Full text of DOI:10.1111/1468-0262.00111

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Econometrica, Vol. 68, No. 2 March, 2000 , 309᎐342
MECHANISM DESIGN WITH COLLUSION AND
CORRELATION
1
BY JEAN-JACQUES LAFFONT AND DAVID MARTIMORT
In a public good environment with positively correlated types, we characterize optimal
mechanisms when agents have private information and can enter collusive agreements.
First, we prove a weak-collusion-proof principle according to which there is no restriction
for the principal in offering weak-collusion-proof mechanisms. Second, with this principle,
we characterize the set of allocations that satisfy individual and coalitional incentive
constraints. The optimal weakly collusion-proof mechanism calls for distortions away from
first-best efficiency obtained without collusion. Allowing collusion restores continuity
between the correlated and the uncorrelated environments. When the correlation be-
comes almost perfect, first-best efficiency is approached. Finally, the optimal collusion-
proof mechanism is strongly ratifiable.
KEYWORDS: Mechanism design, collusion, correlation.
1. INTRODUCTION
THE PROVISION OF PUBLIC GOODS under informational constraints is one of the
leading textbook examples of public economics. How should a society made of
several agents with heterogeneous tastes for a public good design an incentive
mechanism to induce truthful revelation of the agents’ valuations? Does effi-
ciency conflict with incentives? How can this conflict, if any, be solved? Finally,
what is the distribution of informational rents induced by asymmetric informa-
tion?
One striking feature of most previous works on these issues is that they deal
mainly with the case where agents are unable to form coalitions to collectively
manipulate the decision rule.2 By focusing on the role of individual incentive
constraints, an important dimension of resource allocation in society has been
neglected: the formation of groups. It is particularly troublesome when these
1 We thank Jeon Doh-Shin, Antoine Faure-Grimaud, Oliver Hart, Eric Maskin, Patrick Rey,
Mike Riordan, Tomas Sjostrom, Yossef Spiegel, Dimitri Vayanos, Robert Wilson, seminar partici-
pants at Humbolt University-Berlin, Bocconi University, University of Maryland, Paris-Seminaire
´
Roy, University Pompeu Fabra, University of California at Berkeley, University College London,
Stanford Business School, University of Alabama, three referees and a co-editor for helpful
comments. Financial support from INRA’s AIP Contracts is gratefully acknowledged by David
Martimort.
2 Groves 1973 and Green and Laffont 1977 showed that efficiency could be achieved under
asymmetric information on the agents’ valuations with dominant strategy if budget balance is not a
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concern. Arrow 1979 and Aspremont and Gerard-Varet 1979 showed that budget balance could
be achieved under Bayesian implementation. Laffont and Maskin 1982 and Mailath and Postle-
waite 1990 showed that adding the possibility for the agents to veto the mechanism introduces a
real conflict between efficiency and incentives leading to the underprovision of the public good.
Ledyard and Palfrey 1996 show that this conflict also arises in the absence of participation
constraints when incentives conflict with redistribution concerns.
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309

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J.-J. LAFFONT AND D. MARTIMORT
coalitions can significantly reduce the efficiency of the optimal mechanism
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designed in the absence of coalition incenti¨e constraints. As Olson 1965, p. 1
has forcefully emphasized ‘‘groups of indi¨iduals with common interests usually
w
x
attempt to further those common interests иии and are expected to act on behalf of
their common interests much as single indi¨iduals are often expected to act on behalf
of their personal interest.’’ Following this argument, the standard theoretical
framework must be amended to allow also for the formation of groups promot-
ing their own collective goals instead of that of society as a whole. The present
paper offers a framework in which the consequences of collusion under asym-
metric information on both allocative efficiency and the distribution of rents in
society can be assessed.
We model a conflict between a social welfare maximizer and a group of agents
who benefit from the public good but who do not care about its budgetary cost.
Contrary to the standard assumption in mechanism design, the planner has not
a perfect control of the communication technology so that he cannot prevent the
agents from colluding.3 Collusion between these agents is modeled in reduced
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form as in Laffont and Martimort 1997 . A third party proposes to the colluding
agents a side-mechanism to collectively manipulate their sending of messages to
the government. As suggested above, this third party does not internalize the
social cost of the project but maximizes only the sum of the agents’ utilities.
Lastly, no technology for a credible disclosure of information is available to the
colluding partners. The mere forming of a coalition does not change informa-
tional asymmetries between the agents. Coalition formation takes place under
asymmetric information.
When agents do not collude, their Bayesian-Nash behavior does not put any
constraint on the set of interim individually rational and incentive compatible
allocations in a correlated information en¨ironment. Indeed, as shown by Cremer
´
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and McLean 1988 in the case of auction mechanisms, the existence of even a
small amount of correlation between the agents’ valuations for the public good
is enough to allow the principal to elicit this ‘‘almost common’’ information. This
result is in sharp contrast with the case of uncorrelated information since then
efficiency does conflict with incentives. Hence, the optimal levels of public good
exhibit discontinuities when the degree of correlation goes to zero.
This costless extraction of the agents’ surplus by the principal suggests also
that they are likely to form an active coalition in such a correlated information
environment. Nevertheless, a weak collusion-proofness principle holds in this
context: any equilibrium of the overall game of contract offer cum coalition
formation achieves an outcome that can be replicated with a weakly collusion-
proof grand mechanism, i.e., a grand mechanism such that the null side contract
is a continuation equilibrium of the game of coalition formation. Since one
agent’s acceptance of the side contract depends on the status quo utility level
3
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See Palfrey 1992 for a discussion of this assumption and some of its implications in the case of
Bayesian implementation.

MECHANISM DESIGN
311
that he gets from playing noncooperatively the grand mechanism offered by the
principal, the issue of learning from disagreement arises. Weakly collusion-proof
mechanisms are such that the null side mechanism is a continuation equilibrium
of the game of coalition formation sustained with passi¨e beliefs.4
The weak collusion-proofness principle provides a tractable description of the
set of perfect Bayesian equilibria of the overall game of contract offer cum
coalition formation. After having described this set, the principal’s welfare is
optimized subject to participation, individual and coalition incenti¨e constraints.
Generally, the efficient levels of public good can no longer be costlessly
implemented even in a correlated environment. Taking into account coalition
incentive constraints, there exists now a trade-off between efficiency and rent
extraction. Distortions in the quantities of public good that are produced in the
different states of nature reduce the cost of the binding coalition incentive
constraints.
Depending on the degree of correlation, collusion-proofness constraints take
quite different forms. For weak positive correlation, collusion-proofness con-
straints are similar to those that would arise under symmetric information within
the coalition. When the degree of correlation diminishes, coalition incentive
constraints are then less and less binding and the principal prevents more easily
collusion. In the limit of uncorrelated information, the principal costlessly obtains
collusion-proofness and the contractual outcome is the same as if agents had not
been colluding. By adding coalition incentive constraints, one moves then
continuously from the outcome with a strictly positive correlation to the outcome
with no correlation.5
For strong correlation, collusion-proofness constraints under asymmetric in-
formation are instead quite different from those obtained when agents can
credibly disclose their information. When the correlation becomes almost per-
fect, there is only a small probability that agents have different valuations for
the public good. The principal can shutdown production in this state of nature at
almost no social cost. This breaks the coalition agreement and almost achieves
the first-best level of expected welfare.
Since it is implemented with Bayesian strategies, the optimal weakly
collusion-proof contract is sensitive to the exact beliefs that the agents have at
the time of playing this mechanism. Because we focus on the case where
valuations for the public good may take only two values, the equilibrium
correspondence of this optimal mechanism as posterior beliefs change can be
fully described. This step of the analysis allows us to discuss the robustness of
the optimal mechanism to a preplay communication stage in which agents may
veto or ratify the truthful play of this mechanism and thereby signal some
4 Rubinstein 1985 coined this expression for games with asymmetric information in which
equilibrium behavior is sustained with prior beliefs out of the equilibrium path.
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5 Interestingly, this result requires neither risk aversion of the agents’ utility functions nor limited
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liability constraints on transfers Robert 1991 .

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J.-J. LAFFONT AND D. MARTIMORT
information to each other. It is then possible to show that the optimal weakly
collusion-proof mechanism is strongly ratifiable in the sense of Cramton and
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Palfrey 1995 .
Collusion in public good mechanisms has been first analyzed by Green and
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Laffont 1979 who prove that the Groves mechanisms are not robust to
coalitions when agents share freely their information. Still with dominant
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strategy mechanisms but with a continuum of types, Laffont and Maskin 1980
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show then that only pooling decision rules can be implemented. Cremer 1996
´
takes into account asymmetric information within coalitions and shows that the
Groves mechanisms are robust to size-two coalitions but not beyond. Similarly,
there exist some relatively negative results in the case of Nash implementation
when agents can form coalitions Maskin 1979 . The Nash environment can be
seen as an extreme case of perfect correlation between the agents’ types. Our
focus on Bayesian implementation with two types brings more positive results.6
Under asymmetric information within the coalition, the principal can implement
a much larger set of allocations in these strongly correlated environments. More
generally, our approach provides a complete description of the set of imple-
mentable allocations under collusion.
This paper extends Laffont and Martimort 1997 by stressing the role of
correlated information between the agents as a determinant of the strength of
their coalition and by making no restriction on the set of available mechanisms.
In this previous work, we restricted the analysis to the case of anonymous
mechanisms and uncorrelated information.7 With no correlation, there always
exists a costless weakly collusion-proof implementation of the second-best
noncooperative outcome if the principal can offer nonanonymous Bayesian
mechanisms. However, collusion still matters when types are correlated even
without any exogenous restriction on the set of mechanisms. Moreover, working
in a correlated environment, we obtain a full characterization of all transfers in
the optimal weakly collusion-proof mechanism. This characterization allows us
to compute all the ex post rents precisely and to characterize the outcome of the
grand mechanism when beliefs differ from passive ones. This is an important
step towards checking the strong ratifiability of the mechanism.
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Section 2 presents the model. Section 3 discusses the optimal mechanism
when agents do not collude. Section 4 derives the weak collusion-proofness
principle and characterizes weakly collusion-proof allocations. Section 5 de-
scribes the optimal weakly collusion-proof mechanism. Section 6 discusses
strong ratifiability. Section 7 concludes.
6
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See also Laffont and Maskin 1979 .
7 Beside the differences in the informational structures and the set of available mechanisms, this
previous paper was dealing with a model of regulation. This latter difference is without consequence
on the results.

MECHANISM DESIGN
313
2. THE MODEL
2.1. Technology, Preferences, and Information
We consider the provision of public good in a partial equilibrium model. A
Ž . Ž .
positive amount x of public good can be produced at cost c и with cЈ и )0 and
8
Ž .
ꢀ
4
cЉ и )0. There are two agents in the economy denoted by Ai, ig 1, 2 . Each
of them derives utility U s␪i xyti from consuming an amount x of public good
and paying tax ti. Agenti Ai agrees to participate in the public good mechanism
when his participation constraint is satisfied.
ꢀ
4
The agents’ valuations for the public good, ␪i, ig 1, 2 , are drawn from a
common knowledge joint distribution on ⌰ 2 where ⌰s ␪, ␪ is the common
ꢀ
4
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.
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support of ␪ and ␪ ⌬␪s␪y␪ . We refer to the probabilities p ␪i, ␪ of
1
j
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.
2Ž
.
ꢀ
42
each state ␪i, ␪ , for i, j g 1, 2 , as the common knowledge prior beliefs. To
j
Ž
Ž
.
.
Ž
.
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make notation simpler, we also write: p ␪, ␪ sp11, p ␪, ␪ sp ␪, ␪ sp12 ,
Ž
.
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.
p ␪,␪ sp22 , where the equality p ␪, ␪ sp ␪, ␪ is derived from the symmetry
between the two agents. Finally, to capture the congruence of the agents’
interests, ␪ and ␪ are positi¨ely correlated and p12rp11 Fp22rp12. We denote
1
2
2
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by ␳sp11 p22 yp12 the degree of positive correlation ␳s0 for independent
9
.
types . For simplifying technicalities, we also assume that 0-p12 Fp11. The
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conditional beliefs of agent Ai on Aj’s type j/i induced by the joint
distribution above are the same for both agents and, slightly abusing notations,
are denoted by p.
The government, or principal P, only knows the distribution of the agents’
valuations for the public good but is uninformed on the exact realizations of
these shocks at the time of choosing the public good mechanism. His objective is
to maximize the sum of the agents’ utilities knowing that the deficit for the
production of the public good must be covered by distortionary taxation raised
elsewhere in the economy. Formally, the social welfare function is written as
SWsÝ2is1U y 1q␭ c x yÝis1ti , where ␭ is the exogenous cost of public
2
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.Ž Ž .
.
i
funds.10
8 Restricting to two agents avoids considering the formation of subcoalitions and significantly
simplifies notations. However, our methodology could be extended to more than two agents at the
cost of an increase in complexity.
9 This assumption ensures that the optimal collusion-proof mechanism under asymmetric infor-
mation never entails bunching in the case of small correlation. It simplifies significantly the
exposition.
10
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The model is formally equivalent to the Laffont and Tirole 1986 partial equilibrium model of
regulation. It could be possible to build a general equilibrium model endogenizing the value of ␭.
This could be done by introducing a third uninformed agent, say A3, in the analysis. By imposing
that the sum of the contributions made by this agent and the group A1 yA2 covers exactly the cost
of the public good as in Aspremont and Gerard-Varet 1979 , we would be able to endogenize the
´
value of the budget constraint’s multiplier. Moreover, because of complete information on A3’s
valuation for the public good, this agent could be forced to always pay his valuation for the public
good. One would then be interested by the coalition between A1 and A2 against A3.
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J.-J. LAFFONT AND D. MARTIMORT
2.2. Mechanisms
The principal proposes a grand mechanism G to the agents. G maps any pair
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of messages m1, m2 belonging to the product message space M1 =M2 sM
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.
ꢀ
4
where Mi denotes the message space used by agent Ai into a triplet x, t1, t2 .
q
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.
Ž
ꢀ
4.
x denotes the amount of public good produced xgXsޒ and ti ig 1, 2 is
ꢀ
Ž . Ž . Ž .4
the tax paid by agent Ai to the principal. We denote by Gs x и , t1 и , t2 и this
grand mechanism.11
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To make notation simpler in the case of direct mechanisms Ms⌰=⌰ , we
denote by x, x, and x respectively the levels of public good when both agents
ˆ
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claim ␪, when their claims differ ␪, ␪ and when they both claim ␪. We also
ꢀ
4
Ž
denote by tkl for k, lg 1, 2 the tax paid by an agent whose type is ␪i s␪q 2y
.
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k ⌬␪ when the other agent’s type is ␪j s␪q 2yl ⌬␪. Because of symmetry
between the agents, the corresponding taxes are independent of the agents’
identity.12
2.3. Coalition Formation
ꢀ
Ž .
An uninformed third party, T, proposes a side mechanism Ss ␾ и ,
Ž .
4
to the agents to induce their collusive behavior.
yi и
ⅷ
igŽꢀ1,.24
␾ и is a collective manipulation of the messages sent to the principal.
ⅷ
ꢀ
Ž .4
yi и
igꢀ1, 24
is a pair of side transfers. The third party is not a source of
money and therefore the coalition’s budget is balanced: Ý2is1 yi ␪1, ␪ s0 for all
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2
2
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␪1, ␪ g⌰ .
2
From the revelation principle, there is no loss of generality in assuming that S
is a direct mechanism.13 Therefore ␾ и and yk и kg 1, 2 map ⌰ 2 respec-
tively into the set of measures on M and the set of balanced side transfers.
Lastly, T is benevolent and maximizes the sum U1 qU2 of the two colluding
agents’ utilities obtained by playing the composition of the grand and the side
mechanism.14
Ž .
Ž . Ž
ꢀ
4.
11 Note that we do not restrict a priori the set of mechanisms available to the set of direct
mechanisms. Other message spaces than the product of the agents’ type spaces ⌰ 2 can be used by
the principal.
12
The symmetry of the grand mechanism is without loss of generality under noncooperative
behavior as we will see below. We also show in the Appendix that it is without loss of generality in
the case of collusion for a small correlation. The amount of public good does not need to depend on
the identity of who has a high valuation for the public good when claims differ.
13 For any grand mechanism offered by the principal, one can restrict the third party to use direct
side mechanisms at the final stage of the game of contract offer cum coalition formation.
14
Using this third party as a side contract mechanism designer avoids the difficult issue of
informational leakages through contract offers. It eliminates also the problem of finding an extensive
form for describing the collusive game between the agents. This third party paradigm can be seen as
a black box for the repeated interaction by which collusion emerges. This is a modeling shortcut to
justify also our assumption that the side contract is in fact enforceable even if there is no court of
justice available to do so. This modeling characterizes the highest bound that can be achieved by the
coalition.

MECHANISM DESIGN
315
FIGURE 1.ᎏGame tree.
2.4. Timing of the Game
The timing of the overall game of contract offer cum coalition formation is as
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follows see also Figure 1 for the game tree :
1. Agents learn their respective valuations for the public good.
2. P proposes a grand mechanism G. If an agent vetoes the grand mecha-
nism, all agents get their reservation utility normalized exogenously at zero.
3. The third party proposes a side mechanism S to the agents and a
noncooperative continuation play of G if anyone refuses this side contract. If
both agents accept S, agents report their types to the third party who recom-
mends reports into the grand mechanism and who commits to enforce the
corresponding side transfers.
4. Reports are sent into the grand mechanism. The decision on the size of the
public good is made and taxes are paid by the agents. Side transfers, if any, are
implemented.

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J.-J. LAFFONT AND D. MARTIMORT
The third party’s offer of a side mechanism S on top of the grand-mechanism
ˆ
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G induces a two stage game ⌫ G, S . In the first ratification stage, agents
simultaneously accept or refuse the side mechanism and may thereby signal
their types to each other. In the second communication stage, agents send
messages either directly to the principal if at least one of them has refused the
side mechanism or to the third party if both have accepted. The third party
recommends then a collective manipulation of the messages to be sent to the
ˆ
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principal. We denote by E G, S the set of perfect Bayesian equilibria of
ˆ
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⌫ G, S .
We are interested in finding the optimal mechanism G, knowing that the
continuation game of coalition formation consists first of a side mechanism S
optimally chosen by the third party and second of a ratification-communication
ˆ
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game ⌫ G, S .
Note first that we eliminate equilibria based on weakly dominated strategies
at stage 3 of the overall game. Indeed, for any grand-mechanism G, there always
exists a continuation equilibrium of the game of coalition formation in which
each agent refuses any collusive offer he may receive because he expects that
the other agent also refuses this offer anyway. Second, following Ai’s rejection,
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.
Aj j/i may have updated his beliefs on Ai’s type. These beliefs affect the
noncooperative play of the grand mechanism G and therefore the status quo
payoffs that Ai gets following a rejection of the side-mechanism S. Therefore,
there may exist several side-mechanisms offered as continuation equilibria of
the game of coalition formation depending on what is learnt following the
rejection of these side-mechanisms.
ꢀ
4
Let p , p be a belief system where p are agent Ayi’s beliefs on agent Ai if
˜1 ˜2
˜
i
Ayi contemplates Ai’s refusal to play the side-mechanism S. We denote by
Ž
.
the game of asymmetric information induced by the grand mecha-
⌫ G, p , p
˜
i
yi
Ž
.
nism G at stage 4 following Ai’s refusal of playing S. In particular ⌫ G, p, p
denotes this game of asymmetric information when it is played with passive prior
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.
denotes the set of Bayesian-Nash equilibria of
beliefs. E G, p , p
˜
i
yi
Ž
.
Ž
.
⌫ G, p , p . Finally, let us denote by U ␪i, ei the payoff of a ␪ agent Ai in
˜
i
yi
i
i
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.
an equilibrium e gE G, p , p . Note that this interim payoff is computed as
˜
i
an expectation with respect tyoi prior beliefs. Indeed, because joint deviations
have probability zero in a Bayesian-Nash equilibrium, nothing has been learned
on Ayi following Ai’s refusal. Therefore, the deviant agent Ai still continues to
play the grand mechanism with his prior beliefs on Ayi’s type.
i
We are interested in collusive continuation equilibria in which no learning
occurs from the agreement of playing the side-mechanism S. We can thus define
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.
similarly ⌫ G(S, p, p the game of asymmetric information induced by the
composition of the grand-mechanism G and the side mechanism S at stage 4
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.
following acceptance of playing S by both agents. E G(S, p, p denotes simi-
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.
larly the set of Bayesian-Nash equilibria of ⌫ G(S, p, p . Since the revelation
principle applies at the last stage of the game, there is no loss of generality in
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.
Ž .
considering that E G(S, p, p contains the truthful equilibrium e*. Let U ␪
denote agent Ai’s payoff in this equilibrium when his type is ␪i.
i
i

MECHANISM DESIGN
317
3. NO-COLLUSION AND THE FIRST-BEST OUTCOME
It is by now a well known result that the optimal mechanism achieves the
first-best outcome in this correlated environment when the implementation
concept is Bayesian-Nash equilibrium, even with interim individual rationality
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.. 15
constraints Cremer and McLean 1988 . The key is to use agent A2’s report,
´
which is truthful in equilibrium, as a signal correlated with agent A1’s type and
to condition A1’s taxes on this information. The flexibility in the taxes paid in
the different states of nature can then be used to ‘‘stochastically’’ and costlessly
deter any incentive to lie. If agent A1 does not report truthfully his type, the
mechanism is designed so that he gets a negative expected payoff. Since this
ingenious trick can be used for each agent simultaneously, the revelation of both
agents’ types obtains at no cost for the principal.
Obviously, the taxes paid by the agents may be quite large when types become
almost uncorrelated.16 Indeed, the conditional probability that agents have the
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.
same types diminishes and the punishments or rewards in the different states
of nature must increase to induce revelation.
We provide thereafter a simple proof of this first best implementation in our
public good setting. To do that, we first need to describe the set of imple-
mentable allocations in this noncooperative context. Without collusion, the
revelation principle yields a characterization of the set of implementable alloca-
tions with individual incentive compatibility constraints only. Consider the
Bayesian incentive compatibility constraint of agent A1 when he has a high
valuation ␪ for the public good.17 Multiplying it by p ␪ sp11 qp12 )0, this
Ž .
constraint writes as
Ž .
1
p11 yt q␪ x qp yt q␪ x Gp yt q␪ x qp yt q␪ x .
ˆ
ˆ
12 Ž
.
Ž
.
12 Ž
.
11Ž
.
11
12
21
22
Similarly, when A1 has a low valuation ␪ for the public good, his Bayesian
incentive compatibility constraint writes as again multiplying by p ␪ sp12 q
p22 )0
Ž
Ž .
.
Ž .
2
Ž . Ž . Ž . Ž .
p12 yt q␪ x qp yt22 q␪ x Gp12 yt11 q␪ x qp22 yt q␪ x .
ˆ ˆ
21
22
12
Moreover, A1 must be induced to participate to the mechanism without know-
ing A2’s type. The following interim participation constraints must also be
satisfied: For a ␪ agent,
Ž .
3
p11 yt q␪ x qp yt q␪ x G0,
ˆ
Ž
.
12 Ž
.
11
12
and for a ␪ agent,
Ž .
4
Ž . Ž .
p12 yt q␪ x qp yt22 q␪ x G0.
ˆ
21
22
15 See also McAfee and Reny 1991 in the case of a continuum of types and Riordan and
Ž
.
Ž
.
Sappington 1988 for a related model using ex post information in the case of only one agent.
16 See the Appendix for explicit formulas of these taxes.
17 By symmetry A2 faces the same incentive and participation constraints.

318
J.-J. LAFFONT AND D. MARTIMORT
The principal maximizes expected welfare, defined as
Ž
.Ž
Ž ..
SWsp11 2␪ xy2t q 1q␭ 2t yc x
Ž
.
11
11
Ž
.Ž Ž ..
q2 p12 ␪q␪ xyt yt q 1q␭ t qt yc x
ˆ
ˆ
Ž
.
Ž
.
12
21
12
21
Ž
Ž
.Ž Ž ...
qp22 2␪ xy2t22 q 1q␭ 2t22 yc x
Ž .
Ž . 18
subject to constraints 1 to 4 .
PROPOSITION 1: Assume that types are strictly positi¨ely correlated, ␳)0; then
the optimal pro¨ision of public good without collusi¨e beha¨ior entails:
ⅷ
Ž
.
Ž
.
Ž
.
The first-best decision rule x*, x*, x* where cЈ x* s2␪, cЈ x* s␪q␪ and
ˆ
ˆ
Ž
.
cЈ x* s2␪.
ⅷ
Ž .
Ž .
Participation constraints 3 and 4 are binding and both agents’ types get zero
rent.
PROOF: All proofs are in an Appendix.
Ž .
Ž .
To implement the first-best outcome, the participation constraints 3 and 4
Ž .
must be binding and we look for solutions such that the incentive constraints 1
19
Ž .
and 2 are also binding. Then the existence of some strictly positive correla-
Ž .
Ž .
tion imposes that the system of linear binding constraints 1 -to- 4 is in fact
invertible. This invertibility ensures that the first-best schedule of public goods
can be implemented at zero cost for the principal.
The taxes that implement this optimal allocation of public good are highly
dependent on the information structure. When correlation becomes weaker,
taxes become increasingly punishing when both agents claim ␪ and when the
agent who pays the tax claims ␪ and the other claims ␪. On the contrary, agent
Ai is increasingly rewarded when both agents claim being ␪ and when he claims
Ž
.
␪ and Aj j/i claims ␪. Indeed, with positive correlation, it becomes easier to
induce revelation from a ␪ agent Ai if, when he lies, he is heavily punished
Ž
.
when facing a ␪ agent Aj j/i who truthfully reveals.
Even a very small amount of correlation can be used to threaten the agents of
being heavily punished for lying on their types and to achieve the first best
allocation. However, with no correlation, allocative distortions become necessary
to reduce a ␪ agent’s costly informational rent. Therefore, in this noncoopera-
tive setting, the optimal mechanism fails to be continuous with respect to the
information structure.
For notational convenience, let p denote the probability of a high valuation
ˆ
2
2
Ž
.
type in the case of no correlation. We have thus p sp , p s 1yp , and
ˆ
ˆ
11
22
Ž
.
p sp sp 1yp .
ˆ
ˆ
12
21
18
From the symmetry of the model, these incentive and participation constraints are the same for
both agents.
19
Ž
.
Cremer and McLean 1988 show in fact that incentive constraints can be slack.
´

MECHANISM DESIGN
319
PROPOSITION 2: Assume that types are independently distributed, ␳s0; then the
optimal pro¨ision of public good without collusi¨e beha¨ior entails:
U
0
U
ˆ0
U
0
U
ⅷ
Ž
.
Ž
.
The second-best decision rule x , x , x where cЈ x0 s2␪,
␭
p
ˆ
U
Ž
.
cЈ x s␪q␪y
⌬␪
ˆ0
1q␭ 1yp
ˆ
and
␭
p
ˆ
U
Ž
.
cЈ x0 s2␪y2
⌬␪ .
1q␭ 1yp
ˆ
ⅷ
Ž
.
Ž
.
A ␪ resp. ␪ agent gets a strictly positi¨e resp. zero rent.
Ž .
Ž .
Without any correlation, the system of binding equations 1 to 4 can no
Ž .
more be inverted. Only expected taxes are defined from the binding incentive 1
and participation constraints 4 and a ␪ agent’s informational rent becomes
Ž .
costly. Allocative distortions of x and x are needed to reduce this cost.
ˆ
4. COLLUSION
Because the agents get zero rent from the mechanism proposed by the
principal if they play noncooperatively, they are willing to coordinate their
messages to countervail the principal’s power. The optimal grand mechanism
with a noncooperative behavior creates endogenously the stakes for some
collusive behavior.
4.1. The Third-Party’s Problem
Ž
.
Following Cramton and Palfrey 1995 , we say that a side-mechanism S is
Ž
.
unanimously ratified for e , e , p , p if, for all ␪i g⌰ and all i,
˜1 ˜2
1
2
Ž
.
Ž
.
U ␪ GU ␪i , ei
i
i
i
Ž
.
Ž
.
is a noncooperative equilibrium of ⌫ G, p , p and
yi
where e gE G, p , p
˜
˜
i
i
i
yi
where
Ž
.
Ž
< .Ž
␪
yi i
Ž
.
Ž
Ž
yi ..
U ␪ s
p ␪
yi ␪i , ␪ yti ␾ ␪i , ␪
Ý
i
i
yi
⌰
y
i
Ž
Ž
...
q␪i x ␾ ␪i , ␪ ,
yi
᭙ ␪i g⌰ ,
Ž
.
is a ␪ agent Ai’s payoff from playing e*g⌫ G(S, p, p .
i
DEFINITION 1: A continuation collusi¨e equilibrium of the game of coalition
ꢀ
4
formation consists of first, a system of beliefs p , p and associated equilibria
˜1 ˜2
Ž
. Ž
e g⌫ G, p , p
yi
ꢀ
4.
ig 1, 2 and second, a truthtelling direct side mechanism S*
that is unanimously ratified for the quadruplet e , e , p , p and that maxi-
˜
i
i
Ž
.
˜1 ˜2
1
2
mizes the third party’s objective function.

320
J.-J. LAFFONT AND D. MARTIMORT
The continuation equilibrium of the game of coalition formation is thus a
Ž ..
Ž
solution to the third party’s following problem denoted thereafter T :
Ž
.
Ž
.
j
max
pi j yt ␾ ␪ , ␪ yt ␾ ␪ , ␪
1Ž
.
2 Ž
.
Ý
42
i, j g 1, 2
Ž
i
j
i
ꢀ
Ž . Ž .Žk gꢀ1, 24.4
␾ и , yk
и
Ž
.
ꢀ
Ž
.
Ž
.
j
q ␪i q␪ x ␾ ␪ , ␪
Ž
.
.
j
i
subject to
ˆ
ˆ
i
Ž
.
Ž
.
Ž
<
.
BIC
U ␪ G
p ␪
␪
y ␪ , ␪
yt ␾ ␪ , ␪
Ý
i Ž
.
Ž
.
i ž
/
ž
i
i
yi
i
i
yi yi
⌰
y
i
᭙ ␪ , ␪ g⌰ 2 ;
ˆ
ˆ
i
q␪ x ␾ ␪ , ␪
,
Ž
.
Ž
.
ž
/
/
i
i
yi
i
Ž
.
Ž
.
Ž
.
Ž
.
e gE G, p , p
yi
BIR
U ␪ GU ␪i , ei
for some
᭙ ␪i g⌰ ;
˜
i
i
i
i
i
2
2
Ž
.
Ž
.
yi
y
␪i , ␪ s0,
᭙ ␪i , ␪ g⌰ .
Ý
k
yi
k
s1
Along an equilibrium path on which the side-contract S* is unanimously
ratified, no learning occurs since unanimous ratification is a pooling strategy.
Since nothing is revealed at the ratification stage, Bayesian incentive constraints
and final expected utilities are computed with passive beliefs.
4.2. Weakly Collusion-Proof Mechanisms
DEFINITION 2: G is weakly collusion-proof if and only if it is a truthtelling
U
U
ꢀ
Ž
.
4
is
k gꢀ1, 24
direct mechanism and the null side mechanism S0 s ␾*sId, yk s0
Ž
.
unanimously ratified for e*, e*, p, p , where e* is the truthful equilibrium of G
played with passive beliefs.
In other words, G is weakly collusion-proof if and only if the third-party offers
ˆ
Ž
the null side mechanism and there exists a perfect Bayesian equilibrium in E G,
SU0 such that agents accept to play e* sustained with passive beliefs out of the
.
equilibrium path.
PROPOSITION 3: To characterize the outcome of any perfect Bayesian equilibrium
of the game of grand mechanism offer cum coalition formation such that a collusi¨e
continuation equilibrium occurs on the equilibrium path, there is no loss of
generality in restricting the principal to offer weakly collusion-proof mechanisms.
The logic behind this weak collusion-proofness principle is similar to that
underlying the standard revelation principle: any equilibrium of the overall
game of grand mechanism offer cum side contracting gives an allocation that
can be replicated with a direct grand mechanism G offered by the principal
himself. This grand mechanism is such that the coalition still forms at the

MECHANISM DESIGN
321
U
ˆ
Ž
.
ratification stage of ⌫ G, S0 and each agent Ai finds it optimal to report his
valuation ␪ truthfully to the principal thereafter.
i
One could argue that agents’ information is not only restricted to their own
valuations but also includes the knowledge of the side mechanism they use to
collude. The principal could try to elicit this information by also asking the
agents which side mechanism they are actually playing. But the third party could
react by inducing further manipulations of those reports of the side mechanism.
These reactions and counterreactions lead naturally to a problem of infinite
regress. By restricting the principal to use grand mechanisms only contingent on
the agents’ valuations, we cut arbitrarily this process in favor of the colluding
agents. This amounts to an incomplete contracting assumption that fits our
desire to give collusive behavior its best chance.20 Within this incompleteness
contractual framework, we are nevertheless able to generalize the revelation
principle and benefit from the weak collusion-proofness principle to obtain a
simple constructive characterization of the set of implementable allocations.
The next proposition characterizes this set in the case of symmetric grand
mechanisms.
PROPOSITION 4: A symmetric grand mechanism G is weakly collusion-proof if
w
w
and only if there exists ⑀g 0, 1 such that
˜ ˜
˜ ˜
˜ ˜
Ž .
5
y2t q2␪ xGyt ␪ , ␪ yt ␪ , ␪ q2␪ x ␪ , ␪
1ž 2 / ž
/
ž
/
11
1
2
1
2
1
2
᭙ ␪ , ␪ g⌰ 2 ;
˜ ˜
ž
/
1
2
p11
Ž .
6
yt12 yt21 q ␪q␪y
⑀ ⌬␪ x
ˆ
ž
/
p12
p11
p12
˜ ˜
˜ ˜
˜ ˜
Gyt ␪ , ␪ yt ␪ , ␪ q ␪q␪y
⑀ ⌬␪ x ␪ , ␪
1ž 2 / ž
/
ž
/
1
2
1
2
1
2
ž
/
᭙ ␪ , ␪ g⌰ 2 ;
˜ ˜
ž
/
1
2
⑀ p122
Ž .
7
y2t22 q2 ␪y
⌬␪
x
ˆ
2
ž
Ž
.
/
p12 p22 q⑀ p11 p22 yp12
˜ ˜
˜ ˜
1 2
Gyt ␪ , ␪ yt ␪ , ␪
1ž 2 / ž
/
1
2
⑀ p122
˜ ˜
q2 ␪y
⌬␪ x ␪ , ␪
ž
/
1
2
2
ž
Ž
.
/
p12 p22 q⑀ p11 p22 yp12
᭙ ␪ , ␪ g⌰ 2 .
˜ ˜
ž
/
1
2
Ž .
If ⑀)0, the Bayesian incenti¨e compatibility constraint 1 of a ␪ agent is binding.
20
Ž
.
In a related multiprincipal context, Epstein and Peters 1997 show that a similar infinite
regress may converge. This convergence might allow one to define universal sets of types and
universal complete grand mechanisms.

322
J.-J. LAFFONT AND D. MARTIMORT
The weakly collusion-proof mechanisms described above are such that a ␪
agent’s incentive constraint is not binding. Only a ␪ agent’s incentive constraint
may be binding. These mechanisms are the only ones that are of interest as
Proposition 5 will confirm.
The parameter ⑀ is a discount factor less than one which captures the fact
that collusion takes place under asymmetric information. True valuations must
be replaced by ¨irtual valuations21 in the coalition incenti¨e constraints 6 and
Ž .
Ž .
Ž .
7 . The third party problem T is constrained by the reservation utilities that
the agents obtain from playing noncooperatively the grand mechanism and the
incentive compatibility constraint at the coalition formation stage. Virtual valua-
tions are then lower than true valuations to take into account the costly
multipliers of these constraints.
Ž
In the sequel, the downward coalition incentive constraints rewritten after
having used the symmetry of the grand mechanism are of particular interest:
.
Ž .
8
y2t q2␪ xGyt yt q2␪ x,
ˆ
11
12
21
p11
p12
p11
p12
Ž .
9
yt12 yt21 q ␪q␪y
⑀ ⌬␪ xGy2t q ␪q␪y
⑀ ⌬␪ x.
ˆ
22
ž
/
ž
/
Ž .
8 says that a coalition made with two ␪ agents prefers telling collectively the
Ž .
truth to the principal rather than lying and telling that one of the agents is ␪. 9
says that a ␪, ␪ coalition prefers telling the truth rather than claiming that
both agents are ␪.
Ž
.
The logic of the first-best noncooperative implementation described in Sec-
tion 2 is to offer large penalties and large rewards depending on the states of
Ž
.
nature to induce revelation. For instance, the formula for t11 resp. t22 in the
Ž
Appendix shows that this tax may become extremely large and positive resp.
negative when the agents’ types are less and less correlated. This suggests that
the coalition incentive constraint 8 resp. 9 is likely to be binding at the
optimum of the principal’s problem. The extent to which the principal is
restricted in using Bayesian transfers comes from the existence of these coali-
tion incentive constraints.
.
Ž . Ž
Ž ..
Once coalition incentive constraints are characterized, it is useful to derive
necessary monotonicity conditions for the implementability of a schedule of
outputs.
2
Ž
.Ž
.
w
COROLLARY 1: For a weak correlation, ␳F p12 qp22 p12rp11 , the schedule
Ž
.
w
of implementable outputs is increasing xFxFx for all ⑀g 0, 1 . For a strong
correlation, ␳) p12 qp22 p12rp11 , the schedule of implementable outputs is
ˆ
2
Ž
.Ž
.
21
Ž
.
See Myerson 1979 for the standard definition of virtual valuations.

MECHANISM DESIGN
323
Ž
.
non-monotonic xGxGx if and only if
ˆ
⑀ p122
p11
p12
Ž .
␺ ⑀ s1q2
y
⑀-0,
2
Ž
.
p12 p22 q⑀ p11 p22 yp12
i.e., for ⑀ large enough; otherwise it remains increasing.
The striking feature of this corollary is that, in the case of strong correlation
Ž
.
i.e., when p12 is small enough , non-monotonic schedules of outputs can be
implemented by the principal if he chooses ⑀ large enough. The reason for this
non-monotonicity is that virtual valuations coming from the formation of the
coalition under asymmetric information are no longer ranked in the same order
as true valuations. When agents’ types are almost perfectly correlated, the
probability that they both get a high valuation for the public good is large.
Henceforth, the incentive constraint of a ␪ agent willing to mimic a ␪ one at the
coalition formation stage is also very costly to the third-party from an ex ante
point of view. Inducing revelation within the coalition requires therefore large
Ž
.
distortions of the optimal manipulation of reports ␾* ␪, ␪ with respect to what
the third-party could implement under symmetric information. Hence, the sum
Ž
.
of the virtual valuations of a ␪, ␪ coalition may become smaller than that of a
␪, ␪ one.
Ž
.
This nonmonotonicity property will create a substantive difference between
collusion under symmetric and asymmetric information when the correlation is
strong contrary to the case of a weak correlation.
5. THE OPTIMAL WEAKLY COLLUSION-PROOF MECHANISM
We now turn to some normative analysis and optimize the principal’s welfare
subject to Bayesian individual incentive, participation, and coalition incentive
constraints. In the sequel, we focus only on the ␪ agent’s Bayesian incentive
Ž .
Ž .
Ž .
constraint 1 , the downward coalition incentive constraints 8 and 9 , and the
Ž . 22
␪ agent’s participation constraint 5 .
Before writing the principal’s problem, let us first introduce four new vari-
ables usyt q␪ x, u syt q␪ x, u syt q␪ x, usyt q␪ x. These are
ˆ1
ˆ ˆ2
ˆ
11
12
21
22
the ex post rents obtained by both types of agent in each possible state of nature.
Rearranging expected social welfare, individual and coalitional incentive and
participation constraints as functions of these variables, the principal’s problem
Ž .
P rewrites as
ŽŽ .
.Ž
Ž ..
max p11 1q␭ 2␪ xyc x y2␭u
ꢀ Ž .
x и ; u
4
Ž
.
Ž . .
Ž
q2 p12 1q␭ ␪q␪ xyc x y␭ u qu
ˆ
ˆ
ˆ1 ˆ2
Ž
.
Ž
.
ž
/
ŽŽ
.Ž
Ž ..
.
qp22 1q␭ 2␪ xyc x y2␭u ,
22
We check ex post that all other constraints are indeed satisfied.

324
J.-J. LAFFONT AND D. MARTIMORT
subject to
Ž
Ž
.
.
Ž
Ž
.
.
Ž
.
10
11
BIC
CIC
p uqp u Gp u qp uq⌬␪ p xqp12 x ,
12 ˆ1
11ˆ2
11 ˆ
11
12
2uGu qu q⌬␪ x,
1
2
ˆ1 ˆ2
ˆ
p11
p12
Ž
Ž
.
.
Ž
Ž
.
Ž .
⑀ ⌬␪ xyx ,
ˆ
12
CIC
u qu G2uq⌬␪ xq
ˆ1 ˆ2
.
13
IR
p u qp uG0.
12 ˆ2
22
2
ⅷ
Ž
.Ž
.
For a weak correlation, ␳F p12 qp22 p12rp11 , the monotonicity condi-
tion derived from the coalition incentive constraints is
xGxGx.
ˆ
2
ⅷ
Ž
.Ž
.
For a strong correlation, ␳G p12 qp22 p12rp11 , the monotonicity condi-
tion becomes
xGxGx
Ž .
␺ ⑀ G0
if and only if
otherwise.
and
ˆ
xGxGx
ˆ
In the sequel, we focus separately on these two polar cases of weak and strong
correlations.
5.1. Weak Correlation
Solving the principal’s problem with standard Lagrangean techniques yields
Proposition 5.
2
Ž
.Ž
.
PROPOSITION 5: Assuming that 0F␳F p12 qp22 p12rp11 , the symmetric
optimal weakly collusion-proof mechanism G* entails:
A strictly decreasing schedule of outputs xc )xc )xc with ‘‘no distortion at the
ⅷ
ˆ
top’’ xc sx* and downward distortions with respect to the no collusion outcome
otherwise: xc -x* and xc -x* where
ˆ
ˆ
␭
p11
p12
c
Ž
Ž
.
.
Ž
.
14
cЈ x s␪q␪y
⌬␪
1q
,
ˆ
ž
/
1q␭
2 p12
p12 q␳
␭
1
p12 p11
c
Ž
.
15
cЈ x s2␪y
⌬␪
p11 q2 p12 y
.
ž
/
1q␭
p22
p12 q␳
ⅷ
Ž
.
The Bayesian incenti¨e constraint of a ␪ agent 10 is always binding. The
downward coalition incenti¨e constraints 11 and 12 are both strictly binding
when ␳)0. The participation constraint of a ␪ agent 13 is also binding. All
remaining constraints are strictly satisfied. A ␪ agent gets a strictly positi¨e informa-
tional rent.
Ž
.
Ž .
Ž
.

MECHANISM DESIGN
325
Ž
.
Ž
.
Binding Constraints: The fact that both coalitions ␪, ␪ and ␪, ␪ are pre-
vented from misreporting limits the feasible transfers that could be used by the
principal to extract the agents’ information. t11 cannot be made largely positive
as it is in the no-collusion outcome without violating the coalition incentive
Ž
.
Ž
.
constraint 11 . A ␪, ␪ coalition would like to avoid bearing these detrimental
punishments by mimicking a ␪, ␪ coalition. 11 must be binding at the
optimum. Similarly, a ␪, ␪ coalition would like to mimic a ␪, ␪ one to get the
corresponding large rewards requested in the no-collusion outcome since t22 is
then large and negative. 12 must thus also be binding.
Ž
.
Ž .
Ž
.
Ž
.
Ž
.
Because of these constraints on the set of transfers, a ␪ agent must be given a
strictly positive rent contrary to the case without collusion. Large rewards and
punishments can no longer be used by the principal without violating the
coalition incentive constraints.
Coalition Incenti¨e Constraints: For a weak correlation, ⑀s0 at the optimum.
Indeed, there is no gain in having ⑀ strictly positive since this would only
23
Ž
.
increase the cost of the coalition incentive constraint 12 . Interestingly, the
binding collusion-proofness constraints take therefore the same form as if
agents could credibly share their information within the coalition. Everything
happens as if asymmetric information does not really undermine the ability of
the group to form. However, the agents’ participation constraints being the
interim ones, they differ from the case of symmetric information within the
coalition.
Output Distortions: Distorting downward x and x below their first-best values
ˆ
Ž
.
Ž
.
Ž .
reduces the costs of the individual 10 and coalitional incentive 11 and 12
constraints. The size of the public good must be reduced because of asymmetric
information and collusion. Note that this result contrasts with the usual free-rider
phenomenon discussed in the literature. As shown for instance in Mailath and
Ž
.
Postlewaite 1990 , the reduction in the size of the public good is then due to the
conflict between individual incentive and individual participation constraints.
Here, distortions come from the conflict between coalitional incentive and
individual participation constraints. It is because a coalition can form that
individual incentives become costly to provide and that allocative distortions are
needed in this correlated environment.
Role of the Correlation: This conflict between coalition incentive and participa-
tion constraints increases with the amount of positive correlation between the
agents’ types. When types are more positively correlated, the distortion required
c
Ž
.
on x to reduce the cost of the coalition incentive constraint 11 is larger since
ˆ
c
Ž
Ž
..
p12 is smaller see 14 . The distortion on x needed to reduce the cost of the
coalition incentive constraint 12 is instead rather small since p22 is now
relatively large see 15 .
Ž
.
Ž
Ž ..
23
Ž
.
In other words, the Bayesian incentive constraint in the third party problem T is not tight at
Ž
.
the optimum. This does not mean that 10 is not costly for the principal since he has a different
objective function than that of the third party.

326
J.-J. LAFFONT AND D. MARTIMORT
Ex post Rents: Ex post rents in the optimal weakly collusion-proof mechanism
satisfy
Ž
Ž
Ž
Ž
Ž
Ž
Ž
.
.
.
.
.
.
.
16
17
18
19
20
21
22
u-u q⌬␪ xc,
ˆ2 ˆ
u )uq⌬␪ xc,
ˆ1
u -0,
ˆ2
u)u ,
ˆ1
u)0,
u )uy⌬␪ xc,
ˆ2
u)u y⌬␪ xc.
ˆ1
ˆ
Ž
.
Ž .
16 and 17 indicate that the truthful strategy of a ␪ agent is Bayesian.
Ž
.
Indeed 16 shows that a ␪ agent A1 never wants to tell the truth if he is sure of
facing a ␪ agent A2. Instead, 17 shows that he always tells the truth if he is
sure that A2 claims ␪. Contrary to what happens in the no-collusion outcome
Ž
.
where in fact u-u , a ␪ agent is now rewarded when he faces a ␪ agent A
ˆ1
2
and punished when he faces a ␪ agent A2. It is only because these rewards
cover in expectation the penalties that a ␪ agent A1 weakly prefers to tell the
truth to the principal.
Ž
.
Ž .
Instead, 21 and 22 show that the optimal weakly collusion-proof contract is
such that the dominant strategy incentive constraints of a ␪ agent are always
strictly satisfied. This dominant strategy requirement for one type somewhat
simplifies the optimal mechanism since this type’s equilibrium strategy is not
sensitive to the exact beliefs that agents have at the time of playing the
mechanism.
Nevertheless, the ex post rent of a ␪ agent A1 depends also explicitly on A2’s
type. For instance, just as in the no-collusion outcome, two ␪ agents are given
strictly positive ex post rents. These agents are still subsidized for the consump-
tion of the public good. On the contrary, a ␪ agent A1 facing a ␪ agent A2
makes a negative ex post profit. The tax he pays for the public good is very large
and his final utility is negative. Intuitively, by setting u negative, the principal
ˆ2
Ž
.
insures that 11 is not very costly. Satisfying the ␪ agent’s interim participation
constraint requires then to set u strictly above zero.
5.2. The Polar Case of No Correlation
In the degenerate case of no correlation, a rather striking result obtains:
Ž
.
PROPOSITION 6: With no correlation between the agents’ types ␳s0 , the
optimal weakly collusion-proof grand mechanism G* entails the same strictly
decreasing schedule of outputs as without collusion, xc sxU, xc sxU and x0c sxU0 .
ˆ0 ˆ0
0
0

MECHANISM DESIGN
327
Ž
.
Ž .
Only the individual incentive 10 and the participation constraints 13 are
Ž
.
strictly binding. The multipliers of the coalition incentive constraints 11 and
Ž
.
Ž .
12 are zero at the optimum of P . Nonanonymous transfers implement in a
Bayesian and weakly collusion-proof way the optimal second-best contract, i.e.,
the noncooperative outcome obtained when there is no correlation between the
agents’ types. Collusion has no impact when agents do not know more informa-
tion on each other than what is available to the principal. Intuitively, everything
happens as if the principal sells the mechanism to the third party so that the
latter perfectly internalizes the social welfare objective.
Strikingly, the optimal policy moves continuously from the case with positive
correlation to the case without correlation when coalition incentive constraints
are taken into account. Allowing collusion restores continuity of the optimal
contract with respect to the information structure. For any degree of positive
correlation, there are enough binding constraints to pin down the values of the
optimal transfers in all states of nature. When the degree of correlation
converges to zero, these transfers converge and the limiting transfers implement
the second-best outcome in a collusion-proof way. Not only the expected values
of these limiting transfers but also their precise values in all states of nature are
now perfectly determined even in the case of no correlation.24
5.3. The Polar Case of Almost Perfect Correlation
Ž
.
With an almost perfect correlation p12 close to zero , both agents have
almost always the same type.
As a benchmark, suppose first that the agents can credibly exchange informa-
tion on their types because, for instance, the third party is endowed with a
technology making this information verifiable within the coalition.25 Assume also
that agents agree to play the grand mechanism26 before they learn each other’s
types so that their participation constraints remain unchanged with respect to
5.1. Collusion-proofness constraints must now be written with true valuations
instead of virtual ones. This has two implications: First, the binding downward
Ž
.
ˆ
Ž .
coalition-proofness constraints take the same form as 11 and 12 when ⑀s0.
Ž
.
Second, monotonicity of the schedule of outputs xGxGx is now always
necessary for implementability.
If this monotonicity constraint would not be binding at the optimum, the
Ž
.
Ž .
optimal levels of public good would still be given by 14 and 15 . However,
24 Coalition incentive constraints somehow compactify the set of feasible allocations just as risk
Ž
Ž
..
aversion or limited liability does see Robert 1991 .
25 We assume also that information disclosure only takes place when the agents decide to collude.
It implies that the grand-contract must still satisfy the individual Bayesian incentive constraints.
26
Note that we restrict the grand mechanisms to use reports by each agent on his own
information only. This restriction is justified if the disclosure of information does not occur with
probability one. Unrestricted grand mechanisms using both agents’ reports on their own types and
what they have learned on their colluding partner are useless in this case. Indeed, the coalition,
when it forms, can always pretend that it did not so that both agents’ messages on what they have
learned is always uninformative.

328
J.-J. LAFFONT AND D. MARTIMORT
when p12 is small enough, it is easy to check that the monotonicity constraint
xGx would be violated. Hence, we get Proposition 7.
ˆ
Ž
PROPOSITION 7: With almost perfect correlation between the agents’ types p12
.
small enough but positi¨e and symmetric information within the coalition, the
optimal weakly collusion-proof grand mechanism G* entails partial pooling xc sx*
)xc sxc sxcP with xcP defined by
ˆ
p22 ␪qp12 ␪q␪
␭
p11 qp12
2 p12 qp22
Ž
.
y2
cP
Ž
.
Ž
.
s2
23
cЈ x
⌬␪
.
ˆ
ž
/
2 p12 qp22
1q␭
Taking into account collusion-proofness constraints under symmetric informa-
tion in an almost perfectly correlated environment undermines quite signifi-
cantly the achievement of the efficient outcome. The optimal allocation entails
now lots of pooling. When the agents’ types are strongly correlated, we have
seen in Section 2 that rewards and punishments necessary to implement the
first-best outcome are relatively small. However, when agents collude under
symmetric information, their collusion becomes now also harder to prevent. As a
result, the optimal collusion-proof mechanism becomes less responsive to their
messages. In this framework, the optimal contract looks like an incomplete
contract making only partial use of the information.27
Things are quite different under asymmetric information within the coalition.
For a strong correlation, non-monotonic schedules of outputs can be imple-
mented by the principal if ⑀ is large enough. Because now xFx, the collusion-
ˆ
Ž
.
proofness constraint 12 is relaxed when ⑀ is as large as possible. An upper
bound of the principal’s welfare obtains therefore for ⑀s1.
PROPOSITION 8: With almost perfect correlation and asymmetric information
within the coalition, the optimal weakly collusion-proof grand mechanism G*
entails: xc sx*)xc )xc s0 with xc defined by
ˆ
␭
2 p12
c
Ž
.
cЈ x s2␪y
⌬␪ .
1q␭ p22
The expected rent of the ␪ agent is
p uqp u
p11 qp12
p12
12 ˆ1
11
s
⌬␪ xc.
p11 qp12
27 A last interpretation is worth stressing. Assume that the agents’ types are now perfectly
Ž
.
correlated p12 s0 . The optimal pooling allocation becomes
␭
p11
cP
Ž
.
s2␪y2
cЈ x
⌬␪ .
1q␭ p22
This is the optimal distortion in a one principal-agent model where the principal is facing directly a
third party endowed with a utility function being the sum of the agents’ utility functions.

MECHANISM DESIGN
329
Moreo¨er,
p11
u sus0,
ˆ2
u s 1y
⌬␪ xc,
ˆ1
ž
/
p12
and
us⌬␪ xc.
For a strong correlation, the principal can now offer non-monotonic schedules
of outputs in a collusion-proof way. In particular, cancelling the production of
the public good when agents’ types are different and still keeping a positive
production when types are the same becomes a valuable option. This strategy is
not very costly from an ex ante allocative point of view since p12 is small.
Ž
.
However, it relaxes quite significantly 12 when ⑀ is positive. When p12 is
Ž
.
almost zero, the right-hand side of 12 is so largely negative that this coalition
incentive constraint does not matter any more for the principal.28 u can be set
ˆ1
at a very large negative value and still this constraint can be easily satisfied.
Choosing such a punishment in case of different reports also relaxes quite
Ž
.
significantly 11 and the principal’s problem is almost as without collusion.
Since collusion does not affect too much social welfare, the principal can set
xc to a level close to its first best value x*. The punishments for claiming to have
different types are so effective that a ␪ agent’s expected rent is now close to
zero. Therefore, with almost perfect correlation, the optimal collusion-proof
contract achieves an ex ante social welfare close to its full information value.
6. ROBUSTNESS
We focus in this section on the case of a small correlation.
6.1. Multiplicity of Equilibria of G*
When played with passive beliefs, the optimal weakly collusion-proof mecha-
< Ž
.<
nism G* has several pure strategy equilibria, i.e., E G*, p, p )1. The first one
is the truthful symmetric equilibrium e* that we have derived above. However,
there exist also two asymmetric nontruthful pure strategy equilibria. eU1 resp.
Ž
eU2 is one such equlibrium where A1 resp. A2 always claims to be a ␪ agent
.
Ž
.
Ž
.
and A2 resp. A1 reveals truthfully his type.
Ž
.
Ž .
Indeed, from 21 and 22 , a ␪ agent always claims truthfully his type. A ␪
Ž
.
agent A2 anticipates that A1 always claims ␪. Then, from 17 , he reports
truthfully. Thus, agent A2 always reveals his type. Anticipating that A2 always
Ž
.
reveals his type, from the binding constraint 10 , a ␪ agent A1 is indifferent
between lying or not and lies in this equilibrium.
28
When p12 s0, everything happens as if there were a single agent.

330
J.-J. LAFFONT AND D. MARTIMORT
In this asymmetric equilibrium, denoted thereafter eU1 , A1 gets the same ex
U
Ž
.
Ž
.
ante and interim payoffs as in e*, U1 ␪1, e1 sU ␪ for all ␪1. Instead, A2’s
1
U
Ž
.
Ž .
interim payoffs differ from that in e*. Indeed, we have U2 ␪, e1 su -U ␪ s
ˆ1
U
Ž
. Ž
p uqp u r p qp
. Ž
12
Ž
..
Ž
.
Ž .
Ž
Ž ..
from 19 and U2 ␪, e1 su)U ␪ s0 from 20 .
12 ˆ1
11
11
However, using the values of ex post rents given in the Appendix, A2’s ex ante
payoff is the same as in e*.
In fact, in equilibrium e1 , a ␪, ␪ coalition reports ␪, ␪ when it is indiffer-
U
Ž
.
Ž
.
Ž
.
Ž
. ŽŽ .
ent between collectively claiming ␪, ␪ and ␪, ␪ 12 is indeed binding for the
.
ex post rents defined by G* . This indifference of the coalition is broken in favor
of the principal so that, among all payoff equivalent equilibria from the
third-party’s point of view, the agents play the most preferred by the principal.
Ž
.
The fact that E G*, p, p is not a singleton is not a big problem since all these
equilibria yield the same aggregate payoff to the agents.29
6.2. Strong Collusion-Proofness
Ž
.
The fact that E G*, p, p is not a singleton shows also that G* is never
U
ˆ
Ž
.
strongly collusion-proof, i.e., E G*, S0 is not reduced to a singleton either.
Indeed, there exist other equilibria of ⌫ G*, S0 than unanimous ratification
and subsequent play of e*. To show this result, it is enough to exhibit such an
U
ˆ
Ž
.
U
U
ˆ
Ž
.
equilibrium of ⌫ G*, S0 . Unanimous veto of S0 and subsequent play of the
equilibrium eU1 is such an equilibrium of ⌫ G*, S0 sustained with passive
beliefs.
U
ˆ
Ž
.
This observation suggests two things. First, the ratification stage enlarges the
set of equilibria of G*. Second, we should test the robustness of G* to credible
equilibria of the game of coalition formation. We now turn to this issue.
6.3. Strong Ratifiability
U
ˆ
Ž
.
As we have seen above, the ratification stage of ⌫ G*, S0 adds in fact a
preplay communication stage to G*. At this stage, agents are allowed to make
Ž
.
binary preplay announcements ‘‘Veto’’ or ‘‘Accept’’ that may signal some
Ž
.
information on their types. Cramton and Palfrey 1995 have analyzed similar
mechanism design problems and have proposed the notion of strong ratifiability
of a mechanism against itself to test its robustness to such a cheap-talk stage.30
To understand this notion, let us first define credible ¨eto beliefs:
29
Ž
.
This contrasts with Ma, Moore, and Turnbull 1988 where Pareto-dominant noncooperative
and nontruthful equilibria may be a threat to the principal and must therefore be eliminated by
using indirect message games.
30 See Matthews and Postlewaite 1989 and Palfrey and Srivastava 1991 for other analyses
allowing more general cheap-talk stages in mechanism design.
Ž
.
Ž
.

MECHANISM DESIGN
331
ꢀ
4
DEFINITION 3: A belief system p , p on ⌰ is a credible ¨eto system of the
˜1 ˜2
truthful decision rule e* if, for each i, there exist a noncooperative Bayesian
with ei /e* and refusal probabilities ¨i ␪ ,
᭙ ␪i g⌰, ᭙ ig 1, 2 that together satisfy:
Ž
. Ž
.
Ž .
equilibrium e gE G*, p , p
˜
i
i
yi
i
ꢀ
4
Ž .
1. ¨i ␪ )0 for some ␪i g⌰;
i
Ž .
Ž .
Ž .
Ž
.
.
2. ¨i ␪ s1 for all ␪i g⌰ such that U ␪ -U ␪i, ei ;
i
i
Ž .
iŽ
3. ¨i ␪ s0 for all ␪i g⌰ such that U ␪ )U ␪i, ei ;
4. p satisfies Bayes’ rule, given the prior diistribution p and the refusal
i
i
˜
i
Ž .
probabilities ¨i и :
¡
<
Ž .
i
p ␪ ␪ ¨ ␪
Ž
.
i
j
i
Ž
.
i
for ␪ such that ¨i ␪ )0,
i
<
Ž
.
␪
k
p ␪ ␪ ¨
Ž
.
Ý
k
j
k
~
<
p ␪ ␪ s
˜
i Ž
.
i
j
Ž
.
)0
␪kg⌰i, ¨
␪
k
¢
Ž .
for ␪ such that ¨i ␪ s0.
0
i
i
Credibility of beliefs requires that vetoing e* should be interpreted as coming
from the subset of types who are the most likely to benefit from this deviation
when a non-deviant agent’s beliefs put all weight on this particular subset. This
definition captures the rational expectation reasoning that the different types of
the deviant agent make when they envision vetoing the decision rule e*. Those
types who gain from vetoing e* effectively deviate and refuse to play e* with
some probability when the nondeviant agent interprets these deviations in a
rational way. The set of such types is called a credible ¨eto set. Still following
Ž
.
Cramton and Palfrey 1995 , we also have the following definition.
DEFINITION 4: G* is strongly ratifiable if and only if there does not exist a
credible veto system or, if for all credible veto sets and all credible veto beliefs
Ž
.
Ž .
such that U ␪
p , there exists a noncooperative equilibrium e gE G*, p , p
˜
i
˜
i
i
yi
i
Ž
.
sU ␪i, ei for all i and for all ␪ belonging to the credible veto set.
i
i
Strong ratifiability captures the idea that no type of agent is willing to credibly
veto the play of the truthful equilibrium e*. If it is strongly ratifiable, G* is
Ž
.
robust to cheap-talk equilibria such that possibly strict subsets of types may
deviate in a credible way.
Definition 3 is actually a step in the construction of perfect sequential
Ž
.
equilibria made by Grossman and Perry 1986 . Therefore, G* is strongly
U
ˆ
Ž
.
ratifiable if and only if all perfect sequential equilibria of ⌫ G*, S0 are payoff
equivalent to unanimous ratification of e*.
PROPOSITION 9: The optimal weakly collusion-proof grand mechanism G* is
strongly ratifiable.

332
J.-J. LAFFONT AND D. MARTIMORT
Ž
.
.
The proof consists in describing the equilibrium correspondence E G*, p , p
˜
1
2
Ž
1
when p varies. From now on, we abuse notations and denote by p resp. p
˜
1
˜
Ž
.
the probability resp. the conditional prior probability that A2 assigns to A1
U
Ž
.
ꢀ
˜
1
U4
Ž
.
ꢀ
U4
˜
being ␪. Starting from E G*, p, p s e*, e1 , e2 , we get E G*, p , p s e for
˜
1
2
Ž
.
ꢀ
U4
1
optimistic beliefs p )p and E G*, p , p s e*, e for pessimistic beliefs p -
˜
1
1
p.
The characterization of this correspondence makes it easier to isolate the veto
sets that can credibly refuse to play e*. A ␪ agent A1 cannot be a credible veto
set since G* has a unique equilibrium eU2 when A2 holds optimistic beliefs on
A1 and this equilibrium gives to a ␪ agent A1 strictly less utility than e*. On the
contrary, a ␪ agent A1 can be a credible veto set since G* has e* and eU1 for
equilibria when A2 holds pessimistic beliefs on A1. However, by refusing to
ratify e*, this credible veto set cannot obtain more utility than following
Ž
.
unanimous ratification of e* since, with pessimistic beliefs, we have E G*, p , p
˜
s e*, e1 and both equilibria yield payoff U1 ␪, e1 sU ␪ to a ␪ agent1 A1
U
ꢀ
U4
Ž
.
Ž .
anyway.
7. CONCLUSION
When agents collude to influence collective decision on public goods and their
valuations for the public good are positively correlated, there exists a trade-off
between efficiency and rent extraction. The optimal weakly collusion-proof
contract depends on the degree of correlation.
This correlation is also a crucial determinant of the group’s ability to collude.
In particular, a strong positive correlation allows the principal to use asymmetric
information within the coalition to undermine significantly its countervailing
power.
Adding coalition incentive constraints restores also continuity between the
correlated and the uncorrelated information environments. Collusion does not
matter in an uncorrelated environment with risk-neutral agents and the nonco-
operative outcome is implementable in a collusion-proof way.
The benefit from focusing on a discrete two type modeling of asymmetric
information is twofold. First, it has given us a tractable characterization of the
set of collusion-proofness constraints. Second, it is also a key simplification to
test the robustness of the optimal mechanism to some form of preplay commu-
nication since it becomes then possible to fully describe the equilibrium corre-
spondence of the optimal weak collusion-proof mechanism G* when beliefs
change.
Many lessons of this paper are independent of the specifics of the model, like
the principal’s objective function or the preferences of agents within the
coalition. Therefore, these results would also go through in other environments
like auctions, regulation of duopoly, design of incentive schemes within the firm,
and arbitration mechanisms. More generally, our results suggest that collusion
stakes always exist in those correlated environments and that the efficiency of
yardstick mechanisms depends significantly on the amount of correlation.

MECHANISM DESIGN
333
IDEI, GREMAQ, Uni¨ersite des Sciences Sociales, Place Anatole France, 31042
´
Toulouse, France
and
Uni¨ersite de Pau et des Pays de l’Adour, IDEI, GREMAQ, France
´
Manuscript recei¨ed July, 1997; final re¨ision recei¨ed December, 1998.
APPENDIX
Ž .
Ž .
PROOF OF PROPOSITION 1: When 3 and 4 are binding the principal’s expected welfare rewrites
as
Ž
.Ž Ž ...
Ž
Ž
..
ŽŽ
.
Ž
..
ˆ
Ž
SWs 1q␭ p11 2␪ xyc x q2 p12 ␪q␪ xyc x qp22 2␪ xyc x
ˆ
Ž .
Ž .
if there exist transfers such that the incentive compatibility constraints 1 and 2 hold. Optimizing
Ž . Ž .
this expression yields the first-best decision rule. Since we look for transfers such that 1 and 2 are
Ž
.
also binding, t11, t12 , t21, t22 must solve:
Ž
Ž
Ž
Ž
.
.
.
.
Ž
.
ˆ
24
25
26
27
p11 t11 qp12 t12 s␪ p11 xqp12 x ,
Ž
.
p11 t21 qp12 t22 s␪ p11 xqp12 x ,
ˆ
Ž
.
p12 t21 qp22 t22 s␪ p12 xqp22 x ,
ˆ
Ž
.
ˆ
p12 t11 qp22 t12 s␪ p12 xqp22 x .
Those equalities are satisfied for some transfers when the matrix
p11 p12
p
p22
ž
/
12
is invertible. This holds since its determinant is ␳sp11 p22 yp122 )0. Solving for the taxes:
1
Ž
Ž
Ž
Ž
Ž
Ž
Ž
Ž
.
Ž
.
ˆ
.
.
.
.
t11
t12
t21
t22
s
s
s
s
␪
␪
␪
␪
p11 xqp12 x p22 y␪ p12 xqp22 x p
,
,
,
.
ˆ
12
12
␳
1
.
Ž
.
ˆ
p12 xqp22 x p11 y␪ p11 xqp12 x p
ˆ
␳
1
.
Ž
.
p11 xqp12 x p22 y␪ p12 xqp22 x p12
ˆ
ˆ
␳
1
.
Ž
.
p12 xqp22 x p11 y␪ p11 xqp12 x p12
ˆ
ˆ
␳
Ž
.
Ž
These taxes become very large as ␳ goes to zero. Indeed, we have yt11 q␪ xsy 1r␳ ⌬␪ p12 p12
x
.
ˆ
Ž
.
Ž
.
qp22 x , and yt22 q␪ xs 1r␳ ⌬␪ p12 p11 xqp12 x . Hence, t11 goes towards plus infinity and t22
goes towards minus infinity when ␳ goes to zero. Similarly, t21 goes towards plus infinity and t12
goes towards minus infinity when ␳ goes to zero.
ˆ
PROOF OF PROPOSITION 2: The proof is standard and thus omitted.
PROOF OF PROPOSITION 3: Let us consider a perfect Bayesian equilibrium of the overall game of
grand mechanism offer cum coalition formation such that a side contract is unanimously ratified.
ꢀ
Ž
.4
This perfect Bayesian equilibrium is in fact a triplet G*; S*; p , p where:
˜1 ˜2

334
J.-J. LAFFONT AND D. MARTIMORT
G* from MsM1 =M2 into DsX=T 2 maps the messages m1, m2 sent by the agents into
ⅷ
Ž
.
Ž
.
an allocation public good, taxes . G* maximizes the principal’s welfare taking into account the
continuation equilibrium of the game of coalition formation.
ⅷ
S* is a side mechanism that, since the revelation principle applies at the last stage of the game,
can be taken as being a direct mechanism mapping ⌰=⌰ into the set of measures on message
spaces. Let e* be its truthful equilibrium. S* maximizes the sum of the agents’ expected utilities
subject to individual Bayesian incentive constraints, budget balance, and individual rationality
Ž
.
Ž
.
Ž
.
ꢀ
4
and some p , p .
˜1 ˜2
constraints U ␪ GU ␪i, ei for some e gE G, p , p
˜
i
i
i
i
yi
4i
ⅷ
ꢀ
p , p is the system of out-of-equilibrium posterior beliefs used respectively by agent A2 and
A1 to assess respectively A1 and A2’s type following their respective veto of the side mechanism S*.
These are the beliefs used in the noncooperative play of G to compute the status quo payoffs
˜1 ˜2
Ž
.
U ␪i, ei .
i
ⅷ
˜
Consider now the new grand mechanism GsG*(S*. We prove that there exists a perfect
Bayesian equilibrium of the overall game of contract offer cum coalition formation in which the
2
˜
principal offers G, which is a direct mechanism from ⌰=⌰ into DsX=T , the third party offers
U
ꢀ
4
the null side-mechanism S0 s ␾sId, yk s0 and this choice is sustained by passive beliefs,
Ž
.
Ž
.
p sp sp. Because S* solves T with reservation utilities U ␪i, ei , the null side mechanism solves
˜
˜
2
i
Ž 1
.
Ž .
T
with reservation utilities U ␪ . Indeed, suppose it is not the case; then there would exist a side
i i
˜
mechanism S such that the third party can achieve a strictly greater payoff for the coalition than
˜
Ž
.
Ž
.
with S*. Since by definition U ␪ GU ␪i, ei the third party’s payoff from offering S*(S in the first
place would be strictly greater than that achieved with S*. This would contradict that S* is optimal
i
i
i
when G* is offered.
ⅷ
˜
Hence, offering the grand mechanism G insures to the principal that there is a perfect
Bayesian equilibrium of the continuation game sustained with passive beliefs in which the null side
mechanism is unanimously ratified.
PROOF OF PROPOSITION 4 AND COROLLARY 1: We first solve for the third-party’s optimal side
Ž .
mechanism sustained by a reversion to a noncooperative equilibrium of the mechanism G и played
U
ꢀ
4
with passive beliefs. Then, we identify conditions such that S0 s ␾sId, yk s0 , i.e., the null side
mechanism, is the solution to T . We conclude by deriving the monotonicity conditions that must be
Ž
.
satisfied by such a weakly collusion-proof contract.
ⅷ
Since, we are not interested in grand-mechanisms such that the ␪ agent incentive constraint is
Ž
.
Ž
binding this constraint will be satisfied ex post , we write the third-party’s problem as the
lowerscript denotes the index of the agent concerned with the transfer and let ␾ ␪i, ␪ s␾ for
simplicity :
Ž
.
j
i j
.
Ž
Ž
.
Ž
.
Ž . Ž i j ..
q ␪i q␪ x ␾
j
max
pi j yt1
␾
yt2
␾
Ý
42
i, j g 1, 2
i j
i j
ꢀ
Ž . Ž .4
␾ и , y и
Ž
.
ꢀ
subject to:
ⅷ
budget balance:
2
2
Ž
.
Ž
.
Ž
.
j
28
y
␪i , ␪ s0
᭙
␪i , ␪ g⌰ ;
Ý
k
j
k
s1
ⅷ
incentive constraints for respectively the ␪ agents A1 and A2:
Ž
Ž
.
.
Ž
Ž
.
Ž
.
Ž
..
Ž
Ž .
␾
12
Ž
.
Ž 12 ..
29
30
p11 yt1
␾
yy1 ␪ , ␪ q␪ x ␾
qp12 yt1
yy1 ␪ , ␪ q␪ x ␾
11
11
Ž
Ž
.
Ž
.
Ž
..
Ž
Ž
.
Ž
.
Ž
..
yy1 ␪ , ␪ q␪ x ␾ ,
22
Gp11 yt1
␾
yy1 ␪ , ␪ q␪ x ␾
qp12 yt1
␾
21
21
22
Ž
Ž
.
Ž
.
Ž
..
Ž
Ž .
␾
21
Ž
.
Ž 21 ..
p11 yt2
␾
yy2 ␪ , ␪ q␪ x ␾
qp21 yt2
yy2 ␪ , ␪ q␪ x ␾
11
11
Ž
Ž
.
Ž
.
Ž
..
Ž
Ž
.
Ž
.
Ž
..
yy2 ␪ , ␪ q␪ x ␾ ;
22
Gp11 yt2
␾
yy2 ␪ , ␪ q␪ x ␾
qp21 yt2
␾
12
12
22

MECHANISM DESIGN
335
ⅷ
participation constraints for respectively the ␪ agents A1 and A2:
Ž
Ž
.
.
Ž
Ž
.
Ž
.
Ž
..
Ž
Ž .
␾
12
Ž
.
Ž 12 ..
31
32
p11 yt1
␾
yy1 ␪ , ␪ q␪ x ␾
qp12 yt1
yy1 ␪ , ␪ q␪ x ␾
11
11
Ž
.
Ž
.
G
p11 qp12 U1 ␪ , e1 ,
Ž
Ž
.
Ž
.
Ž
..
Ž
Ž .
␾
21
Ž
.
Ž 21 ..
p11 yt2
␾
yy2 ␪ , ␪ q␪ x ␾
qp21 yt2
yy2 ␪ , ␪ q␪ x ␾
11
11
Ž
.
Ž
.
,
G
p11 qp21 U2 ␪ , e2
Ž
. Ž
ꢀ
4.
for some equilibrium ei g⌫ G, p, p ig 1, 2 ;
ⅷ
participation constraints for respectively the ␪ agents A1 and A2:
Ž
Ž
.
.
Ž
Ž
.
Ž
.
Ž
..
Ž
Ž .
␾
22
Ž
.
Ž 22 ..
33
34
p21 yt1
␾
yy1 ␪ , ␪ q␪ x ␾
qp22 yt1
yy1 ␪ , ␪ q␪ x ␾
21
21
Ž
.
Ž
.
G
p21 qp22 U1 ␪ , e1 ,
Ž
Ž
.
Ž
.
Ž
..
Ž
Ž .
␾
22
Ž
.
Ž 22 ..
p12 yt2
␾
yy2 ␪ , ␪ q␪ x ␾
qp22 yt2
yy2 ␪ , ␪ q␪ x ␾
12
12
Ž
.
Ž
.
G
p12 qp22 U2 ␪ , e2 .
Ž
.
Ž
.
Ž
.
Ž
.
Ž .
Let us introduce the following multipliers ␶ ␪i, ␪ for 28 , ␦i for 29 and 30 , ␯ for 31 and
j
i
Ž
.
Ž
.
Ž .
32 , ␯ for 33 and 34 . We write the Lagrangean L of the maximization problem above as
i
2
2
2
Ž
.
Ž
. Ž
.
Ž
. Ž
.
Ž
␯
i
. Ž .
BIR ␪
i
LsE U1 qU2
q
␦ BIC ␪ q
␯
BIR ␪ q
i
i
Ý
Ý
Ý
i
i
i
s1
is1
is1
Ž
.Ž .Ž .
␶ ␪i , ␪ BB ␪i , ␪ .
j
q
Ý
j
Ž
.
␪ i , ␪
j
Ž
.
Ž
.
Optimizing with respect to y1 ␪, ␪ and y2 ␪, ␪ yields respectively:
Ž
Ž
.
.
Ž . Ž
.
1
35
36
␶ ␪ , ␪ yp11 ␦1 q␯ s0,
Ž
.
Ž
.
2
␶ ␪ , ␪ yp11 ␦2 q␯ s0.
Ž
.
Ž
.
Optimizing with respect to y1 ␪, ␪ and y2 ␪, ␪ yields respectively:
Ž
Ž
.
.
Ž . Ž
.
1
37
38
␶ ␪ , ␪ yp12 ␦1 q␯ s0,
Ž
.
␶ ␪ , ␪ qp11 ␦2 yp12 ␯2 s0.
Ž
.
Ž
.
Optimizing with respect to y1 ␪, ␪ and y2 ␪, ␪ yields respectively:
Ž
Ž
.
.
Ž .
␶ ␪ , ␪ qp11 ␦1 yp21␯1 s0,
39
40
Ž
.
Ž
.
2
␶ ␪ , ␪ qp21 ␦2 q␯ s0.
Ž
.
Ž
.
Finally, optimizing with respect to y1 ␪, ␪ and y2 ␪, ␪ yields respectively:
Ž
Ž
.
.
Ž .
␶ ␪ , ␪ qp12 ␦1 yp22 ␯1 s0,
41
42
Ž
.
␶ ␪ , ␪ qp21 ␦2 yp22 ␯2 s0.
ⅷ
Optimizing with respect to ␾ yields
11
U
˜
˜
˜
˜
Ž .
␾
11
˜
Ž 11..
q␪ x ␾
Ž
Ž
.
Ž
Ž
.
Ž
..
Ž
.
Ž
␾
11 garg max p11 yt1
␾
yt2
␾
q2␪ x ␾
q ␯1 q␦1 p11 yt1
11
11
11
˜
␾
11
˜
˜
..
11
Ž
.
Ž
.
Ž
q ␯2 q␦2 p11 yt2
␾
q␪ x ␾
.
11

336
J.-J. LAFFONT AND D. MARTIMORT
Ž
.
Ž .
Taking into account 35 and 36 , ␦1 q␯1 s␦2 q␯2 , and simplifying yields
U
˜
␾
˜
␾
˜
Ž ..
q2␪ x ␾
11
Ž
.
Ž
Ž
.
Ž
.
43
␾
11 garg max yt1
yt2
.
11
11
˜
␾
11
ⅷ
Optimizing with respect to ␾ yields
12
U
˜
␾
˜
␾
˜
Ž 12 ..
q ␪q␪ x ␾
Ž
.
Ž
Ž
.
Ž
.
Ž
.
44
␾
12 garg max p12 yt1
yt2
12
12
˜
␾
12
˜
␾
˜
˜
Ž .
␾
12
˜
Ž 12 ..
q␪ x ␾
Ž
.Ž
Ž
.
Ž
..
Ž
qp12 ␦1 q␯ yt1
q␪ x ␾
y␦2 p11 yt2
1
12
12
˜
␾
˜
..
12
Ž
Ž
.
Ž
q␯ p12 yt2
q␪ x ␾
.
2
12
Ž
.
Ž
.
Ž
.
Ž .
Using 37 and 38 , ␦1 q␯1 sy␦2 p11rp12 q␯2. Inserting into 44 yields
p11
⑀
1
2
U
˜
˜
˜
Ž
.
Ž
.
Ž
Ž
.
.
Ž
.
.
45
␾
12 garg max yt1
␾
yt2
yt2
␾
q
q
␪q␪y
⌬␪ x ␾
12
12
12
ž
ž
/
ž
/
/
p12
˜
␾
12
Ž
.
with ⑀1 s␦2r 1q␦1 q␯ . Similarly
1
p11
⑀
U
˜
␾
˜
˜
Ž
Ž
.
Ž
.
46
␾
21 garg max yt1
␾
␪q␪y
⌬␪ x ␾
21
21
21
/
ž
p21
˜
␾
21
Ž
.
with ⑀2 s␦1r 1q␦2 q␯
.
2
ⅷ
Optimizing with respect to ␾ yields
22
U
˜
␾
˜
␾
˜
˜
␾
˜
Ž 22 ..
q␪ x ␾
Ž
.
Ž
Ž
.
Ž
.
Ž
..
Ž
Ž
.
47
␾
22 garg max p22 yt1
yt2
q2␪ x ␾
qp22
␯
yt1
22
22
22
1
22
˜
␾
22
˜
˜
˜
Ž .
␾
22
˜
Ž 22 ..
q␪ x ␾
Ž
Ž
Ž
.
.
Ž
..
..
Ž
qp22
yp21
␯
␦
yt2
␾
q␪ x ␾
yp12
␦
yt1
2
2
22
22
1
˜
˜
Ž
Ž
yt2
␾
q␪ x ␾
.
22
22
Ž
.
Ž
.
Ž .
Note again that ␦1 q␯1 sy␦2 p11rp12 q␯ and ␦2 q␯2 sy␦ p11rp12 q␯1. Using 41 and
2
Ž
.
42 , one also gets
␦
␳
2
Ž
.
Ž
.
Ž
.
p22 1q␯ yp12 ␦1 sp22 1q␯ yp12 ␦2 sp22 1q␦1 q␯ q
1
1
2
p12
␦1 ␳
Ž
.
q
sp22 1q␦2 q␯
sB.
2
p12
Ž
.
Simplifying into 47 yields then:
Ž
.
␦1 q␦2 p12
U
˜
˜
˜
˜
Ž .
⌬␪ x ␾
22
Ž
.
Ž
.
Ž
.
␾
22 garg max yt1
␾
yt2
yt2
␾
q2␪ x ␾
y
.
22
22
22
ž
/
B
˜
␾
22
Put differently,
U
˜
␾
˜
Ž .
␾
22
Ž
.
Ž
.
48
␾
22 garg max yt1
22
˜
␾
22
ꢁ
⑀1 p12
⑀ p12
2
˜
Ž
.
.
q
2␪y
⌬␪y
⌬␪ x ␾
22
␳
␳
ž
/
p22 q⑀1
p22 q⑀2
0
p12
p12

MECHANISM DESIGN
337
Ž
.
s
In the sequel, we consider symmetric grand mechanisms such that ⑀1 s⑀2 s⑀ and t1 ␪1, ␪
2
31
Ž
.
.
t2 ␪2 , ␪
We then have
1
⑀ p12
U
˜
˜
˜
Ž
.
Ž
.
Ž
.
Ž
.
.
49
␾
22 garg max yt1
␾
yt2
␾
q2 ␪y
⌬␪ x ␾
22
22
22
␳
˜
␾
22
ꢁ
ž
/
p22 q⑀
0
p12
U
ⅷ
Ž
.
Ž
. Ž . Ž
.
Ž
.
In a weakly collusion-proof mechanism ␾i j s ␪i, ␪ . Inserting into 43 , 45 , 46 , and 49
j
Ž . Ž .
Ž .
yields constraints 5 , 6 , and 7 .
ⅷ
Ž
.
w
w
Note that ⑀s␦r 1q␦q␯ g 0, 1 . Moreover, ␦)0 when the Bayesian incentive constraints
Ž
.
Ž
.
Ž .
. Ž . Ž
29 and 30 are binding in T .
ⅷ
Ž
.
Ž .
Note also that 31 , 32 , 33 , and 34 are binding for a weakly collusion-proof mechanism.
Hence, for such a mechanism, the slackness conditions obtained from the Lagrangean optimization
do not give any information on ⑀. Therefore, the principal has some flexibility in choosing this
variable.
ⅷ
We now check for the monotonicity of the schedule of outputs when the mechanism is weakly
U
Ž
.
Ž
.
Ž
.
collusion-proof. From 43 and 44 taken for ␾i j s ␪i, ␪ , we have:
j
p11
p12
y2t11 q2␪ xGyt12 yt21 q2␪ xyt12 yt21
q
␪q␪y
⑀⌬␪ x
ˆ
ˆ
ž
/
p11
Gy2t11
q
␪q␪y
␦⌬␪ x.
ž
/
p12
Summing these two inequalities yields
p11
Ž .
xyx G0,
ˆ
⌬␪ 1q
⑀
ž /
p12
Ž
.
Ž
.
Ž .
which is satisfied for xGx since ⑀G0 . Proceeding in a similar way and using 44 and 49 for the
ˆ
Ž
.
Ž
.
coalitions ␪, ␪ and ␪, ␪ , another revealed preference argument tells us that
2 p122
⑀
p11⑀
Ž .
xyx G0.
ˆ
1q
y
2
ž
Ž
.
/
p12
p12 p22 q⑀ p11 p22 yp12
Ž .
Ž .
since ␺ Љ ⑀ sy4p12 p22 ␳r p12 p22 q⑀␳ -0 . Hence, it is either minimum at 0 or 1. ␺ 0 s1)0
Denoting by ␺ ⑀ the first term on the left-hand-side of the latter inequality, ␺ ⑀ is concave in ⑀
3
3
Ž
Ž .
Ž
.
.
Ž .
2
Ž .
Ž
.Ž
.
Ž
and ␺ 1 )0 if and only if ␳- p22 qp12 p12rp11 . For a weak correlation, namely ␳- p22
q
2
.Ž
correlation, namely ␳) p22 qp12 p12rp11 , ␺ ⑀ is negative for ⑀ close enough to one and the
monotonicity condition becomes then xGx.
.
Ž .
p12 p12rp11 , ␺ ⑀ is always positive and the monotonicity condition becomes xGx. For a strong
ˆ
2
Ž
.Ž
.
Ž .
ˆ
Ž
. Ž . Ž
.
PROOF OF PROPOSITION 5: We denote by ␣, ␤, ␥, ␯, the multipliers respectively of 10 , 11 , 12 ,
and 13 . Optimizing with respect to u yields
Ž
.
Ž
.
50
2␭p11 s␣ p11 q2␤.
Optimizing with respect to u yields
ˆ1
Ž
.
51
2␭p12 s␣ p12 y␤q␥ .
31 Asymmetric mechanisms are discussed in the Proof of Proposition 5.

338
J.-J. LAFFONT AND D. MARTIMORT
Optimizing with respect to u yields
ˆ2
Ž
.
52
2␭p12 sy␣ p11 y␤q␥q␯ p12
.
Optimizing with respect to u yields
Ž
.
53
2␭p22 sy␣ p12 y2␥q␯ p22 .
Ž
.
Ž .
Summing 50 to 53 yields
2␭
Ž
.
54
␯s
)0.
p12 qp22
Ž
.
Inserting this latter expression into 52 yields
1
Ž
.
55
2␭p12 1y
sy␣ p11 y␤q␥ .
ž
/
p12 qp22
Ž
.
Ž .
Subtracting 51 from 55 yields
2␭p12
Ž
.
56
␣s
)0.
Ž
.Ž .
p12 qp22 p11 qp12
Ž
.
Inserting this expression into 50 yields then
p12
Ž
.
57
␤s␭p11 1y
)0.
ž
/
.Ž
Ž
.Ž .
p12 qp22 p12 qp11
Ž
.
Ž .
␣ and ␤ are strictly positive when ␳s p12 qp22 p12 qp11 yp12 )0. Finally, inserting 57 into
Ž
.
51 gives
p12
.Ž
Ž
.
Ž
.
58
␥s␭ p11 q2 p12 1y
)0.
ž
/
Ž
.
p12 qp22 p12 qp11
ⅷ
Since ␥)0 and since the monotonicity condition implies that xyxG0 for a weak correlation,
ˆ
Ž
.
⑀s0 minimizes the cost of constraint 12 .
ⅷ
Ž .
Optimizing with respect to x, x, and x yields cЈ x s2␪,
ˆ
1
1
Ž
.
Ž
.
␣ p11 q␤ ⌬␪ ,
cЈ x s␪q␪y
ˆ
1q␭ 2 p12
and
1
1
Ž
.
Ž
.
␣ p12 q␥ ⌬␪.
cЈ x s2␪y
1q␭ p22
Ž
. Ž
.
Ž
.
Ž
.
Ž .
Using 56 , 57 , and 58 yields 14 and 15 .
Monotonicity of outputs obtains when
␭
p11
p12
␭
1
p12 p11
1y
1q
Gy
p11 q2 p12
y
.
ž
/
ž
/
1q␭ 2 p12
p12 q␳
1q␭ p22
p12 q␳
Because ␳G0, this property holds when
␭
p11
p12
p22
1)
y2
.
ž
/
1q␭ p12