Efficiency and stability of probabilistic assignments in marriage problems

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Contents lists available atScienceDirect

Games

and

Economic

Behavior

www.elsevier.com/locate/geb

Eﬃciency

and

stability

of

probabilistic

assignments

in marriage

problems

✩

Battal Do˘gan

a

,

Kemal Yıldız

b

,∗

a_Faculty_of_Business_and_Economics,_University_of_Lausanne,_Switzerland b_Department_of_Economics,_Bilkent_University,_Turkey

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received20July2014

Availableonline17December2015

JELclassiﬁcation: C60 C71 C78 D61 Keywords: Marriageproblems Probabilisticassignment Eﬃciency Stability

We study marriage problems where two groups of agents, men and women, match each other and probabilistic assignments are possible. When only ordinal preferences are observable, stochastic dominance efficiency (sd-efficiency) is commonly used. First, we provide a characterization of sd-efficient allocations in terms of a property of an orderrelationdefined onthesetofman–womanpairs.Then,usingthischaracterization, we constructivelyprovethatforeachprobabilisticassignmentthatissd-efficientforsome ordinal preferences,there isavon Neumann–Morgensternutility profileconsistentwith the ordinal preferences for whichthe assignment is Pareto efficient. Second, we show that when the preferences are strict, for each ordinal preference profile and each ex-poststableprobabilisticassignment,thereisavon Neumann–Morgensternutilityprofile, consistentwiththeordinal preferences,forwhichtheassignmentbelongstothecoreof theassociatedtransferableutilitygame.

1. Introduction

Thetheoryoftwo-sidedmatchingproblemshasbeenusefulinprovidingsolutionstomanyreal-lifeeconomicproblems (seeRoth andSotomayor,1990).A marriageproblem,whichconstitutesabasisfortwo-sidedmatchingproblems,consists of two equal-sized groups of agents: menand women. Each man has ordinal preferences over women and vice versa. A majorityoftheliteratureisfocused ondeterministicassignments,wheremenandwomenarematchedone-to-one.One canalsothinkofprobabilisticassignments,whicharelotteriesoverdeterministicassignments,andtheseareinterestingfor atleast two reasons: (1) agents maymatchin fractions; forexample, a consultantmay allocatehis time among several ﬁrms,1 (2) probabilisticassignments mayhelpusachieve fairnesswhenit isnot possiblewithdeterministic assignments (e.g. Bogomolnaiaand Moulin, 2001). Here, we consider probabilistic assignments, andin particular their “stability” and “eﬃciency”.

Weconsidera modelwhereonlyordinalpreferenceinformationisavailable.That is,foreach man,all weknowishis preferenceorderingoverwomen,andviceversa,whichisinlinewiththeapplicationsandthetheoreticalliterature.Most allocationrulesthathavebeendiscussedinthisliteratureelicitonlyordinalpreferences,as opposed toutility information

✩ _We_are_grateful_to_Paulo_Barelli,_Srihari_Govindan,_Bettina_Klaus,_Efe_Ok,_Ariel_Rubinstein,_William_Thomson,_Rakesh_Vohra,_two_anonymous_referees,_the

associateeditor,seminarparticipantsatBilkentUniversityandUniversityofRochester,andparticipantsattheBosphorousWorkshoponEconomicDesign andWorkshoponSocialChoiceandMechanismDesignattheUniversityofManchesterfortheirhelpfulcomments.

*

Correspondingauthor.

E-mailaddresses:battaldogan@gmail.com(B. Do˘gan),kemal.yildiz@bilkent.edu.tr(K. Yıldız).

1 _See_{Manjunath (2014)}_and_Bogomolnaia_and_{Moulin (2004)}_for_more_examples_of_fractional_matching_markets. http://dx.doi.org/10.1016/j.geb.2015.12.001

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over possible mates.2 When only ordinal preference information is available, extensively studied efficiency and stability notionsarestochasticdominanceefficiency (sd-efficiency)andex-poststability.We askwhetherprobabilisticassignmentsthat aresd-efficientorex-poststableforordinalpreferencesarepossiblyefficientorstableforcardinalpreferences.

The maincontributionofthispaperpertainstostability,whichisa centralrobustness conditionfortwo-sided match-ings. A deterministicassignment is stableifno pair ofa man and awoman prefers each other to their assignedmates. A probabilisticassignmentisex-poststableifitcanbeexpressedasalotteryoverstabledeterministicassignments.In the deterministiccase,if anassignmentisstable,thenitisalsointhecoreofanassociatednon-transferableutility(NTU)game (RothandSotomayor,1990).It followsthatinanordinalenvironmentwheremonetarytransfersarenotallowed,if an as-signmentisex-poststable,thenthereisnoincentiveforex-post3 _group_deviation._However,_ex-ante,4 _a_group_of_men_and

women maybreakawayfromtherestofsocietyandmayformmatchesamongthemselvesthatmakeeachofthembetter off in terms ofexpected utility.This possibility motivatesus to ask whetherex-post stability implieswelfare properties deduciblefromtheordinalpreferencesthatwouldpreventsuchbreakaways.

Weshowthatwhenpreferencesarestrict(noagentisindifferentbetweentwodifferentagents),foreachex-poststable probabilistic assignment, there is a utility profile consistent with the ordinal preferences such that no group of agents consistingofequalnumbersofmenandwomencandeviatetoaprobabilisticassignmentamongthemselvesandmakeeach memberbetteroff.In fact,we proveanevenstrongerresult(Theorem 2):foreachex-poststableprobabilisticassignment, thereisautilityprofileconsistentwiththeordinalpreferencessuchthatnogroupofagentsconsistingofequalnumbersof menandwomencandeviatetoaprobabilistic assignmentamongthemselvesinwhichthesumoftheir expectedutilities is greater.Put differently,we associatewitheachutility profilea transferableutility (TU)game,as inShapleyandShubik (1971),in whichacoalitionisadmissibleifitconsistsofequalnumbersofmenandwomen,andtheworth ofeachsuch coalitionisthemaximaltotalexpectedutilityitcanachievebyformingmatchesamongthemselves.5_{We show}_that_for_each

ex-poststableassignment,thereisautilityproﬁlesuchthattheassignmentbelongstothecoreoftheassociatedTU-game (Theorem 2).Thatis,evenifweweretoallowmonetarytransfers,therewouldbenoproﬁtableex-antegroupdeviation.

Our other result pertains to efficiency. A natural efficiency requirementfor probabilistic assignments is sd-efficiency, which is based on the first-order stochastic dominance relation. A probabilistic assignment stochastically dominates (sd-dominates) another if for each agent, the probability distribution assigned to that agent in the former first-order stochastically dominatestheprobability distributionassignedtothatagent inthelatterassignment.Assignments thatare undominated inthissensearecalled“sd-efficient”.6 _{We ask}_the_following_question:_Consider _an_ordinal_preference_profile

andan sd-efficient assignment.Does thereexista utility profileconsistent withtheordinal preferencesatwhich the as-signmentisParetoefficient?We showthatforeachordinalpreferenceprofile,andforeachsd-efficientassignment,onecan constructautilityprofileconsistentwiththeordinalpreferencessuchthatthesumoftheexpectedutilitiesoftheagentsis maximizedatthatassignment(Theorem 1).To provetheresult,we characterizethesd-efficiencyofanassignmentinterms ofapropertyofanorderrelationthatwedefineoverthesetofman–womanpairs.7

Our results regarding eﬃciency are intimately related to some recent results. Carroll (2011) proves a counterpart of

Theorem 1inamoregeneralsocialchoicesetup,fromwhichTheorem 1canbeobtainedasacorollary.Azizetal. (2015)

provide an interesting non-expectedutility generalizationof Carroll (2011).Ourresultsare basedon anorder theoretical analysis; we characterize the sd-efficiency ofan assignment in terms ofthe acyclicity of abinary relationon the set of man–womanpairs,whichparallelstheresultsbyBogomolnaiaandMoulin (2001)andKattaandSethuraman (2006).In this vein,Azizetal. (2015)notethatitdoesnotseempossibletoextendthischaracterizationtoCarroll’s(2011)ortheirgeneral socialchoicesetting.Moreover,theutility profileconstructedinTheorem 1clearly relatestotheutilityprofileconstructed inTheorem 2,whichshedsfurtherlightontherelationbetweensd-efficiencyandex-poststability.8

Ourresultsforstability,unlike efficiency,cannotbe directlyrelatedtotherecentfindings onthegeneralsocialchoice setup.To proveTheorem 2,we observeaninterestingpropertyofex-poststableassignmentsrelatedtothelatticestructure of stabledeterministic assignments (see Knuth,1976,pp. 92–93, who attributesthe discovery ofthislattice structure to J.H. Conway). InProposition 3,we showthat each ex-poststableprobabilistic assignment canbe decomposedintoa col-lectionofdeterministic stableassignments,whichcanbeorderedinsuch awaythateachman’swelfareisnon-increasing andeachwoman’swelfareisnon-decreasingaswefollowtheassignments fromthefirsttothelast.Thisresult,whichwe show viatheroundingapproach duetoTeoandSethuraman (1998),playsakeyroleintheproofofTheorem 2.A corollary ofTheorem 2isthatex-poststabilityimpliessd-efficiencywhenpreferencesarestrict,whichisindependentlyshownalso byManjunath (2011).

2 _One_{justiﬁcation}_for_why_such_mechanisms_are_common_is_that_it_is_a_complex_process_for_an_agent_to_formulate_his_utility_information._See_Bogomolnaia andMoulin (2001)foradetaileddiscussion.

3 _After_the_realization_of_a_{deterministic}_assignment.

4 _That_is,_before_the_realization_of_a_particular_{deterministic}_assignment. 5 _We_assume_preferences_to_be_quasilinear_in_money.

6 _This_notion_is_usually _referred_to_as_“ordinal_{eﬃciency”,}_starting_from_Bogomolnaia_and_{Moulin (2001)}_._Here,_{we use}_the_terminology_of_Thomson (2010).

7 _For_the_problem_of_assigning_objects,_similar_{characterizations}_are_given_by_Bogomolnaia_and_{Moulin (2001)}_in_the_case_of_strict_preferences_and_by KattaandSethuraman (2006)inthecaseofweakpreferences.

8 _In_fact,_{as noted}_in_the_proof_of_{Proposition 2}_,_{if the}_ex-post_stable_assignment_has_a_{decomposition}_into_stable_assignments_such_that_no_agent_matches

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1.1. Relatedliterature

Fortheprobabilisticassignmentsetup,a strandoftheliteraturediscussesdifferentstabilitynotionsandtheirrelationto eachother.Attheheartofthisliteratureliesthenotionofex-poststability,whichisalsoatthecenter ofourstudy.Roth etal. (1993),Rothblum (1992),andVande Vate (1989) provideacharacterizationofthesetofex-poststableassignments thatwesummarizeinLemma 2anduseintheproofofProposition 3.Morerecently,Manjunath (2011)investigatesseveral ex-ante stability and corenotions that are based on ﬁrst-order stochastic dominance. Kesten and Ünver (2015)propose ex-ante stability notions forthe school choice problem, in whichschools havepriorityorderings ratherthen preferences, which are not takeninto consideration inthe welfare analysis. Second, schoolsare assumedto behave non-strategically. In allthesestudies,thefocusisonstability andcorenotionsbasedonordinalpreferenceinformation.In contrast,we are interested intherelationship betweenex-post stability basedonordinal preferencesandacorenotionbased oncardinal preferences.

FollowingBogomolnaiaandMoulin (2001),one strandofthe literaturestudies theprobabilistic assignmentofobjects. In this literature, a result similar to our Theorem 1, which is due to McLennan (2002),9 follows from our Theorem 1

(Corollary 1). Manea (2008) gives a constructive proof of McLennan’s result. Ourproof technique is similar to Manea’s: An orderrelation,theacyclicityofwhichcharacterizesanassignment’ssd-efficiency,is usedtoconstructtheutilityprofile. As showninManea (2008),if only onesideofthemarkethaspreferencesovertheother,thentherelationratherdirectly deliversthedesiredutilityprofile.However,in marriageproblems,thepreferencesofbothmenandwomenmustbetaken into account,both in sd-efficiencyand utilitariansocial welfare considerations.We show that doing so requires a rather novelconstructionoftheutilityprofilecomparedtothecaseofone-sidedmarkets.

Inthedeterministicassignmentliterature,thetheoryofstableassignmentswithouttransfers(NTU-model)isﬁrst devel-opedbyGaleandShapley (1962).Givenpreferencerankings,thealgorithmtheyproposeselectsastableassignmentthatis Paretodominantfortheproposingsideamongallstableassignments.Ontheotherhand,thetheoryofstableassignments withtransfers(TU-model)isdevelopedbyShapleyandShubik (1971),whereforgivenvaluationsofagents,thecoreofthe associatedTU-game(theassignmentgame)ischaracterized.10

OurTheorem 2offersa connectionbetweenTUandNTU-models.Echenique (2008) andEcheniqueetal. (2013)offera similarconnectionbetweenTUandNTU-modelsbasedontheobservablecontentofstabilityinbothsetups.Theiranalysis isina revealedpreferenceframework,in whichordinal preferencesarenot observableandare recoveredfromanobserved

aggregatematching.11_Echenique_et_{al. (2013)}_argue_that_in_this_setting,_matching_theory_with_transfers_is_nested_in_matching

theory withouttransfers,that is,foreach aggregatematching,if thereisautility profilesuch thatthematchingisinthe coreof the associated TU-game,then there is an ordinal preference profile such that the matching is theunique stable matching.For ourTheorem 2, we assume that ordinal preferencesand an ex-post stableassignment are observable,but utilityprofilesarenot.We constructautilityprofilethatisconsistentwiththeordinalpreferencessuchthatthematching isinthecoreoftheassociatedTU-game.Sincetheconversestatementdoesnothold,12bycontrastwithEchenique (2008), in oursettingmatchingtheorywithouttransfersisnestedinmatchingtheorywithtransfers.

Anotherrecentstudy,whichprovides severalinteresting insightsontherelationshipbetweenstability inTU and NTU-models,isEcheniqueandGalichon (2014).Oneoftheir goalsistounderstandforwhichstabledeterministicassignments, availabilityofmonetary transferswouldnot affectstability.Theyshowthat, fora particularsubsetofstabledeterministic assignments, whichthey call“isolated” assignments, foranyordinal preferenceproﬁle,one can constructautility proﬁle suchthateachisolatedstabledeterministicassignmentremainsstablewhenmonetarytransfersareintroduced.Attheend ofSection3.3,we showthattheirresultfollowsfromourTheorem 2.

2. Theframework

LetM beasetofn menandW asetofn women.Eachi

∈

M haspreferencesoverW ,andeach j

∈

W haspreferences over M. Let N

=

M

∪

W . For each i

∈

N, the preferences of i, which we denote by Ri, is a weakorder, that is, Ri is transitiveandcomplete.Let Pi denotetheassociatedstrict preferencerelation,and Ii theassociatedindifference relation. Let

R

i denotethesetofall possiblepreferencerelationsfori,and

R

≡ ×

i∈N

R

i denotethesetofall possiblepreference proﬁles.

A deterministicassignment isaone-to-onefunction

μ

:

M

∪

W

→

M

∪

W suchthatforeach

(

m

,

w

)

∈

M

×

W ,we have

μ

(

m

)

∈

W ,

μ

(

w

)

∈

M,and

μ

(

m

)

=

w ifandonlyif

μ

(

w

)

=

m.A deterministicassignmentcanberepresentedbyann

×

n

matrix,withrowsindexedby menandcolumnsindexedbywomen, andhaving entriesin

{

0

,

1

}

,suchthat eachrowand eachcolumnhasexactlyone 1.Sucha matrixiscalleda permutationmatrix.Foreach

(

m

,

w

)

∈

M

×

W ,having1 inthe

(

m

,

w

)

entryindicates thatm isassignedto w.A probabilisticassignment isaprobabilitydistributionoverdeterministic assignments.A probabilisticassignmentcan berepresentedby ann

×

n matrixhavingentriesin

[

0

,

1

]

suchthat thesum

9 _{Athanassoglou (2011)}_approaches_the_problem_in_{McLennan (2002)}_by_using_duality. 10 _This_theory_is_applied_to_the_marriage_problem_by_{Becker (1973)}_.

11 _See_Echenique_et_{al. (2013)}_for_the_deﬁnition_of_an_aggregate_matching. 12 _See_our_{Example 2}_.

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of the entries in each row and each column is 1. Such a matrix is a doublystochasticmatrix. For each probabilistic assignment

π

,andeachpair

(

m

,

w

)

∈

M

×

W ,theentry

π

mw indicatestheprobabilitythatm isassignedtow at

π

.Since eachdoublystochasticmatrixcanberepresentedasaconvexcombinationofpermutationmatrices(Birkhoff,1946andVon Neumann, 1953),the set ofall doubly stochastic matricesprovides another representationforthe set ofall probabilistic assignments.Let

bethesetofalldoublystochasticmatrices.

Wedenotethecollectionofalllotteriesover M by

L(

M

)

,andthecollectionofall lotteriesoverW by

L(

W

)

.Foreach

i

∈

M withpreferences Ri

∈

R

i,a von Neumann–Morgenstern(vNM)utilityfunctionuiisareal-valuedmappingonW ,i.e.

ui

:

W

→ R

.We obtainthecorrespondingpreferencesofi over

L(

W

)

bycomparingexpectedutilities.Foreachi

∈

M with preferences Ri

∈

R

i,a vNMutilityfunction uiisconsistentwith Riifforeachpair

(

w

,

w

)

∈

W wehaveui

(

w

)

≥

ui

(

w

)

if andonlyifwRiw.Foreachwoman,a (vNM)utilityfunctionconsistentwithherordinalpreferencesisdeﬁnedsimilarly.

Foreachutilityproﬁleu

= (

ui

)

i∈N andprobabilisticassignment

π

,the utilitariansocialwelfareat

(

u

,

π

)

isthesumof theexpectedutilitiesoftheagents,thatis:

SW

(

u

,

π

)

=

(m,w)∈M×W

π

mw

(

um

(

w

)

+

uw

(

m

)).

Anassignment

π

isex-anteutilitarian-welfaremaximizingatautilityproﬁle u ifitmaximizesthesocialwelfareat u, i.e.

π

∈

argmaxπ∈SW

(

u

,

π

)

.

Next,we defineawell-known notionofefficiencythatisindependentofanyvNM utilityspecificationconsistentwith theordinal preferences.Let

π

,

π

∈

and R

∈

R

;we saythat

π

ﬁrst-orderstochasticallydominates

π

at R ifforeach pair

(

m

,

w

)

∈

M

×

W :

w:wRmw

π

mw

≥

w:wRmw

π

_mw and

m:mRwm

π

mw

≥

m:mRwm

π

_m_w

such that for atleast one pair, at least one of the inequalities is strict. An assignment

π

∈

issd-eﬃcientat R if no probabilisticassignmentsd-dominates

π

atR.Foreach R

∈

R

,let Psd

(

R

)

denotethesetofsd-eﬃcientassignmentsatR.

3. Results

3.1. Acharacterizationofsd-eﬃciency

Foreachpair

(

π

,

R

)

∈

×

R

,we deﬁnetworelations

∼

(π,R)and

(π,R)onM

×

W inducedby

(

π

,

R

)

,andcharacterize

thesd-eﬃciencyofanassignment

π

atR intermsofapropertyof

∼

(π,R) and

(π,R).

Foreachpair

(

m

,

w

),

(

m

,

w

)

∈

M

×

W ,

(

m

,

w

)

∼

(π,R)

(

m

,

w

)

ifandonlyif

π

mw

>

0,

π

mw

>

0,andwImw,mIwm. For each pair

(

m

,

w

),

(

m

,

w

)

∈

M

×

W ,

(

m

,

w

)

(π,R)

(

m

,

w

)

ifand onlyif

π

m,w

>

0,

π

mw

>

0, and wRmw,mRw

m with at least one strict preference. Let the relation

(π,R) be the union of the two relations just deﬁned. That is,

(π,R)

=∼

(π,R)

∪

(π,R).

Ifassignment

π

isdeterministicandpreferencesarestrict,then

(

m

,

w

)

(π,R)

(

m

,

w

)

impliesthatm andwprefereach

other totheir assignedmates.Furthermore,supposethat

(

m

,

w

)

(π,R)

(

m

,

w

)

(π,R)

(

m

,

w

)

.Then, m andm (or w and

w)arebetteroffbyexchangingmates.In general,if therelation

(π,R)hasacycle,agentsinthecyclecanParetoimprove

byexchangingmatesalongthecycle.Next,we formulateanacyclicityrequirementon

(π,R)withthesameimplicationfor

sd-eﬃciency.

Notethat,if

(

m

,

w

)

(π,R)

(

m

,

w

)

,then

(

m

,

w

)

and

(

m

,

w

)

arenotrelatedaccordingto

∼

(π,R),thatis,

(

m

,

w

)

(π,R)

(

m

,

w

)

.A strongcycleof

(

π,R)isasequenceofpairs

(

m1

,

w1

),

(

m2

,

w2

)

,

. . . ,

(

mk

,

wk

)

∈

M

×

W suchthat

(

m1

,

w1

)

(π,R)

(

m2

,

w2

)

(π,R)

. . .

(π,R)

(

mk

,

wk

)

(π,R)

(

m1

,

w1

)

. The relation

(

π,R) isweaklyacyclic ifand only ifit has no strong cycle. Next,we characterizesd-eﬃcientassignments. Thisresultgeneralizescharacterizationsofsd-eﬃciencyofobject as-signmentsonthestrictpreferencedomain(BogomolnaiaandMoulin,2001)andontheweakpreferencedomain(Kattaand Sethuraman,2006).

Proposition1.Anassignment

π

issd-eﬃcientatapreferenceproﬁleR ifandonlyif

(π,R)isweaklyacyclic. Proof. Let

π

∈

,R

∈

R

.

Only-ifpart: Weprovethecontrapositivestatement.Supposethat

(π,R)isnotweaklyacyclic,thatis,thereisasequence

of pairs

(

m1

,

w1

),

(

m2

,

w2

)

,

. . . ,

(

mk

,

wk

)

∈

M

×

W such that

(

m1

,

w1

)

(π,R)

(

m2

,

w2

)

(π,R)

. . .

(π,R)

(

mk

,

wk

)

(π,R)

(

m1

,

w1

)

.Let

≡

mini∈{1,...,k}

π

miwi.Let

π

∈

bedeﬁnedbysettingforeachi

∈ {

1

,

. . . ,

k

}

,

π

mi wi

=

π

miwi

−

,

π

mi wi₊₁

=

π

miwi+1

+

(withtheconventionthat wk+1

=

w1),andforeachotherpair

(

m

,

w

)

,

π

mw

=

π

mw.Notethat

π

sd-dominates

π

atR.Thus,

π

∈

Psd

(

R

)

.

Ifpart: We provethe contrapositive statement.Suppose that

π

∈

/

Psd

(

R

)

, thatis,there is

π

∈

thatsd-dominates it. Without loss of generality, suppose that there is a man, say m1

∈

M, who is better off at

π

in stochastic dominance

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suchthat m1Rw2m2 and

π

m2w2

<

π

m2w2.Notethat

(

m1

,

w1

)

(π,R)

(

m2

,

w2

)

.Now,therearem3

∈

M, w3

∈

W such that

w3Rm2 w2,

π

m2w3

>

π

m2w3,m2Rw3m3,and

π

m3w3

<

π

m3w3. Then,

(

m1

,

w1

)

(π,R)

(

m2

,

w2

)

(π,R)

(

m3

,

w3

)

.Proceeding inductively,we canaddpairstothissequence,andsincethereareﬁnitelymanyman–womanpairs,thissequenceincludes acycleof

(π,R).

If the cycle includes

(

m1

,

w1

)

, then it is strong and we are done. Suppose otherwise. Let the cycle consist of

(

x1

,

y1

),

(

x2

,

y2

),

. . . ,

(

xk

,

yk

)

∈

M

×

W .Note that

(

x1

,

y1

)

∼

(π,R)

(

x2

,

y2

)

∼

(π,R)

. . .

∼

(π,R)

(

xk

,

yk

)

∼

(π,R)

(

x1

,

y1

)

.

Remem-ber that

π

x1y1

<

π

x1y1

,

π

x2y2

<

π

x2y2

,

. . . ,

π

xkyk

<

π

xkyk, and

π

x1y2

>

π

x1y2

,

π

x2y3

>

π

x2y3

,

. . . ,

π

xky1

>

π

xky1. Let

1

≡

min

{

π

x1y1

−

π

x1y1

,

. . . ,

π

xkyk

−

π

xkyk

}

,

2

≡

min

{

π

x1y2

−

π

x1y2

,

. . . ,

π

xky1

−

π

xky1

}

and

≡

min

{

1 ,

2

}

.Let

π

∈

be de-ﬁned by settingfor each i

∈ {

1

,

. . . ,

k

}

,

π

xiyi

=

π

xiyi

+

,

π

xiyi+1

=

π

xiyi+1

−

(with the convention that yk+1

=

y1), and foreach other pair

(

m

,

w

)

,

π

mw

=

π

mw . Sinceforeach i

∈ {

1

,

. . . ,

k

}

, yiIxi yi+1 and xi−1Iyi xi,

π

alsosd-dominates

π

. Also,by theconstructionof

π

,therearetwo consecutivepairsinthecycle,say

(

xt

,

yt

)

and

(

xt+1

,

yt+1

)

,suchthat either

π

xtyt

=

π

xtyt or

π

xtyt +1

=

π

xtyt+1.

Now,because

π

sd-dominates

π

,as we didabove,we can ﬁndacycleof

(π,R),say

(

x1

,

y1

)

,

(

x2

,

y2

)

,

. . . ,

(

xt

,

yt

)

∈

M

×

W ,suchthatforeach i

∈ {

1

,

. . . ,

t

}

,

π

_x iyi

<

π

x_iy_i and

π

x_iy_i₊₁

>

π

x_iy_i₊₁ (withtheconvention that yt+1

=

y1).Notethat

(

xt

,

yt

)

and

(

xt+1

,

yt+1

)

cannotbepartofthiscycleconsecutively,implyingthatthisnewcyclemustbedifferentfromthe

cyclethatwasidentiﬁedbefore.Continuingsimilarly,we obtainadditionalassignmentsthatsd-dominate

π

andadditional cycles.Noneofthose cyclescaninclude

(

xt

,

yt

)

and

(

xt+1

,

yt+1

)

consecutively;andforeach additionalcycle, we identify

additionalconsecutivepairsinthecyclethatcannotbepartofanycycleinthefuture,implyingthateachcycleisdifferent fromanycyclethat wasidentiﬁedbefore.Sincethenumberofcyclesin

(π,R) isﬁnite,thisprocesseventually leadstoa

strongcycle.Thus,

(π,R)isnotweaklyacyclic.

2

Given R

∈

R

, for each

π

∈

Psd

₍

_R

₎

_, _the _relation

(π,R) can have cycles of the form

(

m1

,

w1

)

(π,R)

(

m2

,

w2

)

(π,R)

. . .

(π,R)

(

mk

,

wk

)

(π,R)

(

m1

,

w1

)

.However,

(π,R)beingweaklyacyclicimpliesthatsuchacycleshouldbelongto

∼

(π,R),

thatis,thecycleshouldbeoftheform

(

m1

,

w1

)

∼

(π,R)

(

m2

,

w2

)

∼

(π,R)

. . .

∼

(π,R)

(

mk

,

wk

)

∼

(π,R)

(

m1

,

w1

)

.

Given

(

π

,

R

)

∈

×

R

, letCπ,R be thebinary relationon M

×

W ,deﬁnedasfollows:Foreach pair

(

m

,

w

),

(

m

,

w

)

∈

M

×

W ,

(

m

,

w

)

Cπ,R

(

m

,

w

)

ifandonlyifthereisacycleof

∼

(π,R)thatcontainsboth,thatis,thereisasequenceofpairs

(not necessarily distinct)

(

m1

,

w1

),

(

m2

,

w2

)

,

. . . ,

(

mk

,

wk

)

∈

M

×

W that includes

(

m

,

w

)

and

(

m

,

w

)

, and is such that

(

m1

,

w1

)

∼

(π,R)

(

m2

,

w2

)

∼

(π,R)

. . .

∼

(π,R)

(

mk

,

wk

)

∼

(π,R)

(

m1

,

w1

)

.Notethat Cπ,R isan equivalence relationon M

×

W , that is, it is reﬂexive, symmetric, and transitive. For each pair

(

m

,

w

)

∈

M

×

W , let

[

m

,

w

]

Cπ,R

≡ {(

m

,

w

)

∈

M

×

W

:

(

m

,

w

)

Cπ,R

(

m

,

w

)

}

denote the equivalenceclass of

(

m

,

w

)

relative to Cπ,R.Let

(π,R) be the relationdeﬁned onthe

setofall equivalenceclassesof C(π,R) asfollows:Foreach pair

(

m1

,

w1

),

(

m2

,

w2

)

∈

M

×

W ,

[

m1

,

w1

]

(π,R)

[

m2

,

w2

]

if

andonlyif

[

m1

,

w1

]

= [

m2

,

w2

]

andthereare

(

m1

,

w1

)

∈ [

m1

,

w1

]

,

(

m2

,

w2

)

∈ [

m2

,

w2

]

suchthat

(

m1

,

w1

)

(π,R)

(

m2

,

w2

)

.

Notethat,if

π

∈

Psd

₍

_R

₎

_,_then

(π,R)isacyclic.

3.2. Aneﬃciencytheorem

In thissection we show that for each probabilistic assignment that is sd-efficient ata givenpreference profile,there isa utilityprofileconsistent withthesepreferences suchthat theprobabilistic assignmentmaximizesthesumofthe ex-pected utilities.First,let usconsider thesimplecasewhere thesd-efficient assignment forwhicha utility functionis to be constructedis anefficient deterministicassignment,

μ

.Let u bea utility proﬁleconsistent with R suchthat foreach

(

m

,

w

)

∈

M

×

W ,if m and w arematchedat

μ

,thenum

(

w

)

=

uw

(

m

)

=

1.Further,foreach w

∈

W suchthatw

mw,let 0

<

um

(

w

)

< δ

forsome

δ >

0 andforeach w

∈

W suchthatw

mw,let1

<

um

(

w

)

<

1

+

forsome

>

0.Letuw be similarlydeﬁned.Notethatforsomesmallenoughselectionof

δ

and

,theeﬃcientassignment

μ

isawelfaremaximizing assignmentatutilityproﬁleu.

This construction would fail even for the simplest probabilistic assignment, which is obtained as a mixture of two eﬃcient deterministic assignments. However, we show that the same conclusion holds foreach sd-eﬃcient probabilistic assignment

π

byprovidinganexplicitconstructionforautilityproﬁleatwhich

π

iswelfaremaximizing.Theconstruction inthenextexampleisinstructivetounderstandthegeneralconstructiontofollowinTheorem 1.

Example1.LetM

= {

1

,

2

,

3

}

andW

= {

a

,

b

,

c

}

.Letthepreferenceproﬁle R beasfollows:

R1 R2 R3 Ra Rb Rc

a b c 3 1 2

c a b 2 3 1

b c a 1 2 3

Let

μ

betheassignmentwhere1 ismatchedwitha,2 with b,and3 withc.Let

μ

betheassignmentwhere1 ismatched withb,2 withc,and3 with a.Let

π

assign0

.

5 probabilitytoeachof

μ

and

μ

.

(6)

Now,letu besuchthateach agentgetsutility 1 fromhis/hertop-rankedagentandutility0 fromhis/herthird-ranked agent. Further,each agentgetsutilityintheopen interval

(

0

,

1

/

2

)

fromhis/hersecond-rankedagent.Clearly,foreachpair

(

m

,

w

)

,if

π

mw

>

0,thenum

(

w

)

+

uw

(

m

)

=

1.Moreover,if

π

mw

=

0,thenum

(

w

)

+

uw

(

m

)

<

1.Hence,thesumofexpected utilities,whichis3 at

π

,cannotexceed3 atanyprobabilisticassignment.

Intheaboveconstruction,foreachman–womanpair,thesumoftheutilitiestheygetfromeachotheristhesame,and thataspectplaysthecriticalrole.Forourgeneralresult,a similarconstructionworks.

Theorem1.Foreachpreferenceprofileandeachassignmentthatissd-efficientatit,thereisautilityprofileconsistentwiththe preferenceprofilesuchthattheassignmentisex-anteutilitarianwelfaremaximizingatthatutilityprofile.

Proof. Let

(

π

,

R

)

∈

×

R

besuchthat

π

∈

Psd

₍

_R

₎

_._Since_we_ﬁx

₍

_π

_,

_R

₎

_throughout_the_proof,_{we remove} _the_reference_to

(

π

,

R

)

indenotingthebinaryrelationswehavedeﬁned,andsimplywrite

∼

,

,and

.

Foreachpairm

∈

M, w

∈

W ,letsmw denotethelengthofthelongestpathof

startingat

[

m

,

w

]

,andletemw denote thelengthofthelongestpathof

endingat

[

m

,

w

]

.13

Step 1: Constructinganauxiliaryutilityproﬁle. Foreachm

∈

M,let vm

:

W

→ R

bedeﬁnedbysetting,foreach w

∈

W , vm

(

w

)

=

emw emw

+

smw

.

Foreachw

∈

W ,letvw

:

M

→ R

bedeﬁnedbysetting,foreachm

∈

M, vw

(

m

)

=

smw emw

+

smw

.

Notethatforeachpair

(

m

,

w

)

∈

M

×

W ,vm

(

w

)

+

vw

(

m

)

=

1.

Lemma1.Let

(

m

,

w

),

(

m

,

w

)

∈

M

×

W .If

[

m

,

w

]

[

m

,

w

]

,thenvw

(

m

)

>

vw

(

m

)

. Proof. Sinceemw

<

emw andsmw

>

smw,then

smw smw

+

emw

>

smw smw

+

emw

,

i.e. 1 1

+

emw_s mw

>

1 1

+

emw smw

.

2

One consequenceof the Lemma is that, foreach pair

(

m

,

w

),

(

m

,

w

)

∈

M

×

W , if

[

m

,

w

]

[

m

,

w

]

, then vm

(

w

)

+

vw

(

m

)

<

1.Letz

((

m

,

w

),

(

m

,

w

))

=

1

−

vm

(

w

)

−

vw

(

m

)

if

[

m

,

w

]

[

m

,

w

]

,and1 otherwise.Let min

(m,w),(m,w)∈M×Wz

((

m

,

w

), (

m

_,

_w

₎₎

_≡

₂

_.

Step 2: Deﬁningtheutilityproﬁle u. Letm

∈

M.Letumbedeﬁnedasfollows(foreachw

∈

W ,we deﬁneuwinasymmetric way):

i. For each w

∈

W suchthat

π

mw

>

0,set um

(

w

)

≡

vm

(

w

)

.We show that um is consistentwith Rm onthe subset of womenforwhom

π

mw

>

0.Letm

∈

M andw

,

w

∈

W besuchthat

π

mw

>

0 and

π

mw

>

0.Withoutlossofgenerality, suppose that wRmw. If w Imw, then note that

(

m

,

w

)

∼ (

m

,

w

)

, and

[

m

,

w

]

∼ [

m

,

w

]

. Thus, um

(

w

)

=

um

(

w

)

, as desired.If wPmw,then

(

m

,

w

)

(

m

,

w

)

,and

[

m

,

w

]

[

m

,

w

]

.Thus,um

(

w

)

>

um

(

w

)

,as desired.

ii. For each w

∈

W such that

π

mw

=

0 and there is no w

∈

W with

π

mw

>

0, wRmw, set um

(

w

)

≤ −

1. Obviously, at this step the utilities can be chosen such that um is consistent with Rm on the subset of women for whom the utilitiesaredeﬁnedsofar.

iii. Foreach w

∈

W suchthat

π

mw

=

0 andthere is w

∈

W with

π

mw

>

0, wRmw, considera bestsuch w,that is,

π

mw

>

0,wRmw,andthereisnosuch w

∈

W with wPmw.Setum

(

w

)

∈ [

vm

(

w

),

vm

(

w

)

+

]

.Obviously,at this steptheutilitiescanbechosensuchthatum isconsistentwithRm ontheentiresetofwomen.

LetthefunctionSW

(

u

,

.)

:

→ R

bedeﬁnedbysetting,foreach

π

∈

, SW

(

u

,

π

)

=

(m,w)∈M×W

[

π

mw

(

um

(

w

)

+

uw

(

m

))

].

13 _A_path_of_length_{k of}_consists_of_{k pairs}₍_m

(7)

Step 3: SW attainsitsmaximumat

π

. For each pair

(

m

,

w

)

∈

M

×

W , if

π

mw

>

0, then um

(

w

)

+

uw

(

m

)

=

1. Thus,

SW

(

u

,

π

)

=

n. We showthat foreach pair

(

m

,

w

)

∈

M

×

W , if

π

mw

=

0, then um

(

w

)

+

uw

(

m

)

≤

1, which implies that themaximalpossibleex-anteutilitariansocialwelfareisn,anditisreachedat

π

.

Let

(

m

,

w

)

∈

M

×

W be such that

π

mw

=

0. Suppose that there is no w

∈

W such that

π

mw

>

0, wRmw. Then,

um

(

w

)

≤ −

1.If thereisnom

∈

M suchthat

π

mw

>

0 andmRwm,thenuw

(

m

)

≤ −

1 andum

(

w

)

+

uw

(

m

)

<

1.If thereis

m

∈

M suchthat

π

mw

>

0 andmRwm,thenuw

(

m

)

≤

uw

(

m

)

+

<

2.Thus,um

(

w

)

+

uw

(

m

)

<

1.Thecasewhenthereis nom

∈

M suchthat

π

mw

>

0,mRwm,is symmetric.

So,thereisonlyonecaselefttoconsider.Supposethatthereisw

∈

W suchthat

π

mw

>

0 and wRmw,andthereis

m

∈

M suchthat

π

mw

>

0 andmRwm.Letw andm bethebestsuch agents.Notethat

(

m

,

w

)

(

m

,

w

)

.If w Imw andmIwm,thenbyLemma 1,um

(

w

)

+

uw

(

m

)

<

1.So,supposethatforatleastoneagent,thepreferenceisstrict.Then,

(

m

,

w

)

(

m

,

w

)

and

[

m

,

w

]

[

m

,

w

]

.Recallthatz

((

m

,

w

),

(

m

,

w

))

=

1

−

vm

(

w

)

−

vw

(

m

)

≥

2

.Now, um

(

w

)

+

uw

(

m

)

≤ [

um

(

w

)

+

] + [

uw

(

m

)

+

] =

vm

(

w

)

+

vw

(

m

)

+

2

whichimpliesum

(

w

)

+

uw

(

m

)

≤

1,as desired.

2

The welfare theorem by McLennan (2002) for the problem of allocating objects is a corollary of Theorem 1. To see this,ﬁrstconsiderthefollowingcounterpartsofsd-eﬃciencyandutilitariansocialwelfareforthatmodel.Letuskeepthe men-womennotation.Anassignment

π

∈

men-sidesd-dominates

π

∈

at R

∈ R

ifforeachagent i

∈

M,thelottery assignedtoi at

π

sd-dominatestheoneassignedat

π

.Thatis,foreachpair

(

m

,

w

)

∈

M

×

W ,

w:wRmw

π

mw

≥

w:wRmw

π

_mw

suchthat foratleastonepairtheinequalityisstrict.Anassignment

π

∈

ismen-sidesd-eﬃcientat R

∈ R

ifno prob-abilisticassignmentmen-sidesd-dominatesitat R.Foreachutilityproﬁleu

= (

ui

)

i∈N andprobabilisticassignment

π

,the men-sideutilitariansocialwelfareat

(

u

,

π

)

isthesumoftheutilitiesofthemen,thatis:

MSW

(

u

,

π

)

=

m∈M

w∈W

π

mwum

(

w

).

Aprobabilistic assignment

π

isex-antemen-sideutilitarianwelfaremaximizingatautilityproﬁle u ifit maximizes themen-sidesocialwelfareatu,thatis,

π

∈

argmaxπ∈MSW

(

u

,

π

)

.

Corollary1.(SeeMcLennan,2002.)Foreachpreferenceprofileandeachassignmentthatismen-sidesd-efficientatit,thereisautility profileconsistentwiththepreferenceprofilesuchthattheassignmentisex-antemen-sideutilitarianwelfaremaximizingatthatutility profile.

Proof. Let

π

∈

bemen-sidesd-eﬃcientatR

∈

R

.LetR

∈

R

besuchthatforeachm

∈

M, R_m

=

Rm,andeachwomanis indifferentbetweenanytwomenat R.Notethat

π

ismen-sidesd-eﬃcientalsoat R.Moreover,

π

∈

Psd

(

R

)

.ByTheorem 1, thereisautilityproﬁleu consistentwithRsuchthat

π

isex-anteutilitarianwelfaremaximizingatu.Sinceeachwoman getsthesameutility fromanytwomenatu,

π

isex-antemen-sideutilitarianwelfaremaximizingatu. Now,let u bea utility proﬁlesuchthatforeachm

∈

M, um

=

um,andforeach w

∈

W ,uw isconsistentwithRw.Notethat

π

isex-ante men-sideutilitarianwelfaremaximizingatu.

2

3.3. Astabilitytheorem

Acentral robustness criterion for deterministic assignments is“stability”,which requires thatthere be no unmatched man–womanpair who prefer each other to their assignedmates. A counterpartofstability forprobabilistic assignments is “ex-post” stability,whichrequires that there be atleastone decompositionof theprobabilistic assignment into stable deterministicassignments.

Let

D

denote theset ofdeterministic assignments. An assignment

μ

∈ D

is stable at R

∈ R

ifthere isno

(

m

,

w

)

∈

M

×

W such that mPw

μ

(

w

)

, wPm

μ

(

m

)

. An assignment

π

∈

is ex-poststable if it can be expressed as a convex combinationofstabledeterministicassignments.

From thispoint on,we restrict ourselvesto strict preferences.Foreach i

∈

N, let

P

i

⊂

R

i bethe setof all transitive, anti-symmetric,andcompletepreferencerelationsfori.Let

P = ×

i∈N

P

i bethesetofallstrictpreferenceproﬁles.

Weshowthatex-poststabilityimplieswelfareproperties,beyondsd-eﬃciency,whicharealsodeduciblefromtheordinal preferencesandcanavoidex-antebreakawaysofmen-womencoalitionsfromthesociety.

Acoalition S

=

M

∪

W

⊆

M

∪

W is admissible if

|

M

|

= |

W

|

.Let

A

bethe setofalladmissiblecoalitions.Foreach

S

∈

A

,let

S _denote_the_set_of_{probabilistic}_assignments_deﬁned_over _S._For_each_utility_proﬁle_u

_{= (}

_u

(8)

transferableutilitygame deﬁned by settingforeach S

=

M

∪

W

∈

A

, Vu

(

S

)

tobe themaximum totalexpectedutility coalition S canachieveamongitsmembers.Thatis,foreachS

=

M

∪

W

∈

A

,

Vu

(

S

)

=

max πS_∈S

(m,w)∈M×W

π

_mwS

(

um

(

w

)

+

uw

(

m

)).

Let E

(

um

|

π

)

=

w∈W

π

mwum

(

w

)

be the expected value of um at

π

and E

(

uw

|

π

)

=

m∈M

π

mwuw

(

m

)

the expected value ofuw at

π

.Givenautility proﬁleu,an assignment

π

∈

isinthe coreof Vu ifnocoalitioncanincreaseits total expected utilityby deviating toanother probabilistic assignmentwherethey are matchedamong themselves.Thatis,for each S

∈

A

, Vu

(

S

)

≤

m∈M E

(

um

|

π

)

+

w∈W E

(

uw

|

π

).

LetC

(

Vu

)

bethe setofallassignmentsthatareinthecoreof Vu.

Let P

∈

P

.LetPM andPW denotethecommonpreferencesofmenandwomenoverdeterministicassignmentsinduced by P ,deﬁnedasfollows:Foreachpair

μ

,

μ

∈

D

,

μ

PM

μ

ifandonlyifforeachm

∈

M,

μ

Rm

μ

withstrictpreferencefor somem.TherelationPW isdeﬁnedsimilarly.

An assignment

π

∈

is well-ordered ex-poststable at P

∈ P

if it has a decomposition into stable assignments

μ1

,

. . . ,

μ

T suchthatforeacht

,

t

∈ {

1

,

. . . ,

T

}

witht

<

t,we have

μ

t PM

μ

t and

μ

tPW

μ

t.

Proposition2.Ifanassignment

π

iswell-orderedex-poststableatastrictpreferenceprofile,thenthereisautilityprofileu consistent withthepreferenceprofilesuchthat

π

∈

C

(

Vu

)

.

Proof. Let

π

∈

be well-orderedex-post stableat P

∈

P

. Suppose that

π

hasthe following decompositioninto stable assignments:

π

= λ

1μ1

+ λ

2μ2

+ · · · + λ

T

μ

T.Supposethatforeachpairt

,

t

∈ {

1

,

2

,

. . . ,

T

}

suchthatt

<

t,

μ

t PM

μ

t and

μ

t PW

μ

t.

Weﬁrstdeﬁnetheutilitieseachagentgetsfromtheagentsthathe/sheismatchedwithpositiveprobability.Let

(

m

,

w

)

∈

M

×

W suchthat

π

mw

>

0.First note thatifm is matchedto w in twodifferentassignments inthe decomposition,say

μ

t,

μ

t,t

<

t,thenthey shouldbematchedinall assignmentsbetween

μ

t and

μ

t,thatis,foreach t

∈ {

t

,

t

+

1

,

. . . ,

t

}

,

μ

t

(

m

)

=

w.So,let

μ

p

,

μ

p+1

,

. . . ,

μ

qbethelistofassignmentsatwhichm ismatchedtow.Let

λ

mw

≡ λ

p

+λ

p+1

+· · ·+λ

q.

Let uw

(

m

)

=

λ

p

λ

mw

·

p T

+

1

+

λ

p+1

λ

mw

·

p

+

1 T

+

1

+ · · · +

λ

q

λ

mw

·

q T

+

1

,

and um

(

w

)

=

λ

p

λ

mw

·

T

−

p

+

1 T

+

1

+

λ

p+1

λ

mw

·

T

−

p T

+

1

+ · · · +

λ

q

λ

mw

·

T

−

q

+

1 T

+

1

.

Ifthereisauniqueassignment

μ

t inthedecompositionsuchthat

μ

t

(

m

)

=

w,we simplyhaveuw

(

m

)

=

T+t1 andum

(

w

)

=

T−t+1

T+1 .

14

Next, we argue that the utilities each agent gets fromthe agents that he/she is matched with zero probability can be deﬁned in such a way that for each such pair

(

m

,

w

)

∈

M

×

W , um

(

w

)

+

uw

(

m

)

<

1. Let

(

m

,

w

)

∈

M

×

W be such that

π

mw

=

0.If there isno w

∈

W suchthat

π

mw

>

0 and wPmw,thenlet um

(

w

)

≤ −

1.The casewhenthere isno

m

∈

W suchthat

π

mw

>

0,mPwmisthesame.So,supposethattherearem

∈

M

,

w

∈

W suchthat

π

mw

>

0,wPmw, and

π

mw

>

0,mPwm.Supposew.l.o.g.that w andmarebestsuchagentsatPw and Pm.We willshowthatum

(

w

)

+

uw

(

m

)

<

1.First,notethatthepairs

(

m

,

w

)

and

(

m

,

w

)

cannotappearinthesameassignmentofthedecomposition,since otherwisethatassignmentwouldnotbestable.Supposethatm and warematchedinassignments

μ

p

,

μ

p+1

,

. . . ,

μ

q,and

mandw arematchedinassignments

μ

p

,

μ

p+1

,

. . . ,

μ

q.Eitherq

<

porq

<

p.In factwecannothaveq

<

p;otherwise,

m wouldpreferhismatein

μ

q,namelyw,tohismatein

μ

p.Butthen,m wouldpreferw tohismatein

μ

p,contradicting the assumption that

μ

p is stable. Thus q

<

p. Since um

(

w

)

≤

T−_T+p+₁1 anduw

(

m

)

≤

q

T+1, um

(

w

)

+

uw

(

m

)

<

1. Then,

by argumentssimilartotheproofofTheorem 1,foreachsuchm, w,m,w,letz

((

m

,

w

),

(

m

,

w

))

=

1

−

um

(

w

)

−

uw

(

m

)

and let 2

be the minimum of z

((

m

,

w

),

(

m

,

w

))

. Now, for the pair

(

m

,

w

)

, let um

(

w

)

∈ [

um

(

w

),

um

(

w

)

+

]

and let

uw

(

m

)

∈ [

uw

(

m

),

uw

(

m

)

+

]

.Thus, utility proﬁleu is consistentwith P andforeach pair

(

m

,

w

)

thatis matchedwith zeroprobabilityum

(

w

)

+

uw

(

m

)

<

1.

Now,we showthat

π

∈

C

(

Vu

₎

_._First,_for_each_admissible_coalition _S

₌

_M

_∪

_W_,_Vu

₍

_S

₎

_{≤ |}

_M

_|

_._Now,_let

₍

_m

_,

_w

₎

_∈

_M

_×

_{W .} We showthat E

(

um

|

π

)

+

E

(

uw

|

π

)

=

1.Observethat

E

(

um

|

π

)

= λ

1 T T

+

1

+ λ

2 T

−

1 T

+

1

+ · · · + λ

T 1 T

+

1

·