Subordinated Hedonic Games

Hindawi Publishing CorporationGameTheoryVolume 2013 Article ID 512742 10 pageshttpdxdoiorg1011552013512742

Research ArticleSubordinated Hedonic Games

Juan Carlos Cesco12

1 Instituto de Matematica Aplicada San Luis (UNSL-CONICET) Avenida Ejercito de Los Andes 950 5700 San Luis Argentina2Departamento de Matematica (UN San Luis) Chacabuco y Pedernera 5700 San Luis Argentina

Correspondence should be addressed to Juan Carlos Cesco jcescounsleduar

Received 24 April 2013 Accepted 9 July 2013

Academic Editor Walter Briec

Copyright copy 2013 Juan Carlos Cesco This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Hedonic games are simple models of coalition formation whose main solution concept is that of core partition Several conditionsguaranteeing the existence of core partitions have been proposed so far In this paper we explore hedonic games where a reducedfamily of coalitions determines the development of the game We allow each coalition to select a subset of it so as to act as itsset of representatives (a distribution) Then we introduce the notion of subordination of a hedonic game to a given distributionSubordination roughly states that any player chosen as a representative for a coalition has to be comfortable with this decisionWithsubordinationwehave a tool within hedonic games to compare howa ldquoconvenientrdquo agreement reached by the sets of representativesof different groups of a society is ldquovaluedrdquo by the rest of the society In our approach a ldquoconvenientrdquo agreement is a core partitionso this paper is devoted to relate the core of a hedonic game with the core of a hedonic game played by the sets of representativesThus we have to tackle the existence problem of core partitions in a reduced game where the only coalitions that matter are thoseprescribed by the distribution as a set of representativesWe also study how a distribution determines thewhole set of core partitionsof a hedonic game As an interesting example we introduce the notion of hedonic partitioning game which resembles partitioninggames studied in the case where a utility transferable or not is present The existence result obtained in this new class of games islater used to provide a nonconstructive proof of the existence of a stable matching in the marriage model

1 Introduction

Coalitional games are models which take into account theinteraction between the players of the game In general anysubgroup of players (a coalition) can influence the result ofthe game However it has been recognized that sometimessome of the coalitions can completely determine the devel-opment of the game In their seminal paper Kaneko andWooders [1] introduce the class of partitioning games as a wayto capture the fact that ldquoIn an 119899-cooperative game it may notbe equally easy to formevery coalitionrdquo In those games only asubset of coalitions play such an essential role that determinesthe behavior of all the other coalitions This characteristicis shared for several of the games studied in the literaturesuch as the marriage game [2] the bridge game [3] theassignment game [4] and the m-sided assignment game [5]among others Kaneko andWooders [1] present a transferableutility version for partitioning games and a nontransferableutility version as well and focus on the nonemptiness of thecore of the games As a key contribution they provide a list

of necessary and sufficient conditions to characterize a classof families of coalitions under which any possible inducedpartitioning game has nonempty core a property motivatedby the behavior exhibited by the assignment game We pointout that the essential coalitions in the well-known marriagegame of Gale and Shapley [2] satisfy those conditionsalthough none of the versions of the games presented byKaneko and Wooders [1] suit well to deal with the existenceof a stable matching Hedonic games [6 7] which constituteanother class of games have received considerable attentionin the last decade and seem to be more appropriate to dealwith matching problems than coalitional games where autility is present A hedonic games (119873 ⪰) is a simple modelof coalition formation determined by a set of 119899 players 119873and a profile of preferences ⪰= (⪰

1 ⪰

119899) where for any

119894 = 1 119899 ⪰119894is an ordering of the coalitions containing

player 119894 The main solution concept for a hedonic game isthat of core partition namely a partition of the set of playersthat is resistant to a certain class of objections raised by thecoalitions in the game (Section 2)There are several sufficient

2 GameTheory

conditions for the existence of core partitions being thoseof top-coalition properties [6] consecutiveness and ordinalbalancedness [7] among themost well known Iehle [8] gives acomplete answer to the existence problem in a general settingby introducing the notion of pivotal balancedness To definepivotal balancedness first it is associated one of its subsetsto each coalition 119878 Iehle [8] calls this family of subsets adistribution Iehlersquos condition states that a hedonic game hasnonempty core if and only if there is a distribution such thatthe game is pivotally balancedwith respect to this distribution(see Section 2) Thus checking this condition is a two-stepprocedure First a distribution has to be considered andthen the balancedness condition has to be verified

On the other hand in many situations different groupsin a society choose a subgroup to act as their representativesThen the agreements (if any) reached by the groups ofrepresentatives are imposed one way or the other to thewhole society An interesting issue is whether a ldquoconvenientrdquoagreement for the representatives is also good for the restof the society With this idea in mind we note that a set ofrepresentatives for a coalition 119878 can be taken as an elementI(119878) of a distribution I a relation that builds a bridgebetween our objective and a well-developed mathematicalframework We also point out that although the notionof pivotally balancedness works very well to deal with thegeneral existence problem of core partitions in hedonicgames it does not reflect towhich extent a family of coalitionsof representatives determines the behavior of the game Forinstance if a family I of coalitions of representatives (adistribution) plays a reduced game where the only admissiblecoalitions are those in I an interesting issue is to look forrelationships if any between the cores of both games Forinstance if 120587 is a core partition in the reduced game (anagreement between the groups of representatives) is also 120587 acore partition (a good agreement for the whole society) in theoriginal game Answers to this problem could help at leastin two dimensions to the treatment of hedonic games Firstsince we start with a distribution the first step in the two-stepprocedure mentioned above is simplified Second in orderto get core partitions in the original game one can solve areduced and possibly simpler game

Unfortunately as several examples show there are nogeneral results What is missing is the description of someconnection apart from inclusion between a coalition andits set of representatives To tackle this point we appeal tothe notion of a hedonic game subordinated to a distributionIn order to deal with the reduced game played by thecoalitions of representatives we present some results about ahedonic game with a restricted family of admissible coalitionsThey are simple extensions of the main existence resultsproved by Iehle [8] Papai [9] has already used the idea ofrestrict the family of feasible coalitions to get uniquenessresults about core partitions in hedonic games while the ideaof subordination (Section 3) seems to be new Subordinationin a game (119873 ⪰) roughly states that any player chosen as arepresentative of a coalition has to be comfortable with thisdecision This is formalized by asking that ifI(119878) stands forthe set of representatives of a coalition 119878 thenI(119878)⪰

119894119878 for any

119894 isin I(119878) Subordination involves some degree of anonymity

on the individual preferences although it is weaker than thecommon ranking property [10] or the top-coalition condition[6]When subordination is present a balancedness conditionhas to be checked only on the family of representatives toguarantee the existence of core partitions Moreover undermild additional conditions the family of representativesdetermines completely the core of the hedonic game

The organization of the paper is as the following In thenext section we present our version of hedonic games wheresome restrictions on the family of admissible coalitions areimposed They are a straightforward extension of hedonicgames as stated by Banerjee et al [6] and Bogomolnaiaand Jackson [7] where there is no restriction on the familyof admissible coalitions For this enlarged class of hedonicgames we state two balancedness conditions extending thoseof ordinal balancedness of Bogomolnaia and Jackson [7] andpivotal balancedness of Iehle [8] Following those authorswe show that both conditions are sufficient to guaranteethe existence of core partitions while pivotal balancednessis also a necessary condition In Section 3 we study theissue of subordinationThe idea behind subordination is thatgiven a reduced family of coalitions along with a profile ofpreferences there are games whose core behavior is deter-mined by this basic information We state some sufficientconditions to guarantee the existence of core partitions and toguarantee that the core of the whole game is fully determinedby the basic information given on a reduced family of coali-tion as well Several examples and counter examples arealso shown there In Section 4 we relate subordination toother sufficient conditions guaranteeing the existence of corepartitions in hedonic games like top-coalition properties[6] and consecutiveness [7] At the end of this section weintroduce the notion of partitioning hedonic game Theyalways have a family of essential coalitions associated andwe prove that every partitioning game has nonempty core ifthe family of essential coalitions satisfies either condition (ii)or the equivalent condition (iii) ofTheorem27 ofKaneko andWooders [1] We use the results about hedonic partitioninggames to elaborate a nonconstructive proof of the existence ofa stable matching in the marriage game [2] different from theproof provided by Sotomayor [11] To this end we associate apartitioning hedonic game to each marriage problem so thatresults in Section 4 can be used We include a final Appe-ndix where we sketch the existence proof of core partitionsin hedonic games with a restricted family of admissiblecoalitions very similar to that of Theorem 3 of Iehle [8]

2 Hedonic Games

To define a hedonic game we start with a nonempty finiteset119873 = 1 119899 the players and its familyN of nonemptysubsets the coalitions Given any family of coalitionsA sube Nand a player 119894 isin 119873 we denote by A(119894) the subfamily ofcoalitions in A containing player 119894 A hedonic game is a 3-tuple (119873 ⪰A) where A sube N is a nonempty family and⪰ = (⪰

119894)119894isin119873

is a preference profile with ⪰119894 for any 119894 isin 119873

being a reflexive complete and transitive binary relation onA(119894) For any 119894 isin 119873 ≻

119894will stand for the strict preference

relation related to ⪰119894(119878≻119894119879 if and only if 119878⪰

119894119879 but not 119879⪰

119894119878)

GameTheory 3

PA(119873) will denote the family of partitions 120587 = 120587

1 120587

119904

of 119873 such that 120587119895

isin A for any 119895 = 1 119904 Given 120587 =

1205871 120587

119904 isin PA

(119873) and 119894 isin 119873 120587(119894) will denote the uniqueset in 120587 containing player 119894

Given a hedonic game (119873 ⪰A) and 120587 isin PA(119873) we

say that 119879 isin A blocks 120587 if for any 119894 isin 119879 119879≻119894120587(119894) The core

119862(119873 ⪰A) of (119873 ⪰A) is the set of partitions in PA(119873)

blocked by no coalition In a game (119873 ⪰A)A is the familyof admissible coalitionsThemost studied case in the literatureis A = N namely when there is no restriction on the set ofadmissible coalitions (see eg [6ndash8]) When this is the casewe will omit A from all the notation and we will use ⪰

119894(≻119894)

instead of ⪰119894(≻119894) to denote the individual (strict) preference

of a player 119894 isin 119873 The case A =N as stated in this paperhas already been used by Cesco [12] to study some existenceresult in many-to-one matching problems

Definition 1 Anonempty collectionA sube N such that 119894 isin Afor any 119894 isin 119873 is called essential

Families with such a characteristic have been used byKaneko and Wooders [1] Quint [5] and Le Breton et al[13] among others to study the existence of core solutions ingames where a restricted family of coalitions determines thebehavior of the whole game Fromnowon when dealingwitha hedonic game (119873 ⪰A) we will assume that the family Ais essential

A family of coalitions B sube N is balanced if thereexists a collection of positive real numbers (120582

119878)119878isinB satisfying

sum119878isinB

119878ni119894

120582119878= 1 for all 119894 isin 119873 The numbers (120582

119878)119878isinB are the

balancing weights for B B is minimal balanced if there isno proper balanced subfamily of it In this case the set ofbalancing weights is unique A family of essential coalitions isA-partitionable [1] if and only if the only minimal balancedsubfamilies that it contains are partitions

Definition 2 Given anonempty family of coalitions119865A sube NI = (I(119860))

119860isinA is anA-distribution if for each coalition119860 isin

AI(119860) isin A andI(119860) sube 119860 In addition anA-distributionis simply called a distribution whenA = N

Given an A-distribution I a family B sube A is I-balanced if the family (I(119861))

119861isinB is balanced

Definition 3 (119873 ⪰A) is ordinally balanced if for each bal-anced family B sube A there is a partition 120587 isin PA

(119873) suchthat for any 119894 isin 119873 120587(119894) ⪰

119894119861 for some 119861 isin B(119894)

Ordinal balancedness is first stated by Bogomolnaia andJackson [7] for the caseA = N

Definition 4 (119873 ⪰A) is pivotally balanced with respect to anA-distribution I if for each I-balanced family B sube A

there is a partition 120587 isin PA(119873) such that for any 119894 isin 119873

120587(119894)⪰119894119861 for some 119861 isin B(119894) The game is pivotally balanced if

it is pivotally balanced with respect to some A-distributionI

This general concept of pivotal balancedness was intro-duced by Iehle [8] for the caseA = N

Remark 5 Ordinal balancedness implies pivotal balanced-ness with respect to theA-distributionI = (120594

119860)119860isinA where

120594119860stands for the indicator vector of the coalition 119860 namely

when 120594119860is the 119899-dimensional vector with 120594

119860(119895) = 1 if player

119895 belongs to119860 and 120594119860(119895) = 0 if player 119895 does not belong to119860

The following result is a simple extension of the maincharacterization proved by Iehle [8 Theorem 3] and whoseproof is carried out in a similar way However for the sake ofcompleteness we sketch the proof in the Appendix

Theorem 6 (see Iehle [8]) Let (119873 ⪰A) be a hedonic gamewith A as its family of admissible coalitions Then the coreof (119873 ⪰A) is nonempty if and only if the game is pivotallybalanced

3 Subordinated Games

As a motivation for the content of this section let us startwith an example which is a slight modification of Game 5 ofBanerjee et al [6]

Example 7 Let (119873 ⪰) be a hedonic game with 119873 = 1 2 3

and where the preferences for players 1 2 and 3 are given inthe three lines below (in this example and in the followingones as well we will omit the subscript 119894 in the symboldescribing the preference of player 119894 to avoid misleading)

1 2 ≻ 1 3 ≻ 1 2 3 ≻ 1

1 2 ≻ 2 3 ≻ 1 2 3 ≻ 2

3 ≻ 1 3 ≻ 2 3 ≻ 1 2 3

(1)

Theonly core partition in (119873 ⪰) is120587 = 1 2 3 Accordingto Iehle [8 Proposition 1] (119873 ⪰) is pivotally balanced withrespect to the distributionI given by

I (1 2 3) = 1 2 I (1 2) = 1 2

I (1 3) = 3

I (2 3) = 3 I (119894) = 119894 forall119894 = 1 2 3

(2)

The familyA = 1 2 3 1 2 describes the set of essen-tially different coalitions in the distribution I Moreover ifwe consider the game (119873 ⪰A) where ⪰ is restriction of ⪰ tothe familyA namely if

1 2 ≻ 1

1 2 ≻ 2

3 ⪰ 3

(3)

are the restricted preferences for players 1 2 and 3 respec-tively then it is easy to check that 120587 is also the only corepartition of this game with a restricted family of admissiblecoalition

In this example the game (119873 ⪰) is pivotally balancedwith respect to the family (119860) and this family determinescompletely the core of the game in the sense that the coreof the reduced game (119873 ⪰ (119860)) coincides with that of (119873 ⪰

) However as the following example shows there can beno relation at all between the core of a hedonic game and

4 GameTheory

the core of a reduced game related to one of the distributionsthe game is pivotally balanced to

Example 8 Let the (119873 ⪰) be a game with119873 = 1 2 3 4 andwith preferences for players 1 2 3 and 4 given by

1 2 3 4 ≻ 1 ≻ 1 2 ≻ 1 3 ≻ 1 4 ≻ 1 2 3 ≻ sdot sdot sdot

1 2 3 3 ≻ 2 3 4 ≻ 2 3 ≻ 2 ≻ sdot sdot sdot

1 2 3 4 ≻ 3 4 ≻ 3 ≻ 2 3 ≻ 3 4 ≻ 2 3 4 ≻ sdot sdot sdot

1 2 3 4 ≻ 3 4 ≻ 2 4 ≻ 4 ≻ 1 2 4 ≻ sdot sdot sdot

(4)

respectively This game is pivotally balanced since 120587 =

1 2 3 4 is clearly a core partition in (119873 ⪰) Moreoveraccording to Iehle [8 Proposition 1] the game is pivotallybalanced with respect to the distributionI whereI(119878) = 119878

for any 119878 with |119878| lt 4 and I(1 2 3 4) = 1 2 3 Inthis case A coincides with the family of all coalitions withcardinality less than four However in the associated game(119873 ⪰A) it is a simple exercise to check that there is ablocking coalition inA for any of the fifteenA-partitions thatthis game have Therefore this game has no core partition

The aim of this section is to explore more deeply thesekind of relationships To this end we start by defining whatwe mean by a subordinated game

Definition 9 A hedonic game (119873 ⪰) is subordinated to adistribution I when for any 119894 isin 119873 and any 119878 isin N(119894)I(119878)⪰

119894119878 wheneverI(119878) contains player 119894

The game (119873 ⪰) in Example 7 is subordinated to thedistributionI showed there

We point out that for any given distribution I in ahedonic game (119873 ⪰) the family

A (I) = 119878 isin N 119878 = I (119879) for some 119879 isin N (5)

is an essential family of coalitions since I(119894) = 119894 andtherefore 119894 isin A for any 119894 isin 119873

When subordination is present the behavior of thecoalitions in a distribution determines to some extent thecore of a hedonic game The next result illustrates this pointGiven a hedonic game (119873 ⪰) and a distribution I let(119873 ⪰A) be the hedonic gamehavingA = A(I) as its familyof admissible coalitions and where ⪰ is the restriction of ⪰ toA

Theorem 10 Let (119873 ⪰) be a hedonic game subordinated to adistributionI withA = A(I) Then

(a) the core of (119873 ⪰A) is included in the core of (119873 ⪰)

(b) If for any 119878 isin N such that 119878 notin A (119878 =I(119879) for any119879 sube 119873)I(119878)≻

119894119878

for any 119894 isin I(119878) then the core of (119873 ⪰A) is the same as thecore of (119873 ⪰)

Proof Let 120587 isin 119862(119873 ⪰A) We will show that 120587 isin 119862(119873 ⪰)

too Clearly there is no 119878 isin A objecting 120587 So let us assumethat there is 119878 notin A objecting 120587 Then for each 119894 isin 119878 we havethat

119878 ≻119894120587 (119894) (6)

But because of subordination

I (119878) ⪰119894119878 (7)

for all 119894 isin I(119878) Then for any 119894 isin I(119878)

I (119878) ≻119894120587 (119894) (8)

and since ⪰ is the restriction of ⪰ toA from (8) we get that

I (119878) ≻119894120587 (119894) (9)

for all 119894 isin I(119878) But this indicates that I(119878) objects 120587 in(119873 ⪰A) a contradiction Thus 119862(119873 ⪰A) sube 119862(119873 ⪰)

To see part (b) let us assume that 120587 = (1205871 120587

119904) isin

119862(119873 ⪰) 119862(119873 ⪰A) Then for at least one 119895 = 1 119904120587119895

notin A Let 119879 = I(120587119895) Then for each 119894 isin 119879 119879≻

119894120587119895

Thus 119879 is an objection to 120587 a contradiction proving that119862(119873 ⪰) = 119862(119873 ⪰A)

In general the reverse inclusion of that stated in (a)does not hold mainly because some core partitions in (119873 ⪰)

are not A-partitions in (119873 ⪰A) On the other hand wenote that the game in Example 7 is a subordinated game forwhich condition (b) ofTheorem 10 holds which supports theequality between the core of (119873 ⪰) and that of 119862(119873 ⪰A)However in general the inclusion in (a) of Theorem 10 isstrict as Example 8 shows Furthermore the inclusion can bestrict even though the core of (119873 ⪰A) is nonemptyThenextexample illustrates this point

Example 11 Let (119873 ⪰) be a hedonic game with119873 = 1 2 3and letI be the distribution given by

I (1 2 3) = 1 3 I (1 2) = 1

I (1 3) = 1

I (2 3) = 3 I (119894) = 119894 for any 119894 = 1 2 3

(10)

Then A = A(I) = 1 2 3 1 3 Let the preferenceprofile ⪰ in (119873 ⪰) have preferences for players 1 2 and 3

given by

1 ≻ 1 2 ⪰ 1 3 ⪰ 1 2 3

1 2 ⪰ 2 3 ⪰ 1 2 3 ⪰ 2

3 ≃ 1 3 ≃ 2 3 ⪰ 1 2 3

(11)

Consequently the preference profile ⪰ defined on A in theassociated game (119873 ⪰A) is

1 ≻ 1 3

2 ≃ 2

3 ≃ 1 3

(12)

GameTheory 5

It is easy to check that the only core partition of (119873 ⪰A)

is 120587 = (1 2 3) Thus while 120587 is still a core partition in(119873 ⪰) it is also simple to check that 120587lowast = (1 2 3) is a corepartition in (119873 ⪰) too

While condition (b) of Theorem 10 is sufficient for thecoincidence of the cores of (119873 ⪰) and (119873 ⪰A) respectivelywhen subordination is present the next example shows thatit is not necessary

Example 12 Let (119873 ⪰) andI be like in Example 11 We nowreplace the individual preference of player 2 by the followingone

2 ≻ 1 2 ⪰ 2 3 ⪰ 1 2 3 (13)

Then although this new game does not satisfy condition (b)of Theorem 10 it is easy to check that 120587 = (1 2 3) is theonly core partition in both games (119873 ⪰) and (119873 ⪰A)

Finally when subordination is not present we can saynothing about the inclusion stated in part (a) of Theorem 10as the two games in the next example show

Example 13 Let us start with the game (119873 ⪰) and the samedistribution I used in Example 7 We now modify theindividual preference of player 2 as follows

1 2 3 ≻ 1 2 ≻ 2 3 ≻ 2 (14)

Then the game with preference profile with this new prefer-ence for player 2 is no longer subordinated to the distributionI since I(1 2 3) = 1 2 but 1 2 997411 1 2 3 forplayer 2 Nevertheless its only core partition is still 120587 =

1 2 3 On the other hand the associated game (119873 ⪰A)

coincides with that of Example 7Then its only core partitionis 120587 = 1 2 3 too and the inclusion stated in part (a) ofTheorem 10 is verified although subordination is not present

Now let119873 = 1 2 3 and let (119873 ⪰) a hedonic game withthe preferences of players 1 2 and 3 given by

1 2 3 ≻ 1 2 ≻ 1 3 ≻ 1

1 2 3 ≻ 1 2 ≻ 2 3 ≻ 2

1 2 3 ≻ 3 ≻ 1 3 ≻ 2 3

(15)

The only core partition in (119873 ⪰) is 1205871015840 = 1 2 3 Now letI be the same distribution used in Example 7 Again sinceI(1 2 3) = 1 2 but 1 2 997411 1 2 3 for player 2 (119873 ⪰) isnot subordinated to I Also the game (119873 ⪰A) is equal tothe corresponding game for the game (119873 ⪰) in Example 7 soits only core partition is 120587 = 1 2 3 Then 119862(119873 ⪰A) isnot included in 119862(119873 ⪰) in this case

In the light of Theorem 10 any condition guaranteeingthe nonemptiness of the core of the game (119873 ⪰A) (egpivotal balancedness) will be a sufficient condition for thenonemptiness of the subordinated game (119873 ⪰) to I Thusthe nonemptiness of the core of (119873 ⪰A) implies the pivotalbalancedness of (119873 ⪰) althoughTheorem 10 does not exhibita family A such that the game (119873 ⪰) be pivotally balancedwith respectA However if we impose on (119873 ⪰A) the morestrict condition of ordinal balancedness then we can alsoexhibit easily such a distribution

Proposition 14 Let (119873 ⪰) be a hedonic game subordinatedto a distribution I with A = A(I) If (119873 ⪰A) is ordinallybalanced then (119873 ⪰) is pivotally balanced with respect to thedistributionI

Proof LetB be anI-balanced family of coalitions in (119873 ⪰)Then the family I(119861)

119861isinB is balanced Since for each 119861 isin

B I(119861) isin A I(119861)119861isinB is also balanced in (119873 ⪰A)

Thus there is an A-partition 120587 such that for each 119894 isin 119873120587(119894)⪰1198941198611015840 for some 1198611015840 isin I(119861)

119861isinB with 1198611015840 containing player119894 However since (119873 ⪰) is subordinated to I we get that120587(119894)⪰119894I(119861)⪰

119894119861 for some 119861 isin B such thatI(119861) = 119861

1015840

Corollary 15 Let (119873 ⪰) be a hedonic game subordinated toa distribution I with A = A(I) If A is partitionable then(119873 ⪰) is pivotally balanced with respect to the distributionI

Remark 16 Theorem 10 Proposition 14 and its Corollary 15are related to a given game (119873 ⪰) and a distribution IHence we derive the familyA = A(I) of essential coalitionsand the reduced game (119873 ⪰A) If we start with a family Aof essential coalitions and (119873 ⪰) is now any hedonic game(119873 ⪰) subordinated to a distributionI such thatA = A(I)those results can be extended to a whole family of games

The result in Proposition 14 is no longer truewhen ordinalbalancedness for the game (119873 ⪰A) is replaced by pivotalbalancedness as Example 17 shows

Example 17 Let us consider Game 5 of Banerjee et al [6]which is the game (119873 ⪰) with 119873 = 1 2 3 and where theindividual preferences for players 1 2 and 3 are given by

1 2 ≻ 1 3 ≻ 1 2 3 ≻ 1

1 2 ≻ 2 3 ≻ 1 2 3 ≻ 2

1 3 ≻ 2 3 ≻ 1 2 3 ≻ 3

(16)

The only core partition in (119873 ⪰) is 120587 = 1 2 3 If I isthe distribution given by

I (1 2 3) = 2 3 I (1 2) = 1 2

I (1 3) = 1 3

I (2 3) = 2 3 I (119894) = 119894 forall119894 = 1 2 3

(17)

thenA = A(I) = 1 2 3 1 2 1 3 2 3 The asso-ciated hedonic game (119873 ⪰A) is not ordinally balanced Infact the family B = 1 2 1 3 2 3 is A-balancedHowever for any 120587 isin PA

(119873) in this game there is always aplayer 119894 that strictly prefers the worst coalition he belongs toinB to 120587(119894) For instance if 120587 = 3 1 2 player 3 strictlyprefers either 1 3 or 2 3 to 120587(3) = 3 On the other handit is easy to check that (119873 ⪰) is subordinated toA but clearlyit is not pivotally balanced with respect to I Neverthelesssince its core is nonempty it is pivotally balancedwith respectto another distribution (see [8])

Even when we do not havemuchmore information aboutthe game (119873 ⪰A) and consequently we are not aware

6 GameTheory

whether Proposition 14 applies or not there is still a proce-dure that can be useful to gather information about the game(119873 ⪰) in many cases

Definition 18 Given a hedonic game (119873 ⪰) and a distributionI we say that a distribution I1015840 is finer than I if I1015840(119878) sube

I(119878) for any 119878 isin NWe point out that when a distribution I satisfies that

I(119878) = 119878 for all 119878 then pivotal balancedness and ordinalbalancedness coincide

In the next result we are going to use the following nota-tion Given a distributionI for any 119878 isin N and for any natu-ral number 119896 let I119896(119878) = I(I119896minus1(119878)) We agree in puttingI0(119878) = 119878

Lemma 19 Given a hedonic game (119873 ⪰) subordinated to adistribution I with A = A(I) there is a finer distributionI1015840 thanI withI1015840(I1015840(119878)) = I1015840(119878) for any 119878 isin N and suchthat (119873 ⪰) is also subordinated to the distributionI1015840

Proof Since for any 119878 isin NI119896(119878) sube I119896minus1(119878) there is a first1198961015840= 1198961015840(119878) 1198961015840 ge 1 such thatI119896

1015840

(119878) = I1198961015840minus1(119878) Now for any

119878 isin N we defineI1015840(119878) = I(I1198961015840minus1(119878)) Then for any 119878 isin N

we have that

I1015840(I1015840(119878)) = I

1015840(I (I

1198961015840minus1(119878)))

= I1015840(I1198961015840

(119878))

= I (I1198961015840minus1(119878))

= I1015840(119878)

(18)

Moreover for any given 119878 isin N and since the game (119873 ⪰)

is subordinated toI we have that

I1015840(119878) ⪰119894I1198961015840minus1(119878) ⪰119894sdot sdot sdot ⪰119894119878 (19)

for any 119894 isin I1015840(119878) so the game (119873 ⪰) is also subordinated toI1015840

In the game of Example 17I1015840 = IThe rationale behind the definition of distribution I1015840

is that the set of representatives of a coalition 119878(I(I(119878)))

can also be a set of representatives of 119878 What subordinationimplies is that if the members of a coalition want to representa set of representatives of a coalition 119878 (I(I(119878))⪰

119894I(119878) for all

119894 isin I(I(119878))) then they are willing to represent the coalition119878 too We point out that since always I(I(119878)) sube I(119878)the distribution I1015840 is endowed with a certain minimalityproperty Namely for any 119878 isin N I1015840(119878) is the smaller set(with respect to inclusion) that it is able to act as a repre-sentative of coalition 119878 according to the rules of selection ofrepresentatives prescribed by the distribution I MoreoverI1015840(119878) is defined in a unique way Under the hypothesis ofLemma 19 we can consider the game (119873 ⪰A1015840) with A1015840 =A(I1015840) We note that for any coalition 119878 isin I1015840 it holds thatI1015840(119878) = 119878 Thus if (119873 ⪰A1015840) is ordinally balanced then

because of Proposition 14 (119873 ⪰) is pivotally balanced withrespect to the distributionI1015840Therefore when subordinationis present wether we know that the game (119873 ⪰A) isordinally balanced or not there is still another instance to getinformation about the core of the original game through thesimpler game (119873 ⪰A1015840) An interesting example is when thefamily A1015840 = 119894 119894 isin 119873 This case illustrates the situationwhere each coalition 119878 is willing to admit a set with onlyone player as its set of representatives Then according toTheorem 10 the partition 120587 = 1 119899 is a core partitionin (119873 ⪰) Moreover if for any 119878 |119878| gt 1 and 119894 isin I1015840(119878) thereis 1 le 119896 le 119896

1015840(119878) such that I119896(119878)≻

119894I119896minus1(119878) then 120587 is the

only core partition in (119873 ⪰) for in this case I1015840(119878) ≻119894119878 for

any 119878 notin A1015840 119894 isin I(119878)Example 8 shows that a game (119873 ⪰) can be pivotally

balanced with respect to a distribution I with A = A(I)while the associated reduced game (119873 ⪰A) is not pivotallybalancedThus in this case there seems to be no informationin the game (119873 ⪰A) about the core of (119873 ⪰) Howeverwhen subordination is present we obtain a positive result

Definition 20 Given a hedonic game (119873 ⪰) we say that adistribution I with A = A(I) is tight if for any partition120587 isin P(119873) there is a partition 120587

lowastisin PA

(119873) such that for any119894 isin 119873 120587lowast(119894)⪰

119894120587(119894)

Theorem 21 Let (119873 ⪰) be a hedonic game subordinated toa tight distribution I with A = A(I) Then the core of(119873 ⪰A) is nonempty if and only if the core of (119873 ⪰) isnonempty

Proof Let us assume first that 120587 is a core partition in (119873 ⪰)Since I is tight there is a partition 120587

lowastisin PA

(119873) such that120587lowast(119894)⪰119894120587(119894) for any 119894 isin 119873 We claim that 120587lowast is a core partition

in 119862(119873 ⪰A) If this was not the case there would be 119878 isin Asuch that 119878⪰

119894120587lowast(119894) ⪰119894120587(119894) for any 119894 isin 119878 but then 119878would block

120587 a contradictionThe other implication follows directly from part (a) of

Theorem 10

Remark 22 From the proof of the latter theorem it followsthat when a game (119873 ⪰) is pivotally balanced with respectto a tight distribution I with A = A(I) the pivotal bala-ncedness of (119873 ⪰) always implies that of (119873 ⪰A) Howeverwhen subordination is not present this condition is notnecessary as the second game in Example 13 illustrates Onthe other hand Example 8 shows a case of a pivotally balancedgame (119873 ⪰) with respect to a distribu tion I where thedistribution I is not tight and the the associated reducedgame is not pivotally balanced Thus without ldquotightnessrdquothe pivotal balancedness of a reduced game (119873 ⪰A) isindependent of that of (119873 ⪰)

Remark 23 When subordination is present in a game (119873 ⪰)for any given partition 120587 and any 119894 isin 119873 I(120587(119894))⪰

119894120587(119894)

However although I(120587(119894)) 119894 = 1 119899 is an A-family ofmutually exclusive sets it is not in general a partitionThustightness provides a related but stronger characteristic thansubordination about the game

GameTheory 7

4 Subordination and OtherSufficient Conditions

Subordination seems to be a strong condition capable oflinking core properties of a hedonic game with core prop-erties of some of its reduced games constructed based on arestricted family of coalitions Many of the games studied inthe literature share the characteristic that its behavior alsodepends on particular subfamilies of coalitions The aim ofthis section is to relate some of these games to subordination

41 Consecutive Games Consecutive games were studied byBogomolnaia and Jackson [7] In order to define them let119873 = 1 2 119899 be a finite set and 119891 119873 rarr 119873 a bijectionnamely an ordering on 119873 A coalition 119878 isin N is consecutivewith respect to 119891 if 119891(119894) lt 119891(119896) lt 119891(119895) with 119894 119895 isin 119878 impliesthat 119896 isin 119878 too

Definition 24 A hedonic game (119873 ⪰) is consecutive if thereis an ordering 119891 on119873 such that 119878⪰

119894119894 for some 119894 isin 119873 implies

that 119878 is consecutive with respect to 119891

Proposition 25 Let (119873 ⪰) be a consecutive game and 119891 anordering on119873 that makes the game consecutive Then

(a) (119873 ⪰) is subordinated to the distributionI defined byI(119878) = 119878 if 119878 is consecutive (with respect to 119891) andI(119878) = 119894 for some 119894 isin 119878 if 119878 is not consecutive

(b) If A = A(I) the core of (119873 ⪰) and the core of(119873 ⪰A) coincide

(c) (119873 ⪰) is pivotally balanced with respect to the distri-butionI

Proof (a) ClearlyI(119878)⪰119894119878 for any 119894 isin 119878when 119878 is consecutive

On the other handI(119878) = 119894≻119894119878 when 119878 is not consecutive

Thus (119873 ⪰) is subordinated toI(b) SinceA = A(I) = 119878 isin N 119878 is consecutive 119878 notin A

implies that 119878 is not consecutive and thus I(119878)≻119894119878 for the

unique element 119894 ofI(119878) Then condition (b) of Theorem 10also applies So the core of (119873 ⪰) is the same as the core of(119873 ⪰A)

(c) First we are going to show that (119873 ⪰A) is ordinallybalanced LetB sube A be a balanced family of coalitions FromGreenberg and Weber [14 Proposition 1] it follows that thefamily A is partitionable so (c) follows from Corollary 15

Remark 26 We point out that part (c) of Proposition 25has already been proven by Iehle [8 Proposition 3] Iehlersquosproof and ours are somehow different but both are based onProposition 1 of Greenberg and Weber [14]

42 Top-Coalition Property The top-coalition property [6] isanother well-known condition guaranteeing the existence ofa core partition in a hedonic game (119873 ⪰) It is also connectedto subordination as the following result showsWe recall that119880 isin N is a top coalition of 119878 isin N if 119880 sube 119878 and for any 119894 isin 119880

and 119879 sube 119878 with 119894 isin 119879 119880⪰119894119879

Definition 27 A hedonic game (119873 ⪰) satisfies the top-coalition property if for any coalition 119880 sube 119873 there is a top-coalition of 119880 One says that a distribution I is a top-coali-tion distribution of (119873 ⪰) if for any 119878I(119878) is a top-coalitionof 119878

Proposition 28 Let (119873 ⪰) be a hedonic game satisfying thetop-coalition propertyThen (119873 ⪰) is subordinated to any top-coalition distributionI

Proof It follows directly from the definition ofI

Many of the examples presented in Banerjee et al [6Section 6] also satisfy the property that the top-coalitiondistribution chosen is ordinally balanced However unlikeconsecutive games a hedonic game satisfying the top-coali-tion property may have core partitions although a top-coalition distribution is not for instance ordinally balancedLet us see the following example

Example 29 Let us consider Game 5 of Banerjee et al [6]again (see Example 13) but now with the distribution Iassigning I(119878) = 119878 for any 119878 with |119878| lt 3 and I(1 2 3) =

1 2 Then I(119878) is a top coalition for any 119878 However forB = 1 2 2 3 1 3 there is no partition 120587 such thatfor any 119894 = 1 2 3 120587(119894)⪰

119894119861 for at least one 119861 isin B whose

members are elements of the family I(119878)119878sube119873

In fact anypartition 120587 has to contain a coalition with only one elementFor this element 119894 it holds that 119894 is not ⪰

119894for any of the

coalitions of B containing it Of course the game is notpivotally balanced with respect to I either However thegame is pivotally balanced with respect to the distributionsI1015840(1 2 3) = 1 2 I1015840(1 2) = 1 2 I1015840(1 3) = 3I1015840(2 3) = 3I1015840(119894) = 119894 119894 = 1 2 3 obtained fromI bythe procedure described by Iehle [8 Proposition 3]

43 Partitioning Hedonic Games Hedonic partitioninggames do not seem to have been studied from this appro-ach before Our motivation to study them comes frompartitioning games as introduced by Kaneko and Wooders[1] in the context of cooperative games where a utility trans-ferable or not is present In addition we also introduce thisnew class of games to have a tool to deal with matchingproblems from a game theoretic point of view Partitioninggames as Kaneko andWooders [1] show in their fundamentalpaper take the idea that the development of a game candepend on a restricted family A of coalitions to the utmostextreme Not only the development of a particular game iscompletely determined byA but also a whole class of gamesrelated to this family has its development prescribed byA Inpartitioning games the behavior of any coalition 119878 outsidethe familyA is determined by itsA-partitions

In order to define a partitioning hedonic game we startwith a hedonic game (119873 ⪰A) with a restricted family Aof admissible coalitions which we will call the germ of thepartitioning hedonic game For any 119878 isin A 119894 isin 119878 let

120574 (119878 119894) = 119878 (20)

8 GameTheory

and for any 119878 notin A 119894 isin 119878 let

120574 (119878 119894) = 119878lowast (21)

with 119878lowast isin 119879 isin A(119894) 119879 sube 119878 satisfying 119879⪰119894119878lowast for all 119879 isin 119879 isin

A(119894) 119879 sube 119878

Definition 30 A partitioning hedonic game is a hedonicgame (119873 ⪰) for which there exists a germ (119873 ⪰A) such thatfor any 119894 isin 119873 ⪰

119894is defined as

for any 119878 119879 isin A (119894) 119878 ⪰119894119879 iff 119878⪰

119894119879

for any 119878 isin A (119894) 119879 isin N (119894)A 119878 ≻119894119879

for any 119878 119879 isin N (119894)A 119878 ⪰119894119879

iff 120574 (119878 119894) ⪰119894120574 (119879 119894)

(22)

Let us use (119873 ⪰| A) to denote a hedonic partitioning gamewith A as its family of essential coalitions We note that theassociated reduced game (119873 ⪰A) to the partitioning game(119873 ⪰| A) coincides with its germ

Theorem 31 Let a partitionable essential family of coalitionsA be given Then every partitioning hedonic game (119873 ⪰| A)

has nonempty core Moreover 119862(119873 ⪰| A) = 119862(119873 ⪰A)where (119873 ⪰A) is any germ of (119873 ⪰| A)

Proof Let I be a distribution such that I(119878) = 119878 for any119878 isin A and I(119878) isin A if 119878 notin A Clearly A = A(I) Weclaim that (119873 ⪰| A) is subordinated to I In fact for any119878 isin A clearly I(119878) = 119878 ⪰

119894119878 for any 119894 isin I(119878) And if 119878 notin A

I(119878)≻119894119878 for any 119894 isin I(119878) according to the construction of

the preference ≻119894 SinceA is partitionable [1] then according

to Corollary 15 the core of (119873 ⪰| A) is nonempty Moreoversince condition (b) of Theorem 10 also holds we have that119862(119873 ⪰| A) = 119862(119873 ⪰A)

44 The Marriage Model A Nonconstructive Proof of theExistence of Stable Matchings As a simple but interestingconsequence ofTheorem 31 we derive a new nonconstructive(See also [11]) proof about the existence of a stable matchingin the marriage model of Gale and Shapley [2] To do this wefirst associate a partitioning hedonic game to each marriageproblem Then we will show that the core of the game isrelated to the set of stable matchings

The marriage model consists of two finite sets of agentsthe sets119872 of ldquomenrdquo and the set119882 of ldquowomenrdquo It is assumedthat any man 119898 isin 119872 is endowed with a preference ⪰119898 overthe set 119882 cup 119898 and that any woman 119908 has a preference ⪰119908on the set 119872 cup 119908 Individual preferences are assumed tobe reflexive complete and transitive on their correspondingdomains Let us denote by (119872119882 ⪰

119872 ⪰119882) a marriage prob-

lem where ⪰119872

= (⪰119898)119898isin119872

and ⪰119882

= (⪰119908)119908isin119882

are thepreference profiles corresponding to the men and women

Amatching is a function 120583 119872cup119882 rarr 119872cup119882 satisfyingthe following

(a) For each119898 isin 119872 if 120583(119898) =119898 then 120583(119898) isin 119882

(b) For each 119908 isin 119882 if 120583(119908) =119908 then 120583(119908) isin 119872(c) 120583(120583(119896)) = 119896 for all 119896 isin 119872 cup119882

A matching 120583 is stable if 120583(119896)⪰119896119896 for all 119896 isin 119872 cup 119882

(individual stability) and if there is no pair 119898 isin 119872 119908 isin 119882

such that 120583(119898) =119908 120583(119908) =119898 and 119908≻119898120583(119898) and 119898≻

119908120583(119908)

(pairwise stability) The pair (119898 119908) is called a blocking pairGiven a marriage problem (119872119882 ⪰

119872 ⪰119882) let us con-

sider the familyA = 119878 sube 119872 cup119882 |119878| = 2 |119878 cap 119872| le 1 |119878 cap

119882| le 1 ClearlyA is essential Moreover it is a partitionablefamily as follows from the result of Kaneko and Wooders [1]about the existence of core points for every assignment game[4]

Given amarriage problem (119872119882 ⪰119872 ⪰119882) let119873 = 119872cup119882

andA as before To such a problem we associate the hedonicgame (119873 ⪰A) with A as its family of admissible coalitionswhere for each 119894 isin 119873 ⪰

119894is defined onA(119894) as follows If 119894 = 119898

for some119898 isin 119872 119878⪰119894119879 if and only if

119878 cap119882⪰119898119879 cap119882 when |119878| = |119879| = 2

119878 cap119882⪰119898119898 when |119878| = 2 and 119879 = 1

119898⪰119898119879 cap119882 when |119878| = 1 and 119879 = 2

If 119894 = 119908 for some 119908 isin 119882 119878⪰119894119879 if and only if

119878 cap119872⪰119898119879 cap119872 when |119878| = |119879| = 2

119878 cap119872⪰119898119908 when |119878| = 2 and 119879 = 1

119908⪰119898119879 cap119872 when |119878| = 1 and 119879 = 2

For any 119894 isin 119873 we also declare that 119878⪰119894119879 when |119878| = |119879| =

1With each partition 120587 in the game (119873 ⪰A) we associate

the matching 120583120587 where for each119898 isin 119872 120583120587(119898) = 120587(119898) cap119882

if |120587(119908)| = 2 and 120583120587(119898) = 119898 if |120587(119898)| = 1 Similarly for each119908 isin 119882 120583120587(119908) = 120587(119908) cap 119882 if |120587(119908)| = 2 and 120583

120587(119908) = 119908 if

|120587(119908)| = 1Now we are ready to state the following result

Theorem 32 Let (119872119882 ⪰119872 ⪰119882) be a marriage problem

Then there is a one-to-one correspondence between the stablematchings of (119872119882 ⪰

119872 ⪰119882) and the core partitions of the

partitioning game (119873 ⪰| A) whose germ is (119873 ⪰A) theassociated game to the marriage problem Therefore the set ofstable matchings of (119872119882 ⪰

119872 ⪰119882) is always nonempty

Proof From Theorem 31 we have that 119862(119873 ⪰| A) =

119862(119873 ⪰A) is nonemptyWe claim that120583120587 is a stablematchingfor each core partition 120587 in (119873 ⪰A) To see this let119898 isin 119872If 120583120587(119898) =119898 then |120587(119898)| = 2 and since 120587 is a core partition119898 cannot be strictly preferred to 120587(119898) Thus 120587(119898)⪰

119898119898 and

according to the definition ⪰119898this implies that 120587(119898) cap 119882 =

120583120587(119898)⪰119898119898 A similar argument shows that for each 119908 isin 119882

120583120587(119908)⪰119908119908 for any 119908 isin 119882 such that 120583120587(119908) = 119908 Then 120583120587 is

individually stableOn the other hand let us assume that there is a blocking

pair (119898 119908) to 120583120587 We claim that the coalition 119878 = 119898119908

objects the partition 120587 leading to a contradiction Indeedfrom 119908≻

119898120583120587(119898) we get that 119878 cap 119882≻

119898120583120587(119898) cap 119882 or

equivalently that 119878≻119898120587(119898) when 120583

120587(119898) =119898 (|120587(119898)| = 2)

GameTheory 9

We also get that 119878≻119898120587(119898) when 120587(119898) = 119898 (|120587(119898)| = 1)In a similar way we obtain that 119898≻

119908120583120587(119908) implies that

119878≻119908120587(119908) showing that 119878 blocks 120587 Thus there is a one-to-one

correspondence between the stablematchings of themarriageproblem and the core partitions of (119873 ⪰A)which coincideswith the core of the partitioning game 119862(119873 ⪰| A)

Remark 33 When all men have identical preferences and allwomen have identical preferences as well (Becker [15] mar-riage game) Banerjee et al [6] give an existence result alsogenerating a hedonic game similar to our partitioning gameFor this particular case they also reach a uniqueness result

5 Conclusions

In this paper we introduce the concept of subordination inhedonic games It is an appropriate tool to relate the corepartitions of a game with the core partitions of a suitablederived subgame We also extend to hedonic games thenotion of partitioning games previously developed in coop-erative games where a utility is present We use results aboutsubordination to show that a particular class of partitioninghedonic games behave pretty much in the same way aspartitioning games behave in the class of cooperative gameswhere a utility is present As an interesting consequencewe derive a new nonconstructive proof of the existenceof stable matchings in the marriage model Finally in thesake of completeness we present a proof of the existence ofcore partitions in hedonic games with a restricted family ofadmissible coalitions

Appendix

Proof of Theorem 6 (a sketch) Let 120587 be a core partition in(119873 ⪰A) Then the game is pivotally balanced with respectto any A-distribution I such that I(119878) sube 119894 isin 119878 120587(119894)⪰

119894119878

(see [8 Proposition 1])To see the converse we will parallel Iehlersquos proof of

Theorem 10 of the latter referenceLet us first take a utility profile for each 119894 isin 119873 namely

a function 119906119894 A(119894) rarr R such that 119906

119894(119878) ge 119906

119894(119879) if and

only if 119878 ⪰119894119879 Such a utility profile always exists Now we

construct an119873119879119880-game (119873119881A) with the restricted familyof admissible coalitionsA where for any 119878 isin A 119878 =119873

119881 (119878) = 119909 isin R119899 119909119894le 119906119894(119878) for any 119894 isin 119878 (A1)

and if Π is the family of allA-partitions of119873

119881 (119873) = ⋃

120587isinΠ

119909 isin R119899 119909119894le 120587 (119894) for any 119894 isin 119873 (A2)

We now show that if (119873 ⪰A) is pivotally balanced withrespect to an A-distribution I then for any I-balancedfamily of coalitions B sube A (119873 ⪰A) satisfies a general-ized version of 119887-balancedness [16] for 119873119879119880-games with arestricted family of coalitions We claim that this conditionimplies that there is 119909 isin 120597119881(119873) int⋃

119878isinA 119881(119878) (given aset 119878 sube 119877

119899 120597119878 and int 119878 stand for the boundary and theinterior of 119878 resp) Such a point 119909 is called a core-point in

(119873119881A) To prove this claim we will parallel the first partof Billerarsquos proof of Theorem 1 in Billera [16] We note thatfor any 119873 = 119878 isin A 119881(119878) is generated in the sense indicatedby (A1) by a unique vector 119906119878 = (119906

119878

119894)119894isin119873

where 119906119878119894= 119906119894(119878)

if 119894 isin 119878 and 119906119878

119894isin R if 119894 notin 119878 We now construct the matrix

1198721 with rows indexed by the players in119873 and the columns

indexed by themembers ofA119873 andwhose entries are givenby

1198981

119894119878=

119906119894(119878) if 119894 isin 119878

119872 if 119894 notin 119878

(A3)

where 119872 is chosen such that 119872 gt 119906119894(119878) for any 119873 = 119878 isin A

119894 isin 119878 On the other hand let 1198722 be another matrix with thesame dimension as1198721 and whose entries are given by

1198981

119894119878=

1 if 119894 isin 119878

0 if 119894 notin 119878

(A4)

Namely the columns of 1198722 are the indicator vectors of thecoalitions inA different from119873 Since the familyA containsthe individual coalitions and if 119898 = |A N| (we note that119898 ge 119899) we have that the convex set

119910 isin R119898 119910 ge 0 119872

2119910 le 120594

119873 (A5)

is bounded So we can apply a well-known result of Scarf[17 Theorem 2] to guarantee the existence of a subbasis (Asubbasis for the linear system 119872

2119910 le 120594

119873is a set 119878

1 119878

119896

of linearly independent columns of 1198722 for which there is avector 119910 such that 1198722119910 le 120594

119873 and 119910

119878= 0 for any 119878 = 119878

1

119878119896) 1198781 119878

119896for the linear system of inequalities so that

if we define

119906119894= min 1198981

119894119878119895 119895 = 1 119896 (A6)

then for each column 119878 of1198721 there is a row 119894 of1198721 satisfyingsum119898

119895=11198982

119894119878119895119910119878119895= 1 such that

119906119894ge 1198981

119894119878119896 (A7)

Therefore the vector 119906 = (119906119894)119894isin119873

does not belong to int 119881(119878)for any 119878 isin A N If we are able to show that 119906 isin 119881(119873)then any Pareto optimal point 119909 ge 119906 will be a core pointin (119873119881A) To see this we first note that by construction119906 isin 119881(119878

119895) for any 119895 = 1 119896 On the other hand since

1198781 119878

119896is a subbasis there is 119910 = (119910

119878)119878isinAN such that

1198722119910 le 120594

119873and 119910

119878= 0 if 119878 = 119878

1 119878

119896 We claim that in

fact 1198722119910 = 120594119873 Then the subbasis 119878

1 119878

119896contains a

balanced subfamilyB Since the game (119873119881A) satisfies thebalancedness property and 119906 isin 119881(119878) for any 119878 isin B weconclude that 119906 isin 119881(119873) We mention that the proof of theclaim is the same given by Billera [16] during the proof ofTheorem 6 for the case that the game is ldquofinitely generatedrdquoso we omit it here

To end the proof of the theorem we have to show that theexistence of a core point 119909 in (119873119881A) implies the existence

10 GameTheory

of a core partition in (119873 ⪰A) Nevertheless since 119909 isin 119881(119873)there is anA-partition 120587 of119873 such that 119909

119894le 120587(119894) for any 119894 isin

119873We claim that120587 is a core partition in (119873 ⪰A) In fact anyblocking coalition 119878 to 120587 would imply that 119906

119894(119878) gt 119906

119894(120587(119894)) ge

119909119894for any 119894 isin 119878 and thus that 119909 isin int119881(119878) a contradiction

Remark 34 The proof of the existence of a core point givenpreviously for a particular case of ldquofinitely generatedrdquo 119873119879119880

game with a restricted family of admissible coalitions can beextended with minor modifications to the general case ofa ldquofinitely generatedrdquo 119873119879119880 game and later following Billera[16] to a general119873119879119880 game (119873119881A)with a restricted familyof admissible coalitions

Acknowledgments

The author would like to thank CONICET and UNSL (Arge-ntina) for their financial support

References

[1] M Kaneko and M H Wooders ldquoCores of partitioning gamesrdquoMathematical Social Sciences vol 3 no 4 pp 453ndash460 1982

[2] D Gale and L Shapley ldquoCollege admissions and the stability ofmarriagerdquoThe American Mathematical Monthly vol 69 pp 9ndash15 1962

[3] M Shubik ldquoThe ldquoBridge Gamerdquo economy an example of indi-visibilitiesrdquo Journal of Political Economy vol 79 pp 909ndash9121971

[4] L S Shapley andM Shubik ldquoThe assignment game I the corerdquoInternational Journal of Game Theory vol 1 no 1 pp 111ndash1301972

[5] T Quint ldquoNecessary and sufficient conditions for balancednessin partitioning gamesrdquoMathematical Social Sciences vol 22 no1 pp 87ndash91 1991

[6] S Banerjee H Konishi and T Sonmez ldquoCore in a simplecoalition formation gamerdquo Social Choice and Welfare vol 18no 1 pp 135ndash153 2001

[7] A Bogomolnaia and M O Jackson ldquoThe stability of hedoniccoalition structuresrdquoGames and Economic Behavior vol 38 no2 pp 201ndash230 2002

[8] V Iehle ldquoThe core-partition of a hedonic gamerdquo MathematicalSocial Sciences vol 54 no 2 pp 176ndash185 2007

[9] S Papai ldquoUnique stability in simple coalition formation gamesrdquoGames and Economic Behavior vol 48 no 2 pp 337ndash354 2004

[10] J Farrell and S Scotchmer ldquoPartnershipsrdquo Quarterly Journal ofEconomics vol 103 pp 279ndash297 1988

[11] M Sotomayor ldquoA non-constructive elementary proof of theexistence of stable marriagesrdquo Games and Economic Behaviorvol 13 pp 135ndash137 1996

[12] J C Cesco ldquoHedonic games related to many-to-one matchingproblemsrdquo Social Choice and Welfare vol 39 pp 737ndash749 2012

[13] M Le Breton G Owen and S Weber ldquoStrongly balancedcooperative gamesrdquo International Journal of Game Theory vol20 no 4 pp 419ndash427 1992

[14] J Greenberg and S Weber ldquoStrong tiebout equilibrium underrestricted preferences domainrdquo Journal of EconomicTheory vol38 no 1 pp 101ndash117 1986

[15] G Becker ldquoA theory of marriage Irdquo Journal of Political Eco-nomics vol 81 pp 813ndash846 1973

[16] L Billera ldquoSome theorems on the core of an n-person gamewithout side paymentsrdquo SIAM Journal on Applied Mathematicsvol 18 no 3 pp 567ndash579 1970

[17] H Scarf ldquoThe core of an n-person gamerdquo Econometrica vol 35pp 50ndash69 1967

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


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International Journal of


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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 GameTheory

conditions for the existence of core partitions being thoseof top-coalition properties [6] consecutiveness and ordinalbalancedness [7] among themost well known Iehle [8] gives acomplete answer to the existence problem in a general settingby introducing the notion of pivotal balancedness To definepivotal balancedness first it is associated one of its subsetsto each coalition 119878 Iehle [8] calls this family of subsets adistribution Iehlersquos condition states that a hedonic game hasnonempty core if and only if there is a distribution such thatthe game is pivotally balancedwith respect to this distribution(see Section 2) Thus checking this condition is a two-stepprocedure First a distribution has to be considered andthen the balancedness condition has to be verified

On the other hand in many situations different groupsin a society choose a subgroup to act as their representativesThen the agreements (if any) reached by the groups ofrepresentatives are imposed one way or the other to thewhole society An interesting issue is whether a ldquoconvenientrdquoagreement for the representatives is also good for the restof the society With this idea in mind we note that a set ofrepresentatives for a coalition 119878 can be taken as an elementI(119878) of a distribution I a relation that builds a bridgebetween our objective and a well-developed mathematicalframework We also point out that although the notionof pivotally balancedness works very well to deal with thegeneral existence problem of core partitions in hedonicgames it does not reflect towhich extent a family of coalitionsof representatives determines the behavior of the game Forinstance if a family I of coalitions of representatives (adistribution) plays a reduced game where the only admissiblecoalitions are those in I an interesting issue is to look forrelationships if any between the cores of both games Forinstance if 120587 is a core partition in the reduced game (anagreement between the groups of representatives) is also 120587 acore partition (a good agreement for the whole society) in theoriginal game Answers to this problem could help at leastin two dimensions to the treatment of hedonic games Firstsince we start with a distribution the first step in the two-stepprocedure mentioned above is simplified Second in orderto get core partitions in the original game one can solve areduced and possibly simpler game

Unfortunately as several examples show there are nogeneral results What is missing is the description of someconnection apart from inclusion between a coalition andits set of representatives To tackle this point we appeal tothe notion of a hedonic game subordinated to a distributionIn order to deal with the reduced game played by thecoalitions of representatives we present some results about ahedonic game with a restricted family of admissible coalitionsThey are simple extensions of the main existence resultsproved by Iehle [8] Papai [9] has already used the idea ofrestrict the family of feasible coalitions to get uniquenessresults about core partitions in hedonic games while the ideaof subordination (Section 3) seems to be new Subordinationin a game (119873 ⪰) roughly states that any player chosen as arepresentative of a coalition has to be comfortable with thisdecision This is formalized by asking that ifI(119878) stands forthe set of representatives of a coalition 119878 thenI(119878)⪰

119894119878 for any

119894 isin I(119878) Subordination involves some degree of anonymity

on the individual preferences although it is weaker than thecommon ranking property [10] or the top-coalition condition[6]When subordination is present a balancedness conditionhas to be checked only on the family of representatives toguarantee the existence of core partitions Moreover undermild additional conditions the family of representativesdetermines completely the core of the hedonic game

The organization of the paper is as the following In thenext section we present our version of hedonic games wheresome restrictions on the family of admissible coalitions areimposed They are a straightforward extension of hedonicgames as stated by Banerjee et al [6] and Bogomolnaiaand Jackson [7] where there is no restriction on the familyof admissible coalitions For this enlarged class of hedonicgames we state two balancedness conditions extending thoseof ordinal balancedness of Bogomolnaia and Jackson [7] andpivotal balancedness of Iehle [8] Following those authorswe show that both conditions are sufficient to guaranteethe existence of core partitions while pivotal balancednessis also a necessary condition In Section 3 we study theissue of subordinationThe idea behind subordination is thatgiven a reduced family of coalitions along with a profile ofpreferences there are games whose core behavior is deter-mined by this basic information We state some sufficientconditions to guarantee the existence of core partitions and toguarantee that the core of the whole game is fully determinedby the basic information given on a reduced family of coali-tion as well Several examples and counter examples arealso shown there In Section 4 we relate subordination toother sufficient conditions guaranteeing the existence of corepartitions in hedonic games like top-coalition properties[6] and consecutiveness [7] At the end of this section weintroduce the notion of partitioning hedonic game Theyalways have a family of essential coalitions associated andwe prove that every partitioning game has nonempty core ifthe family of essential coalitions satisfies either condition (ii)or the equivalent condition (iii) ofTheorem27 ofKaneko andWooders [1] We use the results about hedonic partitioninggames to elaborate a nonconstructive proof of the existence ofa stable matching in the marriage game [2] different from theproof provided by Sotomayor [11] To this end we associate apartitioning hedonic game to each marriage problem so thatresults in Section 4 can be used We include a final Appe-ndix where we sketch the existence proof of core partitionsin hedonic games with a restricted family of admissiblecoalitions very similar to that of Theorem 3 of Iehle [8]

2 Hedonic Games

To define a hedonic game we start with a nonempty finiteset119873 = 1 119899 the players and its familyN of nonemptysubsets the coalitions Given any family of coalitionsA sube Nand a player 119894 isin 119873 we denote by A(119894) the subfamily ofcoalitions in A containing player 119894 A hedonic game is a 3-tuple (119873 ⪰A) where A sube N is a nonempty family and⪰ = (⪰

119894)119894isin119873

is a preference profile with ⪰119894 for any 119894 isin 119873

being a reflexive complete and transitive binary relation onA(119894) For any 119894 isin 119873 ≻

119894will stand for the strict preference

relation related to ⪰119894(119878≻119894119879 if and only if 119878⪰

119894119879 but not 119879⪰

119894119878)

GameTheory 3


1 120587

119904

of 119873 such that 120587119895


1205871 120587

119904 isin PA






119894(≻119894)







sum119878isinB

119878ni119894











119894119861 for some 119861 isin B(119894)











119860(119895) = 1 if player








1 2 ≻ 1 3 ≻ 1 2 3 ≻ 1

1 2 ≻ 2 3 ≻ 1 2 3 ≻ 2

3 ≻ 1 3 ≻ 2 3 ≻ 1 2 3

(1)


I (1 2 3) = 1 2 I (1 2) = 1 2

I (1 3) = 3

I (2 3) = 3 I (119894) = 119894 forall119894 = 1 2 3

(2)


1 2 ≻ 1

1 2 ≻ 2

3 ⪰ 3

(3)




4 GameTheory



1 2 3 4 ≻ 1 ≻ 1 2 ≻ 1 3 ≻ 1 4 ≻ 1 2 3 ≻ sdot sdot sdot

1 2 3 3 ≻ 2 3 4 ≻ 2 3 ≻ 2 ≻ sdot sdot sdot

1 2 3 4 ≻ 3 4 ≻ 3 ≻ 2 3 ≻ 3 4 ≻ 2 3 4 ≻ sdot sdot sdot

1 2 3 4 ≻ 3 4 ≻ 2 4 ≻ 4 ≻ 1 2 4 ≻ sdot sdot sdot

(4)















119894119878




119878 ≻119894120587 (119894) (6)


I (119878) ⪰119894119878 (7)


I (119878) ≻119894120587 (119894) (8)


I (119878) ≻119894120587 (119894) (9)



119904) isin



119894120587119895





I (1 2 3) = 1 3 I (1 2) = 1

I (1 3) = 1

I (2 3) = 3 I (119894) = 119894 for any 119894 = 1 2 3

(10)


given by

1 ≻ 1 2 ⪰ 1 3 ⪰ 1 2 3

1 2 ⪰ 2 3 ⪰ 1 2 3 ⪰ 2

3 ≃ 1 3 ≃ 2 3 ⪰ 1 2 3

(11)


1 ≻ 1 3

2 ≃ 2

3 ≃ 1 3

(12)

GameTheory 5





2 ≻ 1 2 ⪰ 2 3 ⪰ 1 2 3 (13)




1 2 3 ≻ 1 2 ≻ 2 3 ≻ 2 (14)





1 2 3 ≻ 1 2 ≻ 1 3 ≻ 1

1 2 3 ≻ 1 2 ≻ 2 3 ≻ 2

1 2 3 ≻ 3 ≻ 1 3 ≻ 2 3

(15)










1015840





1 2 ≻ 1 3 ≻ 1 2 3 ≻ 1

1 2 ≻ 2 3 ≻ 1 2 3 ≻ 2

1 3 ≻ 2 3 ≻ 1 2 3 ≻ 3

(16)


I (1 2 3) = 2 3 I (1 2) = 1 2

I (1 3) = 1 3

I (2 3) = 2 3 I (119894) = 119894 forall119894 = 1 2 3

(17)




6 GameTheory








1015840



we have that

I1015840(I1015840(119878)) = I

1015840(I (I

1198961015840minus1(119878)))

= I1015840(I1198961015840

(119878))

= I (I1198961015840minus1(119878))

= I1015840(119878)

(18)








119894I(119878) for all



1015840(119878) such that I119896(119878)≻






lowastisin PA


119894120587(119894)



lowastisin PA





Theorem 10



119894120587(119894)


GameTheory 7

































120574 (119878 119894) = 119878 (20)

8 GameTheory


120574 (119878 119894) = 119878lowast (21)


A(119894) 119879 sube 119878


119894is defined as

for any 119878 119879 isin A (119894) 119878 ⪰119894119879 iff 119878⪰

119894119879


for any 119878 119879 isin N (119894)A 119878 ⪰119894119879

iff 120574 (119878 119894) ⪰119894120574 (119879 119894)

(22)












lem where ⪰119872

= (⪰119898)119898isin119872

and ⪰119882

= (⪰119908)119908isin119882







such that 120583(119898) =119908 120583(119908) =119898 and 119908≻119898120583(119898) and 119898≻

119908120583(119908)


119872 ⪰119882) let us con-







119878 cap119882⪰119898119879 cap119882 when |119878| = |119879| = 2

119878 cap119882⪰119898119898 when |119878| = 2 and 119879 = 1

119898⪰119898119879 cap119882 when |119878| = 1 and 119879 = 2


119878 cap119872⪰119898119879 cap119872 when |119878| = |119879| = 2

119878 cap119872⪰119898119908 when |119878| = 2 and 119879 = 1

119908⪰119898119879 cap119872 when |119878| = 1 and 119879 = 2





120587(119908) = 119908 if









119898119898 and







119898120583120587(119898) we get that 119878 cap 119882≻

119898120583120587(119898) cap 119882 or


120587(119898) =119898 (|120587(119898)| = 2)

GameTheory 9


119908120583120587(119908) implies that




5 Conclusions


Appendix


119894119878




119894(119878) ge 119906

119894(119879) if and





119881 (119873) = ⋃

120587isinΠ






119878

119894)119894isin119873

where 119906119878119894= 119906119894(119878)

if 119894 isin 119878 and 119906119878




1198981

119894119878=

119906119894(119878) if 119894 isin 119878

119872 if 119894 notin 119878

(A3)



1198981

119894119878=

1 if 119894 isin 119878

0 if 119894 notin 119878

(A4)


119910 isin R119898 119910 ge 0 119872

2119910 le 120594

119873 (A5)


2119910 le 120594

119873is a set 119878

1 119878

119896


119873 and 119910

119878= 0 for any 119878 = 119878

1

119878119896) 1198781 119878


if we define

119906119894= min 1198981

119894119878119895 119895 = 1 119896 (A6)


119895=11198982

119894119878119895119910119878119895= 1 such that

119906119894ge 1198981

119894119878119896 (A7)




1198781 119878



1198722119910 le 120594

119873and 119910

119878= 0 if 119878 = 119878

1 119878



1 119878

119896contains a



10 GameTheory


119894le 120587(119894) for any 119894 isin


119894(119878) gt 119906

119894(120587(119894)) ge




Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









GameTheory 3


1 120587

119904

of 119873 such that 120587119895


1205871 120587

119904 isin PA






119894(≻119894)







sum119878isinB

119878ni119894











119894119861 for some 119861 isin B(119894)











119860(119895) = 1 if player








1 2 ≻ 1 3 ≻ 1 2 3 ≻ 1

1 2 ≻ 2 3 ≻ 1 2 3 ≻ 2

3 ≻ 1 3 ≻ 2 3 ≻ 1 2 3

(1)


I (1 2 3) = 1 2 I (1 2) = 1 2

I (1 3) = 3

I (2 3) = 3 I (119894) = 119894 forall119894 = 1 2 3

(2)


1 2 ≻ 1

1 2 ≻ 2

3 ⪰ 3

(3)




4 GameTheory



1 2 3 4 ≻ 1 ≻ 1 2 ≻ 1 3 ≻ 1 4 ≻ 1 2 3 ≻ sdot sdot sdot

1 2 3 3 ≻ 2 3 4 ≻ 2 3 ≻ 2 ≻ sdot sdot sdot

1 2 3 4 ≻ 3 4 ≻ 3 ≻ 2 3 ≻ 3 4 ≻ 2 3 4 ≻ sdot sdot sdot

1 2 3 4 ≻ 3 4 ≻ 2 4 ≻ 4 ≻ 1 2 4 ≻ sdot sdot sdot

(4)















119894119878




119878 ≻119894120587 (119894) (6)


I (119878) ⪰119894119878 (7)


I (119878) ≻119894120587 (119894) (8)


I (119878) ≻119894120587 (119894) (9)



119904) isin



119894120587119895





I (1 2 3) = 1 3 I (1 2) = 1

I (1 3) = 1

I (2 3) = 3 I (119894) = 119894 for any 119894 = 1 2 3

(10)


given by

1 ≻ 1 2 ⪰ 1 3 ⪰ 1 2 3

1 2 ⪰ 2 3 ⪰ 1 2 3 ⪰ 2

3 ≃ 1 3 ≃ 2 3 ⪰ 1 2 3

(11)


1 ≻ 1 3

2 ≃ 2

3 ≃ 1 3

(12)

GameTheory 5





2 ≻ 1 2 ⪰ 2 3 ⪰ 1 2 3 (13)




1 2 3 ≻ 1 2 ≻ 2 3 ≻ 2 (14)





1 2 3 ≻ 1 2 ≻ 1 3 ≻ 1

1 2 3 ≻ 1 2 ≻ 2 3 ≻ 2

1 2 3 ≻ 3 ≻ 1 3 ≻ 2 3

(15)










1015840





1 2 ≻ 1 3 ≻ 1 2 3 ≻ 1

1 2 ≻ 2 3 ≻ 1 2 3 ≻ 2

1 3 ≻ 2 3 ≻ 1 2 3 ≻ 3

(16)


I (1 2 3) = 2 3 I (1 2) = 1 2

I (1 3) = 1 3

I (2 3) = 2 3 I (119894) = 119894 forall119894 = 1 2 3

(17)




6 GameTheory








1015840



we have that

I1015840(I1015840(119878)) = I

1015840(I (I

1198961015840minus1(119878)))

= I1015840(I1198961015840

(119878))

= I (I1198961015840minus1(119878))

= I1015840(119878)

(18)








119894I(119878) for all



1015840(119878) such that I119896(119878)≻






lowastisin PA


119894120587(119894)



lowastisin PA





Theorem 10



119894120587(119894)


GameTheory 7

































120574 (119878 119894) = 119878 (20)

8 GameTheory


120574 (119878 119894) = 119878lowast (21)


A(119894) 119879 sube 119878


119894is defined as

for any 119878 119879 isin A (119894) 119878 ⪰119894119879 iff 119878⪰

119894119879


for any 119878 119879 isin N (119894)A 119878 ⪰119894119879

iff 120574 (119878 119894) ⪰119894120574 (119879 119894)

(22)












lem where ⪰119872

= (⪰119898)119898isin119872

and ⪰119882

= (⪰119908)119908isin119882







such that 120583(119898) =119908 120583(119908) =119898 and 119908≻119898120583(119898) and 119898≻

119908120583(119908)


119872 ⪰119882) let us con-







119878 cap119882⪰119898119879 cap119882 when |119878| = |119879| = 2

119878 cap119882⪰119898119898 when |119878| = 2 and 119879 = 1

119898⪰119898119879 cap119882 when |119878| = 1 and 119879 = 2


119878 cap119872⪰119898119879 cap119872 when |119878| = |119879| = 2

119878 cap119872⪰119898119908 when |119878| = 2 and 119879 = 1

119908⪰119898119879 cap119872 when |119878| = 1 and 119879 = 2





120587(119908) = 119908 if









119898119898 and







119898120583120587(119898) we get that 119878 cap 119882≻

119898120583120587(119898) cap 119882 or


120587(119898) =119898 (|120587(119898)| = 2)

GameTheory 9


119908120583120587(119908) implies that




5 Conclusions


Appendix


119894119878




119894(119878) ge 119906

119894(119879) if and





119881 (119873) = ⋃

120587isinΠ






119878

119894)119894isin119873

where 119906119878119894= 119906119894(119878)

if 119894 isin 119878 and 119906119878




1198981

119894119878=

119906119894(119878) if 119894 isin 119878

119872 if 119894 notin 119878

(A3)



1198981

119894119878=

1 if 119894 isin 119878

0 if 119894 notin 119878

(A4)


119910 isin R119898 119910 ge 0 119872

2119910 le 120594

119873 (A5)


2119910 le 120594

119873is a set 119878

1 119878

119896


119873 and 119910

119878= 0 for any 119878 = 119878

1

119878119896) 1198781 119878


if we define

119906119894= min 1198981

119894119878119895 119895 = 1 119896 (A6)


119895=11198982

119894119878119895119910119878119895= 1 such that

119906119894ge 1198981

119894119878119896 (A7)




1198781 119878



1198722119910 le 120594

119873and 119910

119878= 0 if 119878 = 119878

1 119878



1 119878

119896contains a



10 GameTheory


119894le 120587(119894) for any 119894 isin


119894(119878) gt 119906

119894(120587(119894)) ge




Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









4 GameTheory



1 2 3 4 ≻ 1 ≻ 1 2 ≻ 1 3 ≻ 1 4 ≻ 1 2 3 ≻ sdot sdot sdot

1 2 3 3 ≻ 2 3 4 ≻ 2 3 ≻ 2 ≻ sdot sdot sdot

1 2 3 4 ≻ 3 4 ≻ 3 ≻ 2 3 ≻ 3 4 ≻ 2 3 4 ≻ sdot sdot sdot

1 2 3 4 ≻ 3 4 ≻ 2 4 ≻ 4 ≻ 1 2 4 ≻ sdot sdot sdot

(4)















119894119878




119878 ≻119894120587 (119894) (6)


I (119878) ⪰119894119878 (7)


I (119878) ≻119894120587 (119894) (8)


I (119878) ≻119894120587 (119894) (9)



119904) isin



119894120587119895





I (1 2 3) = 1 3 I (1 2) = 1

I (1 3) = 1

I (2 3) = 3 I (119894) = 119894 for any 119894 = 1 2 3

(10)


given by

1 ≻ 1 2 ⪰ 1 3 ⪰ 1 2 3

1 2 ⪰ 2 3 ⪰ 1 2 3 ⪰ 2

3 ≃ 1 3 ≃ 2 3 ⪰ 1 2 3

(11)


1 ≻ 1 3

2 ≃ 2

3 ≃ 1 3

(12)

GameTheory 5





2 ≻ 1 2 ⪰ 2 3 ⪰ 1 2 3 (13)




1 2 3 ≻ 1 2 ≻ 2 3 ≻ 2 (14)





1 2 3 ≻ 1 2 ≻ 1 3 ≻ 1

1 2 3 ≻ 1 2 ≻ 2 3 ≻ 2

1 2 3 ≻ 3 ≻ 1 3 ≻ 2 3

(15)










1015840





1 2 ≻ 1 3 ≻ 1 2 3 ≻ 1

1 2 ≻ 2 3 ≻ 1 2 3 ≻ 2

1 3 ≻ 2 3 ≻ 1 2 3 ≻ 3

(16)


I (1 2 3) = 2 3 I (1 2) = 1 2

I (1 3) = 1 3

I (2 3) = 2 3 I (119894) = 119894 forall119894 = 1 2 3

(17)




6 GameTheory








1015840



we have that

I1015840(I1015840(119878)) = I

1015840(I (I

1198961015840minus1(119878)))

= I1015840(I1198961015840

(119878))

= I (I1198961015840minus1(119878))

= I1015840(119878)

(18)








119894I(119878) for all



1015840(119878) such that I119896(119878)≻






lowastisin PA


119894120587(119894)



lowastisin PA





Theorem 10



119894120587(119894)


GameTheory 7

































120574 (119878 119894) = 119878 (20)

8 GameTheory


120574 (119878 119894) = 119878lowast (21)


A(119894) 119879 sube 119878


119894is defined as

for any 119878 119879 isin A (119894) 119878 ⪰119894119879 iff 119878⪰

119894119879


for any 119878 119879 isin N (119894)A 119878 ⪰119894119879

iff 120574 (119878 119894) ⪰119894120574 (119879 119894)

(22)












lem where ⪰119872

= (⪰119898)119898isin119872

and ⪰119882

= (⪰119908)119908isin119882







such that 120583(119898) =119908 120583(119908) =119898 and 119908≻119898120583(119898) and 119898≻

119908120583(119908)


119872 ⪰119882) let us con-







119878 cap119882⪰119898119879 cap119882 when |119878| = |119879| = 2

119878 cap119882⪰119898119898 when |119878| = 2 and 119879 = 1

119898⪰119898119879 cap119882 when |119878| = 1 and 119879 = 2


119878 cap119872⪰119898119879 cap119872 when |119878| = |119879| = 2

119878 cap119872⪰119898119908 when |119878| = 2 and 119879 = 1

119908⪰119898119879 cap119872 when |119878| = 1 and 119879 = 2





120587(119908) = 119908 if









119898119898 and







119898120583120587(119898) we get that 119878 cap 119882≻

119898120583120587(119898) cap 119882 or


120587(119898) =119898 (|120587(119898)| = 2)

GameTheory 9


119908120583120587(119908) implies that




5 Conclusions


Appendix


119894119878




119894(119878) ge 119906

119894(119879) if and





119881 (119873) = ⋃

120587isinΠ






119878

119894)119894isin119873

where 119906119878119894= 119906119894(119878)

if 119894 isin 119878 and 119906119878




1198981

119894119878=

119906119894(119878) if 119894 isin 119878

119872 if 119894 notin 119878

(A3)



1198981

119894119878=

1 if 119894 isin 119878

0 if 119894 notin 119878

(A4)


119910 isin R119898 119910 ge 0 119872

2119910 le 120594

119873 (A5)


2119910 le 120594

119873is a set 119878

1 119878

119896


119873 and 119910

119878= 0 for any 119878 = 119878

1

119878119896) 1198781 119878


if we define

119906119894= min 1198981

119894119878119895 119895 = 1 119896 (A6)


119895=11198982

119894119878119895119910119878119895= 1 such that

119906119894ge 1198981

119894119878119896 (A7)




1198781 119878



1198722119910 le 120594

119873and 119910

119878= 0 if 119878 = 119878

1 119878



1 119878

119896contains a



10 GameTheory


119894le 120587(119894) for any 119894 isin


119894(119878) gt 119906

119894(120587(119894)) ge




Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









GameTheory 5





2 ≻ 1 2 ⪰ 2 3 ⪰ 1 2 3 (13)




1 2 3 ≻ 1 2 ≻ 2 3 ≻ 2 (14)





1 2 3 ≻ 1 2 ≻ 1 3 ≻ 1

1 2 3 ≻ 1 2 ≻ 2 3 ≻ 2

1 2 3 ≻ 3 ≻ 1 3 ≻ 2 3

(15)










1015840





1 2 ≻ 1 3 ≻ 1 2 3 ≻ 1

1 2 ≻ 2 3 ≻ 1 2 3 ≻ 2

1 3 ≻ 2 3 ≻ 1 2 3 ≻ 3

(16)


I (1 2 3) = 2 3 I (1 2) = 1 2

I (1 3) = 1 3

I (2 3) = 2 3 I (119894) = 119894 forall119894 = 1 2 3

(17)




6 GameTheory








1015840



we have that

I1015840(I1015840(119878)) = I

1015840(I (I

1198961015840minus1(119878)))

= I1015840(I1198961015840

(119878))

= I (I1198961015840minus1(119878))

= I1015840(119878)

(18)








119894I(119878) for all



1015840(119878) such that I119896(119878)≻






lowastisin PA


119894120587(119894)



lowastisin PA





Theorem 10



119894120587(119894)


GameTheory 7

































120574 (119878 119894) = 119878 (20)

8 GameTheory


120574 (119878 119894) = 119878lowast (21)


A(119894) 119879 sube 119878


119894is defined as

for any 119878 119879 isin A (119894) 119878 ⪰119894119879 iff 119878⪰

119894119879


for any 119878 119879 isin N (119894)A 119878 ⪰119894119879

iff 120574 (119878 119894) ⪰119894120574 (119879 119894)

(22)












lem where ⪰119872

= (⪰119898)119898isin119872

and ⪰119882

= (⪰119908)119908isin119882







such that 120583(119898) =119908 120583(119908) =119898 and 119908≻119898120583(119898) and 119898≻

119908120583(119908)


119872 ⪰119882) let us con-







119878 cap119882⪰119898119879 cap119882 when |119878| = |119879| = 2

119878 cap119882⪰119898119898 when |119878| = 2 and 119879 = 1

119898⪰119898119879 cap119882 when |119878| = 1 and 119879 = 2


119878 cap119872⪰119898119879 cap119872 when |119878| = |119879| = 2

119878 cap119872⪰119898119908 when |119878| = 2 and 119879 = 1

119908⪰119898119879 cap119872 when |119878| = 1 and 119879 = 2





120587(119908) = 119908 if









119898119898 and







119898120583120587(119898) we get that 119878 cap 119882≻

119898120583120587(119898) cap 119882 or


120587(119898) =119898 (|120587(119898)| = 2)

GameTheory 9


119908120583120587(119908) implies that




5 Conclusions


Appendix


119894119878




119894(119878) ge 119906

119894(119879) if and





119881 (119873) = ⋃

120587isinΠ






119878

119894)119894isin119873

where 119906119878119894= 119906119894(119878)

if 119894 isin 119878 and 119906119878




1198981

119894119878=

119906119894(119878) if 119894 isin 119878

119872 if 119894 notin 119878

(A3)



1198981

119894119878=

1 if 119894 isin 119878

0 if 119894 notin 119878

(A4)


119910 isin R119898 119910 ge 0 119872

2119910 le 120594

119873 (A5)


2119910 le 120594

119873is a set 119878

1 119878

119896


119873 and 119910

119878= 0 for any 119878 = 119878

1

119878119896) 1198781 119878


if we define

119906119894= min 1198981

119894119878119895 119895 = 1 119896 (A6)


119895=11198982

119894119878119895119910119878119895= 1 such that

119906119894ge 1198981

119894119878119896 (A7)




1198781 119878



1198722119910 le 120594

119873and 119910

119878= 0 if 119878 = 119878

1 119878



1 119878

119896contains a



10 GameTheory


119894le 120587(119894) for any 119894 isin


119894(119878) gt 119906

119894(120587(119894)) ge




Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









6 GameTheory








1015840



we have that

I1015840(I1015840(119878)) = I

1015840(I (I

1198961015840minus1(119878)))

= I1015840(I1198961015840

(119878))

= I (I1198961015840minus1(119878))

= I1015840(119878)

(18)








119894I(119878) for all



1015840(119878) such that I119896(119878)≻






lowastisin PA


119894120587(119894)



lowastisin PA





Theorem 10



119894120587(119894)


GameTheory 7

































120574 (119878 119894) = 119878 (20)

8 GameTheory


120574 (119878 119894) = 119878lowast (21)


A(119894) 119879 sube 119878


119894is defined as

for any 119878 119879 isin A (119894) 119878 ⪰119894119879 iff 119878⪰

119894119879


for any 119878 119879 isin N (119894)A 119878 ⪰119894119879

iff 120574 (119878 119894) ⪰119894120574 (119879 119894)

(22)












lem where ⪰119872

= (⪰119898)119898isin119872

and ⪰119882

= (⪰119908)119908isin119882







such that 120583(119898) =119908 120583(119908) =119898 and 119908≻119898120583(119898) and 119898≻

119908120583(119908)


119872 ⪰119882) let us con-







119878 cap119882⪰119898119879 cap119882 when |119878| = |119879| = 2

119878 cap119882⪰119898119898 when |119878| = 2 and 119879 = 1

119898⪰119898119879 cap119882 when |119878| = 1 and 119879 = 2


119878 cap119872⪰119898119879 cap119872 when |119878| = |119879| = 2

119878 cap119872⪰119898119908 when |119878| = 2 and 119879 = 1

119908⪰119898119879 cap119872 when |119878| = 1 and 119879 = 2





120587(119908) = 119908 if









119898119898 and







119898120583120587(119898) we get that 119878 cap 119882≻

119898120583120587(119898) cap 119882 or


120587(119898) =119898 (|120587(119898)| = 2)

GameTheory 9


119908120583120587(119908) implies that




5 Conclusions


Appendix


119894119878




119894(119878) ge 119906

119894(119879) if and





119881 (119873) = ⋃

120587isinΠ






119878

119894)119894isin119873

where 119906119878119894= 119906119894(119878)

if 119894 isin 119878 and 119906119878




1198981

119894119878=

119906119894(119878) if 119894 isin 119878

119872 if 119894 notin 119878

(A3)



1198981

119894119878=

1 if 119894 isin 119878

0 if 119894 notin 119878

(A4)


119910 isin R119898 119910 ge 0 119872

2119910 le 120594

119873 (A5)


2119910 le 120594

119873is a set 119878

1 119878

119896


119873 and 119910

119878= 0 for any 119878 = 119878

1

119878119896) 1198781 119878


if we define

119906119894= min 1198981

119894119878119895 119895 = 1 119896 (A6)


119895=11198982

119894119878119895119910119878119895= 1 such that

119906119894ge 1198981

119894119878119896 (A7)




1198781 119878



1198722119910 le 120594

119873and 119910

119878= 0 if 119878 = 119878

1 119878



1 119878

119896contains a



10 GameTheory


119894le 120587(119894) for any 119894 isin


119894(119878) gt 119906

119894(120587(119894)) ge




Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









GameTheory 7

































120574 (119878 119894) = 119878 (20)

8 GameTheory


120574 (119878 119894) = 119878lowast (21)


A(119894) 119879 sube 119878


119894is defined as

for any 119878 119879 isin A (119894) 119878 ⪰119894119879 iff 119878⪰

119894119879


for any 119878 119879 isin N (119894)A 119878 ⪰119894119879

iff 120574 (119878 119894) ⪰119894120574 (119879 119894)

(22)












lem where ⪰119872

= (⪰119898)119898isin119872

and ⪰119882

= (⪰119908)119908isin119882







such that 120583(119898) =119908 120583(119908) =119898 and 119908≻119898120583(119898) and 119898≻

119908120583(119908)


119872 ⪰119882) let us con-







119878 cap119882⪰119898119879 cap119882 when |119878| = |119879| = 2

119878 cap119882⪰119898119898 when |119878| = 2 and 119879 = 1

119898⪰119898119879 cap119882 when |119878| = 1 and 119879 = 2


119878 cap119872⪰119898119879 cap119872 when |119878| = |119879| = 2

119878 cap119872⪰119898119908 when |119878| = 2 and 119879 = 1

119908⪰119898119879 cap119872 when |119878| = 1 and 119879 = 2





120587(119908) = 119908 if









119898119898 and







119898120583120587(119898) we get that 119878 cap 119882≻

119898120583120587(119898) cap 119882 or


120587(119898) =119898 (|120587(119898)| = 2)

GameTheory 9


119908120583120587(119908) implies that




5 Conclusions


Appendix


119894119878




119894(119878) ge 119906

119894(119879) if and





119881 (119873) = ⋃

120587isinΠ






119878

119894)119894isin119873

where 119906119878119894= 119906119894(119878)

if 119894 isin 119878 and 119906119878




1198981

119894119878=

119906119894(119878) if 119894 isin 119878

119872 if 119894 notin 119878

(A3)



1198981

119894119878=

1 if 119894 isin 119878

0 if 119894 notin 119878

(A4)


119910 isin R119898 119910 ge 0 119872

2119910 le 120594

119873 (A5)


2119910 le 120594

119873is a set 119878

1 119878

119896


119873 and 119910

119878= 0 for any 119878 = 119878

1

119878119896) 1198781 119878


if we define

119906119894= min 1198981

119894119878119895 119895 = 1 119896 (A6)


119895=11198982

119894119878119895119910119878119895= 1 such that

119906119894ge 1198981

119894119878119896 (A7)




1198781 119878



1198722119910 le 120594

119873and 119910

119878= 0 if 119878 = 119878

1 119878



1 119878

119896contains a



10 GameTheory


119894le 120587(119894) for any 119894 isin


119894(119878) gt 119906

119894(120587(119894)) ge




Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









8 GameTheory


120574 (119878 119894) = 119878lowast (21)


A(119894) 119879 sube 119878


119894is defined as

for any 119878 119879 isin A (119894) 119878 ⪰119894119879 iff 119878⪰

119894119879


for any 119878 119879 isin N (119894)A 119878 ⪰119894119879

iff 120574 (119878 119894) ⪰119894120574 (119879 119894)

(22)












lem where ⪰119872

= (⪰119898)119898isin119872

and ⪰119882

= (⪰119908)119908isin119882







such that 120583(119898) =119908 120583(119908) =119898 and 119908≻119898120583(119898) and 119898≻

119908120583(119908)


119872 ⪰119882) let us con-







119878 cap119882⪰119898119879 cap119882 when |119878| = |119879| = 2

119878 cap119882⪰119898119898 when |119878| = 2 and 119879 = 1

119898⪰119898119879 cap119882 when |119878| = 1 and 119879 = 2


119878 cap119872⪰119898119879 cap119872 when |119878| = |119879| = 2

119878 cap119872⪰119898119908 when |119878| = 2 and 119879 = 1

119908⪰119898119879 cap119872 when |119878| = 1 and 119879 = 2





120587(119908) = 119908 if









119898119898 and







119898120583120587(119898) we get that 119878 cap 119882≻

119898120583120587(119898) cap 119882 or


120587(119898) =119898 (|120587(119898)| = 2)

GameTheory 9


119908120583120587(119908) implies that




5 Conclusions


Appendix


119894119878




119894(119878) ge 119906

119894(119879) if and





119881 (119873) = ⋃

120587isinΠ






119878

119894)119894isin119873

where 119906119878119894= 119906119894(119878)

if 119894 isin 119878 and 119906119878




1198981

119894119878=

119906119894(119878) if 119894 isin 119878

119872 if 119894 notin 119878

(A3)



1198981

119894119878=

1 if 119894 isin 119878

0 if 119894 notin 119878

(A4)


119910 isin R119898 119910 ge 0 119872

2119910 le 120594

119873 (A5)


2119910 le 120594

119873is a set 119878

1 119878

119896


119873 and 119910

119878= 0 for any 119878 = 119878

1

119878119896) 1198781 119878


if we define

119906119894= min 1198981

119894119878119895 119895 = 1 119896 (A6)


119895=11198982

119894119878119895119910119878119895= 1 such that

119906119894ge 1198981

119894119878119896 (A7)




1198781 119878



1198722119910 le 120594

119873and 119910

119878= 0 if 119878 = 119878

1 119878



1 119878

119896contains a



10 GameTheory


119894le 120587(119894) for any 119894 isin


119894(119878) gt 119906

119894(120587(119894)) ge




Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









GameTheory 9


119908120583120587(119908) implies that




5 Conclusions


Appendix


119894119878




119894(119878) ge 119906

119894(119879) if and





119881 (119873) = ⋃

120587isinΠ






119878

119894)119894isin119873

where 119906119878119894= 119906119894(119878)

if 119894 isin 119878 and 119906119878




1198981

119894119878=

119906119894(119878) if 119894 isin 119878

119872 if 119894 notin 119878

(A3)



1198981

119894119878=

1 if 119894 isin 119878

0 if 119894 notin 119878

(A4)


119910 isin R119898 119910 ge 0 119872

2119910 le 120594

119873 (A5)


2119910 le 120594

119873is a set 119878

1 119878

119896


119873 and 119910

119878= 0 for any 119878 = 119878

1

119878119896) 1198781 119878


if we define

119906119894= min 1198981

119894119878119895 119895 = 1 119896 (A6)


119895=11198982

119894119878119895119910119878119895= 1 such that

119906119894ge 1198981

119894119878119896 (A7)




1198781 119878



1198722119910 le 120594

119873and 119910

119878= 0 if 119878 = 119878

1 119878



1 119878

119896contains a



10 GameTheory


119894le 120587(119894) for any 119894 isin


119894(119878) gt 119906

119894(120587(119894)) ge




Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









10 GameTheory


119894le 120587(119894) for any 119894 isin


119894(119878) gt 119906

119894(120587(119894)) ge




Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Subordinated Hedonic Games