Optimal Deterrence of Cooperation

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Optimal Deterrence of Cooperation

Stéphane Gonzalez, Aymeric Lardon

To cite this version:

WP 1621 – June 2016

Optimal Deterrence of Cooperation

Stéphane Gonzalez, Aymeric Lardon

Abstract:

We introduce axiomatically a new solution concept for cooperative games with transferable utility inspired by

the core. While core solution concepts have investigated the sustainability of cooperation among players, our

solution concept, called contraction core, focuses on the deterrence of cooperation. The main interest of the

contraction core is to provide a monetary measure of the robustness of cooperation into the grand coalition.

We motivate this concept by providing optimal fine imposed by competition authorities for the dismantling of

cartels in oligopolistic markets. We characterize the contraction core on the set of balanced cooperative

games with transferable utility by four axioms: the two classic axioms of non-emptiness and individual

rationality, a s uperadditivity principle and a new axiom of consistency.

Keywords:

TU-game, contraction core, optimal fine, Cournot oligopoly, axiomatization

JEL codes:

Optimal Deterren e of Cooperation

∗

Stéphane Gonzalez

†

Aymeri Lardon

‡

June 14,2016 Abstra t

Weintrodu eaxiomati allyanewsolution on eptfor ooperativegameswithtransferable utilityinspiredbythe ore. While oresolution on eptshaveinvestigatedthesustainability of ooperationamongplayers,our solution on ept, alled ontra tion ore, fo useson the deterren eof ooperation. Themaininterestofthe ontra tion oreistoprovideamonetary measureoftherobustnessof ooperationintothegrand oalition. Wemotivatethis on ept byprovidingoptimalneimposedby ompetitionauthoritiesforthedismantlingof artelsin oligopolisti markets. We hara terizethe ontra tion oreonthesetofbalan ed ooperative gameswithtransferableutilitybyfouraxioms: thetwo lassi axiomsofnon-emptinessand individualrationality,asuperadditivityprin ipleandanewaxiomof onsisten y.

Keywords: TU-game, ontra tion ore,optimalne,Cournotoligopoly,axiomatization JELClassi ation: C71,D43

1 Introdu tion

One of the main issues in ooperative game theory on erns the possibility for players to oop-erate alltogether. A well-known solution on ept for ooperative games with transferable utility (hen eforth TU-games) dealing with the existen e of stable ooperative agreements is the ore (Gillies,1953). The lassi Bondareva-Shapley theoremestablishesthatthe non-emptinessof the ore is hara terized by the balan edness property as proved independently in Bondareva (1963) and Shapley(1967). A possible interpretationof the balan ednessproperty whi hwe areinterest in isthe following: ea h playermustdistributeoneunit oftimeamong allthe oalitionsofwhi h sheisamember; thebalan ednesspropertystipulatesthattheoptimaltimeallo ationforplayers is todevote alltheirunit oftime intothe grand oalition,i.e.,thewhole setof players.

Eveninthe ase where the ore isempty,the literaturehas investigated the possibilityto en-for eastable ooperativeagreementby introdu ing other oresolution on epts: thestrong and theweak

ǫ

- ores(ShapleyandShubik,1966),theleast ore(Mas hleretal.,1979), theaspiration

∗

This work was supported by the Programme Avenir Lyon Saint-Etienne of Université de Lyon, within the programInvestissementsd'Avenir (ANR-11-IDEX-0007).

†

Univ Lyon, UJM Saint-Etienne, CNRS, GATE L-SE UMR 5824, F-42023 Saint-Etienne, Fran e. e-mail: stephane.gonzalez

@

univ-st-etienne.fr

‡

Correspondingauthor. Universityof Ni eSophia-Antipolis,CNRS, GREDEGUMR7321, F-06560Valbonne, Fran e. e-mail:aymeri .lardon

@

uni e.fr

thenego iationset(Gonzalez andGrabis h,2015b)andthed-multi oalitional ore(Gonzalez and Grabis h,2016). Allthese oreextensionsarenon-emptywhenappliedtonon-balan edTU-games and oin ide withthe oreon the set ofbalan ed TU-games.

Untilnow, solution on eptsinspiredbythe orehaverestri tedattentiontothesustainability of ooperation. Nevertheless,inmany ompetitiveenvironments, ooperationisnotso ially desir-able,andplayersmustbedis ouragedtoworkalltogether. Forexample,horizontalagreementson pri es betweenrms are punishedby ompetitionauthorities. Similarly,drug artelsare reproved to prote t population. In the same vein, the dismantling of terrorist groups appears to be of primary importan e for nationalse urity. Furthermore, to the best of our knowledge, even when ooperation is e ient, the robustness of stable ooperative agreements has not been studied yet. Forexample, howmu hthe ohesion of ollaborativea tivitieson resear h anddevelopment is sensible to dis overy values? Or does the stability of trade agreements depend ru ially on transportation osts? Tomeetthese hallengesheadon,ageneralsolution on ept spanning sev-eral elds of e onomi s (industrialorganization, innovation, international trade, riminology...) appears fundamentalin orderto provide insightintothe deterren eof ooperation.

Inthisarti le,weinvestigatethedeterren eof ooperationamongplayersforbalan ed ooper-ativeTU-gamesbyimposingmonetarypenaltyonthe grand oalition. Pre isely,we areinterested in nding the minimalamount of ne, alled the optimal ne, under whi h ooperation an no longer be sustained. This leads us to onsider a new solution on ept, alled ontra tion ore, whi h ontains allstable ooperative agreementsfor whi hany ne in rease makesthese agree-ments unstable. Inthis sense,the ontra tion ore ontainsallthe weakest stable ooperative agreements further to the optimal ne imposed on the grand oalition. This ne an be inter-preted as a measure of the robustness of ooperation into the grand oalition. Interms of time allo ation, thismeans that authority detersthe formationof the grand oalitionand that players must devotefra tionsoftheirunit oftimetoany other oalitionasase ond best timeallo ation. This notion will be used for the denition of feasibility and e ien y onditions related to our solution on ept. Unlike the ore solution on epts mentioned above, the ontra tion ore does not ontain the ore, and so it is not a ore extension. Moreover, the ontra tion ore has the advantage of beingasingletonon the set ofbalan ed and symmetri TU-games.

Followinginthefootstepsofpreviousworks(Tro kel,2005;Moulin,2014)whi hdealwith mi roe- onomi sbyusing ooperative on epts,weproposeanillustrativeexampleofoligopolisti markets in order to motivate our solution on ept. In e onomi welfare analysis, it is a well-established and old idea that monopoly power often negatively ae ts so ial welfare. Although oopera-tionon resear hand developmenta tivitiesmay havebene ialwelfareee ts(D'Aspremontand Ja quemin, 1988),mostof horizontalagreementson salespri es are onsidered asharmfulto so- ialwelfare. The ooperativeapproa h ofoligopolygames is ofgreatinterest inorderto analyze the stabilityof artelswhi h areone ofthe mainpreo upationsof ompetition authorities.

1 We pointout thatouranalysisdoesnot pay attentiontothewelfareee tsoftraderestri tionas ad-vo atedbytheruleofreasoninantitrustlaw,butfo usesonthedeterren eofthemonopolypower whi hleads, apriori, towelfarelosses. Thus,the ontra tion ore onstitutesan ee tivetoolto

1

ThedevelopingtheoryofoligopolyTU-games omprisesmany ontributionssu hasZhao(1999),Nordeetal. (2002),DriessenandMeinhardt(2005),Lardon(2012)andLekeasandStamatopoulos(2014)amongothers.

γ

- hara teristi fun tionform(HartandKurz,1983)whi hisplausibleinthe ontextofoligopoly industries. Under this approa h, the worth of any oalition (the artel prot) is enfor ed by a ompetition setting in whi h any artel fa es external rms a ting individually. We assumethat the inverse demandfun tion is linear and rms operateat onstant and identi almarginal osts. These assumptions ensure that the balan edness property holds on this set of Cournot oligopoly TU-games as shown by Lardon (2012),and so the ontra tion oreis well-dened. Afterhaving determinedtheworthofany oalition,we omputethe ontra tion oreandprovideanexpression ofthe optimalneimposedby ompetitionauthoritiesinordertodeter thegrand oalitionwhi h orresponds, inthepresent ase,tothe artel omprisingalltherms. Surprisingly,thisexpression diersdepending on thenumberofrms and leadsto distinguishmarkets ofsmallsize (lessthan ve rms)and thoseof mediumand largesize (morethan sixrms).

Beyondthise onomi appli ation,inordertogetabettergraspofthe ontra tion ore,weprovide an axiomati hara terization ofthisnew solution on ept onthe setof balan edTU-games. We invoke thetwo lassi axiomsofnon-emptinessand individualrationalityaswellastwo newother axioms ofsuperadditivity and onsisten y. Theoriginalsuperadditivity and onsisten yproperties (Peleg, 1986) used to hara terize the ore, impli itlydepend on grand oalition feasibility. We repla e them with similar properties based on a new denition of feasibility derived from non-trivial oalitionformationwhi hrelieson se ondbesttimeallo ationforplayers.

2

Weimposethis feasibilityrequirementon oursuperadditivityprin iple. Consisten yprin ipleisbasedonan appro-priated redu ed games property. Traditionalredu ed games (Davis and Mas hler,1965) used by Peleg(1986)makeanex eptiontothegrand oalitioninordertoensuregrand oalitionfeasibility. Bejan and Gómez (2012) use a more general version (Moldovanu and Winter, 1994) that treats all oalitionsin the sameway. Ournew axiom of onsisten yis basedon a new modied version of redu ed games whi h make again an ex eption to the grand oalition. Pre isely, unlike any other oalition,thegrand oalitionofanyredu edgameisnotallowed to ooperatewiththe om-plementary oalition. Moreover, given any se ond best time allo ation in the originalTU-game, we provide a formulato ompute the orresponding se ond best time allo ation in any redu ed game. Our axioms ofsuperadditivity and onsisten y do not oin ide withthose ofPeleg (1986) and theirgeneralizedversions inBejan and Gómez (2012)on the set ofbalan ed TU-games. Thearti leis organized as follows. Se tion

2

presentsthe ontra tion oreas wellas someofits properties. Se tion

3

gives an illustrative example of oligopolisti markets for the deterren e of the monopoly power. In Se tion

4

, we provide an axiomati hara terization of the ontra tion ore. Se tion

5

dealswith anaturalextensionof the ontra tion oretotheset ofallTU-games.

2 Cooperatives games and the ontra tion ore

2.1 Cooperatives games with transferable utility

A ooperative TU-game is an ordered pair

(N, v)

onsisting of a nite set of players

N

and a hara teristi fun tion

v : 2

N

_{−→ R}

su h that

v(∅) = 0

where

2

N

denotes the power set of

N

. Subsets of

N

are alled oalitions, and we all

v(S)

the worth of oalition

S

. The size of 2

oalition

S

isdenoted by

s = |S|

. Let

Γ

denote the set ofTU-games.

Later in the paper, we willuse both simpleand symmetri TU-games. ATU-game

(N, v)

is simple if for any oalition

S ∈ 2

N

_{\{∅, N }}

, we have

v(S) ∈ {0, 1}

and

v(N ) = 1

. A oalition

S

su h that

v(S) = 1

is alled a winning oalition. A player

i ∈ N

is alled a veto player if she belongs toany winning oalition. ATU-game

(N, v)

is symmetri ifthere existsa mapping

f : N −→ R

su h thatforany oalition

S ∈ 2

N

_\{∅}

,we have

v(S) = f (s)

. Let

B ⊆ 2

N

_\{∅}

bea olle tionof oalitions. Then

B

issaidtobeabalan ed olle tion of oalitionsifforevery

S ∈ B

thereexistsabalan ingweight

δ

S

∈ R

+

su hthat

P

S∈B:i∈S

δ

S

= 1

forevery

i ∈ N

. Consider

δ

S

asanamountoftimeallo atedto oalition

S

byanyofitsmembers. When ea h player has one unit of time, the requirement that

P

S∈B:i∈S

δ

S

= 1

is then a time feasibility ondition. We denote by

Λ(N )

the set of balan ed olle tions and

Λ

∗

_{(N )}

the set of balan ed olle tions not ontaining the grand oalition where

n ≥ 2

. By onvention,

Λ

∗

_{(N ) = Λ(N )}

when

n = 1

. A TU-game

(N, v)

is balan ed if for every balan ed olle tion

B ∈ Λ(N )

it holds that

P

S∈B

δ

S

v(S) ≤ v(N )

. Let

Γ

c

denote the subset of balan ed TU-games. Ontheset

Γ

c

thebesttimeallo ationforplayersistodevotealltheirunitoftimetothe grand oalition.

2.2 Feasibility as se ond best time allo ation

Wenowintrodu etheappropriate notionoffeasibilitywhi hwillbeusefulforthedenitionofthe ontra tion ore. In aTU-game

(N, v) ∈ Γ

,everyplayer

i ∈ N

may re eive apayo

x

i

∈ R

. A ve tor

x ∈ R

n

isa payo ve tor. Forany oalition

S ∈ 2

N

_\{∅}

and any payo ve tor

x ∈ R

n

, wedene

x(S) =

P

i∈S

x

i

and we denoteby

x

S

_{∈ R}

s

theve torsu hthat

x

S

i

= x

i

forall

i ∈ S

. Generallyspeaking,feasibilityisarestri tiononplayers'payosand anbeinterpretedinterms oftimeallo ation. The lassi feasibility ondition, alled thegrand oalitionfeasibility,isdened asthe setofpayove tors, denotedby

X(N, v)

,thatarefeasiblewhenplayersallo atetheirunit of timetothe grand oalition,i.e.:

X(N, v) = {x ∈ R

n

: x(N ) ≤ v(N )}

.

A more relaxed feasibility ondition whi h onsiders non-trivial oalition formation, is dened as thesetofpayove tors,denotedby

X

Λ

(N, v)

,thatarefeasiblewhenplayers andevotefra tions of theirtimeto any oalition,notjust thegrand oalition,i.e.:

X

Λ

(N, v) = {x ∈ R

n

: x(N ) ≤

X

S∈B

δ

S

v(S)

forsome

B ∈ Λ(N )}

.

On the set

Γ

c

, both onditions of feasibility are equivalent sin e the best time allo ation for playersistoformthegrand oalition. Now,supposeauthoritypreventstheformationofthegrand oalition. Thefeasibility onditionwhi hweareinterestedinbe omessomepossiblearrangements of players devotingfra tions oftheir timetoany oalition ex ept thegrand oalition.

Denition 2.1 For any TU-game

(N, v) ∈ Γ

c

, the set of feasible payo ve tors of

(N, v)

, denoted by

X

Λ

∗

(N, v)

,isdened as:

X

Λ

∗

(N, v) = {x ∈ R

n

: x(N ) ≤

X

S∈B

δ

S

v(S)

forsome

B ∈ Λ

∗

_{(N )}}

,

Onthe set

Γ

c

,thisfeasibility onditionrelieson oalitionformationofplayersasse ond besttime allo ation. Thisleads todene the asso iatede ien y ondition.

Denition 2.2 For any TU-game

(N, v) ∈ Γ

c

, the set of e ient payo ve tors of

(N, v)

, denoted by

X

∗

Λ

∗

(N, v)

,isdened as:

X

_Λ

∗

∗

(N, v) = arg max{x(N )

:

x ∈ X

Λ

∗

(N, v)}

.

On the set

Γ

c

,any e ient payove torisexa tly a hieved by a se ondbest timeallo ationfor players.

2.3 Contra tion ore

Thenewfeasibilityande ient onditionsrelatedtose ondbest timeallo ationpermittodene the mainobje t ofour studyon the set

Γ

c

, the ontra tion ore.

Denition 2.3 For any TU-game

(N, v) ∈ Γ

c

,the ontra tion ore, denoted by

CC(N, v)

,is dened as:

CC(N, v) = {x ∈ X

_Λ

∗

∗

(N, v) : ∀S ⊂ N

,

x(S) ≥ v(S)}

.

The ontra tion ore ontains all e ient payo ve tors 3

a hieved by any se ond best time al-lo ation that satisfy a relaxed oalitional stability ondition for whi h the grand oalition is not taken into a ount.

The following are the denitions of the ore and the aspiration ore for whi h we will make omparisons with the ontra tion ore.

4

The ore (Gillies, 1953) of a TU-game

(N, v) ∈ Γ

, denoted by

C(N, v)

,is denedas:

C(N, v) = {x ∈ X(N, v) : ∀S ⊆ N

,

x(S) ≥ v(S)}

.

Bondareva (1963) and Shapley (1967) showed that any TU-game

(N, v) ∈ Γ

c

if and only if

C(N, v) 6= ∅

. The aspiration ore (Albers, 1979; Cross, 1967; Bennett, 1983) of a TU-game

(N, v) ∈ Γ

,denoted by

AC(N, v)

,is denedas:

AC(N, v) = {x ∈ X

Λ

(N, v) : ∀S ⊆ N

,

x(S) ≥ v(S)}

.

Both the ore and the aspiration ore ontainallfeasible payo ve tors (with the understanding thatwe onsidergrand oalitionfeasibilityfortheformerandfeasibilityasrstbesttimeallo ation forthe latter)thatsatisfythe lassi oalitionalstability ondition.

3

Weneedtousethesetofe ientpayove torsinthedenitionofthe ontra tion oreinordertodealwith theone-player ase.

4

Whilethe ontra tion oreisdenedonthesubset

Γ

c

,the oreandtheaspiration orearedenedontheset

We now show that the ontra tion ore is relevant in order to deal with the deterren e of oop-eration. Given any TU-game

(N, v) ∈ Γ

c

and any

t ∈ R

+

, its t- ontra tion is the TU-game, denoted by

(N, v

t

₎

,su hthat

v

t

_{(S) = v(S)}

forany

S ⊂ N

,and

v

t

_{(N ) = v(N ) − t}

. Inparti ular, we assign real number t

(N, v)

, alled the optimal ne, to any TU-game

(N, v) ∈ Γ

c

, whi h is dened as: t

(N, v) =



inf{t ∈ R : ∀k > t

,

(N, v

k

₎

isnot balan ed

}

if

n ≥ 2

;

0

if

n = 1

.

Thet

(N, v)

- ontra tion orrespondstotheoriginalTU-game

(N, v)

forwhi hthegrand oalition must pay optimal ne t

(N, v)

. This optimal ne gives the minimal amount for whi h any ne in rease makes ooperation into the grand oalitionunstable. It an be onsidered as ameasure of the robustness of stable ooperative agreements.

5

An alternative formulaof the optimal ne easier to omputeis the following:

t

(N, v) = v(N ) − max

B∈Λ

∗

_{(N )}

X

S∈B

δ

S

v(S)

.

We show that the ontra tion ore of any TU-game

(N, v) ∈ Γ

c

is equal to the ore of its t

(N, v)

- ontra tion.

Proposition 2.4 Forany TU-game

(N, v) ∈ Γ

c

,itholdsthat

CC(N, v) = C(N, v

t

(N,v)

)

.

Proof: First, we prove that

C(N, v

t

(N,v)

) ⊆ CC(N, v)

. Take any

x ∈ C(N, v

t

(N,v)

)

. Then, it holds that:

x(N ) = v

t

(N,v)

(N )

= v(N ) −

t

(N, v)

= max

B∈Λ

∗

_{(N )}

X

S∈B

δ

S

v(S)

. Hen e

x ∈ X

∗

Λ

∗

(N, v)

. Moreover,forany

S ∈ 2

N

_{\{∅, N }}

we have:

x(S) ≥ v

t

(N,v)

(S)

= v(S)

,

whi hprovesthat

x ∈ CC(N, v)

.

Se ond, we prove that

CC(N, v) ⊆ C(N, v

t

(N,v)

)

. Take any

x ∈ CC(N, v)

. Sin e

x ∈

X

_Λ

∗

∗

(N, v)

the above equalities imply that

x(N ) = v

t

(N,v)

(N )

. Moreover, it holds that for any

S ∈ 2

N

_{\{∅, N }}

,

x(S) ≥ v(S) = v

t

(N,v)

(S)

. Hen e

x ∈ C(N, v

t

(N,v)

)

.



5

optimal ne imposed on the grand oalition. This means that authority deters the formation of the grand oalition whi h ompels players to nd another almost unstable agreement in the ontra tion ore.

2.5 Properties of the ontra tion ore

Oneofthemainadvantagesofthe ontra tion oreistobeasingletononthesetofbalan edand symmetri TU-games. Inordertoprove thisresult,we rstintrodu ethe on ept ofautonomous oalition (Gonzalez and Grabis h, 2015a). Given any TU-game

(N, v) ∈ Γ

c

, a oalition

S ∈

2

N

_\{∅}

isautonomousfor

(N, v)

ifforanypayove tor

x ∈ C(N, v)

,itholdsthat

x(S) = v(S)

. Proposition 2.5 (Gonzalez and Grabis h, 2015a) For any TU-game

(N, v) ∈ Γ

c

, the following statementsare equivalent:

1. There existsa oalition

S ∈ 2

N

_{\{∅, N }}

whi hisautonomous for

(N, v)

. 2. For all

t > 0

,itholds that

C(N, v

t

_{) = ∅}

.

Furthermore, Gonzalez and Grabis h (2015a) prove that the set of autonomous oalitions is a balan ed olle tion.

Proposition 2.6 For any symmetri TU-game

(N, v) ∈ Γ

c

,the ontra tion ore

CC(N, v)

is a singleton.

Proof: It follows from the symmetry of

(N, v)

that its t

(N, v)

- ontra tion

(N, v

t

(N,v)

)

is also symmetri . By Proposition 2.4, it holds that

CC(N, v) = C(N, v

t

(N,v)

)

. It is well-known that payove tor

x ∈ R

n

su hthat

x

i

= v

t

(N,v)

(N )/n

forall

i ∈ N

isa oreelementofanysymmetri andbalan edTU-game. Moreover,itfollowsfromProposition2.5thatthereexistsanautonomous oalition

K ⊂ N

of size

k < n

. The symmetryof

(N, v)

impliesthatany oalition

S

of size

k

is also autonomous. The olle tionof all oalitionsofsize

k

,denotedby

B

,isabalan ed olle tion with weight

δ

S

=

n−1

k−1

forany

S ∈ B

. Wedene payo ve tor

x

′

_{∈ R}

n

su h that

x

′

i

= v(K)/k

forall

i ∈ N

. Hen e,it holdsthat:

x

′

(N ) =

P

S∈B

n−1

k−1

x

′

(S)

=

P

S∈B

n−1

k−1

v(S)

= v

t

(N,v)

(N )

,

where thelastequalityfollowsfromProposition2.5. We on ludethat

x

i

= x

′

i

forall

i ∈ N

,and so

x

′

_{∈ CC(N, v)}

. It remains toshow that

x

′

_{∈ CC(N, v)}

is the unique element of the ontra tion ore. Suppose by ontradi tion that there exists

y ∈ CC(N, v)

su h that

y 6= x

′

. Then, there exists a player

j ∈ N

su h that

y

j

> v(K)/k

and a player

i ∈ N

su h that

y

i

< v(K)/k

. Sin e

k < n

, there exists an autonomous oalition

T

of size

k

su h that

j ∈ T

and

i 6∈ T

. Hen e, it holds that

P

r∈T \{j}

y

r

< (v(K)/k) × (k − 1)

,andso

P

r∈(T ∪{i})\{j}

y

r

< v(K)

,a ontradi tionsin e

(T ∪ {i})\{j}

isalso anautonomous oalition.



Proposition 2.7 Forany simpleTU-game

(N, v) ∈ Γ

withatleasttwo vetoplayers andatleast two winning oalitions,the ontra tion oreis nota singleton.

Proof: Itisknownthatthe oreofanysimpleTU-game ontainsanypayove torthatdistributes allthe gainsofthe grand oalitionamong vetoplayers. Takeany simpleTU-game

(N, v)

with at least two vetoplayers and at leasttwo winning oalitions. It holds that

v(N ) = 1

and

v(S) = 1

for some

S ∈ 2

N

_{\{∅, N }}

whi h implies that

CC(N, v) = C(N, v)

. Moreover, sin e there are at least two veto players, the above-mentioned result on the ore permits to on lude that the

ontra tion oreis nota singleton.



Thefollowingexampleshows thatthe ontra tion oremay not be asingletonevenon the setof allTU-games.

Example 2.8 ConsidertheTU-game

(N, v) ∈ Γ

c

su hthat

N = {1, 2, 3}

,

v({1}) = v({2}) = 0

,

v({3}) = 3

,

v({1, 2}) = 6

,

v({1, 3}) = v({2, 3}) = 0

,and

v({1, 2, 3}) = 15

. Then, itholdsthat t

(N, v) = 6

and

CC(N, v) =

onvexhull

{(6, 0, 3); (0, 6, 3)}

.

3 Illustrative example

In this se tion, we propose to apply the ontra tion ore to oligopolisti markets in order to ompute the optimal ne imposed by ompetition authorities for artel deterren e. We analyze a quantity ompetition between

n

rms. Every rm

i ∈ N

produ es quantity

q

i

∈ R

+

of a homogeneous good. Furthermore, we onsider the linearinverse demand fun tion:

p(Q) = a − bQ

,

where

a

is the inter eptof demand,

b

is the slope of

p

and

Q =

P

i∈N

q

i

is the totaloutput of the market. Ea h rm produ es at onstant average and marginal ost

c ∈ R

+

. Prots for the

i

thprodu erin termsof quantities,

π

i

,isexpressedas:

π

i

(q) = (p(Q) − c)q

i

. Withoutlossof generality,weassumethat

c = 0

.

FollowingHartandKurz(1983)andChanderand Tulkens(1997), we onsiderthesituationin whi hanysubsetofrms

S

forma artel( oalition)whiletheothers ontinuetoa tindependently. Cartel members are assumed to a t as a single rm maximizing their joint prot by orrelating theirstrategies. Thisleads to onsiderthe set ofCournotoligopoly TU-gamesin

γ

- hara teristi fun tion formdened as:

v

γ

(S) =

X

i∈S

π

i

(q

∗

i

, ˜

q

j

)

, where

(q

∗

i

, ˜

q

j

)

is the Cournot-Nash equilibriumbetween

S

and the other players with the under-standingthatea hplayer

i ∈ S

produ esidenti alquantity

q

∗

the samequantity

q

˜

j

. 6

Underthese onsiderations,we an omputethe worthofany oalitionas established in thefollowing proposition.

Proposition 3.1 Let

(N, v

γ

)

be an oligopoly TU-game in

γ

- hara teristi fun tion form. Then forany oalition

S ∈ 2

N

_\{∅}

,itholds that:

v

γ

(S) =

1

b



a

n − s + 2



2

.

Proof: Takeany

S ∈ 2

N

_\{∅}

. Cartelmembersandoutsiders'optimalquantitiesare hara terized by the rst order onditions:

∀i ∈ S

,

∂

∂q

i

X

i∈S

π

i

(q) = 0 ⇐⇒ 2b

X

i∈S

q

i

∗

= a − b

X

j∈N \S

q

j

, and

∀j ∈ N \S

,

∂

∂q

j

π

j

(q) = 0 ⇐⇒ 2b˜

q

j

= a − b

X

k∈N \{j}

q

k

,

respe tively. Sin ethe inverse demand fun tion islinearand rms operateatthe same marginal ost, any Cournot-Nashequilibriumimpliesthatidenti alparties must hooseidenti alstrategies (quantities), i.e., for any

i, k ∈ S

,

q

∗

i

= q

k

∗

and for any

j, l ∈ N \S

,

q

˜

j

= ˜

q

l

. From this remark, the interse tionof the two aboverea tionfun tions yields:

q

_i

∗

=

a

sb(n − s + 2)

and

q

˜

j

=

a

b(n − s + 2)

, whi hpermitsto omputethe worthof oalition

S

as:

v

γ

(S) =

X

i∈S

π

i

(q

∗

i

, ˜

q

j

)

=

1

b



a

n − s + 2



2

.

This on ludesthe proof.



Proposition 3.1 shows that any Cournot oligopoly TU-game

(N, v

γ

)

is symmetri . The worth

v

γ

(S)

ofany oalition

S

is in reasingwith the inter ept ofdemand

a

and the size

s

of oalition

S

. Moreover,itisde reasingwith theslope

b

and the numberof outsiders

n − s

. It follows from Lardon (2012) that

(N, v

γ

) ∈ Γ

c

.

It is now possible to provide the optimal ne imposed by ompetition authorities in order to deter the grand oalition.

6

Proposition 3.2 Forany Cournot oligopolyTU-game

(N, v

γ

)

,itholds that: t

(N, v

γ

) =























1

b

 a(n − 1)

2(n + 1)



2

if

n ≤ 5

;

a

2

b



5n − 9

36(n − 1)



if

n > 6

.

Proof : Sin e the Cournot oligopoly TU-game

(N, v

γ

)

is symmetri , the non-emptiness of the oreis hara terized by thefollowing ondition:

∀S ∈ 2

N

\{∅}

,

v

γ

(S)

s

≤

v

γ

(N )

n

. It followsthatthe optimalpenalty t

(N, v

γ

)

an be omputed as:

t

(N, v

γ

) = v

γ

(N ) − n max

S⊂N

v

γ

(S)

s

=

a

2

4b

− n

s∈{1,...,n−1}

max

a

2

sb(n − s + 2)

2

.

Itremainstondthesize

s

whi hminimizesthefun tion

f (s) = s(n−s+2)

2

denedon

[1; n−1]

. We dedu e from

f

′

_{(s) = (n − s + 2)(n − 3s + 2)}

and

f

′′

_{(s) = −4n + 6s − 8}

that

f

: - attainsits maximumatpoint

s

∗

_{= (n + 2)/3}

where

1 < s

∗

_{< n − 1}

forany

n ≥ 3

; - isstri tly in reasingon

[1; s

∗

_]

and stri tly de reasingon

[s

∗

_{; n − 1]}

.

Hen e itholds that

arg min

s∈[1,...,n−1]

f (s) ⊆ {1; n − 1}

. We distinguishtwo ases: - rst,if

n = 2

ittriviallyholdsthat

f

attainsitsminimum at

s = 1

.

- se ond assumethat

n ≥ 3

. It follows from

f (1) = (n + 1)

2

and

f (n − 1) = 9(n − 1)

that:

arg

min

s∈[1,...,n−1]

f (s) =























{1}

if

3 < n < 5

;

{1; n − 1}

if

n = 5

;

{n − 1}

if

n > 5

. Thus,when

2 ≤ n ≤ 5

itholdsthat:

t

(N, v

γ

) =

a

2

4b

− n

a

2

b(n + 1)

2

=

1

b

 a(n − 1)

2(n + 1)



2

.

t

(N, v

γ

) =

a

2

4b

− n

a

2

9b(n − 1)

=

a

2

b



5n − 9

36(n − 1)



,

whi h on ludes the proof.



Proposition3.2showsthatthe optimalne imposedby ompetitionauthoritiesisin reasingwith theinter eptofdemand

a

and thenumberofrms

n

. Moreover,itisde reasingwiththe slope

b

. Surprisingly,the expressionof the optimal ne leads todistinguishmarkets of smallsize (

n ≤ 5

) and thoseof mediumand largesize (

n ≥ 6

) forthe deterren eof the monopoly power.

We know by Propositions2.6and3.1thatthe ontra tion oreisasingleton. Proposition3.2 permitstogo furtherby providingan expressionof the ontra tion ore.

Corollary 3.3 Forany CournotoligopolyTU-game

(N, v

γ

)

,the ontra tion oreisexpressed as:

CC(N, v

γ

) =































(

1

b



a

n + 1



2

× e

)

if

n ≤ 5

;

(

_

a

2

9b(n − 1)



× e

)

if

n ≥ 6

; where

e = (1, . . . , 1)

.

Inbothmarkettypes,ea hindividualpayointhe ontra tion oreisin reasingwiththeinter ept of demand

a

and de reasingwith the slope

b

and the numberof rms

n

.

4 Axiomatization of the ontra tion ore

In this se tion, we provide an axiomati hara terization of the ontra tion ore on the set of balan ed TU-games.

Let

Γ

0

beanyarbitrarysubsetof

Γ

. Asolutionon

Γ

0

isamapping

σ

thatassignsa(possibly empty)set

σ(N, v) ⊆ X

Λ

∗

(N, v)

toany TU-game

(N, v)

.

4.1 Axioms

We nowpresentthe axioms relevant toour analysis. The rsttwo are lassi inthe literature on oreaxiomatizations.

Denition 4.1 Non-emptiness (NE) A solution

σ

on

Γ

0

satises NE if for any

(N, v) ∈ Γ

0

,

Denition 4.2 Individualrationality(IR)Asolution

σ

on

Γ

0

satisesIRifforany

(N, v) ∈ Γ

0

, every

x ∈ σ(N, v)

,and every

i ∈ N

,

x

i

≥ v({i})

.

Both ofthese axiomsare satisedby all oreextensionsdis ussed inthe introdu tion,andso are useful in hara terizingthem.

Next,weintrodu ethreeversionsofredu edgamesandtheir orresponding onsisten yaxioms in order to make ore omparisons. Take any

(N, v) ∈ Γ

, any

S ∈ 2

N

_\{∅}

and any

x ∈ R

n

. The rst redu ed game type makes a spe ial treatment to the grand oalition and permits to hara terize the ore (Peleg, 1986). The DM-redu ed game (Davis and Mas hler, 1965) of

(N, v)

with respe tto

S

and

x

isthe game

(S, v

S,x

) ∈ Γ

dened forany

T ∈ 2

S

as:

v

S,x

(T ) =







0

if

T = ∅

;

v(N ) − x(N \S)

if

T = S

;

max{v(T ∪ Q) − x(Q) : Q ⊆ N \S}

otherwise.

Denition 4.3 DM- onsisten y (DM-CON) Asolution

σ

on

Γ

0

satises DM-CON iffor any

(N, v) ∈ Γ

,every

S ∈ 2

N

_\{∅}

and every

x ∈ σ(N, v)

,then

(S, v

S,x

) ∈ Γ

0

and

x

S

_{∈ σ(S, v}

S,x

)

. The se ond version is more general and treats all oalitions in the same way and permits to hara terizethe aspiration ore. ThemodiedDM-redu edgame(BejanandGómez,2012)of

(N, v)

with respe tto

S

and

x

isthe game

(S, v

S,x

∗

) ∈ Γ

denedforany

T ∈ 2

S

as:

v

S,x

_∗

(T ) =



0

if

T = ∅

;

max{v(T ∪ Q) − x(Q) : Q ⊆ N \S}

otherwise.

Denition 4.4 MDM- onsisten y (MDM-CON) A solution

σ

on

Γ

0

satises MDM-CON if for any

(N, v) ∈ Γ

, every

S ∈ 2

N

_\{∅}

and every

x ∈ σ(N, v)

, then

(S, v

S,x

∗

) ∈ Γ

0

and

x

S

_∈

σ(S, v

S,x

∗

)

.

We an verifythat the ontra tion oredoesnot satises MDM-CONon

Γ

c

.

Example 4.5 Consider the TU-game

(N, v) ∈ Γ

c

su h that

N = {1, 2, 3}

,

v({1}) = v({2}) =

v({3}) = 0

,

v({1, 2}) = 4

,

v({1, 3}) = v({2, 3}) = 2

, and

v({1, 2, 3}) = 10

. It holds that t

(N, v) = 6

and

CC(N, v) = C(N, v

t

(N,v)

) = {(2, 2, 0)}

. When

S = {1}

and

x = (2, 2, 0)

, the modied DM-redu ed game is given by

v

{1},x

∗

({1}) = v({1, 2, 3}) − 2 − 0 = 8

. Thus,

2 6∈ CC({1}, v

∗

{1},x

) = {8}

sothatthe ontra tion oredoesnot satisedMDM-CON.

The third versionwhi h is relevant for our resultsmakes again aspe ial treatment to the grand oalitionofanyredu edgamewhi hisnotallowedto ooperatewiththe omplementary oalition. This permits to satisfythe feasibility ondition related tose ond best time allo ation. Thenew modiedDM-redu edgameof

(N, v)

withrespe tto

S

and

x

isthegame

(S, v

∗

S,x

) ∈ Γ

dened forany

T ∈ 2

S

as:

v

∗

_S,x

(T ) =







0

if

T = ∅

;

max{v(T ∪ Q) − x(Q) : Q ⊂ N \S}

if

T = S

;

max{v(T ∪ Q) − x(Q) : Q ⊆ N \S}

otherwise.

Denition 4.6 NMDM- onsisten y(NMDM-CON)Asolution

σ

on

Γ

0

satisesNMDM-CON if for any

(N, v) ∈ Γ

, every

S ∈ 2

N

_\{∅}

and every

x ∈ σ(N, v)

, then

(S, v

∗

S,x

) ∈ Γ

0

and

x

S

∈ σ(S, v

∗

S,x

)

.

Observe thatthe three axioms of onsisten y denedabovesatisfy the following logi alequality: DM-CON

∨

NMDM-CON

=

MDM-CON.

Thelastaxiomdiersfromthe lassi superadditivity axiomon the feasibilityrequirement.

Denition 4.7 ConditionalSuperadditivity (C-SUPA)Asolution

σ

on

Γ

0

satisesC-SUPAif forany

(N, v

A

)

,

(N, v

B

) ∈ Γ

0

,every

x

A

∈ σ(N, v

A

)

and every

x

B

∈ σ(N, v

B

)

, then

x

A

+ x

B

∈

σ(N, v

A

+ v

B

)

whenever

(N, v

A

+ v

B

) ∈ Γ

0

and

x

A

+ x

B

is feasible for

(N, v

A

+ v

B

)

, i.e.,

x

A

+ x

B

∈ X

Λ

∗

(N, v

_A

+ v

_B

)

.

While the feasibility requirement related torst best timeallo ationis redundant on the set

Γ

c

, ours isnot triviallysatised sin ethe grand oalition isdeterred.

4.2 Axiomatization

Before hara terizingthe ontra tion ore, werst need the following lemma.

Lemma 4.8 Take any

B ∈ Λ

∗

_{(N )}

where

n ≥ 2

with balan ed weights

(δ

H

)

H

∈B

. For any

S ∈ 2

N

\{∅}

where

s ≥ 2

,dene:

B

S

= {T ⊂ S

:

T = H ∩ S 6= ∅

forsome

H ∈ B}

and forevery

T ∈ B

S

:

ˆ

δ

T

=

1 −

X

H∈B:

H

∩S=S

δ

H

!

−1

X

H∈B:

T

=H∩S

δ

H

. Then,

B

S

_{∈ Λ}

∗

_(S)

with balan edweight

(ˆ

δ

T

)

T

∈B

S

.

Proof: First, it follows from

P

H

∈B:i∈H

δ

H

= 1

,

s ≥ 2

and

N 6∈ B

that

P

H∈B:H∩S=S

δ

H

< 1

. Se ond, forea h

i ∈ S

itholds that:

1 −

X

H∈B:

H∩S=S

δ

H

!

X

T

∈B

S

_:

i∈T

ˆ

δ

T

=

X

T

∈B

S

_:

i∈T

X

H∈B:

T

=H∩S

δ

H

=

X

H∈B:

H∩S⊂S

i∈H

δ

H

=

X

H∈B:

H∩S⊆S

i∈H

δ

H

−

X

H

∈B:

H

∩S=S

i∈H

δ

H

=

X

H∈B:

i∈H

δ

H

−

X

H∈B:

H∩S=S

δ

H

= 1 −

X

H∈B:

H∩S=S

δ

H

,

whi h on ludes the proof.



Givenany se ond best time allo ation inaTU-game, Lemma4.8 provides aformulato ompute the orresponding se ond best time allo ation in any of its redu ed game. It learly implies the result on NMDM-CONinthe followingproposition.

Proposition 4.9 The ontra tion oresatisesNE,IR,NMDM-CONandC-SUPAontheset

Γ

c

. Proof: It is straightforwardto verify thatNE, IRand C-SUPAaresatised. It remainsto prove thatthe ontra tion oresatisesNMDM-CON.Let

(N, v) ∈ Γ

c

,

S ∈ 2

N

_\{∅}

and

x ∈ CC(N, v)

. We distinguishtwo ases:

- assume that

s ≥ 2

. Take

B ∈ Λ

∗

_{(N )}

with balan ed weights

(δ

H

)

H∈B

su h that

x(N ) =

P

H

∈B

δ

H

v(H)

. Then, by Lemma4.8itholds that

B

S

_{∈ Λ}

∗

_(S)

withbalan edweight

(ˆ

δ

T

)

T

∈B

S

. Now, we prove that

x(T ) ≤ v

∗

S,x

(T )

forea h

T ∈ B

S

. Given

T ∈ B

S

, there exists

H ∈ B

su h that

T = H ∩ S

. From

x(N ) =

P

H

∈B

δ

H

v(H)

and

x(S) ≥ v(S)

for ea h

S ∈ 2

N

_{\{∅, N }}

,it holds that

x(H) = v(H)

, hen e

x(T ) = v(H) − x(H\T )

. Sin e

H\T ⊆ N \S

, it holds that

x(T ) ≤ max{v(T ∪ Q) − x(Q) : Q ⊆ N \S} = v

_S,x

∗

(T )

. Then, we prove that

x(T ) ≥ v

∗

S,x

(T )

for ea h

T ∈ 2

S

_{\{∅, S}}

. By ontradi tion, assume that there exists

T ∈ 2

S

_{\{∅, S}}

su h that

x(T ) < v

∗

S,x

(T )

. Hen e there exists

y

T

_{∈ R}

t

su h that

y(T ) = v

_S,x

∗

(T )

and

y(T ) > x(T )

. Thus, it holds that

y(T ) = v(T ∪ Q) − x(Q)

for some

Q ⊆ N \S

. Hen e,

y(T ) + x(Q) = v(T ∪ Q)

and so,

x(T ) + x(Q) < v(T ∪ Q)

,a ontradi tion with

x ∈ CC(N, v)

sin e

T ∪ Q ⊂ N

. We on lude that

x(T ) ≥ v

∗

S,x

(T )

forea h

T ∈ 2

S

_{\{∅, S}}

. Thus,

x(T ) = v

S,x

(T )

for ea h

T ∈ B

S

, and so

x(S) =

P

T

∈B

S

δ

ˆ

_T

x(T ) =

P

T

∈B

S

δ

ˆ

_T

v

_S,x

∗

(T )

. Moreover,

x(T ) ≥ v

∗

S,x

(T )

forea h

T ∈ 2

S

_{\{∅, S}}

impliesthat

x

S

_{∈ CC(S, v}

∗

S,x

)

. - assume that

s = 1

. Take

B ∈ Λ

∗

_{(N )}

with balan ed weights

(δ

H

)

H∈B

su h that

x(N ) =

P

H

∈B

δ

H

v(H)

. Now, we prove that

x

S

_{≤ v}

∗

that

i ∈ H

. From

x(N ) =

P

H

∈B

δ

H

v(H)

and

x(S) ≥ v(S)

for ea h

S ∈ 2

N

_{\{∅, N }}

, it holds that

x(H) = v(H)

, hen e

x(S) = v(H) − x(H\S)

. Sin e

H\S ⊂ N \S

, it holds that

x(S) ≤ max{v(S ∪ Q) − x(Q) : Q ⊂ N \S} = v

_S,x

∗

(S)

. Then, we provethat

x

S

_{≥ v}

∗

S,x

(S)

. By ontradi tion,assumethat

x

S

_{< v}

∗

S,x

(S)

. Hen e,thereis

y

S

_{∈ R}

su hthat

y

S

_{= v}

∗

S,x

(S)

and

y

S

_{> x}

S

. Thus,itholdsthat

y

S

_{= v}

∗

S,x

(S) = v(S ∪Q)−x(Q)

for some

Q ⊂ N \S

. Hen e,

y(S) + x(Q) = v(S ∪ Q)

and so,

x(S) + x(Q) < v(S ∪ Q)

, a ontradi tionwith

x ∈ CC(N, v)

sin e

S ∪ Q ⊂ N

. We on lude that

x

S

_{∈ CC(S, v}

∗

S,x

)

.



Proposition 4.10 Let

σ

be a solution on ept on

Γ

0

⊆ Γ

satisfying IR and NMDM-CON. If

(N, v) ∈ Γ

0

and

x ∈ σ(N, v)

then

x(S) ≥ v(S)

forany

S ∈ 2

N

_{\{∅, N }}

.

Proof: Let

σ

bea solution on ept on

Γ

0

⊆ Γ

satisfying IRand NMDM-CON.Let

x ∈ σ(N, v)

,

S ∈ 2

N

_{\{∅, N }}

and

i ∈ S

. By NMDM-CON,

x

i

∈ σ({i}, v

∗

{i},x

)

. ByIR, itholds that:

x

i

≥ v

_{i},x

∗

({i})

= max{v({i} ∪ Q) − x(Q) : Q ⊂ N \{i}}

≥ v(S) − x(S\{i})

,

whi hprovesthat

x(S) ≥ v(S)

as desired.



Proposition 4.11 If

σ

is a solution on ept dened on

Γ

0

⊆ Γ

c

that satises IR and NMDM-CON, then for any

(N, v) ∈ Γ

0

, any payo ve tor

x ∈ σ(N, v)

is e ient, i.e.,

x(N ) =

max

B∈Λ

∗

_{(N )}

P

_S∈B

δ

_S

v(S)

(or

x ∈ X

∗

Λ

∗

(N, v)

).

Proof: Let

σ

be a solution on ept on

Γ

0

⊆ Γ

c

satisfying IR and NMDM-CON. Assume that

(N, v) ∈ Γ

0

and take any

x ∈ σ(N, v)

and any

y ∈ X

Λ

∗

(N, v)

. Then, there is

B ∈ Λ

∗

_{(N )}

su h that

y(N ) ≤

P

S∈B

δ

S

v(S)

. It followsfrom

B ∈ Λ

∗

_{(N )}

and Proposition4.10 that:

x(N ) =

X

S∈B

δ

S

x(S)

≥

X

S∈B

δ

S

v(S)

≥ y(N )

. We on ludethat

x ∈ X

∗

Λ

∗

(N, v)

.



Proposition 4.12 If

σ

is asolution on ept dened on

Γ

c

satisfying IRand NMDM-CON,then

σ(N, v) ⊆ CC(N, v)

forany

(N, v) ∈ Γ

c

.

Proof: Take any

x ∈ σ(N, v)

. By Proposition 4.10, it holds that

x(S) ≥ v(S)

for every

S ∈ 2

N

_{\{∅, N }}

. Moreover,by Proposition4.11,

x ∈ X

∗

Λ

∗

(N, v)

. So,

x ∈ CC(N, v)

.



Proposition 4.13 Ifasolution on eptdenedon

Γ

c

satisesNE,IR,NMDM-CONandC-SUPA, then

CC(N, v) ⊆ σ(N, v)

forany

(N, v) ∈ Γ

c

.

Proof: 7

Let

x ∈ CC(N, v)

anddene

(N, w) ∈ Γ

c

as:

w(S) =



x(S)

if

|S| ≥ 2

;

v(S)

if

|S| = 1

.

Itholdsthat

C(N, w) = {x}

. ByProposition4.12,

σ(N, w) ⊆ CC(N, w) = C(N, w) = {x}

. By NE,itholds that

x ∈ σ(N, w)

.

Consider the game

(N, z) ∈ Γ

c

dened as:

∀S ∈ 2

N

_{, z(S) = v(S) − w(S)}

.

Hen e,

z(S) ≤ 0

if

2 ≤ |S| < n

,

z({i}) = 0

for every

i ∈ N

and

z(N ) ∈ R

. Note that

0

_{∈ CC(N, z)}

sin e

0 = max

B∈Λ

∗

(N )

P

S∈B

δ

S

z(S) =

P

_i∈N

z({i})

. By Proposition 4.11, for every

y ∈ CC(N, z)

it holds that

y(N ) = 0

. Sin e

z({i}) = 0

for every

i ∈ N

, we have

y

i

≥ 0

by IR and so,

y = 0

. Thus,

CC(N, z) = {0}

. By Proposition

3.8

, it holds that

σ(N, z) ⊆ CC(N, z) = {0}

. ByNE,

0

∈ σ(N, z)

. Note that

x(N ) + 0 = max

B∈Λ

∗

_{(N )}

P

_S∈B

δ

_S

v(S) = max

_B∈Λ

∗

_{(N )}

P

_S∈B

δ

_S

(w + z)(S)

. So,

x + 0 ∈ X

Λ

∗

(N, w + z)

,i.e.,

x + 0

is feasiblefor

(N, w + z)

. Thus,by C-SUPAitfollows from

x ∈ σ(N, w)

and

0

∈ σ(N, z)

that

x + 0 ∈ σ(N, w + z)

, hen e

x ∈ σ(N, v)

.



Theorem 4.14 The ontra tion ore is the only solution on ept on

Γ

c

that satises NE, IR, NMDM-CONand C-SUPA.

Proof: Combine Propositions4.9,4.12 and 4.13.



4.3 Independen e of the axioms

Thefollowingexamplesshowthatthe axiomsused inthe hara terizationof the ontra tion ore are logi allyindependent on theset

Γ

c

,i.e.,noneisimpliedby the others.

Example 4.15 Considerthesolution on ept

σ

1

on

Γ

c

su hthatforany

(N, v) ∈ Γ

c

,

σ

1

(N, v) =

∅

. Obviously,

σ

1

violatesNE butva uouslysatisesIR, NMDM-CONand C-SUPA.

Example 4.16 Considerthesolution on ept

σ

2

on

Γ

c

su hthatforany

(N, v) ∈ Γ

c

,

σ

2

(N, v) =

X

Λ

∗

(N, v)

. We know that

σ

2

satises NE be ause

X

Λ

∗

(N, v) ⊇ CC(N, v) 6= ∅

. It satises C-SUPA by denition. It follows from Lemma 4.8 and a similar argument used in the proof of Proposition4.9thatNMDM-CONis alsosatised. It should be learthat

σ

2

violatesIR.

Example 4.17 Considerthesolution on ept

σ

3

on

Γ

c

su hthatforany

(N, v) ∈ Γ

c

,

σ

3

(N, v) =

{x ∈ X

Λ

∗

(N, v) : x

_i

≥ v({i})}

. Clearly,

σ

3

satises NE, IR and C-SUPA. The previous result impliesthat

σ

3

doesnot satisfyNMDM-CON.

7

OurproofisinspiredfromthatinPelegandSudhölter(2003)inthe asewhere

n ≥

3

.Nevertheless,themain dieren eisthatwedon'tneedtodistinguish ases

n

= 2

and

n ≥

3

.

Example 4.18 For every

(N, v) ∈ Γ

c

, every

S ∈ 2

N

_\{∅}

and every

x ∈ R

n

, the ex ess of

S

from

x

in

(N, v)

is given by the quantity

e(S, x, v) = v(S) − x(S)

. The ex ess

e(S, x, v)

gives the amount of dissatisfa tion of oalition

S

from

x

in

(N, v)

. We dene the ve tor

θ(x) = (θ

1

(x), . . . , θ

2

n−1

(x))

whose omponents are the numbers

(e(S, x, v))

S∈2

N

\{∅}

arranged in non-in reasing order. For any TU-game

(N, v) ∈ Γ

c

, the ontra tion nu leolus,denoted by

CN (N, v)

,is dened as:

CN (N, v) = {x ∈ X

_Λ

∗

∗

(N, v)

:

θ(y) ≥

L

θ(x)

forall

y ∈ X

∗

Λ

∗

(N, v)}

, where

≥

L

is the lexi ographi al ordering. First, sin e

X

∗

Λ

∗

(N, v)

is non-empty, ompa t and onvex, itfollowsfrom orollary

5.1.10

in Pelegand Sudhölter(2003) that

CN (N, v)

onsistsof a single point. Hen e, the ontra tion nu leolus satises NE. Se ond, the ontra tion nu leolus also satises IR sin e it belongs to the ontra tion ore. The proof is left as an exer ise to thereader. Finally,the ontra tion nu leolus omplieswithNMDM-CON. Theproofsareomitted be ausetheyaresimilartothoseinPelegandSudhölter(2003)inordertoshowthatthenu leolus satises DM-CON.Hen e,itfollowsfromouraxiomatizationthatthe ontra tion nu leolusdoes not satisfyC-SUPA.

5 Con luding remarks: extended ontra tion ore

We have introdu ed a new solution on ept, the ontra tion ore, that serves as a basis for the investigation of the deterren e of ooperation. This solution on ept has permitted to provide a measure of the robustness of ooperationwhi h, as far as we know, has not been analyzed in the literature. We have su essfully appliedthe ontra tion oreto oligopolisti markets and we have provided optimal ne imposed by ompetition authorities for arteldeterren e. We an be onvin ed thatthere aremany other potentialappli ationsofthe ontra tion ore.

More generally, we have also provided an axiomati hara terization of the ontra tion ore in ordertobetterunderstandit. Inparti ular,this haspermittedto make omparisonswiththe ore and the aspiration ore. We havedened the ontra tion ore on the set of balan edTU-games inordertobe onsistentwithourobje tivetostudythedeterren eof ooperation. Wearguethat itis possible todenean extended ontra tion oreby applyingthe feasibility onditionrelated to se ond best time allo ation on the set of all TU-games. The extended ontra tion ore is then non-empty on the set of allTU-games and oin ides with the aspiration ore on the set of non-balan ed TU-games. Inthis ase,itwillbeastraightforwardexer i eforthereaderto he k thatthe axiomatization giveninSe tion

4

stillholds onthe set of allTU-games.

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