
=
\F(~]
~
V;
\F{vi)
PROOF: Let G(x)
=
F(x) n
~1
b(vj), then r*s expected utility
Uifabi)
= [
+
''
f
If
.
i
—
-
'
f
T
[*6(:r)
7!
—
with valuation
+
(1
-
fc)6,-])
1
r,-
others will use the strategy
is
dG'(x)
-[kbi +(l-k)b(x))dE(y\x)dG(x)
r
-[*%) +
»
"
y=<>- l (b.)
Jx=b-Ub.)
r=t->(i.)-Jv=b(''.)
/
d-
"
Jx=b-l{bi)Jv=v
+
b,
/-i"
A-]"
conjectures that the
from bidding
(v,
-
-
1
20
(1
-*)&(*)]
dfi"(y
|
x)dG{x),
—
where H(y
n
—
1
=
x)
|
[F(y)/F(x)] n
2
(for
other bids given that the largest
^L =
(„ - 6,)p[6- 1 (6
1
do,
The symmetric
is
x.
The
is
the distribution of the second-largest of the
best response for
^lir[6-
)]^
do,
1
(6
1
therefore solves
;
)]- 2 (r[6-
I
(6 1
n
)]
-
k)
= 0.
-
b{vi)
=
n
b{vi)
—f(^r-
differential equation can be solved using the integrating factor \F(vj)
This linear
is
y<x)
equilibrium 6(17) satisfies
vi
solution
.
—
k]
n
\
the
a one-parameter family satisfying
[v,
-
b(v,)} \F{vi)
-
k)"
=
\F(v)
-
k]
n
du.
J"'
We
to
choose c to make the righthand side equal to zero at
±00
F~
as Vi approaches
1
(k)
from above or below. This choice
*(*)-*-
efficient:
It
= F~
u,-
1
[F(vi)
-
(k); otherwise, bids
of c yields the
tend
symmetric
k]n
V >
Finally, since
(k).
1
—
equilibrium
and truth-telling occurs at
= F~
v,-
0, this
equilibrium
is
ex-post
the partner with the highest valuation receives the good with probability one.
That
remains, then, to verify interim individual rationality.
U'[Vj.b(vi))
—
t;v,
>0
for all v;
r
( Vi
€
\v. vj.
—
-
where
[fc&(x)
C7"[f,-, fc(t-*;)]
-
(1
-
is,
we want
u>(r,-, r,)
=
is
k)b{vi)])
dG(x)
n
Jv
l
+ [
f"
-[kb(vi)
-f
I'
±[kb{y)
Partially differentiating with respect to
for fixed 77,
w
is
minimized at
~~w{ri)
=(l
f"(r,-)
+ (1 +
r,
k)b{x)}
dH(y
\
x)
dG(x)
(l-k)b(x)}dH(y\x)dG(x).
and applying the
= G -1 (r,).
Let w(rj)
- k){-±- fb[x)
dG(x)
first-order condition
=
-
w\vj (r,),r,]; then
ri b{v7)}
+
a-{j>*)"- 2 [i - rK)]6(»D - /''
+
-^r f
21
fc
(*) dGr (*)
b(y)dS(y\x)dG(x)}.
shows that,
—
PROPOSITION
auction
is
The
6.
set of partnerships that can be dissolved efficiently using a k
a nonempty, convex, symmetric subset of the simplex and
is
-f-
1-price
centered around the
equal-shares partnership (£,.-.,£)•
PROOF: Convexity follows from the concavity
of u;(r,),
which holds because
tc'(r,-)
= — v*(r,-)
i
and
= — jf-v*(r,) <
u/'(r,)
since
terms involving
—
1
k.
We
n
= G~
l
F(vl)
n-2
—
x
b(x)dG(x)>T
j[
v-
We
[1-F(v*)}b(v:)>
fc(x) in
l
i
b(v:)
the integral and simplifying.
I
11
./
Z
V
P b{x)dG{x)-—
[
-
[
b(y)dH(y\x)dG(x)
x= „- J v =v-
E
7.
As n
—
do, tie only partnership that can be dissolved efficiently
That
price auction is-ihe equal-shares partnership (1,. ..,£).
PROOF:
It suffices
is.
~-
to
—
1
as
j;
— oc.
show that given
6
>
0.
N
there exists
individual rationality fails for a player with share
(1 -f
such that for
,1-
16.
==-)
r, i
|(
— 6^
<
all
+
(d»-*)(v).
Thus,
is
necessarily false for
and valuation
of winning) (value of winning)
<(!)(!)
(1
Let d(8)
n >
= F \v{
T>(
/
(probability of losing) (value of losing)
)
+ (probability
S)/n.
^—
Interim individual rationality for a player with share
iv(
1-
letting p n be the largest
r
r
by a k-~
partnership that any partner can have such that the partnership can be
in a n-player
dissolved efficiently,
which
second,
(l/n), again by substituting b(v') for b{x) and b{y) in the integrals and simpli-
PROPOSITION
share
And
have
Jv
= G~
First, consider the
V
{\/n) by substituting b[v*) for
consider the terms involving k.
fying.
follows from relabeling the
have
-L_
at v'
Symmetry
).
Finally consider the equal-shares partnership in two steps.
partners.
at v'
= F _1 (r^ -1
t'*
+
±6)
all sufficiently
<
I
+ tuT -1
large n.
19
I
,
:
c
)
N
.
i •*-
ripg
interim
s
i
)
<
implies
1.
References
[l]
Gresik,
Thomas
market becomes
A. and
Mark
efficient as the
A. Satterthwaite:
number
"The rate
of traders increases:
at which a simple
an asymptotic result for
optimal trading mechanisms," Research Paper No. 641. Graduate School of Manage-
ment, Northwestern University, 1985.
[2]
Myerson, Roger
B.:
"Incentive compatibility and the bargaining problem,"
Econometrica Vol. 47, (January 1979), 61-73.
[3]
Myerson, Roger B. and Mark A. Satterthwaite:
bilateral trading," Journal of
[4]
Samuelson, William:
Economic Theory
"A comment on
"Efficient
Wilson, ROBERT:
for
Vol. 28, (April 1983), 265-281.
the Coase theorem," in
Models of Bargaining, edited by Alvin Roth, Cambridge University
[5]
mechanisms
Game
Theoretic
Press, 1985.
"Incentive efficiency of double auctions," Econometrica Vol. 53,
(September 1985), 1101-1116.
23
Figure
1.
Dissolvable partnerships with n
=
3
and F(u)
(0.1,0)
(.7, .3,0)
(1,0.0)
7,0,
•3,0,.7)
[Mrl)
24
.3)
-
u.
I
k
J
J
Date Due
91991
JUL 1*199!
Lib-26-67
c^
N