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Motivation Model Equilibrium Results Conclusion Existence of Equilibrium in The Common Agency Model with Adverse Selection Jos Fajardo 1 Guilherme Carmona 2 1 Economics Department IBMEC Business School 2 Economics Department Universidade

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Motivation Model Equilibrium Results Conclusion

Existence of Equilibrium in The Common Agency Model with Adverse Selection

José Fajardo1 Guilherme Carmona2

1Economics Department

IBMEC Business School

2Economics Department

Universidade Nova de Lisboa

ASSET - Lisbon, November 2-4, 2006.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Outline

1

Motivation Previous Works Contribution

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Outline

1

Motivation Previous Works Contribution

2

Model

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Outline

1

Motivation Previous Works Contribution

2

Model

3

Equilibrium

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Outline

1

Motivation Previous Works Contribution

2

Model

3

Equilibrium

4

Results Main Results Basic Ideas for Proofs

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion Previous Works Contribution

Delegation principle.

Martimort (2006): “What matters per se is not the kind of communication that a principal uses with his agent but the set of options that this principal makes available to the agent.”

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion Previous Works Contribution

Delegation Principle

Common agency problem can be analyzed through a menu game

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion Previous Works Contribution

Delegation Principle

Common agency problem can be analyzed through a menu game Equilibrium must exist!

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion Previous Works Contribution

Related Literature

Page and Monteiro, JME. (2003): Monteiro and Page (2005):

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion Previous Works Contribution

Related Literature

Page and Monteiro, JME. (2003): Monteiro and Page (2005): Normal-form game played by Principals.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion Previous Works Contribution

Related Literature

Page and Monteiro, JME. (2003): Principals’ payoff are not induced by an optimal strategy of the agent. Monteiro and Page (2005): Normal-form game played by Principals.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion Previous Works Contribution

Related Literature

Page and Monteiro, JME. (2003): Principals’ payoff are not induced by an optimal strategy of the agent. Monteiro and Page (2005): Fix an optimal strategy for the agent. Normal-form game played by Principals.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion Previous Works Contribution

Contribution

Sequential Equilibrium (Kreps and Wilson, Ecta. 1982.)

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion Previous Works Contribution

Contribution

Sequential Equilibrium (Kreps and Wilson, Ecta. 1982.) Endogenous Sharing rules (Simon and Zame, Ecta. 1990.)

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion Previous Works Contribution

Contribution

Sequential Equilibrium (Kreps and Wilson, Ecta. 1982.) Endogenous Sharing rules (Simon and Zame, Ecta. 1990.) Existence of Equilibrium

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Menu Games

Principals

Consider a game with m principals and 1 agent.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Menu Games

Principals

Consider a game with m principals and 1 agent. Ki compact metric space: Set of contracts that principal i can offer.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Menu Games

Principals

Consider a game with m principals and 1 agent. Ki compact metric space: Set of contracts that principal i can offer. Ci ⊂ Ki nonempty closed subset: A menu of contracts for principal i ∈ I = {1, . . . , m}.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Menu Games

Principals

Consider a game with m principals and 1 agent. Ki compact metric space: Set of contracts that principal i can offer. Ci ⊂ Ki nonempty closed subset: A menu of contracts for principal i ∈ I = {1, . . . , m}. Pi collection of all nonempty, closed subsets of Ki. (Pi compact metric space w.r.t Hausdorff metric)

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Menu Games

Principals

Consider a game with m principals and 1 agent. Ki compact metric space: Set of contracts that principal i can offer. Ci ⊂ Ki nonempty closed subset: A menu of contracts for principal i ∈ I = {1, . . . , m}. Pi collection of all nonempty, closed subsets of Ki. (Pi compact metric space w.r.t Hausdorff metric) P = P1 × · · · × Pm and C = (C1, . . . , Cm) denote a profile

  • f menus.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Menu Games

Principals

Consider a game with m principals and 1 agent. Ki compact metric space: Set of contracts that principal i can offer. Ci ⊂ Ki nonempty closed subset: A menu of contracts for principal i ∈ I = {1, . . . , m}. Pi collection of all nonempty, closed subsets of Ki. (Pi compact metric space w.r.t Hausdorff metric) P = P1 × · · · × Pm and C = (C1, . . . , Cm) denote a profile

  • f menus.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Menu Games

Agent

T Polish space: set of Agent’s type

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Menu Games

Agent

T Polish space: set of Agent’s type µ probability measure on the set of types.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Menu Games

Agent

T Polish space: set of Agent’s type µ probability measure on the set of types. K compact metric space: the pure action space of the

  • agent. k generic element of K.

v : T × K → R Carathéodory function: Agent’s utility.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Menu Games

Agent

T Polish space: set of Agent’s type µ probability measure on the set of types. K compact metric space: the pure action space of the

  • agent. k generic element of K.

v : T × K → R Carathéodory function: Agent’s utility. ∆(K) space of all Borel probability measures over K.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Menu Games

Agent

T Polish space: set of Agent’s type µ probability measure on the set of types. K compact metric space: the pure action space of the

  • agent. k generic element of K.

v : T × K → R Carathéodory function: Agent’s utility. ∆(K) space of all Borel probability measures over K. ϕ(t, C) ⊆ ∆(K) nonempty compact convex set and ϕ : T × P ⇒ ∆(K) continuous correspondence: Set of available choices

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Particular cases

Contracts are exclusive: KPM = {(i, f) ∈ I × ∪m

i=1Ki : f ∈ Ki},

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Particular cases

Contracts are exclusive: KPM = {(i, f) ∈ I × ∪m

i=1Ki : f ∈ Ki},

Contracts are not exclusive: KMS = K1 × · · · × Km.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Particular cases

Contracts are exclusive: KPM = {(i, f) ∈ I × ∪m

i=1Ki : f ∈ Ki},

Contracts are not exclusive: KMS = K1 × · · · × Km. Ie ⊆ I of principals only allows for exclusive contracts: KH = {(i, f) ∈ Ie × ∪i∈IeKi : f ∈ Ki} ×

i∈Ic

e Ki. Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Particular cases

Contracts are exclusive: KPM = {(i, f) ∈ I × ∪m

i=1Ki : f ∈ Ki},

Contracts are not exclusive: KMS = K1 × · · · × Km. Ie ⊆ I of principals only allows for exclusive contracts: KH = {(i, f) ∈ Ie × ∪i∈IeKi : f ∈ Ki} ×

i∈Ic

e Ki.

KPM, KMS and KH are compact.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Particular cases

Contracts are exclusive: KPM = {(i, f) ∈ I × ∪m

i=1Ki : f ∈ Ki},

Contracts are not exclusive: KMS = K1 × · · · × Km. Ie ⊆ I of principals only allows for exclusive contracts: KH = {(i, f) ∈ Ie × ∪i∈IeKi : f ∈ Ki} ×

i∈Ic

e Ki.

KPM, KMS and KH are compact. We can let some f ∈ Ki denote no contracting

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Particular cases

Three possible specifications for ϕ : ϕPM(t, C) = {λ ∈ ∆(KPM) : λ(∪m

i=1({i} × Ci)) = 1}.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Particular cases

Three possible specifications for ϕ : ϕPM(t, C) = {λ ∈ ∆(KPM) : λ(∪m

i=1({i} × Ci)) = 1}.

ϕMS(t, C) = {λ ∈ ∆(KMS) : λ(C) = 1}

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Particular cases

Three possible specifications for ϕ : ϕPM(t, C) = {λ ∈ ∆(KPM) : λ(∪m

i=1({i} × Ci)) = 1}.

ϕMS(t, C) = {λ ∈ ∆(KMS) : λ(C) = 1} ϕH(t, C) = {λ ∈ ∆(KH) : λ(∪i∈Ie({i} × Ci) ×

  • i∈Ic

e

Ci) = 1}

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Particular cases

Three possible specifications for ϕ : ϕPM(t, C) = {λ ∈ ∆(KPM) : λ(∪m

i=1({i} × Ci)) = 1}.

ϕMS(t, C) = {λ ∈ ∆(KMS) : λ(C) = 1} ϕH(t, C) = {λ ∈ ∆(KH) : λ(∪i∈Ie({i} × Ci) ×

  • i∈Ic

e

Ci) = 1} We show ϕPM, ϕMP and ϕH are continuous with nonempty, convex and compact values.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Optimal problems

Agent

Given t ∈ T and C ∈ P, the agent’s problem is max

λ∈ϕ(t,C)

  • K

v(t, k)dλ(k).

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Optimal problems

Agent

Given t ∈ T and C ∈ P, the agent’s problem is max

λ∈ϕ(t,C)

  • K

v(t, k)dλ(k). Λ : T × P ⇒ ∆(K): correspondence of optimal choices.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Optimal problems

Agent

Given t ∈ T and C ∈ P, the agent’s problem is max

λ∈ϕ(t,C)

  • K

v(t, k)dλ(k). Λ : T × P ⇒ ∆(K): correspondence of optimal choices. A strategy for the agent is then a measurable function σ : T × P → ∆(K).

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Optimal problems

Agent

Given t ∈ T and C ∈ P, the agent’s problem is max

λ∈ϕ(t,C)

  • K

v(t, k)dλ(k). Λ : T × P ⇒ ∆(K): correspondence of optimal choices. A strategy for the agent is then a measurable function σ : T × P → ∆(K). σ is an optimal strategy if and only if it is a selection of Λ.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Optimal problems

Principals

Principals choose simultaneously. For all i ∈ I,

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Optimal problems

Principals

Principals choose simultaneously. For all i ∈ I, ∆(Pi) set of mixed strategies on Pi: Principal i’s choice set.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Optimal problems

Principals

Principals choose simultaneously. For all i ∈ I, ∆(Pi) set of mixed strategies on Pi: Principal i’s choice set. πi : T × K → R bounded Carathéodory function: Profit.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Optimal problems

Principals

Principals choose simultaneously. For all i ∈ I, ∆(Pi) set of mixed strategies on Pi: Principal i’s choice set. πi : T × K → R bounded Carathéodory function: Profit. If the principals offer a menu C = (C1, . . . , Cm) ∈ P and the agent uses σ : T × P → ∆(K), then payoff is Fi(t, C; σ) =

  • K

πi(t, k)dσ(k|t, C).

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Optimal problems

Principals

Principals choose simultaneously. For all i ∈ I, ∆(Pi) set of mixed strategies on Pi: Principal i’s choice set. πi : T × K → R bounded Carathéodory function: Profit. If the principals offer a menu C = (C1, . . . , Cm) ∈ P and the agent uses σ : T × P → ∆(K), then payoff is Fi(t, C; σ) =

  • K

πi(t, k)dσ(k|t, C). if principals choose strategies α = (α1, . . . , αm), Fi(α; σ) =

  • P
  • T

Fi(t, C; σ)dµ(t)dα(C) (1) denotes principal i’s payoff.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Menu Games

A menu game is denoted by G and we use GPM, GMS and GH to denote particular menu games for the corresponding choices

  • f K and ϕ mentioned above. We say that a menu game G is

continuous if it satisfies all the above assumptions.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Sequential Equilibrium

Definition An assessment (ν, (α, σ)) is a sequential equilibrium of a menu game G if and only if

1

νi = µ × α1 × · · · × αi−1 for all i ∈ I,

2

σ is a measurable selection of Λ and

3

Fi(α; σ) ≥ Fi(¯ αi, α−i; σ) for all i ∈ I and ¯ αi ∈ ∆(Pi).

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Sequential Equilibrium

Definition An assessment (ν, (α, σ)) is a sequential equilibrium of a menu game G if and only if

1

νi = µ × α1 × · · · × αi−1 for all i ∈ I,

2

σ is a measurable selection of Λ and

3

Fi(α; σ) ≥ Fi(¯ αi, α−i; σ) for all i ∈ I and ¯ αi ∈ ∆(Pi). Thus, in a sequential equilibrium of G, beliefs are determined by Bayes’ rule, the agent optimizes for all possible types and menus offered, and each principal optimizes given the strategy

  • f the other principals and the strategy of the agent.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Page and Monteiro (2003)

Principal i’s payoff function F PM

i

: P → R is the expected value

  • f π∗:

F PM

i

(C) =

  • T

π∗

i (t, C)dµ(t).

(2)

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Page and Monteiro (2003)

Principal i’s payoff function F PM

i

: P → R is the expected value

  • f π∗:

F PM

i

(C) =

  • T

π∗

i (t, C)dµ(t).

(2) A Page-Monteiro equilibrium is a Nash equilibrium of the normal-form game played by the principals, each of whom, has Pi as his pure strategy set and F PM

i

as his payoff function.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Page and Monteiro (2003)

Principal i’s payoff function F PM

i

: P → R is the expected value

  • f π∗:

F PM

i

(C) =

  • T

π∗

i (t, C)dµ(t).

(2) A Page-Monteiro equilibrium is a Nash equilibrium of the normal-form game played by the principals, each of whom, has Pi as his pure strategy set and F PM

i

as his payoff function. A Page-Monteiro equilibrium is (α1, . . . , αm) ∈ ∆(P) such that F PM

i

(α) ≥ F PM

i

(¯ αi, α−i) for all i ∈ I and ¯ αi ∈ ∆(Pi).

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Page and Monteiro (2003)

Principal i’s payoff function F PM

i

: P → R is the expected value

  • f π∗:

F PM

i

(C) =

  • T

π∗

i (t, C)dµ(t).

(2) A Page-Monteiro equilibrium is a Nash equilibrium of the normal-form game played by the principals, each of whom, has Pi as his pure strategy set and F PM

i

as his payoff function. A Page-Monteiro equilibrium is (α1, . . . , αm) ∈ ∆(P) such that F PM

i

(α) ≥ F PM

i

(¯ αi, α−i) for all i ∈ I and ¯ αi ∈ ∆(Pi). Principals’ beliefs are irrational: in general, there is no agent’s strategy that justifies those beliefs.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Monteiro and Page (2005)

Irrationality of beliefs is corrected: agent will choose the best contract from the point of view of those principals that have

  • ffered a contract in her optimal choice correspondence.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Monteiro and Page (2005)

Irrationality of beliefs is corrected: agent will choose the best contract from the point of view of those principals that have

  • ffered a contract in her optimal choice correspondence.

Let for all t ∈ T and C ∈ P H(t, C) = {i ∈ I : there exists f ∈ Ci such that δ(i,f) ∈ Λ(t, C)}, be the set of principals that have offered a contract in her

  • ptimal choice correspondence.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Monteiro and Page (2005)

Irrationality of beliefs is corrected: agent will choose the best contract from the point of view of those principals that have

  • ffered a contract in her optimal choice correspondence.

Let for all t ∈ T and C ∈ P H(t, C) = {i ∈ I : there exists f ∈ Ci such that δ(i,f) ∈ Λ(t, C)}, be the set of principals that have offered a contract in her

  • ptimal choice correspondence.

Principal i’s payoff function: F MP

i

(C) =

  • T

π∗

i (t, C)

|H(t, C)|dµ(t), ∀i ∈ I and C ∈ P. (3)

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Monteiro and Page (2005)

A Monteiro-Page equilibrium is a Nash equilibrium of the normal-form game played by the principals, each of whom, has Pi as his pure strategy set and F MP

i

as his payoff function.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Monteiro and Page (2005)

A Monteiro-Page equilibrium is a Nash equilibrium of the normal-form game played by the principals, each of whom, has Pi as his pure strategy set and F MP

i

as his payoff function. A Monteiro-Page equilibrium is (α1, . . . , αm) ∈ ∆(P) such that F MP

i

(α) ≥ F MP

i

(¯ αi, α−i) for all i ∈ I and ¯ αi ∈ ∆(Pi).

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Monteiro and Page (2005)

A Monteiro-Page equilibrium is a Nash equilibrium of the normal-form game played by the principals, each of whom, has Pi as his pure strategy set and F MP

i

as his payoff function. A Monteiro-Page equilibrium is (α1, . . . , αm) ∈ ∆(P) such that F MP

i

(α) ≥ F MP

i

(¯ αi, α−i) for all i ∈ I and ¯ αi ∈ ∆(Pi). Implicitly, in a Monteiro-Page equilibrium, the agent’s strategy is fixed exogenously. It assumes that the agent uses strategy σMP

σMP {i}×

  • f ∈ Ci : δ(i,f) ∈ Λ(t, C) and πi(t, i, f) = π∗(t, C)
  • t, C
  • =

1 |H(t, C)|,

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Fajardo and Carmona (2006)

Not all Page and Monteiro (2003) is a Sequential Equilibrium and viceversa.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Fajardo and Carmona (2006)

Not all Page and Monteiro (2003) is a Sequential Equilibrium and viceversa. Without the no-fixed cost property: πi(t, j, f) = 0 for all i, j ∈ I, i = j, t ∈ T and f ∈ Ci.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Fajardo and Carmona (2006)

Not all Page and Monteiro (2003) is a Sequential Equilibrium and viceversa. Without the no-fixed cost property: πi(t, j, f) = 0 for all i, j ∈ I, i = j, t ∈ T and f ∈ Ci. Not all Monteiro and Page (2005) is a Sequential Equilibrium and viceversa

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Fajardo and Carmona (2006)

Remark Let G be a menu game with the no-fixed-cost property. Then, F MP

i

(α) = Fi(α; σMP), for all i ∈ I, α ∈ ∆(P) and all Monteiro-Page strategies σMP. If α is a Monteiro-Page equilibrium, then (α, σMP) is a sequential equilibrium strategy for every Monteiro-Page strategy σMP. If Λ(t, C) is singleton for all t ∈ T and C ∈ P and (α, σ) is a sequential equilibrium strategy, then α is a Monteiro-Page equilibrium (equal to Page-Monteiro equilibrium) and σ is a Monteiro-Page strategy.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion Main Results Basic Ideas for Proofs

Existence of Equilibrium

Theorem A sequential equilibrium exists for all continuous games G.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion Main Results Basic Ideas for Proofs

Existence of Equilibrium

Theorem A sequential equilibrium exists for all continuous games G. Corollary All continuous games GH, GPM and GMS have a sequential equilibrium.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion Main Results Basic Ideas for Proofs

Generalization of Simon and Zame (1990)

Theorem A solution exists for all generalized games with an endogenous sharing rule.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Conclusion

A sequential equilibrium exists in all continuous menu games.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Conclusion

A sequential equilibrium exists in all continuous menu games. We dispense the exclusivity and the no-fixed-cost assumptions.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Conclusion

A sequential equilibrium exists in all continuous menu games. We dispense the exclusivity and the no-fixed-cost assumptions. A simpler proof.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Conclusion

A sequential equilibrium exists in all continuous menu games. We dispense the exclusivity and the no-fixed-cost assumptions. A simpler proof. Informed principals

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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Motivation Model Equilibrium Results Conclusion

Conclusion

A sequential equilibrium exists in all continuous menu games. We dispense the exclusivity and the no-fixed-cost assumptions. A simpler proof. Informed principals Other models with endogenous sharing rules.

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Appendix

References I

Carmona, G. and J. Fajardo On the Definition of Equilibrium in Common Agency Games with Adverse Selection mimeo, Universidade Nova de Lisboa. 2006 Martimort, D. Multi-Contracting Mechanism Design. mimeo, IDEI Toulouse. 2006. Forthcoming in Advances in Economics and Econometrics, Theory and Applications, Ninth World Congress. Edited by Richard Blundell, Persson Torsten and Whitney K. Newey, Econometric Society Monographs, Cambridge University

  • Press. Vol 1. Ch. 2. 2006

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References II

Martimort, D. and Stole, L. The Revelation and Delegation Principles in Common Agency Games. Econometrica, 70:4, 1659–1673. 2002. Page, F . and P . K. Monteiro. Three Principles of Competitive Nonlinear Pricing. Journal of Mathematical Economics, 39:63–109, 2003. P . K. Monteiro and Page, F . Existence of Nash Equilibrium in Competitive Nonlinear Pricing Games with Adverse Selection. mimeo, Fundação Getúlio Vargas and University of Alabama 2005.

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References III

Kreps, D. and Wilson, R. Sequential Equilibria. Econometrica, 50: 863–894, 1982. Simon, L. and Zame, W. Discontinuous Games and Endogenous Sharing Rules. Econometrica, 58: 861–872, 1990.

Fajardo, Carmona Existence of Equilibrium in The Common Agency Model