Common Agency Games with Common ValueExclusion, Convexity and Existence∗

Gwenaël Piaser†

June 27, 2014

Abstract

We consider the model common agency proposed by Biais Martimort and Ro-chet (2000, 2013). We show that in this setting there is no symmetric equilibrium asthe one characterized in those articles. We argue that the equilibrium price sched-ules cannot be simultaneously convex and concave. In particular in the monopolycase, under some classical assumptions, some agents will be excluded from trade.In the other that a price schedule at any symmetric equilibrium must be must beconvex and concave. We conclude that a symmetric equilibrium cannot exist anddiscuss the implications of our result and the links with the existing literature.

∗I would like to thank Tristan Boyer for his help. All errors are mines and copyrighted.†IPAG Business School, Paris

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1 Introduction

Biais, Martimort and Rochet (2000, 2013), BMR thereafter, consider a multi-principalsgame to analyze imperfect competition under adverse selection in financial markets.Strategic liquidity suppliers post nonlinear prices (such as limit order schedules) whichstand ready to trade with a risk-adverse agent who has private information on the funda-mental value of the asset as well as on his hedging needs. BMR show that there existsan unique equilibrium in convex schedules and they analyze its properties.

In the following we argue that such an equilibrium does not exist. We provide atwo steps argument. First, in the third section, we consider the monopolist case. Inthis context we show equilibrium price schedule is never convex (nor concave). Weshow that the optimal price schedule cannot be simultaneously continuous, convex (orconcave) and satisfies the “no distortion at the top” property. The main difference withthe traditional model à la Maskin and Riley (1984) is that quantities can be positive ornegative. In orther word he monopolist can buy or sell the asset. In this framework, wecannot neglect that some agents are excluded: They do not trade with the monopolist.We show that the optimal price schedule is not necessarily continuous and can be convex(or concave) only on some parts.

In a second step, treated in the forth section of the present paper, we consider themultiple principal case. We consider a non-exclusive competition: Agents can deal withmore than one principal, even if it is not compulsory. In this framework we show twothings. First, we provide a characterization of the BMR’s equilibrium. We discuss theproperties of such equilibrium and show that principals offer continuous and convexprice schedules. In the other hand, we show that those schedules at equilibrium mustsatisfied a “no distortion at the top” property, stated in the proper way in a competitionsetting. Using a argument similar to the one given in the monopolistic framework. Weargue that convexity and the “no distortion at the top” property are incompatible. Hence,we conclude that the characterized equilibrium does not exist.

The main proofs are geometrical, based on basic properties of convex functions andthus are quite simple. Our main concern is about “exclusion” which is often neglected.In common agency games it can be a critical issue concerning the existence of somesymmetric equilibria. In the conclusion, we discuss our negative result in the line ofthe existing literature on common agency. Contrarily to the BMR’s approach most of

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the existing paper consider a finite number of types for the agents. In the latter models,existence is problematic and when an equilibrium exists, it is not symmetric in the waywe define symmetry in the present paper. Hence one can see our results as first stone ofa bridge between BMR’s model and discrete type models.

2 Model

We consider a common agency game, denoted Gn, of incomplete information. Thereare (n+ 1) players in the game, n principals and one agent. The principals play first,they offer simultaneously “mechanisms”. A “mechanism” is a mapping from a messagespace (Mi is the set of all possible message spaces for principal i, i ∈ {1, . . . ,n}) to thedecision space. Here a principal takes two decisions, a price Ti and a quantity qi, thedecision space is R2. The principal i’s preferences over qi and Ti are represented by thefollowing profit function:

π(qi,Ti,θ) = Ti −ν(θ)qi. (1)

The function ν is continuous and differentiable, weakly increasing ν ≥ 0.The agent’s preferences are represented by a quadratic utility function.

U((qi,Ti)i=1,...,n ,θ

)= θ ∑

i=1,...,nqi −

γσ2

2

(∑

i=1,...,nqi

)2

− ∑i=1,...,n

Ti. (2)

The variables γ and σ are common knowledge. The variable θ is known only by theagent, principals know only the distribution of that variable over the range of possiblevalues Θ ∈

[θ,θ]. The density function is denoted f , and the repartion function of

the same variable is denoted F . The functions f and F exhibit the usual regularityproperties. These functions are common knowledge.

In this setting optimal quantities, those that would be chosen by an informed benevolantplanner, are given by:

∀θ ∈[θ,θ]

q∗ (θ) =θ−ν(θ)

σ2γ. (3)

We will assume that q∗ (θ) < 0 and q∗(θ)> 0. Principals, if they were perfectly in-

formed, would like to buy share from some agents and to sell share we other agents.

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This feature makes the model different from those considered in general in IndustrialOrganization.

The game follows a classical timing. First each principal i offers a (measurable)mapping from Mi to R2 denoted (Ti(.),qi(.)). We denoted by Γi set of such mappings.The agent can either reject or accept the offer. If he accepts then he sends the mes-sage mi ∈ Mi (we must have mi ∈ Mi ), the agent gets from principal i the decision(Ti (mi) ,qi (mi)). In the BMR model the interpretation of (Ti (mi) ,qi (mi)) is the follow-ing: The agent must trade the quantity qi (mi) at the price Ti (mi). If the agent rejectsthe offer from principal i, he gets (0,0) from him. The agent observes all the offeredmechanisms and he decides to reject or accept some of them. We formalize participa-tion by considering that the agent faces the sets of messages Mi = Mi ∪{ /0}, and weimposes that for any principal i and any mechanism in Γi, Ti ( /0) = 0, and qi ( /0) = 0.Then a strategy, denoted λ, for the agent if a mapping from the set ×i=1,...,nΓi×Θ to theset ×i=1,...,nMi.

Following standard analyses of competing mechanism games, we focus on PerfectBayesian Equilibrium as the relevant solution concept.

3 Monopolist

Since we consider the case with only one principal, we give up the subscript i. Themonopolist want to set the functions q and T in order to maximises its profit. First, by theRevelation Principle we can solve the monopolist problem as if M = Θ, and imposingincentive compatibility for the mapping (T (.),q(.)). From a direct mechanism, we canderive a price schedule (denoted t) which is a more realistic mechanism though indirect.Such a price schedule is defined by

∀θ ∈[θ,θ], t (q(θ)) = T (θ) . (4)

First we characterize the optimal incentive compatible mechanism. Then, we willdiscuss the properties of optimal price schedule t. Since we impose incentive compati-bility, the optimal direct mechanism must satisfy the following conditions:

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The first order condition

θq(θ)− γσ2q(θ) q(θ)− T (θ) = 0 (5)

or equivalentlyU (θ) = q(θ) , (6)

where the function U is defined by

U (θ)≡ θq(θ)− γσ2

2[q(θ)]2 −T (θ) . (7)

Second order condition:q ≥ 0, (8)

or equivalentlyU ≥ 0. (9)

Thus the monopolist maximizes∫θ∈[θ,θ]

[T (θ)−ν(θ)q(θ)]dF (θ) (10)

with respect to T and q under the constraints U ≥ 0, U = q and q ≥ 0.The conditions U (θ) = q(θ) and U ≥ 0 imply that the function U is decreasing

when q is negative, increasing when q is positive, and constant when q is equal to zero.Moreover, since U is convex, U is constant and equal to zero on a connected subset of[θ,θ]. We denote by

[θm

b ,θma]

the subset of[θ,θ]

on which U (θ) = 0, or equivalentlyon which q(θ) = 0. Let us assume that qm (θ)< 0 and qm

(θ)> 0, then the solution has

the following form:Over the set

[θ,θm

b

)the monopolist maximizes∫

θ∈[θ,θmb ][T (θ)−ν(θ)q(θ)]dF (θ) (11)

with respect to T and q under the same constraints: U ≥ 0, U = q and q ≥ 0.

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To solve the problem first we observe that

T (θ) = θq(θ)− γσ2

2[q(θ)]2 −U (θ) , (12)

and since U(θm

b

)= 0, we can state that

U (θ) =∫ θ

θmb

q(δ)dδ (13)

and integrating by parts

∫θ∈[θ,θm

b ]U (θ)dF (θ) =−

∫θ∈[θ,θm

b ]q(θ)

F (θ)f (θ)

dF (θ) . (14)

We neglect the constraint q ≥ 0, and hence we consider that the monopolist maximizes

∫θ∈[θ,θ]

[θq(θ)− γσ2

2[q(θ)]2 +q(θ)

F (θ)f (θ)

−ν(θ)q(θ)]

dF (θ) (15)

with respect to the function q. A pointwise optimization gives qm (θ) = q∗ (θ)+ F(θ)γσ2 f (θ) ,

where q∗ (θ) = θ−ν(θ)γσ2 . Finally, let us remark that the function

θq(θ)− γσ2

2[q(θ)]2 +q(θ)

F (θ)f (θ)

−ν(θ)q(θ) (16)

is concave with respect to q(θ), hence we have characterized a maximum.Over the set

(θm

b ,θ], the optimal quantity is qm (θ) = q∗ (θ)− 1−F(θ)

γσ2 f (θ) .The optimal quantity q with respect to the type as the following form:

• If θ ∈ [θ,θma ) then qm (θ) = q∗ (θ)+ F(θ)

γσ2 f (θ) ,

• if θ ∈(θm

a ,θmb

)then qm (θ) = 0,

• if θ ∈(θm

b ,θ]

then qm (θ) = q∗ (θ)− 1−F(θ)γσ2 f (θ) .

Hence, if θma and θm

b are given, the quantities T(θm

b

)and T (θm

a ) are given by the equa-tions

T (θmb ) = θm

b q(θmb )−

γσ2

2[q(θm

b )]2 , (17)

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and

T (θma ) = θm

a q(θma )−

γσ2

2[q(θm

a )]2 . (18)

Hence, for a given couple(θm

a ,θmb

), the function T is fully characterized.

From that, it follows that one can defined the profit function Π(θm

a ,θmb

)where(

θma ,θm

b

)∈[θ,θ]2 and one can characterize the optimal couple

(θm

a ,θmb

)by maximizing

Π(θm

a ,θmb

), with respect to (θm

a ,θma ). The optimal choice of θm

b do not influence theoptimal choice of θm

a (and vice versa), then the optimizations with respect to θmb and θm

a

can be done separately.We want q

(θm

b

)≥ 0 and q

(θm

b

)≤ 0. We define θ as the type such that

q∗(θ)−

1−F(θ)

γσ2 f(θ) = 0, (19)

i.e. the type θ is such that q(θ)= 0, and is the minimal value for θm

b . In the same

way, we can define θ as the type such that q∗(θ)+

F(θ)γσ2 f(θ)

= 0 and interpret θ as the

maximal value for θmb . Hence the function Π

(θm

a ,θmb

)is defined and continuous over[

θ, θ]×[θ,θ], and hence θm

b is well defined. Since θ < θ, we have θma < θm

b .Now, we want to analyze the price schedule t’s properties. From what precedes, we

can find conditions under which the price schedule t is concave or convex at any pointq. From the agent’s first order condition we have:

T (θ) = θq(θ)− γσ2q(θ) q(θ) , (20)

which can be rewritten

t ′ [q(θ)] q(θ) = θq(θ)− γσ2q(θ) q(θ) (21)

and hencet ′′ [q(θ)] q(θ) = ν(θ)− d

dθ

(1−F (θ)γσ2 f (θ)

). (22)

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If θ ∈(θm

a ,θ]

and t ′′ [q(θ)] q(θ) = ν(θ)+ ddθ

(F(θ)

γσ2 f (θ)

)and if θ ∈

[θ,θm

b

)then implies

that the schedule at the point q(θ), where θ ∈(θm

a ,θ]

is strictly convex if (and only if)

ν(θ)>d

dθ

(1−F (θ)γσ2 f (θ)

), (23)

and similarly, if for any θ ∈[θ,θm

b

)we have ν(θ)>− d

dθ

(F(θ)

γσ2 f (θ)

)then the price sched-

ule t is locally convex around q(θ). It is then natural to define the following condition:

Definition 1 Condition 1 is satisfied if

• ∀θ ∈[θ,θm

b

), ν(θ)>− d

dθ

(F(θ)

γσ2 f (θ)

),

• ∀θ ∈(θm

a ,θ], ν(θ)> d

dθ

(1−F(θ)γσ2 f (θ)

).

At a given point (q(θ) ,T (θ)), the schedule t is convex if the derivative ν(θ) issufficiently high. Intuitively if the difference between the valuation of the monopolistand the agent gets bigger when θ increases, then the monopolist tries to take more andmore surpluses from the agent as his type gets higher.

Condition 1 must be carefully interpreted, in particular, if the price schedule can beconvex on some subset of

[θ,θ], it cannot be globally convex.

Theorem 1 If condition 1 is satisfied then the price schedule t is not continuous.

Proof. We consider the space (q,T ). Since we assumed that q∗ (θ)< 0 and q∗(θ)> 0,

Then t [q∗ (θ)]< 0 and t[q∗(θ)]

> 0.

We must have that t [q∗ (θ)]−ν(θ)q∗ (θ)> 0 and t[q∗(θ)]

−ν(θ)

q∗(θ)> 0.

Due to the definition of q∗ (θ), it turns out that the price schedule t is tangent to theline Π(θ) defined by the equation

T −ν(θ)q = t [q∗ (θ)]−ν(θ)q∗ (θ) (24)

at the point (q∗ (θ) , t [q∗ (θ)]). In the same way, the price schedule T is tangent to theline defined by the equation:

T −ν(θ)q = t [q∗ (θ)]−ν(θ)q∗ (θ) (25)

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T

q

Π(θ)

Π(θ)

t

t[q∗(θ)]

q∗(θ)q∗ (θ)

t [q∗ (θ)]

Figure 1: Convexity of the Price Schedule t

at the point (q∗ (θ) , t [q∗ (θ)]).

If we assume that the schedule t is convex and continuous, then t is above the lineΠ(θ). Moreover, q∗

(θ)> 0. Then it exists a θ ∈

[θ,θ]

such that qm (θ) = 0 andt [qm (θ)]> 0, which is impossible.

The theorem shows that there is a incompatibility between the “no distortion at thetop” property, convexity and continuity of t. The “no distortion at the top” property andthe convexity of t implies that t is above the profit line Π

(θ). This line is itself above

the point (0,0) or cross that point. Since t is strictly above this profit line, it is alsostrictly above (0,0), meaning that at equilibrium there is an agent paying price strictlypositive price for nothing, which is clearly impossible.

A similar argument applies is we consider concavity rather than convexity.

Definition 2 Condition 2 is satisfied if

• ∀θ ∈ [θ,θma ) , ν(θ)<− d

dθ

(F(θ)

γσ2 f (θ)

),

• ∀θ ∈(θm

b ,θ], ν(θ)< d

dθ

(1−F(θ)γσ2 f (θ)

).

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t

Π(θ)

q

Π(θ)

q∗ (θ)

q∗(θ)

t[q∗(θ)]

T

t [q∗ (θ)]

Figure 2: Concavity of the Price Schedule t

In this case (discount prices) one can exclude that the optimal solution is continuous.

Theorem 2 If condition 2 is satisfied then the price schedule t is not continuous.

Proof. At the point (q∗ (θ) , t [q∗ (θ)]), the price schedule t is tangent to the line Π(θ),or in other words ∂t

∂q

∣∣(q∗(θ),T [q∗(θ)]) = ν(θ).

If the schedule t is continuous and concave then

∂t∂q

∣∣(q∗(θ),t[q∗(θ)]) >

∂t∂q

∣∣∣(q∗(θ),t[q∗(θ)]) . (26)

From the characterization of t we have ∂t∂q

∣∣∣(q∗(θ),t[q∗(θ)]) = ν(θ). Since the function ν

is increasing, ν(θ)≥ ν(θ). A contradiction.

An optimal mechanism, if either condition 1 or condition 2 is satisfied, divides theset of possible agents into three categories. Agents with a “hight” θ. As those agentshave a high valuation of the asset, principal sells assets to them. For agents with a “low”θ, as their valuation is low, the principal buys assets from them. There a third category:

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agents with a “medium” valuation. From them any trade is not profitable. This set isnot a singleton because discrimination is costly for the principal. Hence there is a nondegenerate of set of agents for which exchanges with the principal are not interesting.

From the two previous theorems, we can stat the following corollary:

Corollary 1 If the price schedule t is continuous, then it is neither convex, nor concave.

It is worth to note that if the price schedule might be discontinuous, the equilibriumutility is continuous with respect to the agent’s type. In formal terms U is a continuousfunction. Such a property is a direct consequence of the continuity of the utility functionU with respect to θ.

Concerning the monopolist case, our remarks helps to precised the shape of theoptimal price schedule. It turns out that the potential discontinuity that we have stressedin the monopoly case, can be probematic for the existence of equilibria in non-exclusivecompetition games.

4 Non-Exclusive Competition

In our setting of non-exclusive contracting the Revelation Principle does not apply any-more (Peck, 1997). We follow the existing literature on common agency and we applythe “Delegation Principle” (see Peters, 2001; Martimort and Stole, 2002). Neverthelesswe impose more restrictions on the shape of the equilibrium. In particular we wantto restrict attention to symmetric equilibria. In particular at equilibrium we considerthat every principal offers the same price schedule, and the agent buy the same quantityfrom all principals, and hence pays to them the same total price. More specifically, letus denote by ti the equilibrium price schedule offered by principal i.

Definition 3 An equilibrium is fully symmetric if all principals offer the same menus of

contracts: ∀i, j ∈ {1,2, . . . ,n} , ti = t j = t, and when the agent as the type θ, he buy

the quantity q(θ) from all the principals.

Hence, we consider equilibria in which the agent (when is type is θ) owns the quantitynq(θ) and pays nt (q(θ)).

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Let us remark that is an equilibrium is fully symmetric, then principal are offeringmenus (or price schedule here) without “latent contract”. By “latent contracts” we meancontract (or couples price quantity) that are never bought at equilibrium. The role ofsuch contracts is widely studied in the literature in different contexts (see Hellwig, 1983;Bisin and Guaitoli, 2004; Ales and Maziero, 2009; Piaser, 2010; Attar et al., 2011, 2013)

We define the curve (or the set) ζ as the set of all consumption bundle

ζ = (nq(θ) ;nt (q(θ)))θ∈(θ,θ) . (27)

Definition 4 We said that the equilibrium is monotone is the curve ζ is increasing.

It is natural to consider that the total price paid is increasing with the quantity bought.Since we concentrate on the simplest equilibria, restricting attention to monotone equi-libria do not appear problematic.

Lemma 1 If the equilibrium is fully symmetric and monotone, the curve ζ is convex in

the space (q,T ).

Proof. Let us consider two point of ζ , namely A=(nq(θA) ;nt (θA)) and B=(nq(θB) ;nt (θB)),and we assume that between A and B the curve ζ is continous and above the segment[A,B]. Let us onsider the point C defined as C =((n−1)q(θA)+q(θB) ;(n−1) t (θA)+ t (θB)).By contruction the point C belongs to the segment [A,B]. Since indifference curves areconcave, it implies that it exists a type θ ∈ [θA,θB] such that the agent having this type,strictly prefer the point (or contract) C to any point of the curve ζ.

If now, keeping the same notation, we consider the point A and B of the curve ζsuch that for any quantity qn ∈ [nq(θA) ,nq(θB)] there is no T n such that (qn,T n) ∈ ζ.Roughly speaking, we assume that the curve ζ is discontinuous between the point A

and B. Since the curve ζ is the set of optimal choices, and since preferences exhibitthe single crossing property, it must exist a type θ such that the agent having the typeθ is indifferent between A and B. Using the same argument, one can construct a pointC lying on the segment [A,B]. By construction the contract C is feasible and strictlypreferred to the contracts A and B by the agent having the type θ.

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T

C = (q(θB)+q(θA) ;T (θB)+T (θA))

A

(q(θB) ;T (θB))

B

q(q(θA) ;T (θA))

ζ

Figure 3: Convexity of the curve ζ

B = (0,0)

ζ

A =(q2 (θA) ;T 2 (θA)

)C = (q(θA) ;T (θA))

T

q

Figure 4: Continuity of the curve ζ

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To conclude, the curve ζ is continuous and convex.

The convexity of ζ do not rely on assumption on the function ν and F as in themonopoly case. Here convexity comes from a simple arbitrage argument. Convexity isa direct consequence of competition.

These first results do not provide a characterization of the equilibrium of the gameGn. In the next theorem, we argue that the equilibrium price schedules are monotoneand exhibits the “no distortion at the top” properties. Since a proof of this theorem isgiven by BMR, we just sketch a proof in appendix.

Theorem 3 The equilibrium quantities nq(θ) and nq(θ)

are not distorted and equal

to q∗ (θ) and q∗(θ).

Hence, the equilibrium schedule ζ must be convex (and continuous). Using the samearguments as in the monopolist case, we can argue or the geometrical impossibility of acontinuous and convex schedule. From that, we can state our last theorem

Theorem 4 It does not exist any fully symmetric equilibrium in the game Gn.

5 Conclusion

First let us remark that our non-existence problem comes from the restriction to purestrategy equilibria. If we consider mixed strategy equilibria, then Carmona and Fajardo(2009) and Monteiro and Page Jr (2008) show the existence of a Nash equilibrium incommon agency games such as the one considered in the present paper. Characterizationof such equiliibria remains an open problem.

In the existing literature, common agency games with common values do not havefully symmetric equilibria, for example Attar et al. (2011), Ales and Maziero (2009)or Attar et al. (2013). Those papers study model with a finite number of types. Attaret al. (2011) show that in a model with linear preferences equilibria latent contractsplay a crucial role. Ales and Maziero (2009) and Attar et al. (2013) in model withrisk averse agents extend the latter paper’s results. Moreover, in their framework, atequilibrium some agents depending on their type, are excluded from trade. Hence ourpaper a first step toward a clarification of the links between the BMR approach and the

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existing results in games with a discrete number of types. Moreover, in a model ofcomplete information, e.i. a degenerate version of the considered model, Chiesa andDenicolò (2009) show that at equilibrium at least one principal is offering a menu withlatent contract. This suggest that in common agency games with incomplete informationlatent contracts are likely to play an important role.

Let us also insist on the importance of exclusion. In the literature on mechanismdesign, it has been largely neglected, the paper by Jullien (2000) being one of the fewexceptions. In the our context it cannot be seen as a sophistication of the equilibriumas it is the case in the mechanisms design literature. Since we have at equilibriumprincipals trade positive and negative quantities, exclusion plays a fundamental role. Aprincipal, in most situation, would like to divide the agents into three groups. Those towhich he sells, those from which he buys, and a “middle” group: agent from any trade isnot profitable. Due to asymmetric information, discrimination is costly. Thus, excludingsome agents from the trade, those with valuations close to the one of the principal,helps him to sells at higher prices and buy a lower prices. Such a strategy implies thatoptimal price schedule is discontinuous. In the paper we show that competition impliescontinuous price schedule when we restrict attention to some natural equilibria. Sincea price schedule cannot be simultaneously continuous and discontinuous, we have aproblem of existence.

Except theorem 3, all our results do not depend on the shape of the agent’s prefer-ences, as long as those preferences are sufficiently regular (continuity, concavity, etc).Theorem 3 just generalized a basic property of equilibrium schedules, which is standardin the mechanism design literature. Hence we conjecture that our results remains validif we consider agents with more general quasi-linear preferences.

Finally, the shape of equilibria in common agency games with both common valuesand an continuum of type remains an open question. Papers studying similar modelswith a discrete number of types such as Ales and Maziero (2009) and Attar et al. (2013)characterize equilibria in which only one type of agent trades. Moreover, some restrict-ing assumptions are needed. In our context, if such an equilibrium exists, it wouldmean an almost complete collapse of the market. Existence and characterization of anequilibrium in the present context could be a relevant track for future researches.

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A Proof of Theorem 3

We consider the principal i and we consider that all other principals plays the same con-vex price schedule t j. Following Martimort and Stole (2002), we consider that there isno loss of generality in restricting principal i to offer incentive compatible mechanisms.

Hence, first we consider that principal i offers direct mechanisms i.e; a mappings(qi (.) ,Ti (.)) from

[θ,θ]

to R2. If the agent θ reports the vector(

θi,(q j)

j =i

), that is if

he reports θi to principal i, and ask for the quantities q j to principals j, he gets:

U((

θi,(q j)

j =i

)∣∣∣θ)= θ

(qi(θi)+∑

j =iq j

)− γσ2

2

(qi(θi)+∑

j =iq j

)2

−Ti(θi)−∑

j =it j(q j).

(28)We focus on principal i. He considers others principals’ strategies

(t j)

j =i as given. ByQ j ⊂R2 we denote the support of the price schedule t j.

We denote by(q j)

j =i = (q1, . . . ,qi−1,qi+1, . . . ,qn), the set of reports sent by theagent to the other principals. We define the best reports

(q j(θ, θi

))j =i, given the type

(which is θ) of the agent and given an arbitrary report to principal i denoted θi.(q j(θ, θi

))j =i ∈ argmax

(q j) j =i∈× j =iQ j

{U((

θi,(q j)

j =i

)∣∣∣θ)} . (29)

The quantities(q j(θ, θi

))j =i are chosen optimally and they are functions of θ and θi.

We look for an interior the solution, then those quantities must satisfy:

∂U((

θi,(q j(θ, θi

))j =i

)∣∣∣θ)∂q1, . . . ,∂qi−1,∂qi+1, . . . ,∂qn

= 0, (30)

which can be also written

∀ j = i θ− γσ2

(qi (θ)+∑

j =iq j(θ, θi

))= t ′j

(q j(θ, θi

)). (31)

Since the price schedule t j are convex and the agent preferences are concave with respectto the quantity, then second order conditions are straightforwardly satisfied.

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Now, we derive the optimal strategy of the principal i. As we apply the RevelationPrinciple, we assume the principal i offers incentive compatible mechanisms. In otherwords, principal i offers maps Ti(.),qi(.) such that the agent reports to him his true typeθ, given the price schedules offered by the other principals.

The agent reports truthfully his type to principal i if

dU(

θi,(q j(θ, θi

))j =i

∣∣∣θ)dθi

∣∣∣∣∣∣θi=θ

= 0, (32)

applying the envelope theorem we get an equivalent expression:

θ qi (θ)− γσ2

(∑j =i

q j (θ)+qi (θ)

)qi (θ)− Ti (θ) = 0. (33)

To simplify the expressions, we used the notation q j (θ,θ) = q j (θ). We now can definethe rent obtained by the agent. The rent is the utility that the agent gets if his type is θgiven the offers made by all principals

U (θ) = θ

(qi (θ)+∑

j =iq j (θ)

)− γσ2

2

(qi (θ)+∑

j =iq j (θ)

)2

−Ti (θ)−∑j

t j(q j (θ)

),

(34)Applying again the envelope theorem, we get the derivative of U with respect to θ:

∀θ ∈[θ,θ], U (θ) = qi (θ)+∑

j =iq j (θ) . (35)

To get that, a necessary condition is∂2U( θ|θ)(∂θi)

2 < 0. If messages are optimal when θi = θ

then the conditions ∀θ ∈[θ,θ],

∂2U( θ|θ)(∂θi)

2 < 0 becomes

∀θ ∈[θ,θ], θ qi (θ)− γσ2

(qi (θ)+ ∑

j =iq j (θ)

)qi (θ)− γσ2q2

i (θ)− Ti (θ)< 0.

(36)Using standard methods of mechanism design, this last condition can also be written as

17

∀θ ∈[θ,θ], θ qi (θ)≥ 0. (37)

In words, the optimal quantity must be non decreasing with θ. This condition is standardin mechanism design theory.

If the function qi(.) is increasing, it obviously means that it can be first negative, thenequal to zero and finally positive. We denote [θ,θa] the domain on which the functionqi(.) is negative, [θa,θb] on which qi(.) is constant and equal to zero and

[θb,θ

]on

which it is positive. Using these new notation, we can integrate the function U to get anew expression of U .

U (θ) =∫ θ

θb

(qi (θ)+∑

j =iq j (θ)

)dθ (38)

if θ ∈[θb,θ

],

U (θ) =∫ θa

θ

(qi (θ)+∑

j =iq j (θ)

)dθ (39)

if θ ∈ [θ,θa], andU (θ) = 0. (40)

if θ ∈ [θa,θb], where θ ≤ θa ≤ θb ≤ θ.

Integrating by parts these expressions gives

∫ θa

θU (θ)dF (θ) =

∫ θa

θ

(qi (θ)+∑

j =iq j (θ)

)F (θ)f (θ)

dF (θ) (41)

and

∫ θ

θb

U (θ)dF (θ) =∫ θ

θb

(qi (θ)+∑

j =iq j (θ)

)1−F (θ)

f (θ)dF (θ) . (42)

The profit of principal i can be written as

Π =∫[θ,θ]

[Ti (θ)−ν(θ)qi (θ)]dF (θ) , (43)

by using the definition of the utility function we can rewrite the former expression as:

18

Π =∫ θa

θ

θ

(qi (θ)+ ∑

j =iq j (θ)

)− γσ2

2

(qi (θ)+ ∑

j =iq j (θ)

)2dF (θ)

−∫ θa

θ

[U (θ)+ ∑

j =it j(q j (θ)

)+ν(θ)qi (θ)

]dF (θ)

+∫ θ

θb

θ

(qi (θ)+ ∑

j =iq j (θ)

)− γσ2

2

(qi (θ)+ ∑

j =iq j (θ)

)2dF (θ)

−∫ θ

θb

[U (θ)+ ∑

j =it j(q j (θ)

)+ν(θ)qi (θ)

]dF (θ) .

(44)

First, let us consider θa and θb, as given. The problem of the principal is equivalent toa point-wise maximization problem. The principal maximizes the following expressionwith respect to q(θ) if θ ∈

[θb,θ

].

θ

(qi (θ)+ ∑

j =iq j (θ)

)− γσ2

2

(qi (θ)+ ∑

j =iq j (θ)

)2

−

(qi (θ)+ ∑

j =iq j (θ)

)(1−F(θ))

f (θ) − ∑j =i

t j(q j (θ)

)−ν(θ)qi (θ) .

(45)

The first order condition of the Principal’s maximization problem is given by

θ

(1+ ∑

j =i

∂q j(θ)∂qi(θ)

)− γσ2

(qi (θ)+ ∑

j =iq j (θ)

)(1+ ∑

j =i

∂q j(θ)∂qi(θ)

)

−

(1+(n−1) ∑

j =i

∂q j(θ)∂qi(θ)

)(1−F(θ))

f (θ) − ∑j =i

[t ′j(q j (θ)

) ∂q j(θ)∂qi(θ)

]−ν(θ) = 0.

(46)

The second order condition is given

19

θ ∑j =i

∂2q j(θ)∂qi(θ)2 − γσ2

(1+ ∑

j =i

∂q j(θ)∂qi(θ)

)2

− γσ2

(qi (θ)+ ∑

j =iq j (θ)

)∑j =i

∂2q j(θ)∂qi(θ)2

−(n−1) (1−F(θ))f (θ) ∑

j =i

∂2q j(θ)∂qi(θ)2 − ∑

j =i

[t ′′j(q j (θ)

)(∂q j(θ)∂qi(θ)

)2]

− ∑j =i

[t ′j(q j (θ)

) ∂2q j(θ)∂qi(θ)2

]≤ 0.

(47)

As we want to show that a fully symmetric equilibrium does not exist, we will assumethat this condition is satisfied.

To characterize the solution we need the expression of ∑j =i

∂q j(θ)∂qi(θ) . From the self-

selection constraint, we have derived the expressions:

θ− γσ2

(∑j =i

q j (θ)+qi (θ)

)−∑

j =it ′j(q j (θ)

)= 0. (48)

Over the set[θb,θ

], the function qi(.) is increasing. Thus, without loss of generality, we

can rewrite the direct mechanism(q j (.) ,Tj (.)

)as a direct mechanism

(q j (.) , t j

(q j (.)

)).

As we consider a domain in which q j(.) is strictly increasing, we can define a in-verse function θ−1

i (.). Thus the function t j is define for all θ in[θb,θ

]by t j (q(θ)) =

Tj

(θ−1

j (q(θ)))

. The former first order condition (31) becomes:

∀ j = i θ− γσ2

(qi (θ)+∑

j =iq j (θ)

)= t ′j

(q j (θ)

). (49)

Differentiating this equation with respect to qi (θ) gives (as qi (θ) is a parameter in (31),this transformation makes sense):

−γσ2

(1+∑

j =i

∂q j (θ)∂qi (θ)

)= t ′′j

(q j (θ)

) ∂qi (θ)∂qi (θ)

. (50)

By summing these conditions over j = i, we get:

20

−(n−1)γσ2

(1+∑

j =i

∂q j (θ)∂qi (θ)

)= ∑

j =it ′′j(q j (θ)

) ∂q j (θ)∂qi (θ)

. (51)

As we consider a symmetric equilibrium all the principals j, (with j different from i),are offering the same mechanism, and thus the derivative t ′′j

(q j (θ)

)and the quantity

q j (θ) are constant with respect to j and we denote them t ′′ (q(θ)) and q(θ). Thus weget:

− (n−1)γσ2

t ′′ (q(θ))+(n−1)γσ2 = (n−1)∂q(θ)∂qi (θ)

. (52)

Equation (49) at equilibrium can be written

θ−nγσ2q(θ) = t ′ (q(θ)) . (53)

Differentiating this equation with respect to θ gives

1−nγσ2q(θ) = t ′′ (q(θ)) q(θ) , (54)

or1

q(θ)−nγσ2 = t ′′ (q(θ)) . (55)

Thus at equilibrium we have,

(n−1)∂q(θ)∂qi (θ)

=−(n−1)q(θ)γσ2

1− q(θ)γσ2 . (56)

From (49) we gett ′ (q(θ)) = θ−nγσ2q(θ) . (57)

Using the two obtained expressions, from the first order condition (46) can get a equi-librium condition:

21

θ(

1− q(θ)1−γσ2q(θ)γσ2 (n−1)

)− γσ2nq(θ)

(1− q(θ)

1−γσ2q(θ)γσ2 (n−1))

−(

1− q(θ)1−γσ2q(θ)γσ2 (n−1)

)(1−F(θ))

f (θ)

+(n−1)[θ− γσ2nq(θ)

] q(θ)1−γσ2q(θ)γσ2 −ν(θ) = 0.

(58)

Consequently, using the notation q∗ (θ) = θ−ν(θ)γσ2 and qm (θ) = q∗ (θ)− 1−F(θ)

γσ2 f (θ) , we get:

[qm (θ)−nq(θ)]− q(θ)γσ2 [qm (θ)−nq(θ)]

− [qm (θ)−nq(θ)] q(θ)γσ2 (n−1)

+(n−1) [q∗ (θ)−nq(θ)] q(θ)γσ2 = 0,

(59)

and finally

q(θ) =1

γσ2

(n+

(n−1)(q∗ (θ)−nq(θ))nq(θ)−qm (θ)

)−1

, (60)

and if we define the function qn(.) = nq(.), we get equivalently

qn (θ) =1

γσ2

(1+

(n−1)(q∗ (θ)−qn (θ))n(qn (θ)−qm (θ))

)−1

. (61)

If θ = θ, since F(θ)= 1, equation (58) can be rewritten

θ(

1− q(θ)1−γσ2q(θ)

γσ2 (n−1))− γσ2nq

(θ)(

1− q(θ)1−γσ2q(θ)

γσ2 (n−1))

+(n−1)[θ− γσ2nq

(θ)] q(θ)

1−γσ2q(θ)γσ2 −ν

(θ)= 0,

(62)

which is equivalent to

θ− γσ2nq(θ)−ν(θ)= 0. (63)

Hence nq(θ)= q∗

(θ).

22

If θ ∈ [θ,θa], the principal i maximizes the following expression with respect toqi (θ):

θ(qi (θ)+∑ j =i q j (θ)

)− γσ2

2

(qi (θ)+ ∑

j =iq j (θ)

)2

−

(qi (θ)+ ∑

j =iq j (θ)

)F(θ)f (θ) + ∑

j =it j(q j (θ)

)−ν(θ)qi (θ) .

(64)

We can derive the same expression for q(θ), except that qm (θ) = q∗ (θ)− F(θ)γσ2 f (θ) . Using

the same argument as above, we can show that nq(θ) = q∗ (θ).

Given the expressions of q(θ), θa and θb must be such that the function q is contin-uous. As the aggregate supply nq(.) is an increasing function, the form chosen for theutility is justified. Usual conditions on the density f guaranty that q is strictly increasing.Interpretation of these conditions are discussed by Miravete (2002).

Because the price schedule are continuous, at equilibrium we must have nq(θb) = 0,with the function q defined by equation (60). The same argument applies for q(θa).

Convexity of price schedules implies that any agent prefers a bundle belonging to ζrather than a bundle made with the offers of n−1 principals.

Finally, let us remark that one can get the same characterization if we assume thatprincipals play incentive compatible mechanisms (see Piaser, 2007). This observationdoes not contradict the “Delegation Principle”. At equilibrium, principals do not offercontracts with “latent contracts”. Since we consider equilibria in which every principaloffers a menu (which here is called a price schedule) without latent contract, then thisequilibria can be characterized by incentive compatible mechanisms.

ReferencesAles, L. and P. Maziero: 2009, ‘Non-Exclusive Dynamic Contracts, Competition, and

the Limits of Insurance’. mimeo, University of Pennsylvania.

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Attar, A., T. Mariotti, and F. Salanié: 2011, ‘Nonexclusive competition in the marketfor lemons’. Econometrica 79(6), 1869–1918.

Attar, A., T. Mariotti, and F. Salanié: 2013, ‘Nonexclusive competition under adverseselection’. Theoretical Economics. forthcoming.

Biais, B., D. Martimort, and J.-C. Rochet: 2000, ‘Competing mechanisms in a commonvalue environment’. Econometrica 68(4), 799–837.

Biais, B., D. Martimort, and J.-C. Rochet: 2013, ‘Corrigendum to Competing Mecha-nisms in a Common Value Environment’. Econometrica 81(1), 393–406.

Bisin, A. and D. Guaitoli: 2004, ‘Moral hazard and nonexclusive contracts’. RANDJournal of Economics pp. 306–328.

Carmona, G. and J. Fajardo: 2009, ‘Existence of equilibrium in common agency gameswith adverse selection’. Games and Economic Behavior 66(2), 749–760.

Chiesa, G. and V. Denicolò: 2009, ‘Trading with a common agent under complete infor-mation: A characterization of Nash equilibria’. Journal of Economic Theory 144(1),296–311.

Hellwig, M.: 1983, ‘On the moral hazard and non-price equilibria in competitive insur-ance markets’. Discussion Paper 109, Institut fur Gesellschafts.

Jullien, B.: 2000, ‘Participation constraints in adverse selection models’. Journal ofEconomic Theory 93(1), 1–47.

Martimort, D. and L. A. Stole: 2002, ‘The revelation and delegation principles in com-mon agency games’. Econometrica 70(4), 1659–1673.

Maskin, E. and J. Riley: 1984, ‘Monopoly with incomplete information’. The RANDJournal of Economics 15(2), 171–196.

Miravete, E. J.: 2002, ‘Preserving Log-Concavity under convolution: Comment’.Econometrica 70(3), 1253–1254.

Monteiro, P. K. and F. H. Page Jr: 2008, ‘Catalog competition and Nash equilibrium innonlinear pricing games’. Economic Theory 34(3), 503–524.

Peck, J.: 1997, ‘A note on competing mechanisms and the revelation principle’. mimeoOhio State University.

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Piaser, G.: 2007, ‘The Biais-Martimort-Rochet Equilibrium With Direct Mechanisms’.University Ca’Foscari of Venice, Dept. of Economics Research Paper Series 33/06.

Piaser, G.: 2010, ‘Direct Mechanisms, Menus and Latent Contracts’. Modern Economy1, 51–58.

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