Possibilistic preference networks

HAL Id: hal-02191842

https://hal.archives-ouvertes.fr/hal-02191842

Submitted on 23 Jul 2019

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Possibilistic preference networks

Nahla Ben Amor, Didier Dubois, Héla Gouider, Henri Prade

To cite this version:

Nahla Ben Amor, Didier Dubois, Héla Gouider, Henri Prade. Possibilistic preference networks.

Infor-mation Sciences, Elsevier, 2018, 460-461, pp.401-415. �10.1016/j.ins.2017.08.002�. �hal-02191842�

Any correspondence concerning this service should be sent

to the repository administrator:

tech-oatao@listes-diff.inp-toulouse.fr

This is an author’s version published in:

http://oatao.univ-toulouse.fr/22502

To cite this version:

Ben Amor, Nahla and Dubois, Didier

and Gouider, Héla and Prade, Henri Possibilistic preference

networks. (2018) Information Sciences, 460-461. 401-415.

ISSN 0020-0255

Official URL

DOI :

https://doi.org/10.1016/j.ins.2017.08.002

Open Archive Toulouse Archive Ouverte

OATAO is an open access repository that collects the work of Toulouse

researchers and makes it freely available over the web where possible

Possibilistic

preference

networks

s

Nahla Ben

Amor

a

_,

_Didier

_Dubois

b, ∗

_{, Héla Gouider}

a

_,

_{Henri Prade}

b a LARODEC, ISG de Tunis, 41 rue de la Liberté, Le Bardo 20 0 0, Tunisia

b IRIT, UPS-CNRS, 118 route de Narbonne, Toulouse Cedex 09, 31062 France

Keywords: Preference modeling Pareto ordering Graphical models Possibility theory a b s t r a c t

This paper studies the use of product-based possibilistic networks for representing preferences in multidimensional decision problems. This approach uses symbolic possibility weights and defines a partial preference order among solutions to a set of conditional preference statements on the domains of discrete decision variables. In thecase ofBoolean decision variables, this partial ordering isshown to be consistentwith thepreference orderinginducedby theceterisparibus assumptionadopted inCP-nets. Namely,by completngthe possibilistic net ordering with suitable constraints between products of symbolicweights, all CP-net preferences can be recovered. Computing procedures for comparing solutions are provided. The flexibility and representational powerof theapproachisstressed.

1. Introduction

Modelingpreferencesisessentialinanydecisionanalysistask.However,gettingthesepreferencesbecomesnontrivialas soonasalternativesaredescribedbyaCartesianproductofmultiplefeatures.Indeed,thedirectassessmentofapreference relationbetween these alternatives isusually not feasibledueto itscombinatorial nature.Fortunately,thedecision maker can express contextual preferences that exhibit some independence relations,which allowsus to represent her/his prefer-encesinacompact manner.Moreover, graphicalrepresentationsfacilitate preferenceelicitation,aswellastheconstruction ofan orderingfrom these contextuallocal preferences.This useof graphicalpreference representations hasbeen inspired bythesuccessofBayesiannetworksasacomputationallytractableuncertaintymanagementdevice[1].

The useofpossibilisticnetworks forrepresenting conditionalpreference statementson discrete variableshasbeen pro-posedonly recently. The approach uses non-instantiatedpossibility weights to defineconditional preference tables. More-over,additionalinformationabouttherelativestrengthsofthesesymbolic weightscanbetakenintoaccount. Thefactthat atbestwehave someinformationabouttherelativevaluesoftheseweightsacknowledgesthequalitativenature of prefer-encespecification.Theseconditionalpreferencetablesgivebirthtovectorsofsymbolicweights thatreflect thepreferences thataresatisfiedandthosethatareviolatedinaconsideredsituation.Thecomparisonofsuchvectorsmayrelyondifferent orderings:the onesinducedbythe product-based,ortheminimum-based chainrule underlying thepossibilisticnetwork. Athoroughstudy oftherelationsbetweentheseorderings inpresenceofvectorcomponentsthat aresymbolicratherthan numericalis presented.In particular, weestablish that theproduct-basedorderingand the symmetricPareto ordering

co-s Dedication : This paper is dedicated to Janusz Kacprzyk on the occasion of his jubilee. The second and the fourth author, who have a long and fruitful

research companionship with him, are glad to offer him this new piece of work.

∗ _{Corresponding author.}

E-mail addresses: nahla.benamor@gmx.com (N. Ben Amor), dubois@irit.fr (D. Dubois), hela.gouider@gmail.com, gouider.hela@gmail.com (H. Gouider),

prade@irit.fr (H. Prade).

incide in presence of constraintscomparing pairs of symbolic weights. The paper highlights the merits of product-based possibilisticnetworksforrepresentingpreferences,inwhichcasetheyarecalled

π

-prefnets.

Possibilisticpreferencenetworks(

π

-prefnets)belongtothefamilyofmethodsforthemodelingofpreferenceand deci-sionthatstemfromthefuzzysetandpossibilitytheoryliterature,variantsofwhicharefuzzyMarkoviandecisionprocesses studiedvery earlybyKacprzyk[2].JustlikeCP-netsmaybeused forflexiblequerying[3],itseemsthat

π-pref

nets might alsoservethispurpose(followingideasin[4]),atopicthat hasbeenalsomuchinvestigatedbyKacprzyk(e.g.,[5]).

Inthispaper,wealsodiscussexistingrelationshipsbetween

π

-prefnetsandsomepreferencemodelsthat arerelatedto theminsome sense,namely, CP-nets[6],CP-theories [7],and OCF-nets[8].Indeed,

π

-prefnets sharethesamepreference specificationandgraphicalstructureasCP-nets,CP-theoriesareageneralizationofCP-nets,whileOCF-netsarebasedon an additivestructurewhichparallelstheoneof

π

-prefnets.

The paper is organized as follows. Section2 presents a symbolic graphical model for preferences based on possibility theoryandpossibilisticnetworks.Section3comparesvariouswaysoforderingthesolutionstoadecisionproblemexpressed byapossibilistic preferencenetwork. Section4compares possibilisticpreference networks withother qualitative graphical representationsofconditionalpreference,especiallyCP-nets.

Thispaperhas itsroots ina conferencepaper[9] and somewhat borrowsfrom anotherconference paperon the com-parison between different orderings that can be definedbetween configurations [10], and to a lesser extent from a third conferencepaperprovidingacomparativeoverview ofgraphicalpreferencestructures[11].

2. Introducingpossibilisticpreferencenetworks

Possibilisticconditionalpreferencenetworks,

π

-prefnetsforshort, areanovelmodelforrepresentingpreferences.They arebasedonpossibilisticnetworks[12,13].Thelatter areapossibilisticcounterpartofBayesiannetworks[1]inthecontext of possibility theory [14,15], which offers a setting for preferences representation. Weuse a set of conditional preference tablesexpressingpreferencesaboutthevaluesofvariablesconditionaltothevaluesofothervariables.Hereweassumethat such conditionalpreferences are represented byconditional possibility distributions. Moreover, as itis difficultto directly quantifypreference, weshallassumethat possibilityweights remainsymbolic(i.e.,non-instantiated)andthat wemayadd appropriatepreferenceconstraintsbetweensuchweightsiftheyareavailable.Inotherwords,

π

-prefnets,likeCP-nets,are intendedtobeaqualitativepreferencerepresentationframework.

2.1. Conditionalpossibility andpossibilistic networks

Beforedescribing

π

-pref nets in detail, we recall basic notions of possibility theory that will be useful in the sequel. Possibilitytheoryreliesonthenotionofapossibility distribution

π

[15],which isamapping fromauniverseofdiscourse

Ä

totheunitinterval [0,1], orto anybounded totally orderedscale.This possibilisticscalecanbetheunitinterval when possibility values are the result of a clear measurement procedure, or an ordinal scale when values only reflect a total preorder betweenthedifferentelements of

Ä.

Weassume thatthepossibility distributionissuchthat

π (ω)

=1forsome element of

Ä

(

∃

ω

∈

Ä

such that

π (ω)

=1). The possibility distribution

π

is then said to be normalized. When used to representuncertaintyaboutsomevariablextakingvalueson

Ä,

theassignment

π (ω)

=0meansthat

ω

isfullyimpossible asavalueforx,while

π (ω)

=1meansthat

ω

isfullypossible, i.e.,non-surprising.

The occurrence of an event F⊆Ä is then associated with the possibility measure

5(

F

)

=supω∈F

π (ω)

estimating its plausibility, and with the dual necessity measureN

(

F

)

=1−

5(

¯F

)

=1−supω∈/F

π (ω)

estimating itscertainty. The degree

5(

F)evaluatesto whatextentF isconsistentwith theknowledgerepresentedby

π

,whileN(F)evaluates atwhatlevelF is certainlyimpliedby

π

.See[14]foranintroduction topossibilitytheory.

ConditioninginpossibilitytheoryisdefinedfromtheBayesian-likeequation

5(

F∩G

)

=

5(

F

|

G

)



5(

G

)

,

whereisassociative,monotonicallyincreasinginthewidesenseand 1representstheidentityelementsuchthat 1

α

=

α.

Inthispaper,standsfortheproductinaquantitativesetting(numerical),orfortheminimuminaqualitativesetting (ordinal).Namely,

• if istheproduct,wegetastraightforwardcounterpartofconditionalprobability:

5(

F

|

G

)

=

5(

F∩G

)

5(

G

)

providedthat

5(

G

)

>0;

• if istheminimum,wegetaqualitativeversionofconditioning,that makessense onafinitepossibilityscale:

5(

F

|

G

)

=

½

5(

F∩G

)

if

5(

G

)

>

5(

F∩G

)

; 1if

5(

G

)

=

5(

F∩G

)

>0.

Thetwodefinitionsofpossibilisticconditioning leadto twovariants ofpossibilisticnetworks:inthenumericalcontext, wecanexpressproduct-basednetworks,whileinthequalitativecontext,weonlyhavemin-basednetworks(alsoknown as qualitativepossibilisticnetworks) [12].

Let V=

{

A₁,...,An

}

be a set of n variables.Each variable Ai has a finitedomain DAi whose elements areai∈DAi. The universe of discourse

Ä

=

{

ω

1,...,

ω

|Ä|

}

is the Cartesian product DA1× · · · ×DAn of domains of variables in V (so,

|

Ä|

=

|

DA₁

|

× · · · ×

|

DAn

|

, where|T|denotes thecardinality ofafiniteset T). Eachelement

ω

∈

Ä

will becalledaconfiguration.It correspondstoacompleteinstantiationofthevariablesinV.Apossibilisticnetwork hasadefinitionsimilartotheoneofa Bayesiannetwork.

Definition1(Possibilistic networks). [12,16] Apossibilisticnetwork over aset ofvariablesV ischaracterizedbytwo com-ponents:

(i) agraphical componentwhich isaDirectedAcyclicGraph G=(V, E)whereV isasetofnodesrepresenting variablesand

EasetofdirectededgesA_i→A_j encodingconditional(in)dependenciesbetweenvariables.

(ii) avalued componentassociatingalocalnormalizedconditional possibilitydistribution

π

(A_i|p(A_i))toeachvariableA_i∈V inthecontextofeachinstantiationp(A_i)ofitsparentsP(A_i

)

=

{

A_j:A_j→A_i∈E

}

.

Weassumethat

π

(A_i|p(A_i))>0inordertoavoidconditioningonavalueofpossibility0.Italsocomesdowntoassuming thatallconfigurationsaresomewhatpossible.Thisassumptionwillbeinnocuous inthemodelingofpreferences.

Givenapossibilisticnetwork,wecancomputeajoint possibilitydistributionusingthefollowingchainrule:

π

(

A1,...,An

)

=_i=1.n

5(

A_i

|

P(A_i

))

. (1)

When  istheproduct, and no configurationis impossible,the conditionaltables canberetrievedfrom thejoint pos-sibility distribution obtained by thechain rule, usingthe same orderingof variables asin the original network.However this isno longer the case if  is the minimum, as some conditional possibility values may belost when computing the minimuminthechainrule.

Originally,possibilisticnetworksweremeanttomodeluncertaintyandtocomputetheimpactofobservationsassigning values to some variables so as to predict the values of other variables of the network. In this paper, we advocate their interest in preference modeling rather than uncertainty management. Thus here

π

(

ω)

should beunderstood as the level of satisfaction when choosing configuration

ω.

For a set of configurations F,

5(

F) evaluates to what extent satisfying a constraint modeled by F is satisfactory, and N(F) evaluates to what extent this constraint is imperative. As we shall see, beyondtheirgraphicalappeal,conditionalpreferencepossibilisticnetworksprovideanaturalencodingofpreferences.Inthe following,weintroducethekindofpreferenceinformationneededtoconstruct

π

-prefnets.Then,wepresentthedefinition of

π

-prefnetsand explaintheirrepresentationalpower.

2.2. Conditionalpreferencestatements

In qualitative preference models, usersare supposed to express their preferences underthe form ofcomparison state-mentsbetweeninstantiationsofavariable,conditionedbysomeotherinstantiatedvariables.Forinstance, intheparticular caseof Boolean variables, we deal with preferencesof the form:“a ispreferred to ¬a” ifthe preference isunconditional, and for conditional statements,inthe form “in thecontext where cis true, ais preferred to ¬a”, wherec correspondsto theinstantiationof(maybeseveral)othervariables.Moregenerally,

Definition 2. A preferencestatement (A_i, p(A_i), º)is a preference relationbetween values a_ik∈D_A

i of a variableAi, condi-tioned by the instantiationp(A_i) of a set P(A_i

)

of other variables, in the form of a complete preorder º on D_A

i. Namely

∀

a_ik,a_im∈D_A

i,wehave

i) eitherp(A_i):a_ik≻a_im,i.e.,inthecontextp(A_i),a_ikis preferredtoa_im,

ii) orp(A_i):a_ik∼a_im,i.e.,inthecontextp(A_i),oneisindifferentbetween a_im anda_ik,

where≻isthestrictpartofº,and ∼ istheindifference partofº.If P(A_i

)

=∅, thenthepreferencestatementaboutA_i is

unconditional.

Note that wedonot allowincompletepreferencelocal specificationsintheforma_ikºa_im.On eachvariabledomainD_A

i, the user must choose between a_ik≻a_im, a_ik≻a_im and a_ik∼a_im. It comes down to rating each possible instantiation of the variable A_i (whose domaincan benominal) on alocal totally ordered ordinal value scale,which isa usual assumption in multicriteriadecisionmaking.

The runningExample1,inspired from[6],illustratessuchpreferencestatements.

Example1. Considerapreferencespecificationabout aneveningsuit overthreedecisionvariablesV=

{

J,P,S

}

standingfor jacket, pants and shirt respectively, with values in DJ=

{

Red

(

jr

)

, Black

(

jb

)

}

, DP=

{

White

(

pw

)

, Black

(

pb

)

}

and DS=

{

Black

(

s_b

)

, Red

(

sr

)

, White

(

sw

)

}

. Theconditionalpreferencesare giveninTable1.Preferencestatements (s1)and(s2)are unconditional. Note that theuser is indifferent between the values of thecolor of theshirt if hisjacket is blackand his pantsarewhite(inthecontext j_bpw),whichisnotthecaseifhewearsaredjacketandblackpants.Indeed,heprefersred shirttoablackone(inthecontextjrpb).

Table 1

Conditional preference specifi- cation of Example 1 . ( s 1 ) j b ≻j r ( s 2 ) p b ≻ p w ( s 3 ) j b p b : s b ≻ s r ≻ s w ( s 4 ) j b p w : s w ≻ s b ≻ s r ( s 5 ) j r p b : s r ≻ s b ≻ s w ( s 6 ) j r p w : s b ∼ s r ∼ s w

Fig. 1. A possibilistic preference network.

2.3. Introducing

π

-prefnets

Representing the preference statements in a graphical way means that each node in the graph represents a decision variableA_iwhichisassociated toasetoflocalconditionalpreferencestatements,conditionaltothevaluesofvariablesthat areitsparent nodesinthegraph.A(conditional)preferencenetworkcanbedefinedasfollows:

Definition 3. A preference network is a Directed Acyclic Network (DAG) (E, V) with nodes Ai, Aj∈V, s.t. each arc from

A_j→A_i∈Eexpresses that thepreference about A_i depends on A_j. Each nodeA_i is associated witha ConditionalPreference TableCPT_i thatassociates preferencestatements (A_i,p(A_i),º)betweenthevaluesofA_i, conditionaltoeachpossible instan-tiationp(A_i)oftheparents P(A_i

)

ofA_i (ifany).

Ina possibilisticpreference network,for each particularinstantiation p(A_i)of P(A_i

)

, thepreference orderbetween the valuesofA_istatedbytheuserwillbeencodedbyalocalconditionalpossibilitydistributionexpressedbymeansofsymbolic

weights. By a symbolic weight, we mean a symbol representing a strictly positive real number in (0, 1] whose value is unspecified.Werelyonsymbolicweights intheabsenceofavailablenumericalvalues.Moreformally, wehave:

Definition 4. (ConditionalPreferencePossibilisticnetwork(

π

-prefnet))Apossibilisticpreferencenetworkbasedonoperation

 (−

π

-prefnet)

5

G over aset V=

{

A1,...,An

}

of decision variablesis a preference network where each local prefer-encerelation atnodeA_i isassociated witha symbolicconditionalpossibility distribution (

π

i-table forshort), encoding the orderingbetweenthevaluesofA_isuchthat:

(i) If p

(

A_i

)

:a_i≺a′

i then

π (

ai

|

p

(

Ai

))

=

α,

π (

a′i

|

p

(

Ai

))

=

β

where

α

and

β

aresymbolicweights,and 0<

α

<

β

≤1; (ii) If p

(

Ai

)

:ai∼ai′ then

π (

ai

|

p

(

Ai

))

=

π (

a′i

|

p

(

Ai

))

=

α

>0where

α

isasymbolicweightsuchthat

α

≤1;

(iii) For eachinstantiation p(Ai)ofP

(

Ai

)

,

∃

ai∈DAi suchthat

π (

ai

|

p

(

Ai

))

=1.

(iv) Asymbolicdegreeofpossibilityisassignedtoeachconfiguration

ω

usingthechainrule(1)basedon.

LetC₀bethesetstoringtheconstraintsbetweenthesymbolicpossibilityweightspertainingtoeachpreferencestatement (A_i,p(A_i),º),encodingthecompletepreorderingº.Inadditiontothesepreferencesencodedbya

π

-prefnet,additional con-straintscanbetakenintoaccount.Suchconstraints,formingasetdenotedbyC₁,mayexpressthatsomeweightspertaining toonepreferencestatementareequalto,orgreaterthan,weightspertainingtoanotherpreferencestatement.LetC=C₀∪C₁ bethesetofallconstraints.Incaseoneneedstocomparetwoweights

α

and

β,

onecheckifthereisanyrelationbetween themstoredinC,orifonecaninferitbytransitivityfromC;otherwisetheyareconsideredasincomparable.

Example2. Considerthepreferencespecificationabout anevening suitofExample1.Itscorresponding

π

-prefnetand the conditional possibility weights are given by Fig.1. The graph is built based on Definition3. In fact, since P(J

)

=P

(

P

)

=∅ the two variables J and P are independent, while S depends on J and P since the preference statements associated to S

areconditioned by P(S

)

=

{

J,P

}

.The constraints betweensymbolic weights inherent from thepreference specification are representedbythesetC₀ suchthat C₀=

{

(δ

1>

δ

2

)

,

(θ

1>

θ

2

)

,

(λ

1>

λ

2

)

}

.

Table 2

Vectors associated to each configura- tion of Example 2 .

configurations Symbolic vectors J P S jb p b s b (1, 1, 1) jb p b s r (1, 1, δ1 ) jb p b s w (1, 1, δ2 ) jb p w s b (1, β, θ1 ) jb p w s r (1, β, θ2 ) jb p w s w (1, β, 1) jr p b s b (α, 1, λ1 ) jr p b s r (α 1, 1) jr p b s w (α, 1, λ2 ) jr p w s b (α, β, 1) jr p w s r (α, β, 1) jr p w s w ( α, β, 1)

Aset ofconditionalpreferencetablesencodedasa

π

-prefnetdeterminesapartial orderamongconfigurations.Indeed, eachconfigurationhasasatisfactionlevelencodedbyapossibilitydegreecomputedbymeansofthepossibilisticchainrule

(1).Thisleadsustothefollowingdefinitionoftheinducedpreferenceorderingon configurations.

Definition 5 (Preferenceordering). Consider asymbolicpossibilistic preferencenetwork

5

Gand aset C ofconstraints be-tweenthesymbolic weights.Let

ω

and

ω

′ betwo configurationsin

Ä,

and

π

5G(

ω)

(resp.

π

5G(

ω

′))bethesymbolic pos-sibility degree of

ω

(resp.

ω

′_{) computed by(1). Then, configuration}

_ω

_{is weakly preferred to}

_ω

′_{, denoted by}

_ω

_º



ω

′, iff

π

5G(

ω)

≥

π

5G(

ω

′).

Inthedefinition,

π

5G(

ω)

isacombinationofsymbolicweightsusing.So,

π

5G(

ω)

≥

π

5G(

ω

′)(resp.

π

5G(

ω)

>

π

5G(

ω

′),

π

5G

(ω)

=

π

5G

(ω

′

)

) should beunderstood asfollows: this inequality(resp. strict inequality, equality)holds whatever the numerical instantiations of the weights involved in the possibility values, in agreement with constraints in C. This is re-spectively denoted by

ω

º

ω

′,

ω

≻

ω

′ and

ω

∼

ω

′. When it isnot possible to prove aninequality between

π

5G(

ω)

and

π

5G(

ω

′), because it ispossible to have strict inequalities in bothdirections by substituting distinct numerical values, we interpretthissituationintermsofincomparability asalready said,andthisisdenotedby

ω

±

ω

′.

Since we use symbolic weights, adefinite preference between all configurationscannot be established (as long aswe do notinstantiate allsymbolic weights). Each configuration

ω

=a₁...an can alsobeassociated with avector

(α

1,...,

α

n

)

, where

α

i=

π (

ai

|

p

(

Ai

))

and p

(

Ai

)

=

ω

[P(Ai

)]

, where

ω

[P

(

Ai

)]

istherestriction oftheconfiguration

ω

totheparentsofAi. For instance, vectors associated to thepreference possibilistic network of Example2are given in Table2. Thus, comparing configurationsamountstocomparingvectorsofsymbolicweights attachedto configurations,and theuseofthechainrule isjustonewayofcomparingsuchvectors, amongotheronesasdiscussedinthenextsection.However,notethatsymbolic weightsattachedtoavariabledependontheinstantiationsofitsparents.

3. On various ways of ordering configurations induced by conditional preference networks

Intheprevioussection,wehaveshownhowtoencodethepreferencespecificationsinapossibilisticnetworkformatand wehavedefinedapartialorderingonconfigurationsbasedoncomparingexpressions involvingsymbolicweightscombined with operation. In contrast wemay alsocompare vectors ofsymbolic weights representing the local satisfactionof the conditionaltablesfor configurations.Inthissectionwewill firstcompare thepartial orderrelationsbased onproduct and minimum definedabove, and thenclassical vector comparisontechniques such asPareto and symmetric Pareto orderings withthepurposetousethemtogenerateameaningfulorderingoverconfigurations.

3.1. Orderingrelationsforsymbolicvectors

A symbolicvectoron [0, 1] isof theform

α

=

(α

1,. . .,

α

n

)

. Viewed asvectors ofratingsofconfigurations a1...an, the rangeof each component

α

i=

π (

ai

|

p

(

Ai

)

is either the open interval (0, 1), orreduces to {1} (if

π (

ai

|

p

(

Ai

)

=1). In other words,thevectorcontains1’sand/orsymbolsstandingforunknownpositivevaluesstrictlylessthan 1.

Definition 6. Orderingrelationsbetween symbolicvectors

α

=

(α

1,...,

α

n

)

and

β

=

(β

1,...,

β

n

)

based onanoperation , thatcanbetheminimumortheproduct,aredefinedasfollows:

•

α

≻

β

iff n_i=1

α

i>ni=1

β

i •

α

º

β

iff n_i=1

α

i≥ ni=1

β

i •

α

∼

β

iff

α

º

β

and

β

º

α

When=min(resp.product),wewrite

α

≻_min

β

(resp.

α

≻prod

β),

andso onfortheother relations.

In thisdefinition,

α

≻

β

reallymeans that the inequality n_i=1ai>n_j=1bj holds for any choiceof numerical values ai and b_j1 _{in therespective ranges of}

_α

_{i and}

_β

_{i, and likewisefor}

_α

_º

_β

_{usinga weak inequality.Actually we can drop the} terms

α

i and

β

j whose ranges are {1}. Note that the relation

α

≻

β

is then more demanding than the one defined by “αº

β

and not

β

º

α”,

since inthe former we requesta strictinequality for allinstantiationsby numericalvalues. The latteronlyrequires

α

º

β

and

∃

ai,bj:i:αi6=1ai>j:βi6=1bj.Finally,

α

±

β

standsfor

∃

ai,bj,a

′ i,b′ii:αi6=1ai>j:βi6=1bjand _i:α i6=1a ′ i<j:βi6=1b ′ j.

Wealsoconsider thefollowingclassicalrelations,expressedinthesymboliccase:

Definition 7 (Pareto).

•

α

ºP

β

iff

∀

k,

α

k≥

β

k.

•

α

≻P

β

iff

α

ºP

β

and

∃

k,

α

k>

β

k. •

α

∼P

β

iff

α

=

β.

Note that with the type of symbolic vectors that we use,

α

k≥

β

k may hold only if either

α

k=1, or

α

k=

β

k, or 1>

α

k>

β

k inthesameconditionalpreferencetable,and

α

k>

β

k mayholdonlyifeither

α

k=1,

β

k6=1,or1>

α

k>

β

k.

Definition8(SymmetricPareto).

α

ºSP

β

iff thereexistsapermutation

σ

ofthecomponentsof

α,

yieldingavector

α

σ,such

that

α

σºP

β.

Similardefinitionscanbewrittenfor

α

≻SP

β,

α

∼SP

β.

Inthenumericalsettingitiseasytoseethat

α

≻P

β

implies

α

≻SP

β,

which inturn implies

α

≻prod

β.

Besides,

α

ºP

β

implies

α

ºSP

β

which implies both

α

ºprod

β

and

α

ºmin

β.

But

α

≻SP

β

only implies

α

ºmin

β:

for instance, (0.3, 0.8)≻SP(0.7,0.3), whilemin

(0

.3,0.8)=min

(0

.7,0.3).Things changewhen weconsider vectorsofsymbolicweights.

3.2. Comparisonofproduct-basedandmin-basedorderingsinthesymbolicsetting

Inthefollowing,wepresentthepossiblerelationsbetweentheproduct-basedand theminimum-basedorderings inthe particular case where theconstraints known between the symbolic weights only pertainto the expressionof conditional preferences, asallowed byDefinition4. Thus, a constraint ofthis kindmay only compare weights pertainingto the same componentin thevectors, and we have C₁=∅. Namely, weights located in differentcomponents ofavector areassumed tobeincomparable.Underthisassumption,Paretoorderingandsymmetric Paretoyieldthesameordering.Indeed, fortwo vectors

α

=

(α

1,. . .,

α

n

)

and

β

=

(β

1,. . .,

β

n

)

encodingthesymbolingratingsofconfigurations,eachsymbolicweight

α

i6=1 of

α

canonlybecomparedtothesymbolicweight

β

i6=1of

β.

Thus,thereisnoneedtopermutecomponentsastheresult woulddefinitelybeincomparable withanothercomponentweightsinceC₁=∅.

Inthefollowing, wedenote byprod(

α)

the product Qn

i=1

α

i.Then, itcan beshown that inthissituation, the following equivalenceshold:

Proposition 1. The followingequivalencesholdwhenC₁=∅:

•

α≻

prod

β⇔

α≻

P

β

. •

α

∼_prod

β

⇔

α

=

β

•

αº

prod

β⇔

αº

P

β

Proof :

α≻

prod

β

requires that each instantiation of prod(

α)

be at least as great as each instantiation of prod(

α).

As by assumption, symbolsoftheform

α

i and

β

j arenot comparablefori6=j,unless oneof themisequalto 1, theonly wayto have

α≻

prod

β

istohaveaconstraintoftheform

α

i>

β

i(possibly

α

i=1)forsomei’sand

α

j=

β

j fortheothercomponents. Hence

α≻

P

β.

Theothercasesfollowusingthesameapproach. ¤

Wenowcomparethedifferentorderingsinducedbytheuseofproductorminimum,dependingonthechainruleapplied tothepossibilistic network.Theproduct ofsymbolic vectorsoften hasadiscriminatingpower greaterthan theone ofthe minimum operator, in the sense that

αβ

<

α,

while we only have min(

α,

β)

≤

α.

However, with instantiated numerical values,bothproductandminimumleadtototalordersthatmayalsocontradicteachother:forinstance0.1×0.9>0.2×0.2, whilewiththeminwegetmin(0.1,0.9)<min(0.2,0.2).Thefollowingexampleillustratesthedifferencebetweenthepartial ordersobtainedwithproductand minimumincaseofsymbolicweights.

Example 3. Let us consider the possibilistic preference network of Example2. Using the chain rule, we obtain the sym-bolic vectors presentedin Table2, and thefollowing symbolic joint possibilitydistribution:

π (

j_bp_bs_b

)

=1,

π (

j_bp_bsr

)

=

δ

1,

π (

j_bp_bsw

)

=

δ

2,

π (

jbpwsb

)

=

β



θ

1,

π (

jbpwsr

)

=

β



θ

₂,

π (

j_bpwsw

)

=

β,

π (

jrpbsb

)

=

α



λ

1,

π (

jrpbsr

)

=

α,

π (

jrpbsw

)

=

α



_λ

₂,

π (

jrpwsb

)

=

π (

jrpwsr

)

=

π (

jrpwsw

)

=

α



_β

. The product-basedinducedordering basedon inequalityconstraints in C₀ is represented by Fig.2. For instance, j_bp_bsr≻prod jbpbsw because

δ

1>

δ

2 is in C0 and jbpwsw≻prod jrpwsw because

β

>

α



β.

Now, if we use the min-based chain rule, we will not be able to getas many strict preferences as long as no

1 Not to be confused with instantiations a

Fig. 2. Possibilistic product-based order relative to Example 3 .

Fig. _{3. Possibilistic minimum-based order relative to} Example 3 .

otherconstraintisadded.Infact,theonlystrictorderinginformationwecangetatthatstageisthat j_bp_bs_b≻min jbpbsr≻min

j_bp_bsw; jbpbsb≻min jbpwsw;andjbpbsb≻minjrpbsb.But,fortherest,weonlygetweakinequalitiessuchas jrpwsb¹min jbpwsw, since

π

min

(

jrpwsb

)

=min

(α,

β )

≤

π

min

(

jbpwsw

)

=

β

(dottedarrowsonFig.3depictsthismin-based ordering).

The followingresults canbeobserved[10]:

Proposition 2. WhenC₁=∅,

α

∼prod

β⇔

α

∼min

β

.

Indeed,

α

∼_prod

β

ifandonlyif

α

=

β

duetoitscoincidencewithParetoorderingand then

α

∼_min

β.

Conversely,suppose

α

∼min

β,

i.e.,min

α

i=min

β

i.Suppose

α

i6=

β

iforsomei.Theneither

α

i=1anditiseasytoletmin

β

iassmallaspossible, fixing thevalues ofother

α

j’sand setting

β

i to avery smallvalue bi, i.e.,minai>minbi. Wecan dosomething similarif

α

i>

β

i. As the weights arenot comparableacross components of vectors (C1=∅),except for weights 1that do not affect minimumnorproduct, weconclude that

α

and

β

containthesameweightsineachcomponent.

Proposition 3. WhenC₁=∅:

α

±prod

β

⇔

α

±min

β

.

Proof. Indeed,

α

±prod

β

indicates that the vector

α

contains symbolic values that are not comparable to symbolic values in

β.

In that case,

α

±min

β

as well.Conversely, note that no symbolic weight in

α

will be absorbed with theminimum operation since weights appearing in

α

(resp.

β)

are pairwise incomparable.So, in our situation where C₁=∅,

α

±min

β

holdsinthesamecasesaswhen

α

±prod

β

holds. ¤

Example 4. ManycasesofincomparabilitycanbeidentifiedonExample2andthemin-based(resp.product-based)ordering presentedbyFig.3(resp.Fig.2).Forinstance,wehave jrpbsr±prod jbpwsw (resp. jrpbsr±min jbpwsw).

Moreover, usingsymbolic weights, product and minimumprovide consistent orderings, in contrast with thenumerical setting,inthesensethat:

Proposition4. WhenC₁=∅,

α

≻prod

β

⇔

αº

min

β

and

α

6=

β

.

Proof. It is obviousthat

α

≻prod

β

implies

α

6=

β.

That it implies

αº

min

β

comes from theequivalence between Pareto and product orderings when C₁=∅. Conversely since C₁=∅, the only possibility to get

αº

min

β

is that

∀

i,

α

i≥

β

i since the weights are not comparableacross components ofvectors (e.g.

α

i is not comparableto

β

j for i6=j). As

α

6=

β,

there must existisuchthat

α

i>

β

i,and fortheother components,either

α

k>

β

kor

α

k=

β

k.Hence,

α

≻prod

β.

¤

As aconsequence,thestrictpreferencegraphof≻prod willbethesameastheweakpreferencegraphof≻min,aspatent whencomparingFigs.2and 3forExample2.Andwecanseethat

αº

min

β

⇒

αº

prod

β.

Proposition5. WhenC₁=∅,

α≻

min

β

⇒

α≻

prod

β

.

Proof. This is because

α≻

min

β

if and only if

∀

i suchthat

β

i6=1,

α

i>

β

i. Indeed suppose

α

i≤

β

i for some i. Then we can alwayssetboth

α

i and

β

i tothesamevalueascloseto0and possiblesoastomake

α≻

min

β

fail. ¤

For instance, all solid arrows in Fig.3 also appear in Fig.2 for Example2. This indicates that ≻min isa strong form of Paretoordering.Thus,theproductorderingisarefinementoftheminimum-basedorderinginthesymboliccase.

3.3. Constraintsacrosscomponentsofsymbolicvectors

As already mentioned, constraints between symbolic weights, beside those induced from the preference specification, can beadded whenavailable. In thefollowing wewill study therelations betweenthe different orderingrelationsin the presenceofsuch constraints.Ithasbeen shown abovethat,when thereisno constraintbetween symbolicweights in the vectors,theorderinginducedbytheproduct-basedchainrulecorrespondsexactlytotheParetoordering.Thisresultactually holdsreplacingParetobySymmetricParetointhe presenceofinequalityconstraintsbetween symbolicweights.

Proposition6. Givenanyset ofconstraints C oftheform

α

i≥

β

j or

α

i>

β

j between symbolicweights, it holdsthat

α

≻SP

β

iff

α

≻prod

β

and

α

ºSP

β

iff

α

ºprod

β

.

The proof of this result is not trivial and can be found in [10]. It appears in the appendix for the sake of self-containedness.

Letuscomparetheminimumbased-orderingand theproduct-basedordering(equivalently,SP).Itisclear thatwehave asacorollary ofthepreviousresult:

Proposition 7.

α

∼_prod

β

⇒

α

∼_min

β

.

Proof.

α

∼prod

β

iff

α

∼SP

β,

i.e.,

α

∼P

α

σ′ for some permutation

σ

. Thus,

∀

i,

α

i=

β

iσ, where i∈

{

1,...,n

}

. Therefore,

min

(β

1,...,

β

n

)

=min

(α

1,...,

α

n

)

. ¤

Thelast propositionshows that thestrict min-based orderingcan solvesome incomparability cases forthe Symmetric Paretoordering.Indeed:

Proposition8. If

α≻

min

β

wemayeitherhave

α

±SP

β

or

α≻

SP

β

.

Proof. From Proposition7, if

α

∼SP

β

then

α

∼min

β.

Moreover, if

α≺

SP

β

then by definition,

∀

i,

α

i≤

β

σ(i), thus it follows that min

(α

1,...,

α

n,

β

i,...,

β

n

)

=min

(α

1,...,

α

n

)

, this proves that we cannot have

α≻

min

β

in this case.

Proposition8follows. _¤

Itis importantto notice that,ingeneral, symmetric Paretodominance (hencethe product ordering) inthe widesense doesnotrefinetheminimumordering,sincetheformermayyieldincomparabilityinsomecaseswhenminimumsucceeds incomparing.

Theextreme caseis whenassuming a total preorder between allsymbolic weights.In that case, minimumordering is total. However, in the presence of such rich constraints, symmetric Pareto ordering (i.e., product) with symbolic weights maystillleadtoincomparability.Indeed, theonlycase,wheresymmetric Paretoleadstoatotal orderingiswhenthereare enoughconstraintsbetweensubsets ofsymbolicweights(correspondingtothecomparisonofsubproducts).

3.4. Applicationto bestsolutionanddominancequeries in

π

-prefnets

Ina preference model, two typesof queries are commonly used: namely,optimization queries forfinding the optimal configuration(s)(i.e.,thosewhicharenotdominatedbyothers)and dominancequeriesforcomparingconfigurations.

Optimization. Since

π

-pref nets allow theuser toexpress indifference, the optimization querymay return morethan one configuration.Clearly, thebest configurationsare thosehaving ajoint possibility degreeequal to1. Indeed, such a config-urationalwaysexists sincethejointpossibility distributionassociated tothepossibilisticnetwork isnormalized,thanksto the normalization of each conditional possibility table (indeed, for each variable A_i, for each instantiation p(A_i) of P

(

Ai

)

, wehave:max

(π (

a_i

|

p

(

A_i

))

,

π (¬

a_i

|

p

(

A_i

)))

=1where

{¬

a_i

}

=DA_i/

{

ai

}

withai∈DA_i). Thus,wecanalwaysfind anoptimal configuration,startingfromtherootnodeswherewechooseeachtimethemostorone ofthemostpreferredvalue(s)(i.e., with possibility equalto 1). Then, depending on theparents instantiation, eachtime weagain choose analternativewith aconditional possibility equal to1. Atthe endof theprocedure, we getone orseveral completely instantiated configura-tions having apossibility equal to 1. Consequently, partial preference orders with incomparable maximalelements cannot berepresentedbya

π

-prefnet.

Example 5. Letusreconsider Example2anditsproduct-basedjointpossibilitydegreedepicted byFig.2.Then,j_bp_bs_b isthe preferredconfigurationsinceitsjointpossibilityisequalto1,and thisistheonly one.

Thisprocedureislinearinthesizeofthenetwork(usingaforwardsweepalgorithm).Apossiblevariantofthe optimiza-tionproblemistocomputetheMmostpossibleconfigurationsusingavariant oftheMostProbableExplanationalgorithm in [17]. This query can be interesting in

π

-pref nets even if the answer is not always obvious to obtain in presence of incomparableconfigurations.

Algorithm 1: Comparisonbetweentwojoint symbolicpossibilityvectors.

Data:

α,

β,

C

Result:R

1 begin

2 equality

(α,

β,

C);

3 if (empty

(α)

andempty

(β)

) then R←

α

=

β;

4 else s←sort

(α,

β,

C); 5 ifs=truethenR←

α

≻

β;

6 else s←sort

(β,

α,

C); 7 ifs=truethenR←

β

≻

α;

8 else R←

α

±

β;

9 return R

Dominance. The comparison between symbolic possibility degrees can befound usingAlgorithm1 that takes asinput a set of constraintsC between symbolic weights and two ratingvectors. Let usconsider two configurations

ω

and

ω

′ _with

simplifiedrespectivesimplifiedvectors

α

∗₌

_(α

1,...,

α

k

)

and

β

∗=

(β

1,...,

β

m

)

wherethecomponentsequalto1havebeen deleted,withk≤m≤n.Then,thealgorithmproceedsbyfirstdeletingallpairsofequalcomponents,oneineachvector,soto gettotallydifferentsetsofcomponentsineachvector.Second,itchecksifthereexistsaninjectivefunction

ϕ

:

{

1,...,k

}

→

{

1,...,m

}

suchthat

∀

i=1,...,k,

α

i≥

β

ϕ(i) and

∃

ℓ∈

{

1,...,k

}

,

α

ℓ>

β

ϕ(ℓ) (otherwisetheyremainincomparable).

Thusthealgorithmisbasedonthesequentialapplicationof:

(1) Thefunctionequalitythat deletesthecommonvaluesbetween

α

and

β.

(2) Thefunctionsortthatreturnstrueifaninjectionisfoundbetween

α

∗ _and

_β

∗ _{ensuringtheabovedominancecondition.} (3) Thefunctionemptythattestsifavectorofweights

α

isemptyornot.

Example6. Letusconsiderthe

π

-prefnet

5

GofExample2.UsingAlgorithm1,theorderingbetweentheconfigurationsis shownin Fig.2suchthat alink from

ω

to

ω

′ _meansthat

_ω

_{ispreferred to}

_ω

′_{. For instance, considerconfiguration j}

bpwsr suchthat

π (

j_bpwsr

)

=

β

·

θ

2 and configuration jrpwsr suchthat

π (

jrpwsr

)

=

α

·

β.

First, we shoulddelete commonvalues, namely the symbolic weight

β

. Then, we should check if C entails

α

<

θ

2 or the converse. But here,

α

and

θ

2 are not comparable.Thus, j_bpwsr±prod jrpwsr.

Clearly, for

π

-pref nets, the complexity is due to the comparison step in Algorithm1 (since the computation of the possibility degrees is a simple matter using the chain rule), and in particular to the sort function where the matching between thetwo vectors needs the definitionof different possible arrangements, i.e., thealgorithm is oftime complexity O(n!).

4. Comparison of

π

-pref nets with other graphical preference structures

We now compare

π

-prefnets with ConditionalPreference networks (CP-nets) which deal with thesame kind of con-ditional preference statements. Moreover we also discuss the OCF networks that are “semi-qualitative”. For the sake of simplicity,werestricttothecasewherethedecisionvariablesareBoolean.

4.1. CP-nets

CP-nets, initially introduced in[6,18], areconsidered as anefficient model to manage qualitative preferences.They are basedonapreferentialindependencepropertyoftenreferredtoasaceteris paribusassumptionsuchthatapartial configu-rationispreferredtoanotherone,everythingelsebeingequal.Formally,itisdefinedasfollows:

Definition 9 (Preferentialindependence). LetV beasetofvariablesand W beasubsetofV.W issaidtobepreferentially independentfrom itscomplementZ=V

\

W iff foranyinstantiations,z,z′ _{ofvariablesinZ,and w, w}′ _{ofvariablesinW,}

itholdsthat

(

w,z

)

≻

(

w′,z

)

⇔

(

w,z′

)

≻

(

w′,z′

)

.

Preferential independence is asymmetric. Indeed, it might happen, e.g., for disjoint sets X, Y and Z of variables that X is preferentially independent (Definition9) from Y given Z without having Y preferentially independentfrom X. This independenceisataworkinthegraphicalstructureunderlying CP-nets.

Definition 10 (CP-nets). A CP-net is a preference network in the sense of Definition3 where preference statements are interpreted by meansof the ceteris paribus assumption, namely, the preference pertaining to each decision variable Ai at eachnodeonly dependson theparent(s)contextp(Ai),andispreferentiallyindependentfromtherestofvariables.

Using theinformation in the CP-Tables and applying the ceteris paribus principle, we only obtain preferences between configurationsdifferingbyoneflip,i.e.,obtainedbychangingthevalueofasinglevariable.Indeed,whenflippingavalueof

Fig. 4. Preference network for Example 7 .

Fig. _{5. Worsening flip graph for the CP-net of} Fig. 4 up to transitive closure.

one variablein aconfiguration, one obtains eitheranimproved configuration(improving flip),or aworse one (worsening flip).Thesepairsofconfigurationsdifferingbyoneflipcanbeorganizedintoacollectionofworsening(directed)pathswith auniquerootcorrespondingtothebestconfigurationandwheretheother pathextremitiesaretheworstones.ACP-netis saidtobesatisfiableifthereexistsatleastonepartialorderofconfigurationsthatsatisfiesit.NotethateveryacyclicCP-net issatisfiable.

Example 7. Consider a preference specification about a holiday house in terms of four decision variables V=

{

T,S,P,C

}

standing for type, size, place and car park respectively, with values T∈{flat(t₁),house(t₂)}, S∈{big(s₁),small(s₂)}, P∈

{

downtown

(

p₁

)

, outskirt

(

p₂

)

}

and C∈{car(c₁), nocar(c₂)}. Preference on T is unconditional, while all the other prefer-encesareconditional asfollows:t₁≻t₂, t₁:p₁≻p₂, t₂:p₂≻p₁, p₁:c₁≻c₂,p₂:c₂≻c₁,t₁:s₂≻s₁,t₂:s₁≻s₂.Fig.4represents the correspondingCP-net,anditsinducedworseningflipgraphison Fig.5.

Acyclic CP-nets have a unique optimal configuration. Finding it amounts to looking for a configuration where all the conditional preferences are best satisfied. It can be done by asimple forward sweeping procedure where, for eachnode, weassign themost preferredvalue accordingtothe parentscontext. Foracyclic CP-nets,thisprocedure islinearw.r.t. the numberofvariables[6].Incontrast,forcycliconesansweringthisqueryneedsanNP-hardalgorithmand mayleadtomore than oneoptimalconfiguration[19].Dominancequeriesaremorecomplex. Aconfigurationispreferredto anotherif there exists achain(directed path)of worseningflipsbetween them[18].Note that if forany variableA_i∈V,A_i ispreferentially independentfromVA_i,thentheCP-netgraphisdisconnectedandmanyconfigurationscannotbecompared.Thecomplexity ofdominancetestingdependsontheCP-netstructure.IndeedforthecaseofacyclicCP-nets,Boutilieretal.[6]showthat(i) indirectedtreeCP-nets,thecomplexity isquadraticinthenumberofvariables,(ii)inpolytreeCP-nets,itispolynomial in thesizeoftheCP-netdescription(variablesand preferencetablesizes),(iii) insinglyconnectedCP-nets,itisNP-complete. InmultiplyconnectedCP-nets,theproblemisinNPorharder(itremainsanopenproblemuntilnow).Forgeneral CP-nets (allowingcycles)theproblemisPSPACE-complete [19].

Ingeneral,theorderinginducedbyaCP-netisstrictand partial,sinceseveral configurationsmayremainincomparable (i.e.,no worsening flipschainexists betweenthem).Clearly, acyclic CP-netscannotexhibit anyties. Theceteris paribus

as-Fig. 6. Configuration graph of Ex. 7. Thin arrows reflect ≻prod , dotted arrows compare sets S(ω) , and bold arrows reflect additional ceteris paribus compar-

isons, also in bold on Fig. 5 .

sumptionsimplifiespreferenceelicitationforCP-nets;theelicitationcomplexityisequaltoO(nk_{)suchthatnisthenumber} ofnodesand kisthemaximalnumberofparents[20].

However,inCP-nets,preferenceexpressedinaparentnodetendstobemoreimportantthantheoneexpressedinachild one[21].Inotherwords,violatingapreferenceassociatedwithafathernodeismoreimportantthanviolatingapreference associatedwithachildone;thispriorityimplicitlygiven bytheapplicationofceterisparibus assumptionmaybedebatable. Forinstance,intheCP-netofFig.4,configurationt1p1c2s1ispreferredtoconfigurationt2p2c2s2becausethereisasequence ofworseningflipsfromtheformertothelatter,asseeninthegraphofFig.5.Moreover,thiskindofpriorityisnottransitive inthesensethatCP-netscannotalwaysdecidewhetherviolatingpreferencesoftwochildrennodesispreferredtoviolating preferencesassociated withone childand one grandsonnoderespectively (whichmight havebeen expected asbeingless damagingthanviolatingtwochildrenpreferences)[22].Thislimitationisproblematic.Generally,therearepartialpreference orderingsthat CP-netscannotexpress,see[9]foracounterexample.

TheexpressivepowerofCP-netsislimited.Inparticular,weareunabletospecifyimportancerelationsbetweenvariables, besidethoseimplicitlyimposedbetweenparentsand children.Tradeoffs-enhancedCP-nets(TCP-nets)[23]areanextension ofCP-netsthatadds anotionofimportancebetween thevariablesbyenrichingthenetwork withnewarcs.TCP-netsobey thepreferencestatements inducedbyceterisparibusassumption,sincetheorderingobtainedisarefinementoftheCP-nets ordering.Infact, therefinementbroughtbyTCP-netscannotoverridetheimplicitpriorityinfavorofparentsnodes.

4.2.

π

-prefnetsvs.CP-nets

In this section, we show that the configuration graph of any CP-net can be refined using a

π

-pref net without local indifference,basedon thesamepreferencenetwork,providedsomeconstraintson productsofsymbolicweightsareadded tothe

π

-prefnet,inordertorestoresomeceterisparibusassumption-basedpriorities.Precisely,theaddedconstraintsreflect thehigher importanceoffathernodeswithrespecttotheirchildren.

The preferencesexpressed bytheCP-netscan berepresentedbya

π

-prefnetsharingthesamegraphicalstructureand wheretheconditionalpossibilitydistributionsencodethelocalpreferences.

Example 8. Consider the preference network of Fig.4. Encoded as a possibilistic network it reads:

π (

t1

)

=1,

π (

t2

)

=

α,

π (

p1

|

t1

)

=

π (

p2

|

t2

)

=1,

π (

p2

|

t1

)

=

β

1,

π (

p1

|

t2

)

=

β

2,

π (

s1

|

t1

)

=

γ

1,

π (

s2

|

t2

)

=

γ

2,

π (

s2

|

t1

)

=

π (

s1

|

t2

)

=1,

π (

c1

|

p1

)

=

π (

c₂

|

p₂

)

=1,

π (

c₂

|

p₁

)

=

δ

1 and

π (

c1

|

p2

)

=

δ

2. Applying the product-basedchain rule, we can compute the joint possi-bility distribution relative to T,P, Cand S.Thin arrows in Fig.6represent the configurationgraph inducedfrom thisjoint possibilitydistribution,andboldarrowsreflectadditionalceterisparibuscomparisonsforthecorrespondingCP-nets.Clearly, theconfigurationt₁p₁c₁s₂istheroot(sinceitistheuniqueonewithdegree

π (

t₁p₁c₁s₂

)

=1).

Giventheordinalnature ofpreferencetablesofCP-nets,andthefactthatwerestricttoBooleanvariables,italsomakes sensetocharacterizethequalityof

ω

usingthesetS(ω)=

{

A_i:

α

i=1

}

ofsatisfiedpreferencestatements(onepervariable), where

α

=

(α

1,. . .,

α

n

)

.Itisthenclearthat theParetoorderingbetweenconfigurationsinducedbythepreferencetablesis refinedbycomparingthesesatisfactionsets:

α

≻_prod

β

⇒S

(

ω

′

)

⊂ S(

ω

)

(2)

since the only case when

α

i>

β

i is when

α

i=1 if variables are Boolean and ≻prod precisely coincides with the Pareto ordering,after Proposition1.

Example 9. To see that this inclusion-based ordering is stronger than the

π

-pref net ordering, consider Fig.6 where

π (

t1p2c1s2

)

=

β

1

δ

2 with S

(

t1p2c1s2

)

=

{

T,S

}

and

π (

t2p1c2s1

)

=

αβ

2

δ

1 with S(t2p1c2s1

)

=

{

S

}

. We do have that S(t1p2c1s2

)

⊃ S(t2p1c2s1

)

, but

β

1

δ

2 is not comparable with

αβ

2

δ

1. Dotted and thin arrows of Fig.6 represent the con-figurationgraphinducedbycomparingsetsS

(ω)

Theinclusion-basedorderingS(ω′

)

⊂ S(ω)doesnot dependontheparent variablescontexts,but onlyon thefactthat inthe context ofaconfiguration ofits parents,a variable hasapreferred value (we call“good”)or aless preferred value (wecall“bad”).

Inthefollowing,weassumethatthecomponentsofvector

α

arelinearlyorderedinagreementwiththepartialordering ofvariablesinthesymbolicpreference network,namely,if i<jthenA_iisnotadescendentofA_jinthepreferencenet(i.e., topologicalordering). ForinstanceinthepreferencenetofFig.4,wecanusetheordering(T,P,C,S).

Letusfirstprove that,intheconfigurationgraphsinducedbyaCP-net andthecorresponding

π

-prefnet,therecannot beany preferencereversalsbetweenconfigurations.LetC(A

)

denotethechildrenset ofA∈V.

Lemma 1. If

ω

≻CP

ω

′ and these configurations differ by oneflip of some variable Ai, thenthe inclusion S(ω)⊂ S(ω′

)

is not

possible.

Proof. Compare S(ω) and S(ω′

₎

_{. It is clear that A}

i6∈S(ω′

)

(otherwise the flip would not be improving) and S(ω)=

(S(ω

′

₎

_∪

_{

_A

i

}

∪C−+

(

Ai

))

\

C+−

(

Ai

)

,whereC−+

(

Ai

)

isthesetofchildrenvariablesthatswitchfromabadtoagood valuewhen goingfrom

ω

′ _to

_ω,

_andC−

+

(

Ai

)

istheset ofchildrenvariablesthatswitchfromagood toabadvaluewhengoingfrom

ω

′ to

ω.

Itisclearthatitcanneverbethecasethat S(ω)⊂ S(ω′

₎

_,indeedA

iisinS

(ω)

andnotinS(ω′

)

byconstruction.But S(ω′

)

maycontainvariablesnotinS(ω)(thoseinC₊−

(

Ai

)

ifnotempty). ¤

Inthefollowing,given twoconfigurations

ω

and

ω

′,letDω,ω′ _{theset ofvariableswhichbeardifferent valuesin}

_ω

_and

ω

′_.

Proposition 9. If

ω

≻CP

ω

′ thenS(ω′

)

⊂ S(ω)isnotpossible.

Proof. If

ω

≻CP

ω

′,thenthereisachainofimprovingflips

ω

0=

ω

′≺CP

ω

1≺CP· · · ≺CP

ω

k=

ω.

ApplyingtheaboveLemma1, S(ωi

)

=

(S(ω

i−1

)

∪

{

Vi−1

}

∪C−+

(

Vi−1

))

\

(C

+−

(

Vi−1

)

forsomevariableVi−1=Aj.BytheaboveLemma1,wecannothave

ω

i ≺π

ω

i−1.SupposewechoosethechainofimprovingflipsbyflippingateachstepatopvariableAjinthepreferencenet,among theones tobeflipped, i.e., j=min

{

ℓ:A_ℓ∈Dωi−1,ω

_}

_{. Itmeansthat whenfollowing thechainofimproving flips,thestatus}

ofeachflippedvariablewillnotbequestionedbylaterflips,asnoflipped variablewillbeachildofvariablesflipped later on.So S(ω′

₎

_{willcontainsomevariablesnotinS(ω), so}

_ω

_≻

prod

ω

′ isnotpossible. ¤

Thepreviousresultsshowthat itisimpossibletohaveapreference reversalbetweenCP-netorderingand theinclusion ordering,whichimpliesthatno preferencereversalispossiblebetweentheCP-netorderingand the

π

-prefnetordering.It suggeststhatwecantrytoaddceterisparibusconstraintstoa

π

-prefnetandsoastocapturethepreferencesexpressedby aCP-net.Define apreferencerelation≻+_prod betweenconfigurationsasfollows

ω

≻+_prod

ω

′ ⇐⇒

ω

≻prod

ω

′ or

ω

≻CP

ω

′.

Asmentionedearlier,inCP-nets,parentspreferencesaremoreimportantthatchildrenones.Thispropertyisnotensuredby

π

-prefnetswhereallviolationsareconsideredhavingthesameimportance. Inthefollowing,welaybarelocal constraints between productsofsymbolic weights, pertainingto eachnode and itschildren,that enable ceteris paribusassumption to besimulated.

Let

ω

,

ω

′differbyoneflip,andsuchthatnoneof

ω

≻prod

ω

′,

ω

′≻prod

ω

holds,andmoreover,

ω

≻CP

ω

′.Wemustenforcethe condition

π(

ω)

>

π

(

ω

′_{).Suppose theflippingvariableisA.Clearly, A∈S(ω),but A6∈S(ω}′

₎

_.Let

_α

_{bethepossibilitydegree}

of Awhen it takes the bad value in context p(A) (itis 1when it takes the good value). When flipping A from agood to abad value, only thequality ofthechildren variablesC

(

A

)

of Amay change.C

(

A

)

can bepartitioned intoatmost 4sets, C₋−

(

A

)

(resp. C−₊

(

A

)

,C₋+

(

A

)

,C₊+

(

A

)

), which represents theset ofchildren ofA whose values remain bad (resp.change from goodtobad, frombadtogood,and staygood)whenflippingA.Thenitcanbeeasilycheckedthat:

π

(

ω

)

=1· Y Ci∈C+−(A)

γ

i· Y Cj∈C−−(A)

γ

j·

β

π

(

ω

′

)

=

α

· Y Ck∈C+−(A)

γ

k· Y Cj∈C−−(A)

γ

j·

β

where

β

isaproductofsymbols,pertainingtonodesotherthanAanditschildren,that remainunchangedbytheflipofA. Thentheconstraint

π

(

ω

)>

π

(

ω

′)comes downtotheinequality:

Y Ci∈C−+(A)

γ

i>

α

· Y Ck∈C+−(A)

γ

k (3)

wheresymbolsappearing on oneside donot appearon theother side.Suchconstraintsaresufficientto retrievethe pref-erencesoftheCP-net.Notethat thepreferences

ω

≻prod

ω

′ and

ω

≻CP

ω

′ conjointlyholdinbothapproacheswheneverAhas