In this note we show that the equilibrium characterized by Biais, Martimort and Rochet (Econometrica, 78, 2000) could have been characterized by using direct mechanisms.Common Agency, Revelation Principle, Direct Mechanisms, Nonlinear Prices

Gwenael Piaser

English

Research Papers in Economics

DISCUSSION PAPER
2004
The Biais-Martimort-Rochet equilibrium with
direct mechanisms
Gwenaël PIASER1
July 2004
Abstract
In this note we show that the equilibrium characterized by Biais, Martimort,
and Rochet (2000) could have been characterized by using direct mechanisms.
Keywords: Common Agency, Revelation Principle, Direct Mechanisms, Nonlin-
ear Prices.
JEL Classiﬁcation: D82.
1Università Ca’ Foscari di Venezia.
I would like to thank Andrea Attar and Diego Salzman for their comments and their help.1 Introduction
Biais, Martimort, and Rochet (2000) (BMR thereafter) consider a multi-principals game
to analyze imperfect competition under adverse selection in ﬁnancial markets. Strategic
liquidity suppliers post nonlinear prices (such as limit order schedules) which stand
ready to trade with a risk-adverse agent who has private information on the fundamental
value of the asset as well as on his hedging needs. They show that there exists an unique
equilibrium in convex schedules and they analyze its properties.
In order to do that, they do not use standard mechanism design methods. Usually,
in principal-agents games direct mechanisms are sufﬁcient to characterize all equilibria.
Peters (2001) and Martimort and Stole (2002) have shown that restricting the attention
to direct mechanisms may induce a loss of generality. Some equilibria cannot be char-
acterize by direct mechanisms. Nevertheless, if we consider more general mechanisms,
such as menus (or price schedules), one can characterize all equilibria of any common
agency game. The drawback of this approach is that menu (or price schedules) are more
difﬁcult to handle than direct mechanisms.
BMR showed that using calculus of variations one can characterize equilibria even
if we allow principals to use menus. From that point of view BMR is an interesting
contribution to the literature as it provides a clear and rigorous methodology.1 Another
methodology would have been to consider only direct mechanisms. If by doing that
one cannot characterize all equilibria, Peters (2003) has shown that one characterizes
regular equilibria.
In this note, we show that the BMR equilibrium could have been characterized by a
much simpliﬁed approach as the use of direct mechanisms.
2 The Model
We use exactly the BMR’s model. We just quickly present the formal aspects, for a
more complete description of the model and its properties please refer to the original
article.
1It is also a interesting contribution to the ﬁnancial literature as it provides testable predictions.
1There are (n+1) players in the game, n principals and one agent. The principals
playﬁrst, theyoffersimultaneously“mechanisms”. A“mechanism”isamappingfroma
messagespace(Mi isthesetofallpossiblemessagespacesforprincipali, i∈{1,...,n})
to the decision space. Here a principal takes two decisions, a price T and a quantity q,
the decision space is R2
+. Principal i is offering a couple (Mi,(Ti(.),qi(.))), the agent
can either reject or accept the offer. If he accepts then he sends the message m ∈ Mi (we
must have Mi ∈Mi), the agent gets from principal i the decision (Ti(m),qi(m)).
In the BMR model the interpretation of (Ti(m),qi(m)) is the following: the agent
must trade the quantity qi(m) at the price Ti(m). If the agent rejects the offer from
principal i, he gets (0,0) from him. The agent observe all the offered mechanisms and
he/she decides to reject or accept some of them. His preferences are represented by the
following utility function.
U = qåqi−
gs2
2
 
åqi
2−å
i
Ti.
The variables g and s are common knowledge. The variable q is known only by the
agent, principals know only the distribution of that variable over the range of possible
values Q =

q,q

. The density is denoted f. This density is common knowledge.
The principal i’s preferences over qi and Ti are represented by the following utility
function:
Ti−v(q)qi.
We consider Perfect Bayesian Equilibria for that game. The problem is quite com-
plex, the set Mi can be very large (it formalizes all possible communication schemes
between a principal and the agent), and it is difﬁcult to characterize the optimal choice
of Mi.
If we consider a simple principal-agent game (n = 1), the so-called “Revelation
Principle” (Myerson 1979, Myerson 1982) states that one can ignore the choice of Mi,
and consider that the message space is given and equal to Q. One can show that the
unique principal would have chosen (Q,(T∗(.),q∗(.))) even if he would not have been
constrainted to play M = Q The couple (T∗(.),q∗(.)) is called a “direct mechanism”
2An immediate consequence of the revelation principle is that we can retrict our at-
tention to direct “revelating” mechanisms. The direct mechanism (T∗(.),q∗(.)) is “rev-
elating” if it is such that the agent reveals the actual value of q. Considering only “direct
revelating mechanisms” simpliﬁes a lot the game and the optimal values of T∗(.) and
q∗(.) can be then characterized in most of the relevant games.
If we consider a multi-principals game (n > 1), the revelation principle does not ap-
ply: one cannot impose Mi =Q and characterize all equilibria of the game. If we do this
we characterize only a subset of the equilibria of the game.2 If we want to characterize
all the equilibria of the game, we can only consider as possible message space all the
subset of the decision space, and consider that implement the message receive from the
agent (Peters 2001, Martimort and Stole 2002, call that methodology “the Delagation
Principle”). In our particular game, rather than considering any element of the abstract
setM , we can consider only the subsets of R2
+ and the mapping (Ti(.),qi(.)) are deﬁne
by:
∀,
 
ˆ T, ˆ q

∈ Zi,
 
Ti
 
ˆ T, ˆ q

,qi
 
ˆ T, ˆ q

=
 
ˆ T, ˆ q

,
where Zi ⊂ R2
+. Roughly speaking, the agent gets what he asks from any principal, but
he is allow to choose only in a restricted set. These mechanisms are called “menus”, or
sometimes “catalogs”.
Even if this result restricts the possible strategies, it does not simplify a lot the anal-
ysis given that we still have problem with the characterization (considering all subset
of R2
+ is out of reach). BMR restrict the communication set by considering only a par-
ticular class of subset of R2
+: they consider that principals are only allow to choose
continuous and (almost everywhere) differentiable menus. The message space is R+, a
particular message is q ∈ R+, and if the agent sends the message q, he gets (T (q),q),
where T (.) is a continuous function, with a ﬁnite number of non-differentiable points.
In the following section we will show that BMR equilibrium could attained also
using simple direct revelating mechanisms.3
2See Peters (2003).
3Peters (2003) shows that if we modify a the BMR’s model, the revelation applies. If qi is chosen
by agent and observable by prinicipal i, there is no restriction to consider direct mechanisms. But as qi
33 Direct mechanisms equilibria
Principals are using direct mechanisms i.e; mappings (qi(.),Ti(.)) from Q to R2
+. If the
agent q reports the vector ˜ q, he gets
U
 ˜ q,q

= q
 
qi
 ˜ qi

+å
−i
q−i
 ˜ q−i

!
−
gs2
2
 
qi
 ˜ qi

+å
−i
q−i
 ˜ q−i

!2
− Ti
 ˜ qi

−å
−i
T−i
 ˜ q−i

.
We concentrate on principal i (the indices −i represents all other principals). He
considers others principals strategies
 
qj(.),Tj(.)

j6=i as given and known. The agent
reports truthfully his type to principal i if
q ˙ qi(q)−gs2
 
qi(q)+å
−i
q−i
 ˜ q−i

!
˙ qi(q)− ˙ Ti(q) = 0.
The other report ˜ qj, j 6= i are deﬁne by the following ﬁrst order conditions:
q ˙ qj
 ˜ qj

−gs2
 
å
g∈I
qg
 ˜ qg

!
˙ qj
 ˜ qj

− ˙ Tj
 ˜ qj

= 0,
where ˜ qi = q.
From that ﬁrst order conditions, one can deﬁne the rent obtained be the agent.
U (q) = q
 
qi(q)+å
−i
q−i
 ˜ q−i

!
−
gs2
2
 
qi(q)+å
−i
q−i
 ˜ q−i

!2
−Ti(q)−å
−i
T−i
 ˜ q−i

,
is chosen by the agent, direct mechanisms are quite complex, and the revelation principle is not helpful:
for example Laussel and Le Breton (2001) show that even in a complete information setting, observable
actions introduce technical difﬁculties in common agency games. In the folllowing, we keep the BMR’s
model.
4where the ˜ q−i are chosen optimally and then are implicit functions of q. Applying the
envelop theorem, we get:
˙ U = qi+å
−i
q−i.
A necessary second order condition can be obtained by using standard methods. 4 For
any q ∈ Q we have
q ˙ qi(q)−gs2 
qi(q)+å−iq−i
 ˜ q−i

˙ qi(q)− ˙ Ti(q) = 0,
where the ˜ q−i are chosen optimally. Given the deﬁnition of U (q), the necessary condi-
tion
q ¨ qi(q)−gs2 
qi(q)+å−iq−i
 ˜ q−i

¨ qi(q)−gs2 ˙ q2
i (q)− ¨ Ti(q) < 0,
can be written as
q ˙ qi(q) > 0.
Theoptimalquantitymustbeincreasingwithq.5 Thisconditionisstandardinmech-
anism design theory. As the utility of the agent is non monotonic in qi this condition
introduce a technical difﬁculty. To solve the main problem we will assume in the fol-
lowing that the the strategies of all other principals are such that for low value of q the
utility of the agent is decreasing with qi and his utility is increasing with qi when q is
hight enough. We will check that it is the case at equilibrium.
U (q) = −
Z qb
q
 
qi+å
−i
q−i
!
dF
if q > qb,
U (q) =
Z qa
q
 
qi+å
−i
q−i
!
dF
4This necessary condition is not u nique, and clearly not sufﬁcient.
5We have assumed that the functions q(.) and T (.) are twice differenciable. This asssumption is
always made in the literature.
5if q < qa, and
U (q) = 0.
if q ∈ [qa,qb], where q < qa 6 qb < q.
Please note that the function q(q) must be continuous around qa and qb. Otherwise,
by applying a simple argument, it would be possible for the principal to improve his
proﬁt: when q ∈ [qa,qb], q(q) = 0, and the marginal proﬁt for the principal i is equal to
zero. If q(q) does not go to zero when q goes to qa (with q > qa), then by increasing a
little qa, the principal i would increase his proﬁt.
Integrating by parts these expressions gives
Z q
q
U (q)dF =
Z qa
q
 
qi+å
−i
q−i
!
F
f
dF +
Z q
qb
 
qi+å
−i
q−i
!
1−F
f
dF.
The proﬁt of principal i can be written as
P =
Z
Q
[Ti(q)−v(q)qi(q)]dF(q),
by using the deﬁnition of the utility function we can simplify the former expression:
P =
Z
Q

q
 
qi+å
−i
q−i
!
−
gs2
2
 
qi+å
−i
q−i
!2
− U (q)+å
−i
T−i−v(q)qi(q)
#
dF(q).
The problem of the principal is equivalent to a point wise maximization problem. The
principal maximizes the following expression with respect q(q) if q ∈

qb,q

.
q
 
qi+å
−i
q−i
!
−
gs2
2
 
qi+å
−i
q−i
!2
−
 
qi(q)+å
−i
q−i
!
(1−F)
f
+å
−i
T−i−v(q)qi(q).
6As we look for a symmetric equilibrium, we consider that all other principle play
the same strategy, namely qj = q and Tj = T for all j different from i. It simpliﬁes the
expression.
q (qi+(n−1)q) −
gs2
2
(qi+(n−1)q)
2
− (qi(q)+(n−1)q)
(1−F)
f
+(n−1)T −v(q)qi(q).
The ﬁrst order condition is given by
(q−v(q))

1+(n−1)
¶q
¶qi

−gs2nq

1+(n−1)
¶q
¶qi

−

1+(n−1)
¶q
¶qi

(1−F)
f
+(n−1)
¶T
¶qi
+(n−1)v(q)
¶q
¶qi
= 0.
To characterize the solution we need the expression of
¶q
¶qi and ¶T
¶qi. From the self selec-
tion constraint we have derived the expressions:
q˙ qj
 ˜ qj

−gs2 
åg∈Iqg
 ˜ qg

˙ qj
 ˜ qj

− ˙ Tj
 ˜ qj

= 0.
Without loss of generality, we can rewrite the direct mechanism
 
qj(q),Tj(q)

as a
direct mechanism
 
qj(q),tj(q(q))

, then the former ﬁrst oder condition becomes:
q−gs2
 
qi+å
−i
q−i
!
=tj,
where tj ≡ tj. Differentiating this equation with respect to qi(q) gives:
−gs2
 
1+
¶å−iq−i
 ˜ q−i

¶qi
!
=t0¶qj
 ˜ qj

¶qi
.
By summing that conditions over j 6= i, we get:
−(n−1)gs2
 
1+
¶å−iq−i
 ˜ q−i

¶qi
!
=t0å−i¶q−i
 ˜ q−i

¶qi
,
7or equivalently:
−
(n−1)gs2
t0+(n−1)gs2 =å
−i
¶q−i
 ˜ q−i

¶qi
.
It follows that:
(q−v(q))

1−
˙ q(q)
1−gs2 ˙ q(q)
gs2(n−1)

− gs2nq(q)

1−
˙ q(q)
1−gs2 ˙ q(q)
gs2(n−1)

−

1−
˙ q(q)
1−gs2 ˙ q(q)
gs2(n−1)

(1−F(q))
f (q)
+ (n−1)

q−gs2nq(q)
 ˙ q(q)
1−gs2 ˙ q(q)
gs2−(n−1)v(q)
˙ q(q)
1−gs2 ˙ q(q)
gs2 = 0.
Consequently ,
[qm(q)−nq(q)]− ˙ q(q)gs2[qm(q)−nq(q)]
− [qm(q)−nq(q)] ˙ q(q)gs2(n−1)+(n−1)[q∗(q)−nq(q)] ˙ q(q)gs2 = 0,
where q∗(q) =
q−v(q)
gs2 and qm(q) = q∗(q)−
1−F(q)
gs2 f(q). Finally we get:
˙ q(q) =
1
gs2

1+
(n−1)(q∗(q)−qm(q))
nq(q)−qm(q)
−1
,
the expression derived by BMR.6
If q∈[q,qa], the principal maximizes the following expression with respect to q(q):
q
 
qi+å
−i
q−i
!
−
gs2
2
 
qi+å
−i
q−i
!2
−
 
qi(q)+å
−i
q−i
!
F
f
+å
−i
T−i−v(q)qi(q).
6BMR consider aggregate values, we consider individual values. Except that slight difference in the
presentation, the formulas are strictly equivalent.
8We can derive the same expression for ˙ q(q), except that qm(q) = q∗(q)−
F(q)
gs2 f(q).
Given the expressions of ˙ q(q), qa and qb must be such that the function q is contin-
uous. As the aggregate supply nq(.) is an increasing function, the form chosen for the
utility is justiﬁed. Usual conditions on the density f guaranty that q is increasing. 7
4 Conclusion
The theorem suggests four main remarks:
Direct mechanisms are not able to characterize any equilibria in a common agency
game. However, they seem to be quite powerful. It would be very interesting to have a
general theorem giving conditions under which an equilibrium cannot be characterized
by direct mechanisms.
The BMR methodology remains interesting since we do not have this general theo-
rem. We do not have any hints on the generality of our result.
The BMR equilibrium is the unique equilibrium with convex price schedules. It
does not means that is the unique equilibrium of their game. The existence of other
equilibria remains an open question. If it exists other equilibria, we do not know if
direct mechanisms are able to characterize them.
If we consider common agency games some equilibria can be characterized by di-
rect mechanisms, some that cannot be. It would be interesting to know which kind of
equilibria is more likely empirically.
References
BIAIS, B., D. MARTIMORT, AND J.-C. ROCHET (2000): “Competing mechanisms in
a common value environment,” Econometrica, 78(4), 799–837.
LAUSSEL, D., AND M. LE BRETON (2001): “Conﬂict and Cooperation: the structure of
equilibrium payoffs in common agency,” Journal of Economic Theory, 100, 93–128.
7See Miravete (2002) for a discussion of these conditions and their interpretation in the BMR’s model.
9MARTIMORT, D., AND L. A. STOLE (2002): “The revelation and delegation principles
in common agency games,” Econometrica, 70(4), 1659–1673.
MIRAVETE, E. J. (2002): “Preserving Log-Concavity under convolution: Comment,”
Econometrica, 70(3), 1253–1254.
MYERSON, R. B. (1979): “Incentive compatibility and the bargaining problem,”
Econometrica, 47, 61–73.
(1982): “Optimal coordination mechanisms in generalized principal-agent
problems,” Journal of Mathematical Economics, 10, 67–81.
PETERS, M. (2001): “Common Agency and the Revelation Principle,” Econometrica,
69(5), 1349–1372.
(2003): “Negociation and take-it-or-leave-it in common agency,” Journal of
Economic Theory, 111(1), 88–109.
10

Common Agency and the Revelation Principle,”

Competing mechanisms in a common value environment,”

Conﬂict and Cooperation: the structure of equilibrium payoffs in common agency,”

for a discussion of these conditions and their interpretation in the BMR’s model.

Incentive compatibility and the bargaining problem,”

Negociation and take-it-or-leave-it in common agency,”

Preserving Log-Concavity under convolution:

The revelation and delegation principles in common agency games,”

The Biais-Martimort-Rochet equilibrium with direct mechanisms

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The Biais-Martimort-Rochet equilibrium with direct mechanisms

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