Existence of Equilibrium in The Common Agency Model with Adverse - PowerPoint PPT Presentation

Motivation Model Equilibrium Results Conclusion Existence of Equilibrium in The Common Agency Model with Adverse Selection José Fajardo 1 Guilherme Carmona 2 1 Economics Department IBMEC Business School 2 Economics Department Universidade Nova de Lisboa ASSET - Lisbon, November 2-4, 2006. Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

Motivation Model Equilibrium Results Conclusion Outline Motivation 1 Previous Works Contribution Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

Motivation Model Equilibrium Results Conclusion Outline Motivation 1 Previous Works Contribution Model 2 Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

Motivation Model Equilibrium Results Conclusion Outline Motivation 1 Previous Works Contribution Model 2 Equilibrium 3 Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

Motivation Model Equilibrium Results Conclusion Outline Motivation 1 Previous Works Contribution Model 2 Equilibrium 3 Results 4 Main Results Basic Ideas for Proofs Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

Motivation Model Previous Works Equilibrium Contribution Results Conclusion 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

Motivation Model Previous Works Equilibrium Contribution Results Conclusion Delegation Principle Common agency problem can be analyzed through a menu game Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

Motivation Model Previous Works Equilibrium Contribution Results Conclusion 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

Motivation Model Previous Works Equilibrium Contribution Results Conclusion Related Literature Page and Monteiro, JME. (2003): Monteiro and Page (2005): Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

Motivation Model Previous Works Equilibrium Contribution Results Conclusion 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

Motivation Model Previous Works Equilibrium Contribution Results Conclusion 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

Motivation Model Previous Works Equilibrium Contribution Results Conclusion 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

Motivation Model Previous Works Equilibrium Contribution Results Conclusion Contribution Sequential Equilibrium (Kreps and Wilson, Ecta. 1982.) Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

Motivation Model Previous Works Equilibrium Contribution Results Conclusion 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

Motivation Model Previous Works Equilibrium Contribution Results Conclusion 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

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

Motivation Model Equilibrium Results Conclusion Menu Games Principals Consider a game with m principals and 1 agent. K i compact metric space: Set of contracts that principal i can offer. Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

Motivation Model Equilibrium Results Conclusion Menu Games Principals Consider a game with m principals and 1 agent. K i compact metric space: Set of contracts that principal i can offer. C i ⊂ K i nonempty closed subset: A menu of contracts for principal i ∈ I = { 1 , . . . , m } . Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

Motivation Model Equilibrium Results Conclusion Menu Games Principals Consider a game with m principals and 1 agent. K i compact metric space: Set of contracts that principal i can offer. C i ⊂ K i nonempty closed subset: A menu of contracts for principal i ∈ I = { 1 , . . . , m } . P i collection of all nonempty, closed subsets of K i . ( P i compact metric space w.r.t Hausdorff metric) Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

Motivation Model Equilibrium Results Conclusion Menu Games Principals Consider a game with m principals and 1 agent. K i compact metric space: Set of contracts that principal i can offer. C i ⊂ K i nonempty closed subset: A menu of contracts for principal i ∈ I = { 1 , . . . , m } . P i collection of all nonempty, closed subsets of K i . ( P i compact metric space w.r.t Hausdorff metric) P = P 1 × · · · × P m and C = ( C 1 , . . . , C m ) denote a profile of menus. Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

Motivation Model Equilibrium Results Conclusion Menu Games Principals Consider a game with m principals and 1 agent. K i compact metric space: Set of contracts that principal i can offer. C i ⊂ K i nonempty closed subset: A menu of contracts for principal i ∈ I = { 1 , . . . , m } . P i collection of all nonempty, closed subsets of K i . ( P i compact metric space w.r.t Hausdorff metric) P = P 1 × · · · × P m and C = ( C 1 , . . . , C m ) denote a profile of menus. Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

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

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

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

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

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

Motivation Model Equilibrium Results Conclusion Particular cases Contracts are exclusive: K PM = { ( i , f ) ∈ I × ∪ m i = 1 K i : f ∈ K i } , Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

Motivation Model Equilibrium Results Conclusion Particular cases Contracts are exclusive: K PM = { ( i , f ) ∈ I × ∪ m i = 1 K i : f ∈ K i } , Contracts are not exclusive: K MS = K 1 × · · · × K m . Fajardo, Carmona Existence of Equilibrium in The Common Agency Model

Motivation Model Equilibrium Results Conclusion Particular cases Contracts are exclusive: K PM = { ( i , f ) ∈ I × ∪ m i = 1 K i : f ∈ K i } , Contracts are not exclusive: K MS = K 1 × · · · × K m . I e ⊆ I of principals only allows for exclusive contracts: K H = { ( i , f ) ∈ I e × ∪ i ∈ I e K i : f ∈ K i } × � e K i . i ∈ I c Fajardo, Carmona Existence of Equilibrium in The Common Agency Model