eio.usc.eseio.usc.es/pub/julio/slides/tesis.pdf · General Overview General Overview Part I:...

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

General Overview

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

General Overview

General Overview

Part I: Noncooperative Game Theory

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


General Overview

General Overview


A Silent Battle over a Cake (Chapter 1)



General Overview

General Overview



A Noncooperative Approach to Bankruptcy Problems(Chapter 4)



General Overview

General Overview




Repeated Games (Chapters 2 and 3)



General Overview

General Overview






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


General Overview

General Overview







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


General Overview

General Overview





Repeated Games





General Overview

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




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





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





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






Repeated Games


Repeated Games



Repeated Games


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)


F :=



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 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) :=

1

T

T∑

t=1

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

T∑

t=1

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

T∑

t=1

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

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)

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


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

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

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

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

R

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

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

R


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

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

R


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

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

R


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

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

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

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

R

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

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

R


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

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

R


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

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

R

r

1,-1,-1

0,-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

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

R

r

1,-1,-1

0,-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

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

R

r

1,-1,-1

0,-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

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

R

r

1,-1,-1

0,-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

δ )

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

∏

i∈N Si

(Si := AHi )

πTδ



Repeated Games




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

∏

i∈N Ai

π := (π1, . . . , πn)


δ )

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

∏

i∈N Si

(Si := AHi )

πTδ

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



Repeated Games




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

∏

i∈N Ai

π := (π1, . . . , πn)


δ )

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

∏

i∈N Si

(Si := AHi )

πTδ


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)


δ )

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

∏

i∈N Si

(Si := AHi )

πTδ


AU :=∏



such that1 ∅ ( Ac

i ⊆ Ai,



Repeated Games




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

∏

i∈N Ai

π := (π1, . . . , πn)


δ )

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

∏

i∈N Si

(Si := AHi )

πTδ


AU :=∏



such that1 ∅ ( Ac

i ⊆ Ai,2 αi :

∏

j∈N 2Aj −→ Ai



Repeated Games




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

∏

i∈N Ai

π := (π1, . . . , πn)


δ )

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

∏

i∈N Si

(Si := AHi )

πTδ


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)


δ )

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

∏

i∈N Si

(Si := AHi )

πTδ


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)


δ )

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

∏

i∈N Si

(Si := AHi )

πTδ


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












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

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

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)






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)


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






UC lead to weaker assumptions for the folk theorems

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)








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

Mj(v)



Brief Overview

The τ -value

Previous concepts





v(S) −∑

j∈S\i

Mj(v)

Core cover:



Brief Overview

The τ -value

Previous concepts





v(S) −∑

j∈S\i

Mj(v)

Core cover:

CC(v) := x ∈ Rn :



Brief Overview

The τ -value

Previous concepts





v(S) −∑

j∈S\i

Mj(v)

Core cover:

CC(v) := x ∈ Rn :∑

i∈N

xi = v(N),



Brief Overview

The τ -value

Previous concepts





v(S) −∑

j∈S\i

Mj(v)

Core cover:

CC(v) := x ∈ Rn :∑

i∈N

xi = v(N), m(v) ≤ x ≤ M(v)



Brief Overview

The τ -value

Previous concepts





v(S) −∑

j∈S\i

Mj(v)

Core cover:

CC(v) := x ∈ Rn :∑

i∈N

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) −∑

j∈N

mj(v)



Brief Overview

Results

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

Mi(v) − mi(v) v(N) −∑

j∈N

mj(v)



Brief Overview

Results


Mi(v) − mi(v) ≤ v(N) −∑

j∈N

mj(v)



Brief Overview

Results


Mi(v) − mi(v) ≤ v(N) −∑

j∈N

mj(v)

TheoremIf v satisfies P1,



Brief Overview

Results


Mi(v) − mi(v) ≤ v(N) −∑

j∈N

mj(v)

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



Brief Overview

Results


Mi(v) − mi(v) ≤ v(N) −∑

j∈N

mj(v)


Quant et al. (2004):



Brief Overview

Results


Mi(v) − mi(v) ≤ v(N) −∑

j∈N

mj(v)



Bankruptcty Problem



Brief Overview

Results


Mi(v) − mi(v) ≤ v(N) −∑

j∈N

mj(v)



Bankruptcty Problem −→ Bankruptcty Game



Brief Overview

Results


Mi(v) − mi(v) ≤ v(N) −∑

j∈N

mj(v)



Bankruptcty Problem −→ Bankruptcty GameProportional Rule



Brief Overview

Results


Mi(v) − mi(v) ≤ v(N) −∑

j∈N

mj(v)



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



Brief Overview

Results


Mi(v) − mi(v) ≤ v(N) −∑

j∈N

mj(v)




Adjusted Proportional Rule



Brief Overview

Results


Mi(v) − mi(v) ≤ v(N) −∑

j∈N

mj(v)




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)







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 :∑

i∈N

xi = v(N) and, for each S ⊆ N,∑

i∈S

xi ≥ v(S)







i=1 xi = v(N)



i∈S xi ≥ v(S)



C(v) := x ∈ Rn :∑

i∈N

xi = v(N) and, for each S ⊆ N,∑

i∈S

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):









v









C(v)









U(C(v))









E(

U(C(v)))









µ(v) := E(

U(C(v)))




Motivation

v =

v(1) = 0 v(2) = 0 v(3) = 0v(12) = 1 v(13) = 4 v(23) = 7

v(123) = 15




Motivation

v =

v(1) = 0 v(2) = 0 v(3) = 0v(12) = 1 v(13) = 4 v(23) = 7

v(123) = 15

Core = Core Cover = Weber Set




Motivation

v =

v(1) = 0 v(2) = 0 v(3) = 0v(12) = 1 v(13) = 4 v(23) = 7

v(123) = 15


Weber Set




Motivation

v =

v(1) = 0 v(2) = 0 v(3) = 0v(12) = 1 v(13) = 4 v(23) = 7

v(123) = 15


Weber Set & Shapley Value




Motivation

v =

v(1) = 0 v(2) = 0 v(3) = 0v(12) = 1 v(13) = 4 v(23) = 7

v(123) = 15





Motivation

v =

v(1) = 0 v(2) = 0 v(3) = 0v(12) = 1 v(13) = 4 v(23) = 7

v(123) = 15


Core-Cover




Motivation

v =

v(1) = 0 v(2) = 0 v(3) = 0v(12) = 1 v(13) = 4 v(23) = 7

v(123) = 15


Core-Cover & τ -value




Motivation

v =

v(1) = 0 v(2) = 0 v(3) = 0v(12) = 1 v(13) = 4 v(23) = 7

v(123) = 15





Motivation

v =

v(1) = 0 v(2) = 0 v(3) = 0v(12) = 1 v(13) = 4 v(23) = 7

v(123) = 15


Core




Motivation

v =

v(1) = 0 v(2) = 0 v(3) = 0v(12) = 1 v(13) = 4 v(23) = 7

v(123) = 15


Core & Core-Center




Motivation

v =

v(1) = 0 v(2) = 0 v(3) = 0v(12) = 1 v(13) = 4 v(23) = 7

v(123) = 15





Motivation

v =

v(1) = 0 v(2) = 0 v(3) = 0v(12) = 1 v(13) = 4 v(23) = 7

v(123) = 15


Weber Set & Shapley Value

Core-Cover & τ -value

Core & Core-Center




The Core-Center: Basic Properties

Basic Properties





Basic Properties

• Efficiency





Basic Properties

• Efficiency • Individual rationality





Basic Properties

• Efficiency • Individual rationality• Stability





Basic Properties

• Efficiency • Individual rationality• Stability • Dummy player





Basic Properties

• Efficiency • Individual rationality• Stability • Dummy player• Symmetry





Basic Properties

• Efficiency • Individual rationality• Stability • Dummy player• Symmetry • Translation invariance





Basic Properties

• Efficiency • Individual rationality• Stability • Dummy player• Symmetry • Translation invariance• Scale invariance





Basic Properties

• Efficiency • Individual rationality• Stability • Dummy player• Symmetry • Translation invariance• Scale invariance • . . .




The Core-Center: Continuity





Continuity





Continuityϕ : Ω ⊆ R2n






−→ Rn






−→ Rn

v 7−→ ϕ(v)






−→ Rn

v 7−→ ϕ(v)

(0, 1)

(−a, 0) (a, 0)






−→ Rn

v 7−→ ϕ(v)

(0, 1)

(−a, 0) (a, 0)

(0, 13)






−→ Rn

v 7−→ ϕ(v)

(0, 1)

(−a, 0) (a, 0)

(0, 13)

(−a

2, 0) (a

2, 0)






−→ Rn

v 7−→ ϕ(v)

(0, 1)

(−a, 0) (a, 0)

(0, 13)

(−a

2, 0) (a

2, 0)

(0, 12)






−→ Rn

v 7−→ ϕ(v)

(0, 1)

(−a, 0) (a, 0)

(0, 13)

(−a

2, 0) (a

2, 0)

(0, 12)

Continuity??






−→ Rn

v 7−→ ϕ(v)

(0, 1)

(−a, 0) (a, 0)

(0, 13)

(−a

2, 0) (a

2, 0)

(0, 12)

Continuity??

TheoremThe Core-Center is a continuous allocation rule




The Core-Center: Monotonicity

• Take a pair of games v and w






Strong monotonicity






Strong monotonicity

Let i ∈ N . If for each S ⊆ N\i,

w(S ∪ i) − w(S) ≥ v(S ∪ i) − v(S),






Strong monotonicity


w(S ∪ i) − w(S) ≥ v(S ∪ i) − v(S), then ϕi(N, w) ≥ ϕi(N, v)






Strong monotonicity NOT SATISFIED











• w(T ) > v(T ) and for each S 6= T , w(S) = v(S)










Coalitional monotonicity











For each i ∈ T ,











For each i ∈ T , ϕi(N, w) ≥ ϕi(N, v)










Coalitional monotonicity NOT SATISFIED













Aggregate mononicity













T = N implies that for each i ∈ N ,













T = N implies that for each i ∈ N , ϕi(N, w) ≥ ϕi(N, v)












Aggregate mononicity NOT SATISFIED















Weak coalitional monotonicity














Weak coalitional monotonicity∑

i∈T ϕi(w) ≥∑

i∈T ϕi(v)














Weak coalitional monotonicity SATISFIED!!!∑

i∈T ϕi(w) ≥∑

i∈T ϕi(v)














Weak coalitional monotonicity SATISFIED!!!∑

i∈T ϕi(w) ≥∑

i∈T ϕi(v)Core-Center ⇐⇒ Nucleolus




The Core-Center: An Additivity Property





Superadditivity:





Superadditivity: If S ∩ T = ∅, then





Superadditivity: If S ∩ T = ∅, then v(S ∪ T ) ≥ v(S) + v(T )






Let v be a balanced game. Let T ( N .






Let v be a balanced game. Let T ( N . Let k ∈ [v(T ), v(N) − v(N\T )]







v(S) =

k T = S

v(S) otherwise







v(S) =

maxv(S), v(S\T ) + k T ⊆ S

v(S) otherwise







v(S) =


v(S) otherwise

v(S) =

maxv(S), v(S\(N\T )) + v(N) − k N\T ⊆ S

v(S) otherwise







v(S) =


v(S) otherwise

v(S) =


v(S) otherwise

“CUT”







v(S) =


v(S) otherwise

v(S) =


v(S) otherwise

Definitionϕ is a T -solution







v(S) =


v(S) otherwise

v(S) =


v(S) otherwise

Definitionϕ is a T -solution if for each pair v, v







v(S) =


v(S) otherwise

v(S) =


v(S) otherwise


ϕ(v) = αϕ(v) + (1 − α)ϕ(v)

where α ∈ [0, 1]






ϕ(v) = αϕ(v) + (1 − α)ϕ(v)

where α ∈ [0, 1]






ϕ(v) = αϕ(v) + (1 − α)ϕ(v)

where α ∈ [0, 1]

DefinitionDissection of a game v:






ϕ(v) = αϕ(v) + (1 − α)ϕ(v)

where α ∈ [0, 1]

DefinitionDissection of a game v: G(v) = v1, v2, . . . , vr






ϕ(v) = αϕ(v) + (1 − α)ϕ(v)

where α ∈ [0, 1]


Definitionϕ is an RT -solution if:






ϕ(v) = αϕ(v) + (1 − α)ϕ(v)

where α ∈ [0, 1]



1 ϕ is a T -solution






ϕ(v) = αϕ(v) + (1 − α)ϕ(v)

where α ∈ [0, 1]



1 ϕ is a T -solution

2 Translation Invariance




The Core-Center:

Balanced Games




The Core-Center:

Balanced Games

C(v)

21

3




The Core-Center:

Balanced Games

C(v)

k21

3




The Core-Center:

Balanced Games

3

s

C(v)

k21

3




The Core-Center:

Balanced Games

3

s

C(v)

k21

3 C(v)




The Core-Center:

Balanced Games

3

s

C(v)

k21

3 C(v)

C(v)




The Core-Center:

Balanced Games

3

s

C(v)

k21

3 C(v)

C(v)

CUT!!!




The Core-Center:

Balanced Games

3

s

C(v)

k21

3 C(v)

C(v)Definition




The Core-Center:

Balanced Games

3

s

C(v)

k21

3 C(v)

C(v)DefinitionLet v be a balanced game




The Core-Center:

Balanced Games

3

s

C(v)

k21

3 C(v)

C(v)DefinitionLet v be a balanced gameLet v′ and v′′ be two balanced games such that belong to somedissection of v




The Core-Center:

Balanced Games

3

s

C(v)

k21

3 C(v)


ϕ satisfies fair additivity with respect to the core if:




The Core-Center:

Balanced Games

3

s

C(v)

k21

3 C(v)



1 ϕ is a RT -solution




The Core-Center:

Balanced Games

3

s

C(v)

k21

3 C(v)



1 ϕ is a RT -solution

2 C(v′) = C(v′′) implies that αv(v′) = αv(v

′′)


Table of PropertiesShapley Nucleolus Core-Center


Efficiency

Individual Rationality

Continuity

Dummy Player

Symmetry

Translation and Scale Invariance


Efficiency X X X

Individual Rationality X X X

Continuity X X X

Dummy Player X X X

Symmetry X X X

Translation and Scale Invariance X X X


Efficiency X X X


Continuity X X X

Dummy Player X X X

Symmetry X X X


Stability


Efficiency X X X


Continuity X X X

Dummy Player X X X

Symmetry X X X


Stability X X X


Efficiency X X X


Continuity X X X

Dummy Player X X X

Symmetry X X X


Stability X X X

Strong Monotonicity

Coalitional Monotonicity

Aggregate Monotonicity

Weak Coalitional Monotonicity


Efficiency X X X


Continuity X X X

Dummy Player X X X

Symmetry X X X


Stability X X X

Strong Monotonicity X X X

Coalitional Monotonicity X X X

Aggregate Monotonicity X X X

Weak Coalitional Monotonicity X X X


Efficiency X X X


Continuity X X X

Dummy Player X X X

Symmetry X X X


Stability X X X





Additivity


Efficiency X X X


Continuity X X X

Dummy Player X X X

Symmetry X X X


Stability X X X





Additivity X X X


Efficiency X X X


Continuity X X X

Dummy Player X X X

Symmetry X X X


Stability X X X





Additivity X X X

Consistency (Davis/Maschler)

Consistency (Hart/Mas-Collel)


Efficiency X X X


Continuity X X X

Dummy Player X X X

Symmetry X X X


Stability X X X





Additivity X X X

Consistency (Davis/Maschler) X X X

Consistency (Hart/Mas-Collel)


Efficiency X X X


Continuity X X X

Dummy Player X X X

Symmetry X X X


Stability X X X





Additivity X X X


Consistency (Hart/Mas-Collel) X X X


Efficiency X X X


Continuity X X X

Dummy Player X X X

Symmetry X X X


Stability X X X





Additivity X X X



Fair Additivity w.r.t. the core


Efficiency X X X


Continuity X X X

Dummy Player X X X

Symmetry X X X


Stability X X X





Additivity X X X



Fair Additivity w.r.t. the core X X X



The Characterization





Fair Additivity ⇐⇒ Volume






Theorem






TheoremLet ϕ be an allocation rule satisfying







Efficiency







Efficiency

Translation Invariance







Efficiency


Weak Symmetry







Efficiency


Weak Symmetry

Continuity







Efficiency


Weak Symmetry

Continuity

Fair Additivity with respect to the core







Efficiency


Weak Symmetry

Continuity


Then, for each v ∈ BG,







Efficiency


Weak Symmetry

Continuity


Then, for each v ∈ BG, ϕ(v) coincides with the core-center







Efficiency


Extended Weak Symmetry

Continuity









Efficiency


Extended Weak Symmetry

Continuity



The axioms are independent




The Core-Center and the Shapley Value

Convex Games





Convex Games

3

1

2

4





Convex Games

3

1

2

4

I(v)





Convex Games

3

1

2

4

u1

I(v)





Convex Games

3

1

2

4

u1u2

I(v)





General ideas:





General ideas:

We define games u1, u2, u3, u4





General ideas:


Whose cores decompose I(v) inC(v), C(u1), C(u2), C(u3), C(u4)





General ideas:



The corresponding volumes arew0, w, w1, w2, w3, w4





General ideas:




We define the game v∗





General ideas:





We combine the two additivity properties





General ideas:






µ(N, v)





General ideas:






µ(N, v)Fair Add.

=





General ideas:






µ(N, v)Fair Add.

= µ(N, u0) −∑

i∈Nµ(N, ui)





General ideas:






µ(N, v)Fair Add.

=w0

wµ(N, u0) −

∑

i∈N

wi

wµ(N, ui)





General ideas:






µ(N, v)Fair Add.

=w0

wµ(N, u0) −

∑

i∈N

wi

wµ(N, ui)

=w0

wSh(N, u0) −

∑

i∈N

wi

wSh(N, ui)





General ideas:






µ(N, v)Fair Add.

=w0

wµ(N, u0) −

∑

i∈N

wi

wµ(N, ui)

=w0

wSh(N, u0) −

∑

i∈N

wi

wSh(N, ui)

Shap Add.=





General ideas:






µ(N, v)Fair Add.

=w0

wµ(N, u0) −

∑

i∈N

wi

wµ(N, ui)

=w0

wSh(N, u0) −

∑

i∈N

wi

wSh(N, ui)

Shap Add.= Sh(N, v∗)





3

1

2

4





3

1

2

4

u1





3

1

2

4

u1u2





3

1

2

4

u1u2u12




Cooperative Game TheoryConclusions

Conclusions





Conclusions

We have shown a geometric characterization of the τ value





Conclusions


We have studied a new allocation rule: the core-center





Conclusions



We have carried out an axiomatic analysis





Conclusions




We have presented an axiomatic characterization





Conclusions




We have presented an axiomatic characterization

We established a connexion between the core-center and theShapley value




Cooperative Game TheoryFuture Research

Future Research





Future Research

Try to find different characterizations of the core-center





Future Research


Deepen in the relation between the core-center and theShapley value





Future Research



Look for noncooperative foundations for the core-center(implementation)





Future Research




Look for a consistency property for the core-center





Future Research




Look for a consistency property for the core-center

Look for extensions to different classes of games




Cooperative Game TheoryReferences

References

Gonzalez-Dıaz, J., P. Borm, R. Hendrickx, M. Quant (2005):

“A Geometric Characterisation of the Compromise Value,” Mathematical

Methods of Operations Research 61

Gonzalez-Dıaz, J., E. Sanchez-Rodrıguez (2003):

“From Set-Valued Solutions to Single-Valued Solutions: The Centroidand the Core-Center,” Reports in Statistics and Operations Research03-09, University of Santiago de Compostela


“The Core-Center and the Shapley Value: A Comparative Study,”Reports in Statistics and Operations Research 03-10, Universidade deSantiago de Compostela


“A Characterization of the Core-Center,” Preprint


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





Part II: Cooperative Game Theory



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