The Role of Commitment in Repeated Games Julio Gonz alez-D az - - PowerPoint PPT Presentation

The Role of Commitment in Repeated Games

Julio Gonz´ alez-D´ ıaz Ignacio Garc´ ıa-Jurado

Department of Statistics and Operations Research School of Mathematics Universidade de Santiago de Compostela

Introduction Outline

Motivation

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

Introduction Outline

Motivation

Commitment

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

Introduction Outline

Motivation

Commitment Repeated games

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

Introduction Outline

Motivation

Commitment Repeated games Unilateral commitments in repeated games

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

Introduction Outline

Motivation

Commitment Repeated games Unilateral commitments in repeated games Delegation games

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

Introduction Outline

Outline

1

Virtually Subgame Perfect Equilibrium Some Examples Formal Definitions Discussion

2

Unilateral Commitments Definitions Delegation Models and Unilateral Commitments Results

3

Conclusions

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Some Examples Formal Definitions Discussion

Outline

1

Virtually Subgame Perfect Equilibrium Some Examples Formal Definitions Discussion

2

Unilateral Commitments Definitions Delegation Models and Unilateral Commitments Results

3

Conclusions

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Some Examples Formal Definitions Discussion

Virtually Subgame Perfect Equilibrium

First Example

U1 D1 U2 D2 U2 D2 1 2 1 2 (1,1) (1,0) (0,1) (2,-2) (-2,2) (-2,2) (2,-2) a1

1

a2

1

a1

2

a2

2

a1

2

a2

2

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Some Examples Formal Definitions Discussion

Virtually Subgame Perfect Equilibrium

First Example

Pure Strategies !!! U1 D1 U2 D2 U2 D2 1 2 1 2 (1,1) (1,0) (0,1) (2,-2) (-2,2) (-2,2) (2,-2) a1

1

a2

1

a1

2

a2

2

a1

2

a2

2

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Some Examples Formal Definitions Discussion

Virtually Subgame Perfect Equilibrium

First Example

Pure Strategies U1 D1 U2 D2 U2 D2 1 2 1 2 (1,1) (1,0) (0,1) (2,-2) (-2,2) (-2,2) (2,-2) a1

1

a2

1

a1

2

a2

2

a1

2

a2

2

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Some Examples Formal Definitions Discussion

Virtually Subgame Perfect Equilibrium

First Example

Pure Strategies U1 D1 U2 D2 U2 D2 1 2 1 2 (1,1) (1,0) (0,1) (2,-2) (-2,2) (-2,2) (2,-2) a1

1

a2

1

a1

2

a2

2

a1

2

a2

2

E1 E2 E2 E2

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Some Examples Formal Definitions Discussion

Virtually Subgame Perfect Equilibrium

First Example

Pure Strategies U1 D1 U2 D2 U2 D2 1 2 1 2 (1,1) (1,0) (0,1) (2,-2) (-2,2) (-2,2) (2,-2) a1

1

a2

1

a1

2

a2

2

a1

2

a2

2

E1 E2 E2 E2 a1

2

a2

2

(-5,-5) (-5,-5) (-5,-5) (-5,-5) (-5,-5)

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Some Examples Formal Definitions Discussion

Virtually Subgame Perfect Equilibrium

First Example

Pure Strategies U1 D1 U2 D2 U2 D2 1 2 1 2 (1,1) (1,0) (0,1) (2,-2) (-2,2) (-2,2) (2,-2) a1

1

a2

1

a1

2

a2

2

a1

2

a2

2

E1 E2 E2 E2 a1

2

a2

2

(-5,-5) (-5,-5) (-5,-5) (-5,-5) (-5,-5)

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Some Examples Formal Definitions Discussion

Virtually Subgame Perfect Equilibrium

Second Example

Pure Strategies 1 2 1 2 (1,1) (1,0) (0,1) (1,-1) (-1,1) (-1,1) (1,-1) U1 D1 U2 D2 U2 D2 a1

1

a2

1

a1

2

a2

2

a1

2

a2

2

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Some Examples Formal Definitions Discussion

Virtually Subgame Perfect Equilibrium

Second Example

Pure Strategies 1 2 1 2 (1,1) (1,0) (0,1) (1,-1) (-1,1) (-1,1) (1,-1) U1 D1 U2 D2 U2 D2 a1

1

a2

1

a1

2

a2

2

a1

2

a2

2

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Some Examples Formal Definitions Discussion

Virtually Subgame Perfect Equilibrium

Second Example

Pure Strategies 1 2 1 2 (1,1) (1,0) (0,1) (1,-1) (-1,1) (-1,1) (1,-1) U1 D1 U2 D2 U2 D2 a1

1

a2

1

a1

2

a2

2

a1

2

a2

2

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Some Examples Formal Definitions Discussion

Virtually Subgame Perfect Equilibrium

Second Example

Pure Strategies 1 2 1 2 (1,1) (1,0) (0,1) (1,-1) (-1,1) (-1,1) (1,-1) U1 D1 U2 D2 U2 D2 a1

1

a2

1

a1

2

a2

2

a1

2

a2

2

Not Subgame Perfect

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Some Examples Formal Definitions Discussion

Virtually Subgame Perfect Equilibrium

Second Example

Pure Strategies 1 2 1 2 (1,1) (1,0) (0,1) (5,5) (5,5) (5,5) (5,5) U1 D1 U2 D2 U2 D2 a1

1

a2

1

a1

2

a2

2

a1

2

a2

2

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Some Examples Formal Definitions Discussion

Virtually Subgame Perfect Equilibrium

Second Example

Pure Strategies 1 2 1 2 (1,1) (1,0) (0,1) (5,5) (5,5) (5,5) (5,5) U1 D1 U2 D2 U2 D2 a1

1

a2

1

a1

2

a2

2

a1

2

a2

2

Subgame Perfect!!!

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Some Examples Formal Definitions Discussion

Virtually Subgame Perfect Equilibrium

Formal Definitions

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Some Examples Formal Definitions Discussion

Virtually Subgame Perfect Equilibrium

Formal Definitions

Let Γ be an extensive-form game and let x and σ be a single-node information set and a strategy profile, respectively. Then, Γx denotes the subgame of Γ that begins at node x and σx the restriction of σ to Γx.

Example

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Some Examples Formal Definitions Discussion

Virtually Subgame Perfect Equilibrium

Formal Definitions

Let Γ be an extensive-form game and let x and σ be a single-node information set and a strategy profile, respectively. Then, Γx denotes the subgame of Γ that begins at node x and σx the restriction of σ to Γx.

Example

Now, let Γ be an extensive-form game, σ a strategy profile of Γ, and x a single-node information set. Then, the subgame Γx is σ-relevant if either (i) Γx = Γ, or (ii) there are a player i, a strategy σ′

i, and a single-node information set y such that Γy

is σ-relevant and node x is reached by (σ−i, σ′

i)y.

Example

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Some Examples Formal Definitions Discussion

Virtually Subgame Perfect Equilibrium

Formal Definitions

Let Γ be an extensive-form game and let x and σ be a single-node information set and a strategy profile, respectively. Then, Γx denotes the subgame of Γ that begins at node x and σx the restriction of σ to Γx.

Example

Now, let Γ be an extensive-form game, σ a strategy profile of Γ, and x a single-node information set. Then, the subgame Γx is σ-relevant if either (i) Γx = Γ, or (ii) there are a player i, a strategy σ′

i, and a single-node information set y such that Γy

is σ-relevant and node x is reached by (σ−i, σ′

i)y.

Example

Let Γ be an extensive-form game. The strategy profile σ is a virtually subgame perfect equilibrium of Γ if for each σ-relevant subgame Γx, then σx is a Nash equilibrium of Γx.

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Some Examples Formal Definitions Discussion

Virtually Subgame Perfect Equilibrium

Discussion

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Some Examples Formal Definitions Discussion

Virtually Subgame Perfect Equilibrium

Discussion

Subgame Perfect Vs Virtually Subgame Perfect

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Some Examples Formal Definitions Discussion

Virtually Subgame Perfect Equilibrium

Discussion

Subgame Perfect Vs Virtually Subgame Perfect

Why do we need VSPE?

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Some Examples Formal Definitions Discussion

Virtually Subgame Perfect Equilibrium

Discussion

Subgame Perfect Vs Virtually Subgame Perfect

Why do we need VSPE?

In our model, we face very large trees

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Some Examples Formal Definitions Discussion

Virtually Subgame Perfect Equilibrium

Discussion

Subgame Perfect Vs Virtually Subgame Perfect

Why do we need VSPE?

In our model, we face very large trees There can be subgames with no Nash Equilibrium

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Some Examples Formal Definitions Discussion

Virtually Subgame Perfect Equilibrium

Discussion

Subgame Perfect Vs Virtually Subgame Perfect

Why do we need VSPE?

In our model, we face very large trees There can be subgames with no Nash Equilibrium Hence,

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Some Examples Formal Definitions Discussion

Virtually Subgame Perfect Equilibrium

Discussion

Subgame Perfect Vs Virtually Subgame Perfect

Why do we need VSPE?

In our model, we face very large trees There can be subgames with no Nash Equilibrium Hence, We cannot use the classic results for the existence of SPE

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Outline

1

Virtually Subgame Perfect Equilibrium Some Examples Formal Definitions Discussion

2

Unilateral Commitments Definitions Delegation Models and Unilateral Commitments Results

3

Conclusions

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Unilateral Commitments

Definitions

The stage game:

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Unilateral Commitments

Definitions

The stage game: G := (N, A, ϕ)   

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Unilateral Commitments

Definitions

The stage game: G := (N, A, ϕ)    N := {1, . . . , n}

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Unilateral Commitments

Definitions

The stage game: G := (N, A, ϕ)    N := {1, . . . , n} A :=

i∈N Ai

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Unilateral Commitments

Definitions

The stage game: G := (N, A, ϕ)    N := {1, . . . , n} A :=

i∈N Ai

ϕ := (ϕ1, . . . , ϕn)

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Unilateral Commitments

Definitions

The stage game: G := (N, A, ϕ)    N := {1, . . . , n} A :=

i∈N Ai

ϕ := (ϕ1, . . . , ϕn) The repeated game: GT

δ := (N, S, ϕδ)

      

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Unilateral Commitments

Definitions

The stage game: G := (N, A, ϕ)    N := {1, . . . , n} A :=

i∈N Ai

ϕ := (ϕ1, . . . , ϕn) The repeated game: GT

δ := (N, S, ϕδ)

       N := {1, . . . , n} S :=

i∈N Si

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Unilateral Commitments

Definitions

The stage game: G := (N, A, ϕ)    N := {1, . . . , n} A :=

i∈N Ai

ϕ := (ϕ1, . . . , ϕn) The repeated game: GT

δ := (N, S, ϕδ)

       N := {1, . . . , n} S :=

i∈N Si

(Si := AH

i )

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Unilateral Commitments

Definitions

The stage game: G := (N, A, ϕ)    N := {1, . . . , n} A :=

i∈N Ai

ϕ := (ϕ1, . . . , ϕn) The repeated game: GT

δ := (N, S, ϕδ)

       N := {1, . . . , n} S :=

i∈N Si

(Si := AH

i )

ϕδ

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Unilateral Commitments

Definitions

The stage game: G := (N, A, ϕ)    N := {1, . . . , n} A :=

i∈N Ai

ϕ := (ϕ1, . . . , ϕn) The repeated game: GT

δ := (N, S, ϕδ)

       N := {1, . . . , n} S :=

i∈N Si

(Si := AH

i )

ϕδ The UC-extension: U(G) := (N, AU, ϕU)

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Unilateral Commitments

Definitions

The stage game: G := (N, A, ϕ)    N := {1, . . . , n} A :=

i∈N Ai

ϕ := (ϕ1, . . . , ϕn) The repeated game: GT

δ := (N, S, ϕδ)

       N := {1, . . . , n} S :=

i∈N Si

(Si := AH

i )

ϕδ The UC-extension: U(G) := (N, AU, ϕU) AU :=

i∈N AU i , where AU i is the set of all couples (Ac i, αi)

such that

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Unilateral Commitments

Definitions

The stage game: G := (N, A, ϕ)    N := {1, . . . , n} A :=

i∈N Ai

ϕ := (ϕ1, . . . , ϕn) The repeated game: GT

δ := (N, S, ϕδ)

       N := {1, . . . , n} S :=

i∈N Si

(Si := AH

i )

ϕδ The UC-extension: U(G) := (N, AU, ϕU) AU :=

i∈N AU i , where AU i is the set of all couples (Ac i, αi)

such that

1

∅ Ac

i ⊆ Ai,

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Unilateral Commitments

Definitions

The stage game: G := (N, A, ϕ)    N := {1, . . . , n} A :=

i∈N Ai

ϕ := (ϕ1, . . . , ϕn) The repeated game: GT

δ := (N, S, ϕδ)

       N := {1, . . . , n} S :=

i∈N Si

(Si := AH

i )

ϕδ The UC-extension: U(G) := (N, AU, ϕU) AU :=

i∈N AU i , where AU i is the set of all couples (Ac i, αi)

such that

1

∅ Ac

i ⊆ Ai,

2

αi :

j∈N 2Aj −

→ Ai

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Unilateral Commitments

Definitions

The stage game: G := (N, A, ϕ)    N := {1, . . . , n} A :=

i∈N Ai

ϕ := (ϕ1, . . . , ϕn) The repeated game: GT

δ := (N, S, ϕδ)

       N := {1, . . . , n} S :=

i∈N Si

(Si := AH

i )

ϕδ The UC-extension: U(G) := (N, AU, ϕU) AU :=

i∈N AU i , where AU i is the set of all couples (Ac i, αi)

such that

1

∅ Ac

i ⊆ Ai,

2

αi :

j∈N 2Aj −

→ Ai and, for each Ac ∈

j∈N 2Aj,

αi(Ac) ∈ Ac

i.

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Unilateral Commitments

Definitions

The stage game: G := (N, A, ϕ)    N := {1, . . . , n} A :=

i∈N Ai

ϕ := (ϕ1, . . . , ϕn) The repeated game: GT

δ := (N, S, ϕδ)

       N := {1, . . . , n} S :=

i∈N Si

(Si := AH

i )

ϕδ The UC-extension: U(G) := (N, AU, ϕU) AU :=

i∈N AU i , where AU i is the set of all couples (Ac i, αi)

such that

1

∅ Ac

i ⊆ Ai,

2

αi :

j∈N 2Aj −

→ Ai and, for each Ac ∈

j∈N 2Aj,

αi(Ac) ∈ Ac

i.

Commitments are Unilateral

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Unilateral Commitments

Definitions

The stage game: G := (N, A, ϕ)    N := {1, . . . , n} A :=

i∈N Ai

ϕ := (ϕ1, . . . , ϕn) The repeated game: GT

δ := (N, S, ϕδ)

       N := {1, . . . , n} S :=

i∈N Si

(Si := AH

i )

ϕδ The UC-extension: U(G) := (N, AU, ϕU) AU :=

i∈N AU i , where AU i is the set of all couples (Ac i, αi)

such that

1

∅ Ac

i ⊆ Ai,

2

αi :

j∈N 2Aj −

→ Ai and, for each Ac ∈

j∈N 2Aj,

αi(Ac) ∈ Ac

i.

Commitments are Unilateral Complete Information

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Delegation Models and Unilateral Commitments

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Delegation Models and Unilateral Commitments

Fershtman et al (1991)

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Delegation Models and Unilateral Commitments

Fershtman et al (1991) Players: 2 principals, 2 agents

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Delegation Models and Unilateral Commitments

Fershtman et al (1991) Players: 2 principals, 2 agents Compensation Monotonic Function Schemes:

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Delegation Models and Unilateral Commitments

Fershtman et al (1991) Players: 2 principals, 2 agents Compensation Monotonic Function Schemes:

depend on the payoffs

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Delegation Models and Unilateral Commitments

Fershtman et al (1991) Players: 2 principals, 2 agents Compensation Monotonic Function Schemes:

depend on the payoffs

Contracts: Public

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Delegation Models and Unilateral Commitments

Fershtman et al (1991) Our Model Players: 2 principals, 2 agents Compensation Monotonic Function Schemes:

depend on the payoffs

Contracts: Public

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Delegation Models and Unilateral Commitments

Fershtman et al (1991) Our Model Players: 2 principals, 2 agents 2 principals, 2 agents Compensation Monotonic Function Schemes:

depend on the payoffs

Contracts: Public

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Delegation Models and Unilateral Commitments

Fershtman et al (1991) Our Model Players: 2 principals, 2 agents 2 principals, 2 agents Compensation Monotonic Function Proportional within Ac Schemes:

depend on the payoffs

Contracts: Public

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Delegation Models and Unilateral Commitments

Fershtman et al (1991) Our Model Players: 2 principals, 2 agents 2 principals, 2 agents Compensation Monotonic Function Proportional within Ac Schemes:

depend on the payoffs depend on the strategies

Contracts: Public

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Delegation Models and Unilateral Commitments

Fershtman et al (1991) Our Model Players: 2 principals, 2 agents 2 principals, 2 agents Compensation Monotonic Function Proportional within Ac Schemes:

depend on the payoffs depend on the strategies

Contracts: Public Public

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Delegation Models and Unilateral Commitments

Fershtman et al (1991) Our Model Players: 2 principals, 2 agents 2 principals, 2 agents Compensation Monotonic Function Proportional within Ac Schemes:

depend on the payoffs depend on the strategies

Contracts: Public Public Complete Information!!!

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Delegation Models and Unilateral Commitments

Fershtman et al (1991) Our Model Players: 2 principals, 2 agents 2 principals, 2 agents Compensation Monotonic Function Proportional within Ac Schemes:

depend on the payoffs depend on the strategies

Contracts: Public Public

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Unilateral Commitments

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Unilateral Commitments

Objectives

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Unilateral Commitments

Objectives Results:

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Unilateral Commitments

Objectives Results: The Folk Theorems

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Unilateral Commitments

Objectives Results: The Folk Theorems

Minmax Payoffs:

vi = min

a−i∈A−i max ai∈Ai ϕi(ai, a−i)

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Unilateral Commitments

Objectives Results: The Folk Theorems

Minmax Payoffs:

vi = min

a−i∈A−i max ai∈Ai ϕi(ai, a−i)

Feasible and Individually Rational Payoffs:

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Unilateral Commitments

Objectives Results: The Folk Theorems

Minmax Payoffs:

vi = min

a−i∈A−i max ai∈Ai ϕi(ai, a−i)

Feasible and Individually Rational Payoffs:

F := co{ϕ(a) : a ∈ ϕ(A)}

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

Unilateral Commitments

Objectives Results: The Folk Theorems

Minmax Payoffs:

vi = min

a−i∈A−i max ai∈Ai ϕi(ai, a−i)

Feasible and Individually Rational Payoffs:

F := co{ϕ(a) : a ∈ ϕ(A)} ¯ F := F ∩ {u ∈ Rn : u ≥ v}

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

The Folk Theorems

Finite Horizon

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

The Folk Theorems

Finite Horizon

Nash Folk Theorem (without UC)

G must have a Nash equilibrium in which some player gets more than his minmax payoff

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

The Folk Theorems

Finite Horizon

Nash Folk Theorem (without UC)

G must have a Nash equilibrium in which some player gets more than his minmax payoff

Theorem 1 (Garc´ ıa-Jurado et al., 2000)

No assumption is needed for the Nash folk theorem with UC.

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

The Folk Theorems

Finite Horizon

Nash Folk Theorem (without UC)

G must have a Nash equilibrium in which some player gets more than his minmax payoff

Theorem 1 (Garc´ ıa-Jurado et al., 2000)

No assumption is needed for the Nash folk theorem with UC.

Proof

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

The Folk Theorems

Finite Horizon

Nash Folk Theorem (without UC)

G must have a Nash equilibrium in which some player gets more than his minmax payoff

Theorem 1 (Garc´ ıa-Jurado et al., 2000)

No assumption is needed for the Nash folk theorem with UC.

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

The Folk Theorems

Finite Horizon

Nash Folk Theorem (without UC)

G must have a Nash equilibrium in which some player gets more than his minmax payoff

Theorem 1 (Garc´ ıa-Jurado et al., 2000)

No assumption is needed for the Nash folk theorem with UC. Moreover, the Nash equilibrium of the repeated game with UC can be chosen such that

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

The Folk Theorems

Finite Horizon

Nash Folk Theorem (without UC)

G must have a Nash equilibrium in which some player gets more than his minmax payoff

Theorem 1 (Garc´ ıa-Jurado et al., 2000)

No assumption is needed for the Nash folk theorem with UC. Moreover, the Nash equilibrium of the repeated game with UC can be chosen such that the subgame that begins after the commitments has a unique Nash payoff

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

The Folk Theorems

Finite Horizon

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

The Folk Theorems

Finite Horizon

Subgame Perfect Folk Theorem (without UC)

G must have a pair of Nash equilibra in which some player gets different payoffs

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

The Folk Theorems

Finite Horizon

Subgame Perfect Folk Theorem (without UC)

G must have a pair of Nash equilibra in which some player gets different payoffs

Proposition 1

The counterpart of Theorem 1 for VSPE does not hold.

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

The Folk Theorems

Finite Horizon

Subgame Perfect Folk Theorem (without UC)

G must have a pair of Nash equilibra in which some player gets different payoffs

Proposition 1

The counterpart of Theorem 1 for VSPE does not hold.

Proposition 2

Let ¯ a ∈ A be a Nash equilibrium of G. Then, the game U(G) has a VSPE with payoff ϕ(¯ a).

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

The Folk Theorems

Finite Horizon

Subgame Perfect Folk Theorem (without UC)

G must have a pair of Nash equilibra in which some player gets different payoffs

Proposition 1

The counterpart of Theorem 1 for VSPE does not hold.

Proposition 2

Let ¯ a ∈ A be a Nash equilibrium of G. Then, the game U(G) has a VSPE with payoff ϕ(¯ a).

Theorem 2

No assumption is needed for the VSPE folk theorem when we have two stages of commitments.

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

The Folk Theorems

Finite Horizon

Theorem 1 No assumptions for the Nash folk theorem with UC. Proposition 2 Let ¯ a ∈ A be a Nash equilibrium of G. Then, the game U(G) has a VSPE with payoff ϕ(¯ a).

Theorem 2

No assumption is needed for the VSPE folk theorem when we have two stages of commitments.

Proof

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

The Folk Theorems

Finite Horizon

Theorem 1 No assumptions for the Nash folk theorem with UC. Proposition 2 Let ¯ a ∈ A be a Nash equilibrium of G. Then, the game U(G) has a VSPE with payoff ϕ(¯ a).

Theorem 2

No assumption is needed for the VSPE folk theorem when we have two stages of commitments.

Proof

1 Apply Theorem 1 to G(δ, T) Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

The Folk Theorems

Finite Horizon

Theorem 1 No assumptions for the Nash folk theorem with UC. Proposition 2 Let ¯ a ∈ A be a Nash equilibrium of G. Then, the game U(G) has a VSPE with payoff ϕ(¯ a).

Theorem 2

No assumption is needed for the VSPE folk theorem when we have two stages of commitments.

Proof

1 Apply Theorem 1 to G(δ, T)

= ⇒ U(G(δ, T)) has a Nash

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

The Folk Theorems

Finite Horizon

Theorem 1 No assumptions for the Nash folk theorem with UC. Proposition 2 Let ¯ a ∈ A be a Nash equilibrium of G. Then, the game U(G) has a VSPE with payoff ϕ(¯ a).

Theorem 2

No assumption is needed for the VSPE folk theorem when we have two stages of commitments.

Proof

1 Apply Theorem 1 to G(δ, T)

= ⇒ U(G(δ, T)) has a Nash

2 Apply Proposition 2 to U(G(δ, T)) Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

The Folk Theorems

Finite Horizon

Theorem 1 No assumptions for the Nash folk theorem with UC. Proposition 2 Let ¯ a ∈ A be a Nash equilibrium of G. Then, the game U(G) has a VSPE with payoff ϕ(¯ a).

Theorem 2

No assumption is needed for the VSPE folk theorem when we have two stages of commitments.

Proof

1 Apply Theorem 1 to G(δ, T)

= ⇒ U(G(δ, T)) has a Nash

2 Apply Proposition 2 to U(G(δ, T))

= ⇒ U(U(G(δ, T))) has a VSPE

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

The Folk Theorems

Finite Horizon

Theorem 1 No assumptions for the Nash folk theorem with UC. Proposition 2 Let ¯ a ∈ A be a Nash equilibrium of G. Then, the game U(G) has a VSPE with payoff ϕ(¯ a).

Theorem 2

No assumption is needed for the VSPE folk theorem when we have two stages of commitments.

Proof

1 Apply Theorem 1 to G(δ, T)

= ⇒ U(G(δ, T)) has a Nash

2 Apply Proposition 2 to U(G(δ, T))

= ⇒ U(U(G(δ, T))) has a VSPE Moreover, the VSPE can be chosen such that

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

The Folk Theorems

Finite Horizon

Theorem 1 No assumptions for the Nash folk theorem with UC. Proposition 2 Let ¯ a ∈ A be a Nash equilibrium of G. Then, the game U(G) has a VSPE with payoff ϕ(¯ a).

Theorem 2

No assumption is needed for the VSPE folk theorem when we have two stages of commitments.

Proof

1 Apply Theorem 1 to G(δ, T)

= ⇒ U(G(δ, T)) has a Nash

2 Apply Proposition 2 to U(G(δ, T))

= ⇒ U(U(G(δ, T))) has a VSPE Moreover, the VSPE can be chosen such that the subgame that begins after the first stage of commitments has a unique Nash payoff

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

The Folk Theorems

Discussion

Theorem 2

No assumption is needed for the VSPE folk theorem when we have two stages of commitments.

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

The Folk Theorems

Discussion

Theorem 2

No assumption is needed for the VSPE folk theorem when we have two stages of commitments.

Remarks

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

The Folk Theorems

Discussion

Theorem 2

No assumption is needed for the VSPE folk theorem when we have two stages of commitments.

Remarks

Are two stages of commitments natural??

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

The Folk Theorems

Discussion

Theorem 2

No assumption is needed for the VSPE folk theorem when we have two stages of commitments.

Remarks

Are two stages of commitments natural?? We “allow for” commitments on commitments

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions Definitions Delegation Models and Unilateral Commitments Results

The Folk Theorems

Discussion

Theorem 2

No assumption is needed for the VSPE folk theorem when we have two stages of commitments.

Remarks

Are two stages of commitments natural?? We “allow for” commitments on commitments President − → Manager − → Director

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

The State of Art

Necessary and Sufficient Conditions for the Folk Theorems

Without UC 1 stage of UC 2 stages

• f UC

Nash Theorem None Infinite Horizon

(Fudenberg and Maskin, 1986)

(Virtual) Perfect Th. Non-Equivalent Utilities Infinite Horizon

(Abreu et al., 1994)

Nash Theorem Minimax-Bettering Ladder Finite Horizon

(Gonz´ alez-D´ ıaz, 2003)

(Virtual) Perfect Th. Recursively-distinct Finite Horizon Nash payoffs (Smith, 1995)

The State of Art

Necessary and Sufficient Conditions for the Folk Theorems

Without UC 1 stage of UC 2 stages

• f UC

Nash Theorem None None Infinite Horizon

(Fudenberg and Maskin, 1986) (Prop. 2)

(Virtual) Perfect Th. Non-Equivalent Utilities None Infinite Horizon

(Abreu et al., 1994) (Prop. 2)

Nash Theorem Minimax-Bettering Ladder Finite Horizon

(Gonz´ alez-D´ ıaz, 2003)

(Virtual) Perfect Th. Recursively-distinct Finite Horizon Nash payoffs (Smith, 1995)

The State of Art

Necessary and Sufficient Conditions for the Folk Theorems

Without UC 1 stage of UC 2 stages

• f UC

Nash Theorem None None Infinite Horizon

(Fudenberg and Maskin, 1986) (Prop. 2)

(Virtual) Perfect Th. Non-Equivalent Utilities None Infinite Horizon

(Abreu et al., 1994) (Prop. 2)

Nash Theorem Minimax-Bettering Ladder None Finite Horizon

(Gonz´ alez-D´ ıaz, 2003) (Garc´ ıa-Jurado et al., 2000)

(Virtual) Perfect Th. Recursively-distinct Finite Horizon Nash payoffs (Smith, 1995)

The State of Art

Necessary and Sufficient Conditions for the Folk Theorems

Without UC 1 stage of UC 2 stages

• f UC

Nash Theorem None None Infinite Horizon

(Fudenberg and Maskin, 1986) (Prop. 2)

(Virtual) Perfect Th. Non-Equivalent Utilities None Infinite Horizon

(Abreu et al., 1994) (Prop. 2)

Nash Theorem Minimax-Bettering Ladder None Finite Horizon

(Gonz´ alez-D´ ıaz, 2003) (Garc´ ıa-Jurado et al., 2000)

(Virtual) Perfect Th. Recursively-distinct Minimax-Bettering Ladder Finite Horizon Nash payoffs (Smith, 1995)

(Prop. 2, only sufficient)

The State of Art

Necessary and Sufficient Conditions for the Folk Theorems

Without UC 1 stage of UC 2 stages

• f UC

Nash Theorem None None None Infinite Horizon

(Fudenberg and Maskin, 1986) (Prop. 2) (Prop. 2)

(Virtual) Perfect Th. Non-Equivalent Utilities None None Infinite Horizon

(Abreu et al., 1994) (Prop. 2) (Prop. 2)

Nash Theorem Minimax-Bettering Ladder None None Finite Horizon

(Gonz´ alez-D´ ıaz, 2003) (Garc´ ıa-Jurado et al., 2000) (Prop. 2)

(Virtual) Perfect Th. Recursively-distinct Minimax-Bettering Ladder None Finite Horizon Nash payoffs (Smith, 1995)

(Prop. 2, only sufficient) (Th. 2)

VSPE Unilateral Commitments Conclusions

Outline

1

Virtually Subgame Perfect Equilibrium Some Examples Formal Definitions Discussion

2

Unilateral Commitments Definitions Delegation Models and Unilateral Commitments Results

3

Conclusions

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions

Conclusions

Our contribution

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions

Conclusions

Our contribution

UC lead to weaker assumptions for the folk theorems.

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions

Conclusions

Our contribution

UC lead to weaker assumptions for the folk theorems. Nonetheless, some assumptions are still needed for some VSPE folk theorems.

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

VSPE Unilateral Commitments Conclusions

Thanks

Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

1 2 1 2 (1,1) (1,0) (0,1) (1,-1) (-1,1) (-1,1) (1,-1) U1 D1 U2 D2 U2 D2 a1

1

a2

1

a1

2

a2

2

a1

2

a2

2

Return

1 2 1 2 (1,1) (1,0) (0,1) (1,-1) (-1,1) (-1,1) (1,-1) U1 D1 U2 D2 U2 D2 a1

1

a2

1

a1

2

a2

2

a1

2

a2

2

Return

1 2 1 2 (1,1) (1,0) (0,1) (1,-1) (-1,1) (-1,1) (1,-1) U1 D1 U2 D2 U2 D2 a1

1

a2

1

a1

2

a2

2

a1

2

a2

2

Return

Return

1 2 1 2 2 1

Return

1 2 1 2 2 1 σ

Return

1 2 1 2 2 1 σ

Return

1 2 1 2 2 1 σ

Return

1 2 1 2 2 1 σ

Return

Return

Proof

Return

Proof

Let u ∈ ¯ F and let ¯ a ∈ A be such that ϕ(¯ a) = u

Return

Proof

Let u ∈ ¯ F and let ¯ a ∈ A be such that ϕ(¯ a) = u Strategy for a player i

Return

Proof

Let u ∈ ¯ F and let ¯ a ∈ A be such that ϕ(¯ a) = u Strategy for a player i

1 Commitment: ¯

Sc

i := “If ¯

a is played in the first stage, then I play ¯ ai forever”

Return

Proof

Let u ∈ ¯ F and let ¯ a ∈ A be such that ϕ(¯ a) = u Strategy for a player i

1 Commitment: ¯

Sc

i := “If ¯

a is played in the first stage, then I play ¯ ai forever”

2 Strategy:

Return

Proof

Let u ∈ ¯ F and let ¯ a ∈ A be such that ϕ(¯ a) = u Strategy for a player i

1 Commitment: ¯

Sc

i := “If ¯

a is played in the first stage, then I play ¯ ai forever”

2 Strategy:

If Sc = ¯ Sc: Return

Proof

Let u ∈ ¯ F and let ¯ a ∈ A be such that ϕ(¯ a) = u Strategy for a player i

1 Commitment: ¯

Sc

i := “If ¯

a is played in the first stage, then I play ¯ ai forever”

2 Strategy:

If Sc = ¯ Sc:

i plays ¯ ai in the first stage

Return

Proof

Let u ∈ ¯ F and let ¯ a ∈ A be such that ϕ(¯ a) = u Strategy for a player i

1 Commitment: ¯

Sc

i := “If ¯

a is played in the first stage, then I play ¯ ai forever”

2 Strategy:

If Sc = ¯ Sc:

i plays ¯ ai in the first stage If someone deviates i punishes him forever

Return

Proof

Let u ∈ ¯ F and let ¯ a ∈ A be such that ϕ(¯ a) = u Strategy for a player i

1 Commitment: ¯

Sc

i := “If ¯

a is played in the first stage, then I play ¯ ai forever”

2 Strategy:

If Sc = ¯ Sc:

i plays ¯ ai in the first stage If someone deviates i punishes him forever

If someone has deviated from the commitment i punishes him forever Return

Return