Asymmetric Partnerships
Nicolás Figueroa, Pontificia Universidad Catolica de Chile
Vasiliki Skreta, NYUz
y
December 6, 2011
Abstract
We study asymmetric partnerships and show that the surplus (gains from trade minus incentive
costs minus participation costs) is maximized if all agents’ payo¤s at their critical valuations (where
participation constraints bind) are equal. JEL classi…cation codes: C72, D82, L14. Keywords: e¢ cient
mechanism design, ownership structure, partnerships.
We are grateful to Roger Myerson and to Phil Reny for useful discussions. We also bene…ted from comments of the
audiences at Northwestern University and the University of Chicago. Nicolás Figueroa thanks Millenium Nucleus Information
and Coordination in Networks, ICM/FIC P10-024-F.
y
Instituto de Economía, Av. Vicuña Mackenna 4860, Santiago, Chile.; [email protected];
z
Leonard Stern School of Business, Kaufman Management Center, 44 West 4th Street, KMC 7-64, New York, NY 10012,
Email:[email protected]
1
1. Introduction
An important economic insight is that the presence of asymmetric information seriously hinders the ability
of negotiating parties to achieve mutually bene…cial agreements. The seminal paper by Myerson and Satterthwaite (1983) considers a bilateral trading environment with double-sided asymmetric information and
shows that no feasible ex-post e¢ cient negotiation procedure exists when gains from trade are uncertain.
For this reason, asymmetric information is viewed as a serious form of transaction costs in Coase’s tradition. However, Cramton, Gibbons and Klemperer (1987) show that if a group of ex-ante identical agents
jointly own an asset in equal (or close to equal) shares, then it is possible to give complete control to the
partner with the highest ex-post valuation. They conclude that similar property rights are a key factor in
determining whether or not e¢ ciency is achievable.
In this paper, we consider a partnership environment in which partners’valuations are private information but are drawn from di¤erent distributions. We investigate which ownership structures make e¢ cient
dissolution possible, why they do so, and what the relationship is between the degree of asymmetries across
partners and the ownership structures that make e¢ cient dissolution possible. Along the way, we shed new
light to the impossibility result of Myerson and Satterthwaite (1983) and the possibility result of Cramton
et al..
We show that e¢ cient dissolution is possible if all agents’critical valuations–the valuations where gains
of trade are minimal–are equal: vi = vj : For the case of symmetric distributions (as in Cramton et al.),
the condition vi = vj holds with equal property rights. However, for asymmetric environments, we show
that the property rights that guarantee vi = vj can be extremely unequal. Moreover, we show that agents
who most likely have the highest valuation, must initially own a bigger share of the asset.
The analysis proceeds as follows: First, we show that the marginal cost of increasing his ownership share
while maintaining voluntary participation is an agent’s critical type. Therefore, an ownership structure
is transfer-minimizing if vi = vj for any pair of agents. We then show that the transfer-minimizing
partnership is e¢ ciently dissolvable.
Our results shed light on the problem of e¢ cient allocation of new technologies among various …rms.
In a patent race, initial shares can be seen as the probability that each agent will win the race. Then,
our …ndings indicate that to have e¢ cient trade, the …rms that are better at inventing (that is, have the
bigger “initial share”) should also have a higher capacity for developing applications after the technology
is discovered (a better distribution of valuations).1 This could provide a rationale for the integration of
research departments into big …rms. Integration helps avoid lost pro…ts due to transaction costs associated
1
This is often not true, as can be seen in the case of the technology used in BlackBerry mobile devices. Research in Motion
(RIM), the developer of BlackBerry, did not own the rights to the technology and fought a costly litigation for more than
three years with NTP. NTP owned the rights to the technology, but it is primarily a patent-owning company, with no ability
to directly develop products and pro…t from the patents. There is extensive press coverage of this lawsuit. For a sample, see
“Detractors of BlackBerry See Trouble Past Patents,” The New York Times, March 6, 2006.
2
with incomplete information.
2. Trading Mechanisms with Co-ownership (Partnerships)
There are I risk-neutral agents. Agent i’s payo¤ from owning a fraction q of the asset is q vi .2 Types vi
are independently distributed according to Fi on Vi = [v i ; v i ], with 0 v i v i < 1: The partnership is
characterized by the initial property rights, Q = (r1 ; :::; rI ) with i2I ri = 1. Agent i’s payo¤ at the status
quo Q = (r1 ; :::; rI ), is U i (vi ) = ri vi :
At an ex-post e¢ cient assignment, the agent with the highest valuation is awarded exclusive ownership
of the asset, and the total social welfare is W (v) = maxi vi . Then, the expected payo¤ for agent i; when
Q
Q
his valuation is vi ; is Ui (vi ) =
Fj (vi ) vi + Ev i [xi (v)]; where
Fj (vi ) is the probability that vi is the
j6=i
j6=i
highest valuation and xi is i’s payment. Voluntary participation requires that Ui (vi ) ri vi :
We ask: When can we expect to …nd ex-post e¢ cient and incentive-feasible mechanisms that satisfy
voluntary participation without outside transfers? As in Makowski and Mezzeti (1993), Williams (1999),
and Krishna and Perry (2000), we proceed as follows: By the revelation principle,3 it su¢ ces to restrict
attention to incentive-compatible direct mechanisms. From the revenue-equivalence theorem (Myerson
(1981)), we know that all incentive-compatible mechanisms that implement the same allocation rules
generate the same expected payo¤ for each agent up to a constant. Therefore, the interim information
rent of an agent is identical for all e¢ cient mechanisms up to a constant. A Vickrey-Clarke-Groves (VCG)
mechanism is e¢ cient and incentive-compatible.4 Hence, when we are interested in properties of e¢ cient
mechanisms, we can, without loss, restrict attention to VCG mechanisms. Moreover, at a VCG mechanism
(see, for example, Krishna (2002)), i’s interim payo¤ is equal to the expected gains from trade plus a
constant; that is,
Ui (vi ) = Ev i [W (v)] + Ki :
(1)
The transfer-minimizing ownership structure Consider
Ki (ri ) = max [ri vi
vi 2[v i ;v i ]
Ev i W (vi ; v i )]:
(2)
Then, …nding the ownership structure that minimizes the sum of transfers necessary to guarantee voluntary
participation for all i, is equivalent to solving
min
fri gi2I
i2I Ki
(ri ) subject to
2
i2I ri
= 1:
All of our results generalize in a straightforward way to the case where agent i’s payo¤ from owning a share q is q
where i is strictly increasing and convex in vi :
3
See, for example, Myerson (1981).
4
See Vickrey (1961), Clarke (1971), or Groves (1973).
3
(3)
i
(vi ),
The maximizer, vi ; is di¤erentiable w.r.t. ri because the objective function is strictly concave. Ki (ri ) is
the smallest constant that must be added to Ev i [W (v)] to ensure voluntary participation.
In some sense, designing property rights that enable the e¢ cient dissolution of the partnership can be
seen as a problem of optimally allocating resources: The more property rights that are given to an agent
(higher ri ), the higher Ki must be to ensure his participation. The next result, which comes from an
envelope condition, derives the marginal cost (in terms of transfers) of increasing the property rights of an
agent.
Lemma 1 Suppose that the ownership share of agent i is marginally increased. Then, the marginal increase
in the transfers necessary to guarantee voluntary participation is
dKi (ri )
= vi :
dri
Proof. Using (2) ; we see that the marginal increase in the transfers necessary to guarantee voluntary
participation is the derivative of
dKi (ri )
dri
@Ev i W (vi ; v i ) @vi
@vi
@ri
@vi
@ri
@Ev i W (vi ; v i )
@vi
= vi
ri
@ri
@vi
= vi ; for all i
= vi + ri
where the last equality follows from the fact that at an (interior) vi , we have
@vi
@ri
@Ev
i W (vi
@vi
;v
i)
ri = 0; if vi
is not interior, then
= 0:
With this, we can shed light on the impossibility result in Myerson and Satterthwaite (1983) and the
possibility result in Cramton et al.. The next Proposition shows that at the transfer-minimizing ownership
structure, we have that vi = vj for all i; j 2 I.
Proposition 1 Consider a partnership. If all valuation functions have the same range (v i = v; and vi = v
for all i 2 I); then, the transfer-minimizing property rights (r1 ; :::; rI ) are such that vi = vj for all i; j 2 I.
Moreover, such property rights exist.
dK (r )
j j
Proof. In any interior solution of (3), the …rst-order conditions imply dKdri (ri i ) = dr
; which from Lemma
j
1 becomes vi = vj .
We now establish that for any distributions Fi ; i 2 I; there exists an initial ownership structure
(r1 ; :::; rI ); with ri 2 (0; 1) ; that implies vi = vj . First, note that if ri 2 (0; 1), then vi is interior5 and
5
It is easy to see that
dUi (vi )
dvi
=
Y
Fj (vi ). Therefore, if ri 2 (0; 1); the tangency condition is satis…ed at vi 2 (v i ; vi ).
j6=i
4
participation and non-participation payo¤s must be tangent, implying that
Y
Fj (vi ) = ri :
(4)
j6=i
De…ning Gi (s) =
Y
Fj (s), and noting that it is invertible (since it is strictly increasing), (4) can be
j6=i
rewritten as vi = Gi 1 (ri ): Therefore, for vi = vj to be true, we must have
Gi (Gj 1 (rj )) = ri for all i;
which, for a given ri ; determines rj . Now, we only have to check the consistency requirement that
(5)
I
P
ri = 1;
i=1
which, using (5), becomes
r1 +
I
i=2 Gi
G1 1 (r1 ) = 1:
(6)
Equation (6) has an interior solution since the LHS is equal to 0 at r1 = 0; it is greater than 1 at r1 = 1;
and it is a continuous function of r1 . Using (5), we conclude that every ri is interior.
The intuition of this result is as follows: Lemma 1 tells us that the marginal transfer that must be paid
to agent i if his property rights are increased, is his “critical” valuation vi . Then, if vi 6= vj , there is an
obvious way to reduce the transfers: by reducing the property rights of an agent with a high valuation at
the critical type, and redistributing it to an agent with a lower one.
In a symmetric linear environment (like the one in Cramton et al.), equality of payo¤s at the critical
types is equivalent to equality of property rights ri . This is one way to explain what is going on behind
the possibility result of Cramton et al.: In a symmetric environment, equal property rights minimize the
transfers needed for voluntary participation by equating critical types.6 However, extreme property rights,
as in Myerson and Satterthwaite (1983), imply that the critical type for the seller is his highest valuation,
while the critical type for the buyer is his lowest valuation, thus maximizing the di¤erence in agents’payo¤s
at their critical types.
From Proposition 1, it follows directly that in order to design e¢ ciently dissolvable partnerships, one
should aim for environments in which the valuations of critical types are the same across agents. Our next
Proposition shows that when this is true, e¢ ciency is indeed feasible.
Proposition 2 Consider a partnership. If property rights (r1 ; :::; rI ) are such that
vi = vj ; for all i and j;
(7)
then there exists a feasible and ex-post e¢ cient mechanism.
6
In Proposition 1, we considered the case in which the range of valuation functions is the same. When this is not true, we
may be at a ‘corner’solution where the marginal costs of participation of each agent are not equal.
5
Proof. From Schweizer’s (2006) analysis, we know that it is possible to design an ex-post e¢ cient mechanism i¤ the maximized sum of all agents’utilities minus information rents (incentive costs): maxi2I vi
P
P
E [maxi2I fvi g maxfvi ; v i g] and outside options (participation costs):
ri vi is non-negative:
i2I
i2I
E
"
(I
1) maxfvi g +
i2I
X
i2I
maxfvi ; v i g
#
X
ri vi
X
ri vi :
0:
(8)
i2I
A su¢ cient condition for (8) is that for all v
(I
1) maxfvi g +
i2I
X
i2I
maxfvi ; v i g
(9)
i2I
We just need to verify that (9) is satis…ed for all v whenever (7) is satis…ed.
Suppose that, for some vector of valuations v; we have that vk = maxi2I fvi g: Then, for this vector of
valuations, (9) holds if the following holds:
(I
1)vk + (I
1)vk + maxfvk ; v
kg
X
ri vi ;
i2I
which always does whenever vi = vj . Since this holds for any v; (9) always holds.
Proposition 2 is related to Proposition 2 in Schweizer (2006), which states that if the social surplus is
a linear function of the information pro…le and a convex function of the collective decision, there exists a
default option such that e¢ cient negotiations are possible. In relation to Schweizer (2006), we not only
state that there exist property rights that guarantee the e¢ cient dissolution of an asymmetric partnership,
but we also identify the critical condition as vi = vj .7 More importantly, we discover (cf. Lemma 1) why
it is crucial: An agent’s payo¤ at the critical type is the marginal cost of increasing his ownership share
while maintaining voluntary participation. Finally, below, we also explain the economic consequences of
that condition: Ex-ante property rights have to be well aligned with the distribution of the ex-post realized
valuation of the asset.
If all types are distributed according to the same distribution F , then condition vi = vj is satis…ed if
ri I1 , exactly as in Cramton et al.. However, when distributions are asymmetric, the property rights that
guarantee vi = vj can be extremely unequal:
1
Example 1 Consider a partnership with two agents, vi = vi , F1 (v1 ) = v1n and F2 (v2 ) = v2n . Then, it is
1
easy to see that v1 = F2 1 (r1 ) = r1n and v2 = F1 1 (r2 ) = r2n . From Proposition 2, we know that if v1 = v2 ;
1
e¢ cient dissolution is possible. For this example, this condition is equivalent to r1n = r2n . Recalling that
r1 + r2 = 1; this reduces to
1
r1n = (1 r1 ) n :
7
This condition was discovered concurrently and independently by Che (2006).
6
For n = 3; we obtain r1 = 0:8243 and r2 = 0:1757; which give us that v1 = v2 = 0:56009. Moreover, a
simple calculation shows that for these distributions, with property rights of ri = 12 ; there is no possibility
of e¢ cient dissolution. For n = 99, optimal property rights are even more extreme: r1 = 0:99926 and
r2 = 0:00074. For this case, the corresponding critical types are v1 = v2 = 0:92933:
Example 1 shows that, for certain distributions, very extreme property rights are needed in order to
have e¢ cient dissolution of the partnership, quite contrary to the intuition one gets from the discussion
of symmetric environments. In fact, when n = 99, property rights very close to (1; 0) are needed. However, from Myerson and Satterthwaite (1983), we know that (r1 ; r2 ) = (1; 0) will never allow an e¢ cient
dissolution. In fact, our example shows that arbitrarily close-to-extreme property rights can be needed
for e¢ cient dissolution, but property rights that are extreme will never allow it. Moreover, this example
suggests that agent 1, whose valuation is more likely to be higher, must own a higher proportion of the
asset and vice versa. This turns out to be a general result with interesting economic consequences:
Corollary 1 Suppose that distributions of valuations can be ranked according to …rst-order stochastic dominance
F1 F2 ::: FI :
(10)
Then, the property rights that guarantee the possibility of e¢ cient dissolution satisfy r1
r2
:::rI .
Proof. We know that at the transfer-minimizing ownership structure critical types are interior and thus
Q
satisfy
Fj (vi ) = ri : Moreover, for the property rights that guarantee dissolution, we have vi
x for all
j6=i
i 2 I. Therefore, we have ri =
Q
Fj (x); from which we obtain that
j6=i
ri
rj
=
Fj (x)
Fi (x)
> 1:
Our …ndings generalize straightforwardly to the case where a partnership owns multiple assets.8
References
[1] Che, Y. K. (2006): “Beyond the Coasian Irrelevance: Asymmetric Information.”Unpublished Notes.
[2] Clarke, E. (1971): “Multipart Pricing of Public Goods,” Public Choice, 2, 19-33.
[3] Cramton, P., R. Gibbons, and P. Klemperer(1987): “Dissolving a Partnership E¢ ciently,”
Econometrica, 55, 615-632.
[4] Groves (1973): “Incentives in Teams,” Econometrica, 41, 617-631.
[5] Krishna, V. (2002): “Auction Theory,” Academic Press.
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Details are available from the authors upon request.
7
[6] Krishna, V., and M. Perry (2000): “E¢ cient Mechanism Design,” working paper.
[7] Makowski, L. and C. Mezzetti (1993): “The Possibility of E¢ cient Mechanisms for Trading an
Indivisible Object,” Journal of Economic Theory, Vol. 59, p. 451–465.
[8] Myerson, R. (1981): “Optimal Auction Design,” Mathematics of Operations Research, 6, 58-73.
[9] Myerson R. B. and M. A. Satterthwaite (1983): “E¢ cient Mechanisms for Bilateral Trading,”
Journal of Economic Theory, 29(2):p. 265–281.
[10] Schweizer, U. (2006): “Universal Possibility and Impossibility Results,” Games and Economic
Behavior, 57, 73-85.
[11] Vickrey, W. (1961): “Counterspeculation, Auctions and Competitive Sealed Tenders,” Journal of
Finance, 16, 8-37.
[12] Williams S. R. (1999): “A Characterization of E¢ cient, Bayesian Incentive Compatible Mechanisms,” Economic Theory, Volume 14, Issue 1, p. 155 - 180.
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