the role of commitment in repeated games - usceio.usc.es/pub/julio/slides/commitments.pdf · the...

The Role of Commitment in Repeated Games

Julio Gonzalez-Dıaz Ignacio Garcıa-Jurado

Department of Statistics and Operations ResearchSchool of Mathematics

Universidade de Santiago de Compostela

IntroductionOutline

Motivation

Julio Gonzalez-Dıaz, Ignacio Garcıa-Jurado The Role of Commitment in Repeated Games

IntroductionOutline

Motivation

Commitment


IntroductionOutline

Motivation

Commitment

Repeated games


IntroductionOutline

Motivation

Commitment

Repeated games

Unilateral commitments in repeated games


IntroductionOutline

Motivation

Commitment

Repeated games

Unilateral commitments in repeated games

Delegation games


IntroductionOutline

Outline

1 Virtually Subgame Perfect EquilibriumSome ExamplesFormal DefinitionsDiscussion

2 Unilateral CommitmentsDefinitionsDelegation Models and Unilateral CommitmentsResults

3 Conclusions


VSPEUnilateral Commitments

Conclusions

Some ExamplesFormal DefinitionsDiscussion

Outline



3 Conclusions



Conclusions


Virtually Subgame Perfect EquilibriumFirst 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)

a11

a21

a12

a22

a12

a22



Conclusions



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)

a11

a21

a12

a22

a12

a22



Conclusions



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)

a11

a21

a12

a22

a12

a22



Conclusions



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)

a11

a21

a12

a22

a12

a22

E1

E2

E2

E2



Conclusions



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)

a11

a21

a12

a22

a12

a22

E1

E2

E2

E2

a12

a22

(-5,-5)

(-5,-5)

(-5,-5)

(-5,-5)

(-5,-5)



Conclusions


Virtually Subgame Perfect EquilibriumSecond 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

a11

a21

a12

a22

a12

a22



Conclusions



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

a11

a21

a12

a22

a12

a22

Not Subgame Perfect



Conclusions



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

a11

a21

a12

a22

a12

a22



Conclusions



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

a11

a21

a12

a22

a12

a22

Subgame Perfect!!!



Conclusions


Virtually Subgame Perfect EquilibriumFormal Definitions



Conclusions



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



Conclusions




Now, let Γ be an extensive-form game, σ a strategy profile ofΓ, 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, astrategy 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



Conclusions




Now, let Γ be an extensive-form game, σ a strategy profile ofΓ, 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, astrategy 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 avirtually subgame perfect equilibrium of Γ if for eachσ-relevant subgame Γx, then σx is a Nash equilibrium of Γx.



Conclusions


Virtually Subgame Perfect EquilibriumDiscussion



Conclusions



Subgame Perfect Vs Virtually Subgame Perfect



Conclusions




Why do we need VSPE?



Conclusions





In our model, we face very large trees



Conclusions






There can be subgames with no Nash Equilibrium



Conclusions







Hence,



Conclusions







Hence,

We cannot use the classic results for the existence of SPE



Conclusions

DefinitionsDelegation Models and Unilateral CommitmentsResults

Outline



3 Conclusions



Conclusions


Unilateral CommitmentsDefinitions

The stage game:



Conclusions



The stage game: G := (N, A, ϕ)



Conclusions




N := {1, . . . , n}



Conclusions




N := {1, . . . , n}A :=

∏

i∈N Ai



Conclusions




N := {1, . . . , n}A :=

∏

i∈N Ai

ϕ := (ϕ1, . . . , ϕn)



Conclusions




N := {1, . . . , n}A :=

∏

i∈N Ai

ϕ := (ϕ1, . . . , ϕn)

The repeated game: GTδ := (N, S, ϕδ)



Conclusions




N := {1, . . . , n}A :=

∏

i∈N Ai

ϕ := (ϕ1, . . . , ϕn)


N := {1, . . . , n}S :=

∏

i∈N Si



Conclusions




N := {1, . . . , n}A :=

∏

i∈N Ai

ϕ := (ϕ1, . . . , ϕn)


N := {1, . . . , n}S :=

∏

i∈N Si

(Si := AHi )



Conclusions




N := {1, . . . , n}A :=

∏

i∈N Ai

ϕ := (ϕ1, . . . , ϕn)


N := {1, . . . , n}S :=

∏

i∈N Si

(Si := AHi )

ϕδ



Conclusions




N := {1, . . . , n}A :=

∏

i∈N Ai

ϕ := (ϕ1, . . . , ϕn)


N := {1, . . . , n}S :=

∏

i∈N Si

(Si := AHi )

ϕδ

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



Conclusions




N := {1, . . . , n}A :=

∏

i∈N Ai

ϕ := (ϕ1, . . . , ϕn)


N := {1, . . . , n}S :=

∏

i∈N Si

(Si := AHi )

ϕδ


AU :=∏

i∈N AUi , where AU

i is the set of all couples (Aci , αi)

such that



Conclusions




N := {1, . . . , n}A :=

∏

i∈N Ai

ϕ := (ϕ1, . . . , ϕn)


N := {1, . . . , n}S :=

∏

i∈N Si

(Si := AHi )

ϕδ


AU :=∏



such that1 ∅ ( Ac

i ⊆ Ai,



Conclusions




N := {1, . . . , n}A :=

∏

i∈N Ai

ϕ := (ϕ1, . . . , ϕn)


N := {1, . . . , n}S :=

∏

i∈N Si

(Si := AHi )

ϕδ


AU :=∏



such that1 ∅ ( Ac

i ⊆ Ai,2 αi :

∏

j∈N 2Aj −→ Ai



Conclusions




N := {1, . . . , n}A :=

∏

i∈N Ai

ϕ := (ϕ1, . . . , ϕn)


N := {1, . . . , n}S :=

∏

i∈N Si

(Si := AHi )

ϕδ


AU :=∏



such that1 ∅ ( Ac

i ⊆ Ai,2 αi :

∏

j∈N 2Aj −→ Ai and, for each Ac ∈∏

j∈N 2Aj ,αi(A

c) ∈ Aci .



Conclusions




N := {1, . . . , n}A :=

∏

i∈N Ai

ϕ := (ϕ1, . . . , ϕn)


N := {1, . . . , n}S :=

∏

i∈N Si

(Si := AHi )

ϕδ


AU :=∏



such that1 ∅ ( Ac

i ⊆ Ai,2 αi :

∏


j∈N 2Aj ,αi(A

c) ∈ Aci .

Commitments are UnilateralJulio Gonzalez-Dıaz, Ignacio Garcıa-Jurado The Role of Commitment in Repeated Games


Conclusions




N := {1, . . . , n}A :=

∏

i∈N Ai

ϕ := (ϕ1, . . . , ϕn)


N := {1, . . . , n}S :=

∏

i∈N Si

(Si := AHi )

ϕδ


AU :=∏



such that1 ∅ ( Ac

i ⊆ Ai,2 αi :

∏


j∈N 2Aj ,αi(A

c) ∈ Aci .

Commitments are Unilateral Complete Information



Conclusions


Delegation Models and Unilateral Commitments



Conclusions



Fershtman et al (1991)



Conclusions




Players: 2 principals, 2 agents



Conclusions





Compensation Monotonic FunctionSchemes:



Conclusions





Compensation Monotonic FunctionSchemes: depend on the payoffs



Conclusions






Contracts: Public



Conclusions



Fershtman et al (1991) Our Model



Contracts: Public



Conclusions




Players: 2 principals, 2 agents 2 principals, 2 agents


Contracts: Public



Conclusions





Compensation Monotonic Function Proportional within Ac

Schemes: depend on the payoffs

Contracts: Public



Conclusions






Schemes: depend on the payoffs depend on the strategies

Contracts: Public



Conclusions







Contracts: Public Public



Conclusions







Contracts: Public PublicComplete Information!!!



Conclusions







Contracts: Public Public



Conclusions


Unilateral Commitments



Conclusions



Objectives



Conclusions



Objectives

Results:



Conclusions



Objectives

Results:

The Folk Theorems



Conclusions



Objectives

Results:

The Folk Theorems

Minmax Payoffs:

vi = mina−i∈A−i

maxai∈Ai

ϕi(ai, a−i)



Conclusions



Objectives

Results:

The Folk Theorems

Minmax Payoffs:


maxai∈Ai

ϕi(ai, a−i)

Feasible and Individually Rational Payoffs:



Conclusions



Objectives

Results:

The Folk Theorems

Minmax Payoffs:


maxai∈Ai

ϕi(ai, a−i)


F := co{ϕ(a) : a ∈ ϕ(A)}



Conclusions



Objectives

Results:

The Folk Theorems

Minmax Payoffs:


maxai∈Ai

ϕi(ai, a−i)


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



Conclusions


The Folk TheoremsFinite Horizon



Conclusions



Nash Folk Theorem (without UC)

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



Conclusions





Theorem 1 (Garcıa-Jurado et al., 2000)

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



Conclusions







Proof



Conclusions









Conclusions







Moreover, the Nash equilibrium of the repeated game with UCcan be chosen such that



Conclusions







Moreover, the Nash equilibrium of the repeated game with UCcan be chosen such that the subgame that begins after thecommitments has a unique Nash payoff



Conclusions





Conclusions



Subgame Perfect Folk Theorem (without UC)

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



Conclusions





Proposition 1

The counterpart of Theorem 1 for VSPE does not hold.



Conclusions





Proposition 1


Proposition 2

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



Conclusions





Proposition 1


Proposition 2

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

Theorem 2No assumption is needed for the VSPE folk theorem when we havetwo stages of commitments.



Conclusions



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


Proof



Conclusions





Proof

1 Apply Theorem 1 to G(δ, T )



Conclusions





Proof

1 Apply Theorem 1 to G(δ, T ) =⇒ U(G(δ, T )) has a Nash



Conclusions





Proof


2 Apply Proposition 2 to U(G(δ, T ))



Conclusions





Proof


2 Apply Proposition 2 to U(G(δ, T ))=⇒ U(U(G(δ, T ))) has a VSPE



Conclusions





Proof



Moreover, the VSPE can be chosen such that



Conclusions





Proof



Moreover, the VSPE can be chosen such that the subgamethat begins after the first stage of commitments has a uniqueNash payoff



Conclusions


The Folk TheoremsDiscussion




Conclusions




Remarks



Conclusions




Remarks

Are two stages of commitments natural??



Conclusions




Remarks


We “allow for” commitments on commitments



Conclusions




Remarks


We “allow for” commitments on commitments

President −→ Manager −→ Director


The State of ArtNecessary and Sufficient Conditions for the Folk Theorems

Without UC 1 stage of UC2 stagesof UC

Nash Theorem NoneInfinite Horizon (Fudenberg and Maskin, 1986)

(Virtual) Perfect Th. Non-Equivalent UtilitiesInfinite Horizon (Abreu et al., 1994)

Nash Theorem Minimax-Bettering LadderFinite Horizon (Gonzalez-Dıaz, 2003)

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



Nash Theorem None NoneInfinite Horizon (Fudenberg and Maskin, 1986) (Prop. 2)

(Virtual) Perfect Th. Non-Equivalent Utilities NoneInfinite Horizon (Abreu et al., 1994) (Prop. 2)

Nash Theorem Minimax-Bettering LadderFinite Horizon (Gonzalez-Dıaz, 2003)






Nash Theorem Minimax-Bettering Ladder NoneFinite Horizon (Gonzalez-Dıaz, 2003) (Garcıa-Jurado et al., 2000)






Nash Theorem Minimax-Bettering Ladder NoneFinite Horizon (Gonzalez-Dıaz, 2003) (Garcıa-Jurado et al., 2000)

(Virtual) Perfect Th. Recursively-distinct Minimax-Bettering LadderFinite Horizon Nash payoffs (Smith, 1995) (Prop. 2, only sufficient)



Nash Theorem None None NoneInfinite Horizon (Fudenberg and Maskin, 1986) (Prop. 2) (Prop. 2)

(Virtual) Perfect Th. Non-Equivalent Utilities None NoneInfinite Horizon (Abreu et al., 1994) (Prop. 2) (Prop. 2)

Nash Theorem Minimax-Bettering Ladder None NoneFinite Horizon (Gonzalez-Dıaz, 2003) (Garcıa-Jurado et al., 2000) (Prop. 2)

(Virtual) Perfect Th. Recursively-distinct Minimax-Bettering Ladder NoneFinite Horizon Nash payoffs (Smith, 1995) (Prop. 2, only sufficient) (Th. 2)


Conclusions

Outline



3 Conclusions



Conclusions

Conclusions

Our contribution



Conclusions

Conclusions

Our contribution

UC lead to weaker assumptions for the folk theorems.



Conclusions

Conclusions

Our contribution

UC lead to weaker assumptions for the folk theorems.

Nonetheless, some assumptions are still needed for someVSPE folk theorems.



Conclusions

Thanks


1

2

1

2

(1,1)

(1,0)

(0,1)

(1,-1)

(-1,1)

(-1,1)

(1,-1)

U1

D1

U2

D2

U2

D2

a11

a21

a12

a22

a12

a22

Return

Return

1

2

1

2

21

Return

1

2

1

2

21

σ

Return

Return

Proof

Return

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

Return


Strategy for a player i

Return



1 Commitment: Sci := “If a is played in the first stage, then I

play ai forever”

Return




play ai forever”2 Strategy:

Return





If Sc = Sc:

Return





If Sc = Sc:

i plays ai in the first stage

Return





If Sc = Sc:


If someone deviates i punishes him forever

Return





If Sc = Sc:


If someone deviates i punishes him forever

If someone has deviated from the commitment i punishes himforever

Return

Return

the role of commitment in repeated games - usceio.usc.es/pub/julio/slides/commitments.pdf · the...

Documents