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Essays on Competition and Cooperationin Game Theoretical Models

Julio Gonzalez Dıaz

Department of Statistics and Operations ResearchUniversidade de Santiago de Compostela

Thesis DissertationJune 29th, 2005

General Overview

Competition and Cooperation in Game Theoretical Models Julio Gonzalez Dıaz 1/60

General Overview

Part I: Noncooperative Game Theory

Part II: Cooperative Game Theory(On the Geometry of TU games)

General Overview

A Silent Battle over a Cake (Chapter 1)

General Overview

A Noncooperative Approach to Bankruptcy Problems(Chapter 4)

General Overview

Repeated Games (Chapters 2 and 3)

General Overview

A Geometric Characterization of the τ -value (Chapter 8)

General Overview

The Core-Center (Chapters 5, 6, and 7)

General Overview

Repeated Games

General Overview

Repeated Games

The Core-Center

A Silent Battle over a CakeBankruptcy Problems

Repeated Games

Part I

Noncooperative Game Theory

Repeated Games

What is a Strategic Game?And a Nash Equilibrium?

Repeated Games

Strategic game

Repeated Games

Strategic game

A strategic game G is a triple (N, A, π) where:

Repeated Games

Strategic game

N = 1, . . . , n is the set of players

Repeated Games

Strategic game

A =∏n

i=1 Ai, where Ai denotes the set of actions for player i

Repeated Games

Strategic game

A =∏n

π =∏n

i=1 πi, where πi : A → R is the payoff function of player i

Repeated Games

Strategic game

A =∏n

π =∏n

Nash equilibrium

Repeated Games

Strategic game

A =∏n

π =∏n

Nash equilibrium

An action profile a ∈ A is a Nash equilibrium of (N, A, π) if

Repeated Games

Strategic game

A =∏n

π =∏n

Nash equilibrium

An action profile a ∈ A is a Nash equilibrium of (N, A, π) if foreach i ∈ N

Repeated Games

Strategic game

A =∏n

π =∏n

Nash equilibrium

An action profile a ∈ A is a Nash equilibrium of (N, A, π) if foreach i ∈ N and each bi ∈ Ai,

Repeated Games

Strategic game

A =∏n

π =∏n

Nash equilibrium

πi( , )

Repeated Games

Strategic game

A =∏n

π =∏n

Nash equilibrium

πi(bi, a−i)

Repeated Games

Strategic game

A =∏n

π =∏n

Nash equilibrium

πi(bi, a−i) ≤

Repeated Games

Strategic game

A =∏n

π =∏n

Nash equilibrium

πi(bi, a−i) ≤ πi(a)

Repeated Games

Strategic game

A =∏n

π =∏n

Nash equilibrium

πi(bi, a−i) ≤ πi(a)

Unilateral deviations are not profitable

Repeated GamesBrief Overview

A Silent Battle over a Cake

1 A Silent Battle over a CakeBrief Overview

2 A Noncooperative Approach to Bankruptcy ProblemsBrief Overview

3 Repeated GamesDefinitions and Classic ResultsA Generalized Nash Folk TheoremUnilateral Commitments

Timing Games

Patent race

Timing Games

“Non-Silent” timing games

Patent race

Timing Games

Patent race

“Silent” timing games

Timing Games

Patent race

J. Reinganum, 1981(Review of Economic Studies)

Timing Games

Patent race

H. Hamers, 1993(Mathematical Methods of OR)

Timing Games

Patent race

H. Hamers, 1993(Mathematical Methods of OR) −→ Cake Sharing Games

Results

Hamers (1993) introduces the cake sharing games

Results

Existing Results

Results

Existing Results

Hamers (1993) proves the existence and uniqueness of theNash equilibrium of any two player cake sharing game

Results

Existing Results

Koops (2001) finds several properties that Nash equilibria ofthree player cake sharing games must satisfy

Results

Existing Results

Our Contribution

Results

Existing Results

Our Contribution

Alternative proof of the existence and uniqueness result of theNash equilibrium in the two player case

Results

Existing Results

Our Contribution

Alternative proof of the existence and uniqueness result of theNash equilibrium in the two player case

Proof of the existence and uniqueness result of the Nashequilibrium in the general case (n-players)

A Noncooperative Approach to Bankruptcy Problems

Bankruptcy Problems and Bankruptcy Rules

Bankruptcy Problem

(E, d)

Bankruptcy Problem

(E, d)

E ∈ R+ Amount to be divided

Bankruptcy Problem

(E, d)

d ∈ Rn+ Claims of the agents

Bankruptcy Problem

(E, d)

∑ni=1 di > E

Bankruptcy Problem

(E, d)

∑ni=1 di > E

Bankruptcy Rules

Bankruptcy Problem

(E, d)

∑ni=1 di > E

Bankruptcy Rules

ϕ : Ω −→ Rn

(E, d) 7−→ ϕ(E, d)

Bankruptcy Problem

(E, d)

∑ni=1 di > E

Bankruptcy Rules

ϕ : Ω −→ Rn

(E, d) 7−→ ϕ(E, d)

ϕi(E, d) ∈ [0, di]

Bankruptcy Problem

(E, d)

∑ni=1 di > E

Bankruptcy Rules

ϕ : Ω −→ Rn

(E, d) 7−→ ϕ(E, d)

ϕi(E, d) ∈ [0, di]∑

i∈N ϕi(E, d) = E

Bankruptcy Problem

(E, d)

∑ni=1 di > E

Foundations for rules

Bankruptcy Rules

ϕ : Ω −→ Rn

(E, d) 7−→ ϕ(E, d)

ϕi(E, d) ∈ [0, di]∑

i∈N ϕi(E, d) = E

Bankruptcy Problem

(E, d)

∑ni=1 di > E

Axiomatic approach

Bankruptcy Rules

ϕ : Ω −→ Rn

(E, d) 7−→ ϕ(E, d)

ϕi(E, d) ∈ [0, di]∑

i∈N ϕi(E, d) = E

Bankruptcy Problem

(E, d)

∑ni=1 di > E

Axiomatic approach

Noncooperative approach

Bankruptcy Rules

ϕ : Ω −→ Rn

(E, d) 7−→ ϕ(E, d)

ϕi(E, d) ∈ [0, di]∑

i∈N ϕi(E, d) = E

Bankruptcy Problems and Bankruptcy RulesAn Example

A strategic game

Bankruptcy problem (E, d)

A strategic game

Each player i announces a portion of di

A strategic game

Each player i announces a portion of di (admissible for him)

A strategic game

The lower the portion you claim,

A strategic game

The lower the portion you claim, the higher your priority

A strategic game

Unique Nash payoff

A strategic game

Unique Nash payoff

Coincides with the proposal of the proportional rule

Bankruptcy Problems and Bankruptcy RulesResults

Our contribution

Our contributionWe define a family, G, of strategic games such that:

Each G ∈ G has a unique Nash payoff N(G)

For each bankruptcy rule ϕ

and each bankruptcy problem (E, d),

and each bankruptcy problem (E, d),there is G ∈ G such that N(G) = ϕ(E, d)

Repeated Games

Definitions and Classic ResultsA Generalized Nash Folk TheoremUnilateral Commitments

Repeated Games

“The model of a repeated game is designed to examine the logic

of longterm interaction. It captures the idea that a player will take

into account the effect of his current behavior on the other players’

future behavior, and aims to explain phenomena like cooperation,

revenge, and threats”.

(Martin J. Osborne, Ariel Rubinstein)

Repeated Games

The Stage Game

Repeated Games

The Stage Game

A =∏n

π =∏n

i=1 πi, where πi : A → R is the utility function of player i

Repeated Games

The Stage Game

A =∏n

π =∏n

Minmax Payoffs:

vi := mina−i∈A−i

maxai∈Ai

πi(ai, a−i)

Repeated Games

The Stage Game

A =∏n

π =∏n

Minmax Payoffs:

maxai∈Ai

πi(ai, a−i)

Feasible and Individually Rational Payoffs:

Repeated Games

The Stage Game

A =∏n

π =∏n

Minmax Payoffs:

maxai∈Ai

πi(ai, a−i)

Repeated Games

The Stage Game

A =∏n

π =∏n

Minmax Payoffs:

maxai∈Ai

πi(ai, a−i)

F := coπ(a) : a ∈ π(A)

Repeated Games

The Stage Game

A =∏n

π =∏n

Minmax Payoffs:

maxai∈Ai

πi(ai, a−i)

F := coπ(a) : a ∈ π(A) ∩ u ∈ Rn : u ≥ v

Repeated Games

The Stage Game

A =∏n

π =∏n

Minmax Payoffs:

maxai∈Ai

πi(ai, a−i)

F := coπ(a) : a ∈ π(A) ∩ u ∈ Rn : u ≥ v

Repeated Games

The Repeated Game

Repeated Games (with complete information)

Repeated Games

The Repeated Game

Let G = (N, A, π)

Repeated Games

The Repeated Game

Let G = (N, A, π)

GTδ denotes the T -fold repetition of the game G with discount

parameter δ

Repeated Games

The Repeated Game

Let G = (N, A, π)

parameter δ

GTδ := (N, S, πT

δ ) where

Repeated Games

The Repeated Game

Let G = (N, A, π)

parameter δ

GTδ := (N, S, πT

δ ) where

N := 1, . . . , n

Repeated Games

The Repeated Game

Let G = (N, A, π)

parameter δ

GTδ := (N, S, πT

δ ) where

N := 1, . . . , nS :=

i∈N Si,

Repeated Games

The Repeated Game

Let G = (N, A, π)

parameter δ

GTδ := (N, S, πT

δ ) where

N := 1, . . . , nS :=

i∈N Si, Si := AHi

Repeated Games

The Repeated Game

Let G = (N, A, π)

parameter δ

GTδ := (N, S, πT

δ ) where

N := 1, . . . , nS :=

i∈N Si, Si := AHi

Discounted payoffs in the repeated game,

Repeated Games

The Repeated Game

Let G = (N, A, π)

parameter δ

GTδ := (N, S, πT

δ ) where

N := 1, . . . , nS :=

i∈N Si, Si := AHi

πTδ (σ) :=

Repeated Games

The Repeated Game

Let G = (N, A, π)

parameter δ

GTδ := (N, S, πT

δ ) where

N := 1, . . . , nS :=

i∈N Si, Si := AHi

πTδ (σ) :=

π(at)

Repeated Games

The Repeated Game

Let G = (N, A, π)

parameter δ

GTδ := (N, S, πT

δ ) where

N := 1, . . . , nS :=

i∈N Si, Si := AHi

πTδ (σ) :=

1 − δ

1 − δT

δt−1π(at)

Repeated Games

The Repeated Game

Let G = (N, A, π)

parameter δ

GTδ := (N, S, πT

δ ) where

N := 1, . . . , nS :=

i∈N Si, Si := AHi

πTδ (σ) :=

1 − δ

1 − δT

δt−1π(at)

Folk Theorems

Repeated Games

General Considerations

Our framework:

Repeated Games

Our framework:

The sets of actions are compact

Repeated Games

Our framework:

Continuous payoff functions

Repeated Games

Our framework:

Finite Horizon

Repeated Games

Our framework:

Finite Horizon

Nash Equilibrium

Repeated Games

Our framework:

Finite Horizon

Nash Equilibrium

Complete Information

Repeated Games

Our framework:

Finite Horizon

Nash Equilibrium

Complete Information

Perfect Monitoring (Observable mixed actions)

Repeated Games

The State of ArtThe Folk Theorems

Nash Subgame Perfect

InfiniteHorizon

FiniteHorizon

Repeated Games

InfiniteHorizon

The “Folk Theorem” (1970s)Fudenberg and Maskin (1986)Abreu et al. (1994)Wen (1994)

FiniteHorizon

Benoıt and Krishna (1987)Benoıt and Krishna (1985)Smith (1995)Gossner (1995)

Repeated Games

InfiniteHorizon

FiniteHorizon

Necessary and Sufficient conditions

Repeated Games

InfiniteHorizon

FiniteHorizon

Necessary and Sufficient conditions

Repeated Games

(Benoıt and Krishna, 1987)

Assumption for the game G

Result

Repeated Games

Existence of strictly rational Nash payoffs

Result

Repeated Games

Existence of strictly rational Nash payoffsFor each player i there is a Nash Equilibrium ai of G such that

πi(ai) > vi

Result

Repeated Games

πi(ai) > vi

Result

Every payoff in F can be approximated in equilibrium

Repeated Games

πi(ai) > vi

Result

Every payoff in F can be approximated in equilibriumFor each u ∈ F and each ε > 0, there are T0 and δ0 such that for each

T ≥ T0 and each δ ∈ [δ0, 1], there is a Nash Equilibrium σ of G(δ, T )

satisfying that ‖πT

δ (σ) − u‖ < ε

Repeated Games

πi(ai) > vi

Result

Every payoff in F can be approximated in equilibriumFor each u ∈ F and each ε > 0, there are T0 and δ0 such that for each

T ≥ T0 and each δ ∈ [δ0, 1], there is a Nash Equilibrium σ of G(δ, T )

satisfying that ‖πT

δ (σ) − u‖ < ε

Repeated Games

Our ContributionMinmax Bettering Ladders

Repeated Games

Examplel m r

T 0,0,3 0,-1,0 0,-1,0M -1,0,0 0,-1,0 0,-1,0B -1,0,0 0,-1,0 0,-1,0

l m r0,3,-1 0,-1,-1 1,-1,-1-1,0,-1 -1,-1,-1 0,-1,-1-1,0,-1 -1,-1,-1 0,-1,-1

Repeated Games

Examplel m r

T 0,0,3 0,-1,0 0,-1,0M -1,0,0 0,-1,0 0,-1,0B -1,0,0 0,-1,0 0,-1,0

l m r0,3,-1 0,-1,-1 1,-1,-1-1,0,-1 -1,-1,-1 0,-1,-1-1,0,-1 -1,-1,-1 0,-1,-1

Minmax Payoff (0,0,0)

Repeated Games

Examplel m r

T 0,0,3 0,-1,0 0,-1,0M -1,0,0 0,-1,0 0,-1,0B -1,0,0 0,-1,0 0,-1,0

l m r0,3,-1 0,-1,-1 1,-1,-1-1,0,-1 -1,-1,-1 0,-1,-1-1,0,-1 -1,-1,-1 0,-1,-1

Nash Equilibrium (T,l,L), Payoff (0,0,3)

Repeated Games

Examplel m r

T 0,0,3 0,-1,0 0,-1,0M -1,0,0 0,-1,0 0,-1,0B -1,0,0 0,-1,0 0,-1,0

l m r0,3,-1 0,-1,-1 1,-1,-1-1,0,-1 -1,-1,-1 0,-1,-1-1,0,-1 -1,-1,-1 0,-1,-1

Nash Equilibrium (T,l,L), Payoff (0,0,3) (B-K not met)

Repeated Games

Examplel m r

T 0,0,3 0,-1,0 0,-1,0M -1,0,0 0,-1,0 0,-1,0B -1,0,0 0,-1,0 0,-1,0

l m r0,3,-1 0,-1,-1 1,-1,-1-1,0,-1 -1,-1,-1 0,-1,-1-1,0,-1 -1,-1,-1 0,-1,-1

Nash Equilibrium (T,l,L), Payoff (0,0,3) (B-K not met)

Player 3 can be threatened

Repeated Games

Examplel m r

T 0,0,3 0,-1,0 0,-1,0M -1,0,0 0,-1,0 0,-1,0B -1,0,0 0,-1,0 0,-1,0

l m r0,3,-1 0,-1,-1 1,-1,-1-1,0,-1 -1,-1,-1 0,-1,-1-1,0,-1 -1,-1,-1 0,-1,-1

Repeated Games

Examplel m r

T 0,0,3 0,-1,0 0,-1,0M -1,0,0 0,-1,0 0,-1,0B -1,0,0 0,-1,0 0,-1,0

l m r0,3,-1 0,-1,-1 1,-1,-1-1,0,-1 -1,-1,-1 0,-1,-1-1,0,-1 -1,-1,-1 0,-1,-1

Player 3 is forced to play R

Repeated Games

Examplel m r

T 0,0,3 0,-1,0 0,-1,0M -1,0,0 0,-1,0 0,-1,0B -1,0,0 0,-1,0 0,-1,0

l m r0,3,-1 0,-1,-1 1,-1,-1-1,0,-1 -1,-1,-1 0,-1,-1-1,0,-1 -1,-1,-1 0,-1,-1

The profile α3 =(T,l,R) is a Nash Equilibrium of the reducedgame with Payoff (0,3,-1)

Repeated Games

Examplel m r

T 0,0,3 0,-1,0 0,-1,0M -1,0,0 0,-1,0 0,-1,0B -1,0,0 0,-1,0 0,-1,0

l m r0,3,-1 0,-1,-1 1,-1,-1-1,0,-1 -1,-1,-1 0,-1,-1-1,0,-1 -1,-1,-1 0,-1,-1

The profile α3 =(T,l,R) is a Nash Equilibrium of the reducedgame with Payoff (0,3,-1)

Now player 2 can be threatened

Repeated Games

Example

l m rT 0,0,3 0,-1,0 0,-1,0M -1,0,0 0,-1,0 0,-1,0B -1,0,0 0,-1,0 0,-1,0

l m0,3,-1 0,-1,-1-1,0,-1 -1,-1,-1-1,0,-1 -1,-1,-1

1,-1,-1

0,-1,-1

Repeated Games

Example

l m rT 0,0,3 0,-1,0 0,-1,0M -1,0,0 0,-1,0 0,-1,0B -1,0,0 0,-1,0 0,-1,0

l m0,3,-1 0,-1,-1-1,0,-1 -1,-1,-1-1,0,-1 -1,-1,-1

1,-1,-1

0,-1,-1

Player 3 is forced to play R and player 2 to play r

Repeated Games

Example

l m rT 0,0,3 0,-1,0 0,-1,0M -1,0,0 0,-1,0 0,-1,0B -1,0,0 0,-1,0 0,-1,0

l m0,3,-1 0,-1,-1-1,0,-1 -1,-1,-1-1,0,-1 -1,-1,-1

1,-1,-1

0,-1,-1

The profile α32 =(T,r,R) is a Nash Equilibrium of the reducedgame with Payoff (1,-1,-1)

Repeated Games

Example

l m rT 0,0,3 0,-1,0 0,-1,0M -1,0,0 0,-1,0 0,-1,0B -1,0,0 0,-1,0 0,-1,0

l m0,3,-1 0,-1,-1-1,0,-1 -1,-1,-1-1,0,-1 -1,-1,-1

1,-1,-1

0,-1,-1

The profile α32 =(T,r,R) is a Nash Equilibrium of the reducedgame with Payoff (1,-1,-1)

Now player 1 can be threatened

Repeated Games

Minmax Bettering LaddersFormal Definition

Repeated Games

reliable players ∅

Repeated Games

game G

Repeated Games

game G

“Nash equilibrium” σ1

Repeated Games

reliable players ∅ N1

game G

Repeated Games

game G G(aN1)

Repeated Games

game G G(aN1)

“Nash equilibrium” σ1 σ2

Repeated Games

reliable players ∅ N1 . . .

game G G(aN1) . . .

“Nash equilibrium” σ1 σ2 . . .

Repeated Games

reliable players ∅ N1 . . . Nh−1

game G G(aN1) . . .

Repeated Games

game G G(aN1) . . . G(aNh−1)

Repeated Games

“Nash equilibrium” σ1 σ2 . . . σh

Repeated Games

reliable players ∅ N1 . . . Nh−1 Nh

“Nash equilibrium” σ1 σ2 . . . σh

Repeated Games

game G G(aN1) . . . G(aNh−1) −−−

“Nash equilibrium” σ1 σ2 . . . σh −−−

Repeated Games

game G G(aN1) . . . G(aNh−1) −−−

A minimax-bettering ladder of a game G is a tripletN ,A, Σ

Repeated Games

game G G(aN1) . . . G(aNh−1) −−−

N := ∅ = N0 ( N1 ( · · · ( Nh subsets of N

Repeated Games

game G G(aN1) . . . G(aNh−1) −−−

A := aN1 ∈ AN1 , . . . , aNh−1∈ ANh−1

Repeated Games

game G G(aN1) . . . G(aNh−1) −−−

A := aN1 ∈ AN1 , . . . , aNh−1∈ ANh−1

Σ := σ1, . . . , σh

Repeated Games

game G G(aN1) . . . G(aNh−1) −−−

A := aN1 ∈ AN1 , . . . , aNh−1∈ ANh−1

Σ := σ1, . . . , σh

Nh is the top rung of the ladder

Repeated Games

game G G(aN1) . . . G(aNh−1) −−−

A := aN1 ∈ AN1 , . . . , aNh−1∈ ANh−1

Σ := σ1, . . . , σh

Nh is the top rung of the ladderNh = N → complete minimax-bettering ladder

Repeated Games

The New Folk Theorem(Gonzalez-Dıaz, 2003)

Result

Repeated Games

Existence of a complete minmax bettering ladder

Result

Repeated Games

Result

Repeated Games

Result

RemarkUnlike Benoıt and Krishna’s result, this theorem provides anecessary and sufficient condition

Repeated Games

Result

RemarkUnlike Benoıt and Krishna’s result, this theorem provides anecessary and sufficient condition

Why the word generalized?

Repeated Games

Unilateral Commitments

Motivation

Repeated Games

Motivation

Commitment

Repeated Games

Motivation

Commitment

Repeated games

Repeated Games

Motivation

Commitment

Repeated games

Unilateral commitments in repeated games

Repeated Games

Motivation

Commitment

Repeated games

Unilateral commitments in repeated games

Delegation games

Repeated Games

Unilateral CommitmentsDefinitions

The stage game: G := (N, A, π)

N := 1, . . . , nA :=

i∈N Ai

π := (π1, . . . , πn)

The repeated game: GTδ := (N, S, πT

N := 1, . . . , nS :=

i∈N Si

(Si := AHi )

Repeated Games

N := 1, . . . , nA :=

i∈N Ai

π := (π1, . . . , πn)

N := 1, . . . , nS :=

i∈N Si

(Si := AHi )

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

Repeated Games

N := 1, . . . , nA :=

i∈N Ai

π := (π1, . . . , πn)

N := 1, . . . , nS :=

i∈N Si

(Si := AHi )

AU :=∏

i∈N AUi , where AU

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

such that

Repeated Games

N := 1, . . . , nA :=

i∈N Ai

π := (π1, . . . , πn)

N := 1, . . . , nS :=

i∈N Si

(Si := AHi )

AU :=∏

such that1 ∅ ( Ac

i ⊆ Ai,

Repeated Games

N := 1, . . . , nA :=

i∈N Ai

π := (π1, . . . , πn)

N := 1, . . . , nS :=

i∈N Si

(Si := AHi )

AU :=∏

such that1 ∅ ( Ac

i ⊆ Ai,2 αi :

j∈N 2Aj −→ Ai

Repeated Games

N := 1, . . . , nA :=

i∈N Ai

π := (π1, . . . , πn)

N := 1, . . . , nS :=

i∈N Si

(Si := AHi )

AU :=∏

such that1 ∅ ( Ac

i ⊆ Ai,2 αi :

j∈N 2Aj −→ Ai and, for each Ac ∈∏

j∈N 2Aj ,αi(A

c) ∈ Aci

Repeated Games

N := 1, . . . , nA :=

i∈N Ai

π := (π1, . . . , πn)

N := 1, . . . , nS :=

i∈N Si

(Si := AHi )

AU :=∏

such that1 ∅ ( Ac

i ⊆ Ai,2 αi :

j∈N 2Aj ,αi(A

c) ∈ Aci

Commitments are UnilateralCompetition and Cooperation in Game Theoretical Models Julio Gonzalez Dıaz 24/60

Repeated Games

N := 1, . . . , nA :=

i∈N Ai

π := (π1, . . . , πn)

N := 1, . . . , nS :=

i∈N Si

(Si := AHi )

AU :=∏

such that1 ∅ ( Ac

i ⊆ Ai,2 αi :

j∈N 2Aj ,αi(A

c) ∈ Aci

Commitments are Unilateral Complete Information

Repeated Games

Virtually Subgame Perfect EquilibriumMotivation

Repeated Games

Subgame Perfect Equilibrium

Repeated Games

Eliminates Nash equilibria based on incredible threats

Repeated Games

A strategy σ is a SPE if it prescribes a Nash equilibrium foreach subgame

Repeated Games

Definition (Informal)

Let σ be a strategy profile.

Repeated Games

Let σ be a strategy profile. A subgame is σ-relevant

Repeated Games

Let σ be a strategy profile. A subgame is σ-relevant if it can bereached by a sequence of unilateral deviations of the players

Repeated Games

Virtually Subgame Perfect Equilibrium

Repeated Games

A strategy σ is a VSPE if it prescribes a Nash equilibrium foreach σ-relevant subgame

Repeated Games

Virtually Subgame Perfect EquilibriumDiscussion

Repeated Games

Subgame Perfect Vs Virtually Subgame Perfect

Repeated Games

Why do we need VSPE?

Repeated Games

In our model, we face very large trees

Repeated Games

There can be subgames with no Nash Equilibrium

Repeated Games

Hence,

Repeated Games

Hence,

We cannot use the classic results for the existence of SPE

Repeated Games

The Folk TheoremsFinite Horizon

Repeated Games

Nash Folk Theorem (without UC)

Repeated Games

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

No assumption is needed for the Nash folk theorem with UC

Repeated Games

Subgame Perfect Folk Theorem (without UC)

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

Repeated Games

Subgame Perfect Folk Theorem (without UC)

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

Proposition 1

The counterpart of Theorem 1 for VSPE does not hold

Repeated Games

Proposition 1

Repeated Games

Proposition 1

Proposition 2

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

Repeated Games

Proposition 1

Proposition 2

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

Repeated Games

Proposition 1

Proposition 2

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

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)

(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)

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)

Repeated Games

Noncooperative Game TheoryConclusions

Conclusions

Repeated Games

Conclusions

We have studied a special family of timing games, extendingthe results in Hamers (1993)

Repeated Games

Conclusions

We have presented a noncooperative approach to bankruptcyproblems

Repeated Games

Conclusions

We have extended the result in Benoıt and Krishna (1987)

Repeated Games

Conclusions

We have generalized the result in Benoıt and Krishna (1987)

Repeated Games

Conclusions

Our main result establishes a necessary and sufficientcondition for the finite horizon Nash folk theorem

Repeated Games

Conclusions

UC lead to weaker assumptions for the folk theorems

Repeated Games

Conclusions

Nonetheless, some assumptions are still needed for someVSPE folk theorems

Repeated Games

Conclusions

Nonetheless, some assumptions are still needed for someVSPE folk theorems

Repeated Games

Noncooperative Game TheoryFuture Research

Future Research

Repeated Games

Future Research

Study the extent up to which our results for timing games canbe extended

Repeated Games

Future Research

Study whether the games in our family are natural fordifferent bankruptcy rules

Repeated Games

Future Research

Try to find decentralized results (where the regulator needsnot to have complete information)

Repeated Games

Future Research

Study whether the general folk theorems like our’s can beobtained in other families of repeated games (infinite horizon,subgame perfection, incomplete information)

Repeated Games

Future Research

Study whether the general folk theorems like our’s can beobtained in other families of repeated games (infinite horizon,subgame perfection, incomplete information)

Study Unilateral Commitments in models with incompleteinformation

Repeated Games

Noncooperative Game TheoryReferences

References

Gonzalez-Dıaz, J., P. Borm, H. Norde (2004):

“A Silent Battle over a Cake,” Tech. rep., CentER discussion paper series

Garcıa-Jurado, I., J. Gonzalez-Dıaz, A. Villar (2004):

“A Noncooperative Approach to Bankruptcy Problems,” Preprint

Gonzalez-Dıaz, J. (2003):

“Finitely Repeated Games: A Generalized Nash Folk Theorem,” Toappear in Games and Economic Behavior

Garcıa-Jurado, I., J. Gonzalez-Dıaz (2005):

“Unilateral Commitments in Repeated Games”, Preprint

A Geometric Characterization of the τ -valueThe Core-Center

Part II

Cooperative Game Theory

What is a Cooperative Game?And an Allocation Rule?

Cooperative game (with transferable utility)

A cooperative TU game is a pair (N, v) where:

v is the characteristic function,

v is the characteristic function,v : 2n −→ R

S 7−→ v(S)

Allocation rule

S 7−→ v(S)

Allocation rule

An allocation rule on a domain Ω is a function ϕ such that

S 7−→ v(S)

Allocation rule

ϕ : Ω −→ Rn

(N, v) 7−→ ϕ(N, v)

S 7−→ v(S)

Allocation rule

ϕ : Ω −→ Rn

(N, v) 7−→ ϕ(N, v)

An allocation x ∈ Rn is efficient if∑n

i=1 xi = v(N)

Brief Overview

A Geometric Characterization of the τ -value

4 A Geometric Characterization of the τ -valueBrief Overview

5 The Core-CenterThe Core-Center: Definition and PropertiesA Characterization of the Core-CenterThe Core-Center and the Shapley Value

Brief Overview

The τ -value

Let v denote a cooperative game (N is fixed)

Brief Overview

The τ -value

Previous concepts

Brief Overview

The τ -value

Previous concepts

Utopia vector, M(v) ∈ Rn:

Brief Overview

The τ -value

Previous concepts

for each i ∈ N, Mi(v) := v(N) − v(N\i)

Brief Overview

The τ -value

Previous concepts

Minimum right vector, m(v) ∈ Rn:

Brief Overview

The τ -value

Previous concepts

for each i ∈ N, mi(v) := maxS⊆N, i∈S

v(S) −∑

j∈S\i

Brief Overview

The τ -value

Previous concepts

v(S) −∑

j∈S\i

Core cover:

Brief Overview

The τ -value

Previous concepts

v(S) −∑

j∈S\i

Core cover:

CC(v) := x ∈ Rn :

Brief Overview

The τ -value

Previous concepts

v(S) −∑

j∈S\i

Core cover:

CC(v) := x ∈ Rn :∑

xi = v(N),

Brief Overview

The τ -value

Previous concepts

v(S) −∑

j∈S\i

Core cover:

CC(v) := x ∈ Rn :∑

xi = v(N), m(v) ≤ x ≤ M(v)

Brief Overview

The τ -value

Previous concepts

v(S) −∑

j∈S\i

Core cover:

CC(v) := x ∈ Rn :∑

xi = v(N), m(v) ≤ x ≤ M(v)

A game v is compromise admissible if CC(v) 6= ∅

Brief Overview

The τ -value

The τ -value or compromise-value (Tijs, 1981):

Brief Overview

The τ -value

τ(v) := “point on the line segment between m(v) and M(v) thatis efficient”,

Brief Overview

The τ -value

τ(v) = λm(v) + (1 − λ)M(v),

Brief Overview

The τ -value

τ(v) = λm(v) + (1 − λ)M(v), λ ∈ [0, 1] is such thatXi∈N

τi = v(N)

Brief Overview

The τ -value

τi = v(N)

By definition, CC(v) is a convex polytope

Brief Overview

The τ -value

τi = v(N)

The τ ∗-value, Gonzalez-Dıaz et al. (2003):

Brief Overview

The τ -value

τi = v(N)

τ∗(v) := “center of gravity of the edges of the core-cover”

Brief Overview

The τ -value

τi = v(N)

τ∗(v) := “center of gravity of the edges of the core-cover”(multiplicities for the edges have to be taken into account)

Brief Overview

The τ -value

τi = v(N)

τ∗(v) := “center of gravity of the edges of the core-cover”(multiplicities for the edges have to be taken into account)

By definition, τ(v) ∈ CC(v) and τ∗(v) ∈ CC(v)

Brief Overview

Results

Brief Overview

Results

P1: Let v be such that

Brief Overview

Results

P1: Let v be such that

v(N) −∑

Brief Overview

Results

P1: Let v be such that for each i ∈ N ,

Mi(v) − mi(v) v(N) −∑

Brief Overview

Results

Mi(v) − mi(v) ≤ v(N) −∑

Brief Overview

Results

Mi(v) − mi(v) ≤ v(N) −∑

TheoremIf v satisfies P1,

Brief Overview

Results

Mi(v) − mi(v) ≤ v(N) −∑

TheoremIf v satisfies P1, then τ(v) = τ∗(v)

Brief Overview

Results

Mi(v) − mi(v) ≤ v(N) −∑

Quant et al. (2004):

Brief Overview

Results

Mi(v) − mi(v) ≤ v(N) −∑

Bankruptcty Problem

Brief Overview

Results

Mi(v) − mi(v) ≤ v(N) −∑

Bankruptcty Problem −→ Bankruptcty Game

Brief Overview

Results

Mi(v) − mi(v) ≤ v(N) −∑

Bankruptcty Problem −→ Bankruptcty GameProportional Rule

Brief Overview

Results

Mi(v) − mi(v) ≤ v(N) −∑

Bankruptcty Problem −→ Bankruptcty GameProportional Rule −→ τ -value

Brief Overview

Results

Mi(v) − mi(v) ≤ v(N) −∑

Adjusted Proportional Rule

Brief Overview

Results

Mi(v) − mi(v) ≤ v(N) −∑

Adjusted Proportional Rule −→ τ∗-value

The Core-Center: Definition and PropertiesA Characterization of the Core-CenterThe Core-Center and the Shapley Value

The Core-Center

Some More Background

Fix a game v and an allocation x ∈ Rn

x is efficient if∑n

i=1 xi = v(N)

x is individually rational if

i=1 xi = v(N)

x is individually rational if for each i ∈ N ,

i=1 xi = v(N)

x is individually rational if for each i ∈ N , xi ≥ v(i)

i=1 xi = v(N)

x is stable if

i=1 xi = v(N)

x is stable if for each S ⊆ N ,

i=1 xi = v(N)

x is stable if for each S ⊆ N ,∑

i∈S xi ≥ v(S)

i=1 xi = v(N)

i∈S xi ≥ v(S)

The Core (Gillies, 1953):

i=1 xi = v(N)

i∈S xi ≥ v(S)

The core of v is the set of all efficient and stable allocations

i=1 xi = v(N)

i∈S xi ≥ v(S)

C(v) := x ∈ Rn :∑

xi = v(N) and, for each S ⊆ N,∑

xi ≥ v(S)

i=1 xi = v(N)

i∈S xi ≥ v(S)

C(v) := x ∈ Rn :∑

xi = v(N) and, for each S ⊆ N,∑

xi ≥ v(S)

A game v is balanced if C(v) 6= ∅

The Core-Center: Definition

Let U(A) be the uniform distribution defined over A

Let E(P) be the expectation of P

The Core-Center (Gonzalez-Dıaz and Sanchez Rodrıguez, 2003):

Let v be a balanced game

Let v be a balanced gameThe core-center, µ(v), is center of gravity of C(v):

U(C(v))

U(C(v)))

µ(v) := E(

U(C(v)))

Motivation