Using inconsistency measures for estimating reliability

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2220

To cite this version:

Cholvy, Laurence and Perrussel, Laurent

and Thévenin, Jean-Marc

Using

inconsistency measures for estimating reliability.

(2017) International Journal of

Approximate Reasoning, 89. 41-57. ISSN 0888-613X.

Official URL:

https://doi.org/10.1016/j.ijar.2016.10.004

Using

inconsistency

measures

for

estimating

reliability

L. Cholvy

a

,

∗

,

L. Perrussel

b

,

J-M. Thévenin

b a_{ONERA,Toulouse,France}

b_{IRIT,UniversitéToulouse1Capitole,Toulouse,France}

a

b

s

t

r

a

c

t

Keywords:

Logic Inconsistency Reliabilityassessment

Any decision taken by an agent requires some knowledge of its environment. Communica-tion with other agents is a key issue for assessing the overall quality of its own knowledge. This assessment is a challenge itself as the agent may receive information from unknown agents. The aim of this paper is to propose a framework for assessing the reliability of unknown agents based on communication. We assume that information is represented through logical statements and logical inconsistency is the underlying notion of reliability assessment. In our context, assessing consists of ranking the agents and representing reliability through a total preorder.

The overall communication set is first evaluated with the help of inconsistency measures. Next, the measures are used for assessing the contribution of each agent to the overall inconsistency of the communication set. After stating the postulates specifying the expected properties of the reliability preorder, we show through a representation theorem how these postulates and the contribution of the agent are interwoven. We also detail how the properties of the inconsistency measures influence the properties of the contribution assessment. Finally we describe how to aggregate different reliability preorders, each of them may be based on different inconsistency measures.

1. Introduction

To be able to act ordeliberate, any rational agent must acquire knowledge of its environment. It gets it by merging informationprovidedbyitsownsensorsand/orbymerginginformationcommunicatedbyother agents.Mergingbasic in-formationisakeyissueforanyagent asitistheunderlyingrationalfordecisionmakinganditcontributestojustifythe agent’sepistemicstate. Techniquesformergingrawinformationhavebeenstudiedinan extensiveway.Thesetechniques usually assume thatall informationprovided by thesources (i.e.agents) should beconsidered asa whole.Twodifferent approacheshave been studied: thefirst one considers sources in an equal wayandhas led tomerging techniques such asmajoritymerging,negotiation,arbitrationmergingordistance-basedmergingforsolvingconflictbetweencontradicting information[21,8,27,9].Thesecond onedistinguishessources throughareliability criterion.Taking sourcesreliabilityinto accountprovidesrationalesfordiscountingorignoringpiecesofinformationwhosesourceisnotconsideredassufficiently reliable.Somepromoteaquantitativemodelofreliability:informationsourcesareassociatedwithareliabilitylevel repre-sentedbyanumberusedbythemergingoperator.Accordingtothebelieffunctiontheory,thereliabilitylevelofasourceis anumberbetween0and1.Thisnumberisthenusedbythediscountingruleinordertoweakentheimportanceof

infor-*

Correspondingauthor.

mationprovidedbythissource[31].Someotherspromoteaqualitativeapproachtoreliabilityandconsiderthatinformation sources are rankedaccordingtotheir reliability.Thisorderorpre-order isthenusedby themergingoperator. In[5],the authordefinesamergingoperatorwhichassumesthatthesourcesaretotallyordered:ifs issaidtobemorereliablethan

sandtogetherprovidecontradictinginformation,theninformationprovidedbys isprivileged;whileinformationprovided by s whichdoes notcontradict informationof s isalso considered asacceptable.The sameidea is followedby [22] for reasoningaboutmorecomplexbeliefsandin[24] forrevisingabeliefbase.Alltheseworksassumethat thereliabilityof thesources isgivenasaparameter(quantitativeorqualitative),they donotaddressthequestionofhowtobuild upthis reliability.

Inthispresentpaperouraimistoaddressthekeyquestionofhowtobuildareliabilitypreorderofinformationsources, in a context where sources are unknown: no extra information aboutsources is available and information provided by the sources isonlyqualitative (i.e.,statements). Weadopta qualitativepoint ofview to representreliability:therelative reliabilityofinformationsourcesisrepresentedbyatotalpreorder.Weproposetoconsideraphase,beforetheinformation mergingphase,duringwhichinformationsourcesareobservedinordertoobtainareliabilitypreorder.Thepurposeofthis phaseistoanalyzetheinconsistencyofinformationreportedbythedifferentsourcesw.r.t.sometrustedknowledge.

Ourmaingoalisthustoshowthattherelativereliabilityofinformationsourcescanbeestimatedfromtheinconsistency ofreportedinformation.Twodifferentapproachescanbefollowed.Thefirstapproachconsistsinusinganad-hocmodelfor reported informationandindevelopingnewinconsistencymeasures. The second approachconsistsinmodeling reported informationinaconventionalwayinordertousewellknowninconsistencymeasures.

Inarecentpaper[6],wefollowedthefirstapproach.Reportedinformationwasmodeledbypairs:

<

agent

,

f ormula

>

,

f ormula representingapiece ofinformationcommunicated byagent. Forinstance,theset

{<

a

,

p

>,

<

b

,

¬

p

>,

<

b

,

q

>

}

represented the fact that agent a had reported p, agent b had reported

¬

p andhad alsoreported q. The main notions (inconsistency, minimalinconsistentsubsets,inconsistencymeasures...)availableintheliteraturehavebeenadaptedtothis model.

Inthispaper,ourverymotivationistoshow anoriginalapplicationofinconsistencymeasures,i.e.reliabilityestimation.

Ourstartingpointistheexistinginconsistencymeasures.Hereafter,wesimplifytherepresentationofreportedinformation sothatwecanre-usetheseexistinginconsistencymeasuresforelaboratingagent’sreliability.

Our original contributions consist in (i) characterizing the individual contribution of each agent to the overall incon-sistency ofasetofreportedinformationand(ii)introducing postulateswhichcharacterizetheexpectedpropertiesofthe reliabilitypreorder;Basedontheseaxiomaticperspectiveonreliability assessment,weshow (i)howthepropertiesofthe inconsistencymeasureinfluencethepropertiesofthecontributionsmeasures and(ii)howpostulates aboutreliabilityand propertiesofagent contributionare relatedthrougha representationtheorem.Finally,we show howtoaggregate several preorderspossiblyobtainedthroughdifferentinconsistencymeasures;namelyweshowhowtheoverallaggregatedpreorder maysatisfythereliabilitypostulatesiftheinitialpreordersalsosatisfythesepostulates.

This paperisorganized asfollows.Section 2 andSection 3introduce themain notions neededtoassessreliability of agents.Theyintroduceinconsistentcommunicationsets andfocusonmeasuringtheinconsistencyincommunicationsets. Basedontheinconsistencymeasures,Section4showshowtoassesstheindividualcontributionofanagent totheoverall inconsistency ofacommunicationset.Contribution isfirstcharacterized inanaxiomatic wayandnext two possible con-tributionfunctionsinstantiatingtheexpectedpropertiesaredetailed. Someimplementationandcomplexityconsiderations are alsoaddressed. Section5 proposesa setofpostulates whichaxiomaticallycharacterize reliabilitypreorders andshow through two representationtheoremshowthesepostulates andtheagent contributions arerelated. StillinSection 5,we presenttwo possiblesolutions forbuildingareliabilitypreorder compliantwiththesepostulates.Section 6considers the aggregationofseveralreliabilitypreorders andshowshowArrow’sconditionforaggregationandourpostulates interplay. Finally,Section7concludesthepaperanddiscussesfuturework.

2. Inconsistentcommunicationsets

Thissectionintroducescommunicationsetsandfocusesontheirinconsistency.

2.1. Preliminaries

Let

L

be apropositional language offormulasdefinedovera finitesetofpropositionalsymbols

P

,propositional con-stants



,

⊥

andthelogicalconnectives

∧

,

∨

,

¬

.Weuse p

,

q

,

r

,

...

to denotethepropositionalsymbolsandGreek letters

φ,

ψ,

...

todenoteformulasoftheclassicalpropositional logicdefinedover

L

.Aninterpretation i isa totalfunctionfrom

P

to

{

0

,

1

}

fromwhichanassignmentto

{

0

,

1

}

isgeneratedforalltheformulasof

L

definedintheusual wayofclassical logic.As usual,i

(

)

=

1 andi

(

⊥)

=

0.Interpretation i isamodel offormula

φ

iffi

(φ)

=

1.Tautologies areformulaswhich are interpretedby 1 inanyinterpretation.We write

|= φ

when

φ

is atautology. Aformulaisconsistent iffithasatleast onemodel.Otherwiseitisinconsistent.

Acommunicationbase1 K isafinite(possiblyempty)setofformulasof

L

. At

(

K

)

denotesthesetofpropositional sym-bols appearing informulas whichbelong to K . Acommunication baseis consistent ifftheconjunction ofits formulas is

consistent. Otherwise, it is inconsistent. For a communication base K , M I

(

K

)

is the set of minimal inconsistent subsets of K , i.e., M I

(

K

)

= {

K

⊆

K

|

Kis inconsistent and

∀

K

⊂

K Kis consistent

}

. MC

(

K

)

is the set of maximal consistent subsetsof K ,i.e., MC

(

K

)

= {

K

⊆

K

|

Kis consistent and

∀

Ks.t. K

⊂

K K is inconsistent

}

.IfM I

(

K

)

= {

M1

,

...,

Mn

}

then

P roblematic

(

K

)

=

M1

∪ ...

∪

Mn,and F ree

(

K

)

=

K

\

P roblematic

(

K

)

.ThesetofformulasinK thatareinconsistentisgiven

bythefunction Self contradiction

(

K

)

= {φ ∈

K

| φ

is inconsistent

}

.Noticethatthesedefinitionsareprovidedby[10]. Finally,asshownin[10],athree-valuedlogiccanbeusedtogiveasemanticstoinconsistentformulae.Thethreevalues are T

,

F

,

B whereT and F correspondtotheclassicalvalues1,0 respectivelyandtheadditionaltruthvalue B stands for

both and representsinconsistency.Assumingthat thethreevaluesare orderedby: F

<

tB

<

tT , thevaluationofformulae

in an interpretation i is given by: i

(

)

=

T , i

(

⊥)

=

F , i

(

¬φ)

=

B

⇐⇒

i

(φ)

=

B, i

(

¬φ)

=

T

⇐⇒

i

(φ)

=

F , i

(φ

∧ ψ)

=

min_≤t

(

i

(φ),

i

(ψ))

, i

(φ

∨ ψ)

=

max≤t

(

i

(φ),

i

(ψ))

.Interpretation i isa modelof K if no formulain K isassigned thetruth

value F .Wewrite i

|=

3K . Binar ybase

(

i

)

= {

p

∈

P |

i

(

p

)

=

T ori

(

p

)

=

F

}

andConf lictbase

(

i

)

= {

p

∈

P |

i

(

p

)

=

B

}

.Notice

thatfori

|=

3K ,Conf lictbase

(

i

)

returnsthesubsetofpropositionalsymbolsdirectlyinvolvedininconsistencyaccordingto i.

2.2. Communicationsets

From now,we assumea finitesetofagents A

= {

a1

,

...,

an

}

(n

≥

1). Foreachagent ai, K

(

ai

)

isan

L

-formulawhichis

the conjunctionof all the pieces ofinformation reportedby ai. K

(

ai

)

iscalled reportofagentai. The report of an agent which provides no information is any tautology or, for short,



. Given K

(

a1

),

...,

K

(

an

)

, the multi-set of formulas



=

{

K

(

a1

),

...,

K

(

an

)

}

iscalledcommunicationset.

Given



= {

K

(

a1

),

...,

K

(

an

)

}

andC

= {

ai1

,

..,

aim

}

⊆

A,wedefine

(

C

)

by

(

C

)

= {

K

(

ai1

),

..,

K

(

aim

)

}

.

Consider now two situations, one in which agent reports are K

(

a1

)...

K

(

an

)

and a second one in which they report

K

(

a1

)...

K

(

an

)

.Let



= {

K

(

a1

),

...,

K

(

an

)

}

and





= {

K

(

a1

),

...,

K

(

an

)

}

bethecorrespondingcommunicationsets.Wewrite



≡ 

 (



and



areequivalent)iff

∀

a

∈

A,

|=

K

(

a

)

↔

K

(

a

)

.Thatis,a’sreportin



isequivalenttoa’sreportin



.We write



 

₍

_

_and

_

 _{areweaklyequivalent)iff}

_∀

_a

_∈

_A,

_∃

_b

_,

_∃

_c

_∈

_{A suchthat}

_|=

_K

₍

_a

₎

_↔

_K

₍

_b

₎

_and

_|=

_K

₍

_a

₎

_↔

_K

₍

_c

₎

_.That

is,werelaxheretheconstraintthatreportofagenta shouldbeequivalentbothin



and



;insteadweonlyrequiresome otheragent,possiblydifferentfroma,reportsequivalentinformation.

2.3. I C -inconsistentcommunicationsets

Inthe contextof acommunicationset



,consistency willbe evaluated withrespect tosome integrity constraints I C

which is a consistent formulaof

L

. I C has tobe viewedas informationtaken forgranted or certain. Thus we say that



= {

K

(

a1

),

...,

K

(

an

)

}

isI C -inconsistent iff



∪ {

I C

}

isinconsistent;otherwise



isI C -consistent.Inthefollowing,weadapt

thedefinitionsgiveninthepreliminariestothecaseofcommunicationsets.

Definition1.

•  ⊥

I C is the multi-set of minimal I C -inconsistent subsets of



i.e the multi-set of X

⊆ 

such that X is

I C -inconsistentand

∀

X X

⊂

X

,

XisI C -consistent.

• 

I C

= {

X

⊆  :

X is I C -consistent and

∀

X X

⊂

X

,

X is I C -inconsistent

}

istheset ofmaximal I C -consistent

sub-multisetsof



.

•

P roblematic

()

=



M∈⊥I CM.

•

F ree

()

=  \

P roblematic

()

.

•

Self contradiction

()

= {

K

(

a

)

∈ 

:K

(

a

)

∧

I C is inconsistent

}

.

2.4. I C -inconsistentsetsofagents

WefinallyintroducethenotionofminimalI C -inconsistentsubsetsofagentsandthenotionofproblematicagentswhich areagentswhosereportsareproblematic:

Definition2.

•

A

⊥

I C

= {

X

⊆

A

: (

X

)

∈  ⊥

I C

}

.

•

P roblematic

(

A

)

= {

a

∈

A

:

K

(

a

)

∈

P roblematic

()

}

.

Example1.Consider A

= {

a

,

b

,

c

}

, I C

= ¬

q,K

(

a

)

=

p

,

K

(

b

)

= ¬

p

∧

q

,

K

(

c

)

=

r

∧

s.i.e.,a hasreported p,b hasreported

¬

p

andq,andc hasreportedr ands.Here,



= {

p

,

¬

p

∧

q

,

r

∧

s

}

.Thus,



⊥

I C

= {{¬

p

∧

q

}}

,





I C

= {{

p

,

r

∧

s

}}

,A

⊥

I C

= {{

b

}}

andP roblematic

(

A

)

= {

b

}

.

Example 2. Assume A

= {

a

,

b

,

c

}

, I C

= ¬(

p

∧

q

)

, K

(

a

)

=

K

(

b

)

=

p

,

K

(

c

)

=

q. Here,



= {

p

,

p

,

q

}

. Thus,



⊥

I C

=

3. MeasuringtheI C -inconsistencyofcommunicationsets

Understanding the nature ofinconsistency of a set of formulas isan importanttopic which aroused a great amount of research during the pastdecadeThe purpose isto analyze towhichdegree a setof formulasis inconsistent.A lotof inconsistency measures havebeen proposed according to different points of view: some of them are based on minimal inconsistent subsets[15,16,25,20] oron maximal consistentsubsets [10,1],some considerparaconsistent models such as three valued logic [17,10], some consider probabilistic functions over the underlying propositional language [29], some considermodeldistance[11]andfinally someareproof-based[19].Foradetailedreviewoftheseinconsistencymeasures we refer thereader to[30].Inthissection, we firstreview some well-known inconsistency measuresthen we show our requirementstoadaptthemtoourcontext.

3.1. Inconsistencymeasuresforsetsofformulas

Accordingto[10],an inconsistencymeasureonsetsofformulasisafunction I whichassigns anysetofformulas K to

anelementof

R

₊satisfyingatleastthefollowingthreeproperties:

•

Consistency: I

(

K

)

=

0 iff K isconsistent.

•

Monotony: if K

⊆

K

,

then I

(

K

)

≤

I

(

K

)

.

•

Freeformulaindependence: If

φ

∈

F ree

(

K

)

then I

(

K

)

=

I

(

K

\ {φ})

.

Thatis,themeasureofinconsistencyofasetofformulasisnulliffthissetisnotinconsistent.Themeasureof inconsis-tencyofasetofformulasdoesnotdecreaseifweaddmoreformulas.Finally,removingaformulathatdoesnotcauseany contradictiondoesnotchangetheinconsistencymeasure.

Letusnowbrieflyreviewsomeinconsistencymeasuresforsetsofformulas,introducedintheliterature[15–17,10]

•

ID

(

K

)

=

0 ifK isconsistent;1 otherwise.

•

IC

(

K

)

= |

M I

(

K

)

|

.

•

IM

(

K

)

= (|

MC

(

K

)

|

+ |

Self contradictions

(

K

)

|)

−

1.

•

IP

(

K

)

= |

P roblematic

(

K

)

|

.

•

IQ

(

K

)

=

0 if K isconsistent;



K∈M I(K) 1 |K| otherwise.

•

IB

(

K

)

=

min

{|

Conf lictbase

(

i

)

|

|

i

|=

3K

}

.

ID

(

K

)

isa trivialmeasure whichassigns0to anyconsistent setofformulasand1toanyinconsistentset.Itdoesnot

quantify how much K is inconsistent. IC

(

K

)

countsthe numberofminimal inconsistent subsetsof K . IM

(

K

)

countsthe numberofmaximalconsistentsubsetstogetherwiththenumberofcontradictoryformulasbut1tomakeIM

(

K

)

=

0 when

K isconsistent.IP

(

K

)

countsthenumberofformulasinminimalinconsistentsubsetsof K .IQ

(

K

)

computestheweighted sumoftheminimalinconsistentsubsetsofK ,wheretheweightistheinverseofthesizeoftheminimalinconsistentsubset, sothatsmallerinconsistentsubsetsareregardedasmoreinconsistentthanlargerones.IB

(

K

)

returnstheminimumnumber

ofpropositional symbolsthathavetobesettovalue B inordertogetathreevaluedmodelofK .Measures IC

(

K

),

IP

(

K

)

and IQ

(

K

)

assume that inconsistency is rooted inminimal inconsistent subsets: removing any formulafrom a minimal inconsistentsubsetinsufficienttoproduceamaximalconsistentsubset.Tosomeextent IM

(

K

)

canbeviewedasavariant oftheseinconsistencymeasures.ItisinterestingtonotethatwhileIC

(

K

)

givethesamescoretosetsofformulascontaining

the same numberof minimal inconsistent subsets, IQ

(

K

)

is able todifferentiate their levelof inconsistencyconsidering

that thesmalleris thesizeofa minimalinconsistent subset,thebigger isthe amountofinconsistency.Measures IB and

IL Pm followadifferentapproachassessingtheseverityofinconsistencybyconsideringthenumberofpropositionalsymbols

involvedininconsistency.

J. GrantandA.Huntershowedin[10]that inconsistencymeasurescanbe usedtoorderseveralsetsofformulasfrom theleastinconsistentonetothemostinconsistentone.Theydefinedtwoinconsistencymeasures Ix andIy asbeing order-compatible ifforallsetsofformulas K1 andK2,Ix

(

K1

) <

Ix

(

K2

)

iffIy

(

K1

)

<

Iy

(

K2

)

.Theyprovedthat IC

,

IM

,

IP,IQ andIB

arepairwiseorder-incompatibleandweaddthat ID isalsoorder-incompatiblewitheachofthem.Thisresultisinteresting asitshowsthateachinconsistencymeasuregivesaparticularinsightontheinconsistencyofasetofformulas.

3.2. I C -inconsistencymeasuresforcommunicationsets

In ordertoassign a degreeof I C -inconsistency toa communicationset,weadapt theinconsistencymeasures. ID, IC,

IM, Ip and IQ are typically syntax-based inconsistency measures in the sense they mainly take into consideration the

formulascomposingK .Thisaspectisquiteimportantinourcontext.Howeverweneedtointroduceanextrapropertythat

I C -inconsistencymeasuresshouldsatisfy.Thispropertyisnamedsyntaxweak-independence.Itstatesthat I C -inconsistency

measuresshouldnotbedependentofthesyntaxofI C andthattwoweaklyequivalentcommunicationsetsshouldhavethe same I C -inconsistencymeasure.Inotherwords,themeasuredoesnotdependonthesourcesofthecommunication.

Definition3. Let I C and I C be two integrity constraint and



and



 be two communication setson A.Function II C

:

2A×L

_{→ R}

+ isasyntaxweak-independentI C -inconsistencymeasure iffitsatisfiesthefollowingproperties:

•

Consistency: II C

()

=

0 iff



is I C -consistent.

•

Monotony: If



⊆ 

 _{thenII C}

_(ψ)

_≤

_{II C}

_(ψ



₎

_.

•

Freeformulaindependence: If

φ

∈

F ree

()

then II C

()

=

II C

(

\ {φ})

.

•

Syntaxweak-independence:

1. forallI C if

|=

I C

↔

I C thenII C

()

=

II C

()

. 2. forall



if



 

_{then II C}

_()

₌

_{II C}

_(



₎

_.

According to thisdefinition, the measure of I C -inconsistency of a communication set is null iff thiscommunication set is not I C -inconsistent. The measure of I C -inconsistency of a communication set does not decrease if we add more communications.Removingareport whichdoesnot causeanycontradictiondoesnotchangethe I C -consistencymeasure ofthecommunicationset.Finally,themeasureof I C -inconsistencyofacommunicationsetdoesnotdependonthesyntax ontheintegrityconstraintsandtwoweaklyequivalentcommunicationsetsgetthesamemeasureofI C -inconsistency.

Thedifferentmeasuresintroducedforsetsofformulascannowberedefinedforcommunicationsetsasfollows:

Definition4.Let



beacommunicationsetandI C anintegrityconstraint.The I C -inconsistencymeasures ID_{I C},IC_{I C},IM_{I C},IP_{I C},

IQ_{I C} andIB_{I C} aredefinedasfollows:

•

ID_{I C}

()

=

0 if



isI C -consistent;1 otherwise.

•

IC I C

()

= | ⊥

I C

|

.

•

IM_{I C}

()

= (|

I C

|

+ |

Self contradictions

()

|)

−

1.

•

IP I C

()

= |

P roblematic

()

|

.

•

IQ_{I C}

()

=

0 if



isI C -consistent;



_{K∈⊥I C|K}1_| otherwise.

•

IB_{I C}

()

=

min

{|

Conf lictbase

(

i

)

|

|

i

|=

3



K∈K

∧

I C

}

.

Proposition1.ID_{I C},IC_{I C},IM_{I C},I_{I C}P,IQ_{I C}andIB_{I C}aresyntaxweak-independentinconsistencymeasures.

(Allproofsaredetailedintheappendixsection,beforereferences)

ID_{I C}

()

is still a trivial measure, while each other measure gives insight on the contribution of reports to inconsis-tency. IC

I C

()

countsthenumberofminimal I C -inconsistentsubsetsofreports. IMI C

()

countsthenumberofthemaximal I C -consistent subsetsofreports together withthe numberof selfcontradictory reportsbut 1to make IM_{I C}

()

=

0 when



is I C -consistent. IP_{I C}

()

countsthenumberofproblematicreports. I_{I C}Q

()

addstheinverseofthesizesoftheminimal

I C -inconsistentsubsetsofreports,sothatsmallerI C -inconsistentsubsetsofreportsareregardedasmoreinconsistentthan largerones. IB

I C

()

returnstheminimumnumberofpropositionalsymbolsthathavetobesettovalue B inordertogeta

threevaluedmodelof



∪

I C .

4. Contributionofanagenttoinconsistency

Inthissection, weaimatcharacterizing thecontributionofan agent totheoverall inconsistencyofacommunication set.

4.1. Contributionfunction

Evaluating thecontribution of an agent to the overall inconsistency ofa communication setshould be relative to an inconsistencymeasure.Aspreviouslystressed,inconsistencymeasuresprovidesdifferentperspectivesoninconsistencyand thecontributionofanagent maythen differw.r.t.some inconsistencymeasure.Severalpossiblesolutionsare possiblefor assessing this contributionbut, asfor the inconsistencymeasure, some constraintsshould also be satisfied. At first, the contributionofanagentshouldbenullifitdoesnotcontributetoanyinconsistency.Second,anagentnotinvolvedinany inconsistencyshould not influencethe assessmentofthecontribution oftheother agents; finally contributionshould be syntaxindependent.Syntax-weak independenceisnot relevanthereasillustratedbythefollowingexample: supposetwo sets



and



 suchthatthey areweaklyequivalent;inthatcontext,itmaybethecasethatsome agenta isproblematic w.r.t.



whileitisnotw.r.t.



.Hencetheircontributionshouldnotbethesame.

Definition5.Considerasetofagents A,a communicationset



on A,anintegrityconstraint I C andan I C -inconsistency

measure II C.FunctionCont,II C isasyntaxindependentcontributionfunctionifitassociatestoanyagenta

∈

A apositive

•

Consistency: Cont,II C

₍

_a

₎

=

_{0 iffa}

_/

∈

_{P roblematic}

₍

_A

₎

_.

•

Freeagentindependence: ifb

/

∈

P roblematic

(

A

)

thenCont,II C

₍

_a

₎

=

_Cont\{K(b)},II C

₍

_a

₎

_.

•

Syntaxindependence:

1. forall I Cif

|=

I C

↔

I CthenCont,II C

₍

_a

₎

=

_Cont,II C 

₍

_a

₎

_.

2. forall



if



≡ 

_thenCont,II C

₍

_a

₎

=

_Cont,II C

₍

_a

₎

_.

4.2. Assessingthecontribution

Hereafter, weadapttwowell knownmetricsinordertodefine thecontributionofanagent tothecommunicationset inconsistency, namely theShapley value andthe Banzhafindex. They are well known measures forassessing the power of an agentin avotingprocedure. Ourcontextis similar:Shapley value enablesto assesthe personal contributionofan agent to overall inconsistency of a communication base; in other words it measures the importance of thisagent in a coalitionalgamedefinedbyfunctionII C [17].TheBanzhafindexassessesthepivotalroleofanagentinthedefinitionofan inconsistentset.ItcorrespondstotheBanzhafscoreasshownin[18].Thesesfunctionsarerespectivelydenoted Cont,II C

s

andCont,II C

b andwheneverit’sclear,wedenotethemContsandContb forshort.TheShapleybasedcontributionfunction

isdefinedasfollows.

Definition6.Considera setofagents A,acommunicationset



on A,anintegrity constraintI C andan I C -inconsistency

measure II C.FunctionContsassociatesanyagenta withapositiverealnumberConts

(

a

)

sothat:

Conts

(

a

)

=



C⊆A C=∅

(

|

C

| −

1

)

!(|

A

| − |

C

|)!

|

A

|!

(

II C

((

C

))

−

II C

((

C

\ {

a

}))).

Thefollowingpropositionconnectstheinconsistencymeasureandthecontributionfunction.Itstatesthatifthe incon-sistencymeasureissyntaxweakindependentthenthecontributionfunctionissyntaxindependent.

Proposition2.FunctionContsisasyntaxindependentcontributionfunctionifII C isasyntax-weakindependentI C -inconsistency measure.

LetusnowconsidertheBanzhafbasedcontribution:

Definition7.Consider asetofagents A,acommunicationset



on A,anintegrityconstraint I C andan I C -inconsistency

measure II C.FunctionContbassociatesanyagenta withapositiverealnumberContb

(

a

)

sothat:

Contb

(

a

)

=| {

C

:

II C

((

C

∪ {

a

})) =

0 and II C

((

C

))

=

0

} | .

As for the Shapleyvalue, function Contb is syntax independent aslong asthe inconsistency measure issyntax weak

independent.

Proposition3.FunctionContbisasyntaxindependentcontributionfunctionifII Cisasyntax-weakindependentI C -inconsistency measure.

4.3. Implementationandcomplexityconsiderations 4.3.1. ImplementationofConts

Algorithm 1implementsfunctionConts.Itsparametersareacommunicationset



,anintegrityconstraintI C andalist ofinconsistencymeasures II Clist.Itshowsthatifweonlyconsiderinconsistencymeasuresbasedonminimalinconsistent subsets, then theShapleyvaluesforall theinconsistencymeasures canbe computedwithoutaddingextra callsto aSAT solver.Itiscomposedoftwosteps.Firststep(Line3)isacalltoFunction InconsistencyMeasures (detailedinAlgorithm 2) which computes,foreach coalition ofagents,the inconsistencymeasures requested in II Clist and storesthe resultsin a tupleofarraysindexedbythecoalitionnumber.Secondstep(Lines4to16)computes,foreach agent,theShapleyvalues for each inconsistency measure by applyingsimple operationson the arrayscomputed atthe previous step. Clearly, the second step isnot computationally hard except that it contains an internal loop on each coalition whichis exponential with respectto thenumberofagents.This second stepis preciselyin O

(

n

∗

2n−1

/

2

)

,n being the numberofagentsand consideringthateachagentisinvolvedinhalfofthepossiblecoalitions.Itdoesn’trequireanycalltoaSATsolver.

Algorithm 2iscomposedofapreparationstepandanexecutionstep.Thepreparationstep(Lines3to5)isexecutedonly ifinconsistencymeasuresbasedonminimalinconsistentsubsetsarerequestedandcomputesthesetofminimalinconsistent subsetsforthecoalitioncomposedofthewholesetofagents(



⊥

I C ).Computingthesetofminimalinconsistentsubsetsof abeliefbaseisahardproblemwhichrequirestouseaSATsolver.Itisshownin[26,12] thatthisproblemisDP-complete.

Algorithm1ContributionfunctionConts.

1: function Conts(,I C,II Clist) 2: n= ||

3: (IC

I C,IPI C,IBI C,Ratio)=InconsistencyMeasures(,I C,II Clist)

4: for j=0 ton−1 do foreachagent

5: for i=1 to2n_{−1 do foreachcoalition}

6: if Bit(j,BinString(i))=1 then 7: switch II Clist do 8: case IC I C 9: ContICI C s [j] =Cont IC I C s [j] +Ratio[i] ∗ (ICI C[i] −ICI C[i−2j]) 10: case IP I C 11: ContII CP s [j] =Cont IP I C s [j] +Ratio[i] ∗ (IPI C[i] −I P I C[i−2 j_]) 12: case IB I C 13: ContIBI C s [j] =Cont IB I C s [j] +Ratio[i] ∗ (IBI C[i] −I B I C[i−2 j_]) 14: end if 15: end for 16: end for 17: return(ContI C I C s ,Cont IP I C s ,Cont IB I C s ) 18: end function

Algorithm2Computinginconsistencymeasuresforeachcoalition.

1: function InconsistencyMeasures(,I C,II Clist) 2: n= || 3: switch II Clist do 4: case IC I Cor IPI Cor I Q I C 5: ⊥I C=Compute⊥I C(,I C)

6: for i=0 to2n_{−1 do foreachcoalition}

7: switch II Clist do 8: case IC I C or IPI Cor I Q I C 9: M I[i] =M I C(i,⊥I C) 10: switch II Clist do 11: case IC I C 12: IC I C[i] = |M I[i]| 13: case IP I C 14: IP I C[i] =ComputeIP(M I[i]) 15: case IB I C 16: IB I C[i] =ComputeIB(Ci) 17: Ratio[i] =(|Ci|−1)!(n−|Ci|)! n! 18: end for 19: return(IC I C,IPI C,IBI C,Ratio) 20: end function

Fortunately, algorithmssuch asthosepublished in[12]make thecomputation ofminimalinconsistentsubsetsfeasiblein real-lifeapplications.Theexecutionstep(Lines6to18)computes,foreachcoalitionofagents,therequestedinconsistency measures plusthe ratiobasedonthe coalitionsize.We usefunctionMIC (detailedlater)whichallows toextract forany coalitionofagentsthecorrespondingsetofminimalinconsistentsubsetsoutof



⊥

I C withoutanymorecalltoaSATsolver. Thanks to thisresultcomputing IC

I C, II CP or I Q

I C isvery easy atthislevel. Onthe other handcomputing IBI C stillrequires

tocompute thethreevaluedmodels ofthecurrentcoalition,which iscomputationallyhard.The executionsub-stepisin

O

(

2n−1

)

andisnotcomputationallyhardifweonlycomputeinconsistencymeasuresbasedonminimalinconsistentsubsets. Note that in [12], the set of maximal consistent subsets is computed before producing the set ofminimal inconsistent subsets. Consequently,inconsistency measure IM

I C can also be computed without extra complexity. It is not the case for

inconsistencymeasure IB_{I C}

In summary, computing Conts is Cpreparationstep

+

Cexecution step

+

Csecond step where Cpreparation step is DP-complete, Cexecution step is in O

(

2n−1

)

and Csecond step is in O

(

n

∗

2n−1

/

2

)

. In [16] the authors show that inthe specific caseusing

inconsistencymeasure IC

(

K

)

, itis possibleto compute the Shapleyvalue in polynomial time.This resultcan directlybe appliedforcomputingConts usinginconsistencymeasure IC_{I C}.

Letusdetailsome hintsused by Algorithm 1.Suppose thereare n agents named0

,

...

n

−

1. Then thereare 2n

possi-blecoalitions ofagents we name0

,

1

...

2n

₋

_{1 with 0 being the emptycoalition and2}n

₋

_{1 thewholeset ofagents.We}

introducetwofunctionsBinString andBit.BinString

(

i

)

returnsabinarystringcorrespondingtothebinaryrepresentation ofcoalition i. Bit

(

j

,

BinString

(

i

))

returnsthe bitof weight2j in BinString

(

i

)

. Thus,agent j isinvolvedincoalition i iff Bit

(

j

,

BinString

(

i

))

=

1.

Letusnowdetailsome hintsusedbyAlgorithm 2.Line5calls functionCompute⊥I C(

,

I C )whichreturns



⊥

I C .As

any



includedin



i.eremoving thereportofsomeagentfrom



doesnotcreatenewinconsistentsubset.Thisremark allows usto drasticallyreducethe complexity forproducingthe minimalinconsistent subsetsofeach coalition.Line9is a call to function M I C

(

i

,



⊥

I C

)

which returnsthe set ofminimal inconsistent subsetsof



which aremade ofreports emitted byagentsincoalitioni,that is

{

X

|

X

⊆ (

Agents

(

i

))

and X

∈  ⊥

I C

}

( Agents

(

i

)

returnstheagentsinvolvedin coalitioni,i.e.agentswhosebitissetto1 in BinString

(

i

)

).

4.3.2. ImplementationofContb

Algorithm 3providesanimplementationoffunctionContb.Ittakesasparameteracommunicationset



andanintegrity

constraint I C . This algorithm is similar to Algorithm 1 and hasthe same computational complexity.Again, most of the complexity is inthe computation ofthe inconsistency measures for each possible coalition.However it is interesting to notice that the Banzhaf-based contribution uses only two inconsistency values,namely

>

0 and

=

0. Consequently, the drasticinconsistencymeasure,whosecomputation isthecheapestone,shouldbepreferredtocomputetheBanzhaf-based contributionoftheagents.

Algorithm3ContributionfunctionContb.

1: function Contb(,I C,II C)

2: n= ||

3: (II C)=InconsistencyMeasures(,I C,II C)

4: for j=0 ton−1 do foreachagent

5: for i

=

1 to2n_{−1 do foreachcoalition}

6: if Bit(j,BinString(i))=1 then

7: if then(II C[i] >0and II C[i

−

2j] =0) 8: ContII C b [j] =Cont II C b [j] +1 9: end if 10: end if 11: end for 12: end for 13: return(ContII C b ) 14: end function 5. Assessingreliability

We nowconsider thequestion ofassessingreliability. Inthefollowing, werepresentreliability asa preorderover the setofagentsandstatementa

≤

b standsforb isatleastasreliableasa.Thefollowingpostulatesaxiomaticallycharacterize anyreliabilitypreorderbasedonwhattheagentshavereportedinthecontextofagivenintegrityconstraint.

Given a set of agents A, an integrity constraint I C and a communication set



,the total preorder representing the relative reliabilityofagentsin A isdenoted

≤

_{I C}A,.Thispreorder ischaracterized bythefollowing postulateswhichshow thatreliabilityshouldberootedininconsistency:

P1

≤

_{I C}A, isatotalpreorderon A. P2 If



≡ 

_then

_≤

A, I C

=≤

A, I C . P3 If

|=

I C

↔

I Cthen

≤

_{I C}A,

=≤

A_{I C}, .

P4 Ifa

/

∈

P roblematic

(

A

)

then

∀

b

,

c

∈

A,ifb

≤

A_{I C}\{a},\{K(a)}c thenb

≤

A_{I C},c. P5 If



is I C -consistentthen

≤

A_{I C}, istheequalitypreorder.

P6 If



is I C -inconsistentthen

∀

a

∈

P roblematic

(

A

)

,

∀

b

/

∈

P roblematic

(

A

)

,a

<

_{I C}A,b. P7 If

{

a1

,

...,

ak

}

∈

A

⊥

I C fork

≥

2,then

∃

i

,

j suchthat j

=

i,ai

<

I CA,aj.

Postulate P1 specifiesthat thereliabilitypreorder isatotalpreorder. P2 and P3 deal withsyntaxindependence. More precisely, ifwe consider two equivalent communication sets orif we considertwo equivalent integrity constraints,then we get the same total preorder on agents. P4 states that reliability is assessed with respect to inconsistency: in other words, agentswhichdo not causeinconsistencyissueshave noinfluence.A typicalexample isan agent whichreports a tautology orwhich reports no information, then it should not haveinfluence on the relative reliability of other agents.

P5, P6 and P7 focuson consistencyofinformationprovidedby agentsin A.Postulate P5 considersthecasewhen



set isnot I C -inconsistent.Insucha case,theagentsare consideredasequallyreliable. P6 and P7 considerthecasewhere



is I C -inconsistent.According to P6,anyagent whichisresponsible ofthe I C inconsistencyof



is considered asstrictly less reliable than anyother agent which is not. According to P7, theagents ofa minimal I C -inconsistent subset cannot be equally reliable: at least one of these agents is strictly less reliable than the others. This is coherent with the way we understandreliability:ifsomeagentsareequallyreliable,then aftermergingwe willbelieve,withthesamestrength, information they willprovide. However, itis generallyassumed[7,23] that graded beliefsatisfies themodal logicaxiom whichstatesthatbeliefshouldbeconsistent:thatis,twocontradictorypieces ofinformationcannotbe believedwiththe samestrength.Consequently,agentswhoareinvolvedinaminimalI C -inconsistentsetcannotbeequallyreliable.

5.1. Buildingreliabilityfromcontribution

It isclear that themore an agent contributes tothe overall inconsistencyof a communicationset,the lessis should bereliable.Thatis,asourceisconsideredstrictlymore(resp,equally)reliablethanasecondoneiffitscontributiontothe globalinconsistencyisstrictlysmallerthan(respequalto)thecontributionoftheother.Noticethatthisprincipleonlytakes careofthefirstsixpostulatesasshownbythefollowingpostulates.

Theorem1.Givenasetofagents A,anintegrityconstraintI C andacommunicationset



,thereliabilitypreorder

≤

_{I C}A,satisfies P1–P6 iffthereexistsasyntaxindependentContributionFunctionCont suchthatforanytwoagentsa andb:

a

≤

A_{I C},b iff Cont

(

a

)



Cont

(

b

).

Thefollowingexampleillustratesthat P7 doesnothold.

Example3.Consider



= {<

a

,

p

>,

<

b

,

¬

p

>,

<

c

,

q

>

}

andI C

= 

.Then

{

a

,

b

}

isaminimal I C -conflicting setofagents butShapley-basedContributionFunctiongives:Cont,II C

s

(

a

)

=

Conts,II C

(

b

)

andconsequentlya

=

b whichviolates P7.

Thisexampleshowsthatatie-breakingruleismissing:ifallagentsinvolvedinthedefinitionofaminimalinconsistent subsethaveanequalcontributionthenone ofthemshould stillbeconsideredaslessreliable.Forvotingrules, aclassical waytohandletie-breakingistoconsideran additionallexicographic orderoverthesetofagentsinordertoalwaysgeta winner[2].Inourcontext,thecontributionfunctionshould preventthecaseswhereallagent involvedinaninconsistent coalitionhavesimilarcontribution.Thefollowingconstrainttranslatesthisprincipleinformalterms:

Definition8.LetCont beasyntaxindependentContributionFunction.Cont issaidtobetie-free ifitsatisfiesthefollowing constraint:

∀

C

∈

A

⊥

I C

,

∃

a

,

b

∈

C

,

s.t. Cont

(

a

)

=

Cont

(

b

).

Theimmediatequestionis“doessuch functionexist?”.Togettheanswer,letusrevisitourShapleyandBanzhaf based contributionfunctionsandbuiltontopofthematie-free function.Theideaissimilartie-breaking:foranyinconsistentset ofagents,one agentcontributionisslightlyincreasedsothatthereisnomoreinconsistentsetofagentswhereallagents havethesamecontribution.Aswehavenoinformationabouttheagentsinvolvedinthecommunicationset,themodified contributionmaybechosen inanarbitraryway.Hereafterwejustconsideralexicographic orderforthischoice(asinthe votingrules).

Definition9.LetCont beaContribution functiondefinedeitherby Definitions 6 and7.FunctionContt isthendefinedas follows:

•

IfP roblematic

(

A

)

= ∅

thenContt

=

Cont.

•

Otherwise,let



beapositiverealnumberwhichisstrictlysmallerthanthesmallestdifferencebetweentwo contribu-tions:0

<



<

min

(

{

Cont

(

a

)

−

Cont

(

b

)

|

Cont

(

a

)



Cont

(

b

)

})

.

1. Let Stie bethesetofagentswithequalcontributiont:

Stie

= {

a

| ∃

C

∈

A

⊥

I C such that

|

C

| >

1 and a

∈

C and

∀

x

,

y

∈

C

,

Cont

(

x

)

=

Cont

(

y

)

}.

Let S be the minimal subset of Stie w.r.t. the lexicographic order such that (i)

∀

C

∈

A

⊥

I C ,

∃

a

∈

S

∩

C and (ii)

∀

C

∈

A

⊥

I C ,C

\

S

= ∅

.SetContt

(

a

)

=

Cont

(

a

)

+



foranya

∈

S.

2. Contt

(

b

)

=

Cont

(

b

)

foranyagentb

/

∈

S.

Thefollowingpropositionshowsthatthisfunctionisatie-free one.

Proposition4.Conttis tie-free.

WehaveprovedthatthereexistContributionfunctionswhicharetie-free,wecanrephrasetheprevioustheoremsothat

P7 holds.

Theorem2.GivenasetofagentsA,anintegrityconstraintI C andacommunicationset



,thereliabilitypreorder

≤

_{I C}A,satisfies P1–P7 iffthereexistsa syntaxindependentContributionFunctionCont whichis tie-freesuchthatforanytwoagentsa andb:

Example4.Takeagain A

= {

a

,

b

}

andK

(

a

)

=

p, K

(

b

)

= ¬

p and I C

= 

.

{

a

,

b

}

∈

A

⊥

I C and,accordingtotheShapleybased Contribution function,itholdsthatConts

(

a

)

=

Conts

(

b

)

.Letusconsiderthetie-freeContribution functionbasedonConts.

Accordingtothelexicographicagenta then,Contt

(

a

) >

Contt

(

b

)

andweget:a

<

b.

This second theorem shows atfirst that there exists a reliability preorder which satisfy all thepostulates. Second, it shows that oneof thekey issue inthe inconsistencydefinitionof reliabilityis Postulate P7.It forcesto rankthe agents involved ina minimal I C -inconsistentsubset. Wepropose to handlethisissueby constraining thecontributionfunction. Remind that all agents are unknown andwe do not have extra information for setting the choice. Hence thisfunction may leadto arbitrarychoices. However, evenifwe do notknow anyextrainformation,we haveseen that we canbuild up a reliabilitypreorder by first assessing the overall inconsistencyof a communicationset; secondly by computing the “responsibility”degreeorcontributionofeachagentintheoverallinconsistencyandthirdlybyrootingthereliabilityofthe agentsintotheircontributions.

6. Reliabilityaggregation

In the previous section, we haveshown that assessing reliabilityof agentsmay be achievedby choosing a particular inconsistencymeasureandaparticularcontributionfunction,eachspecificpair



measure,contribution



assessingthesources ina differentway.Thechoiceofaspecific pairisachallengesincewe havenoinformationaboutthesources.Asolution maybetoassessreliabilitywithrespecttoseveralpairsandtomergealltheresultingpreorderstoobtaina“consolidated” one. Theimmediatequestion isthen:ifeach ofthereliabilitypreorders satisfiespostulates P1–P7, canwethenobtainan aggregatedpreorderwhichalsosatisfiesthesepostulates?Theaimofthissectionistotacklethisissue.

Thisproblemisclearly connectedtothefield ofpreferencesaggregation.Arrowimpossibilitytheorem[3]statesthatit isnotpossibletoaggregatepreferenceswhileguaranteeingUniversality,Non-Dictatorship,UnanimityandIndependenceto IrrelevantAlternatives.ThequestionisthentoevaluatehowArrow’sconditionsinterplaywiththereliabilitypostulates.

6.1. Arrow’sconditions

Inthefollowing,wefirstdefineouraggregationoperatorandnextrevisittheArrow’sconditions.Let

⊕

beour aggrega-tionoperatordefinedasfollows:

Definition10.An-aryreliabilityaggregationoperator

⊕

isafunctionwhichassociatesn totalpreorderson A,respectively

≤

1

,

...,

≤

n,withatotalpreorderon A denoted

≤

⊕.

Next,werephrasetheclassicalArrow’sconditions:

Definition11.Let

⊕

bean-aryreliabilityaggregationoperator.Weconsiderthefollowingproperties:

Universality thedomainof

⊕

isthesetofallpossiblen-tupleofreliabilitytotalpreorders.

Non-dictatorship



i

∈ {

1

...

n

}

suchthat

∀

≤

1

...

≤

n totalpreorderson A,

≤

⊕

=≤

i. Unanimity Leta

∈

A andb

∈

A.If

∀

i

∈ {

1

...

n

}

a

≤

ib thena

≤

⊕b.

Independence of irrelevant alternatives (IIA) Leta andb betwoagents.Let

(

≤

1

,

...,

≤

n

)

and

(

≤

1

,

...,

≤

n

)

betwosetsofn

totalpreorders.If

∀

i

∈ {

1

...

n

},

(

a

≤

ib

⇔

a

≤

ib

)

then

(

a

≤

⊕b

⇔

a

≤

⊕b

)

. 6.2. Influenceofarrow’sconditions

ThefollowingpropositionsexhibittheinterplaybetweenArrow’sconditionsandthesevenpostulatescharacterizingthe consistency-basedreliabilityassessment.Namely,assumingthateach

≤

i satisfiesourpostulates,whichArrow’scondition(s)

arerequiredsothatthepostulatesarealsosatisfiedbytheresultingpreorder.

Propositions 5to11allusethefollowingelements:let



beacommunicationseton A and I C anintegrityconstraint; let Ii

I C

(

i

=

1

...

n

)

ben syntaxweak-independent I C -inconsistencymeasures.Forthesakeofconciseness,

≤

i(i

=

1

...

n)stands

for

≤

_{I C}A, ithreliabilitypreorder.

Proposition5(P1⊕).If

⊕

satisfiestheconditionofUniversalitythen

Ifall

≤

isatisfiesP1,then

≤

⊕satisfiesP1(i.e.,

≤

_⊕isatotalpreorder).

Thispropositionstatesthatiftheinputisitselfatupleoftotalpreorders Universality willguaranteethatthe aggregation operator

⊕

willreturnatotalpreorder.

Proposition6(P2⊕).Ifall

≤

isatisfyP2then

≤

⊕satisfiesP2(i.e.,if



≡ 

_then

_≤

_⊕

_=≤

 ⊕).

Table 1

InterplaybetweenpostulatesandArrow’sconditions. Sufficient conditions Satisfied postulates Universality {P1}

Dictatorship {P7}

Unanimity {P5, P6}

IIA {P4}

ThispropositionisanimmediateconsequenceofPostulate P2:ifeachreliabilityassessmentoperatorissyntax indepen-dent(w.r..t.communicationset)thentheresultingpreorderisalsosyntaxindependent.

Proposition7(P3⊕).Ifall

≤

isatisfyP3then

≤

⊕satisfiesP3(i.e.,foranyI Cs.t.

|=

I C

↔

I C,

≤

_⊕

=≤

_⊕).

ThispropositionisaconsequenceofPostulate P3:ifeach

≤

i issyntaxindependentw.r.t.someintegrityconstraintthen

theresultingpreorderisalsosyntaxindependent.NoticethatthelasttwopropositionsdonotinvolveanyArrowconditions asopposedtotheotherones.

Proposition8(P4⊕).If

⊕

satisfiestheconditionofIndependenceofIrrelevantAlternativesthen

Ifall

≤

isatisfyP4then

≤

⊕satisfiesP4(i.e.,if

≤

−adenotesthepreorder

≤

I CA\{a},\K(a),ifa

/

∈

P roblematic

(

A

)

,then

∀

b

,

c

∈

A,if b

≤

−_⊕ac thenb

≤

_⊕c).

Thispropositionstatesthat P4 (noinfluenceofagenta whoonlyreportstautologies)canonlybepreservedif IIA holds. Thatis,ifall preorders areunchangedafterexcluding agent a, thenaggregationalso producesa similar resultif IIA also holds.

Proposition9(P5⊕).If

⊕

satisfiestheconditionofUnanimitythen

Ifall

≤

isatisfyP5then

≤

⊕satisfiesP5(i.e.,if



isI C -consistentthen

≤

⊕istheequalitypreorder).

Postulate P5 statesthatconsistencyleadstoaflatordering.Consequently,ifthereisnoconflictamongagents,every

≤

i

hastobeflatandtheaggregationproducesaflatorderaslongas Unanimity holdsfortheoperator

⊕

.

Proposition10(P6⊕).If

⊕

satisfiestheconditionofUnanimitythen

Ifall

≤

isatisfyP6then

≤

⊕satisfiesP6(i.e.,if



isI C -inconsistentthen

∀

a

∈

P roblematic

(

A

)

,

∀

b

/

∈

P roblematic

(

A

)

,a

<

⊕b).

Proposition 10stressesup asecond timethekey roleof Unanimity condition.IfPostulate P6 holds forevery

≤

i then

preorders

≤

i unanimously statesthat for anya

∈

P roblematic

(

A

)

andb

∈

A

\

P roblematic

(

A

)

, b ismore reliable than a. a

<

⊕b alsoholdsonlyif Unanimity holds.

Proposition11(P7⊕).If

⊕

doesnotsatisfytheconditionofNon-dictatorship then

Ifall

≤

isatisfyP7then

≤

⊕satisfiesP7(i.e.,if

{

a1

,

...,

ak

}

∈

A

⊥

I C fork

≥

2,then

∃

i

,

j suchthatj

=

i andai

<

⊕aj).

Thispropositionshowstheroleofthe Non-dictatorship condition.Postulate P7 statesthatatleastoneagentinvolved inaconflictingsetofagentsshouldberankedwithalowerreliability.Ifeach

≤

ihavedecreasedthereliabilityofadifferent

agentthen,ifthe Non-dictatorship propertyholds,noagentwillbedecreasedbytheaggregationprocedure

⊕

.Hence,the constraintenforcesby P7 willholdonlyifthe Non-dictatorship propertyisnotsatisfied,i.e. thereisadictator.

Tosumup,asshownby Table 1fourArrow’sconditionshaveaninfluenceon thefulfillmentofthepostulates. Notice thattheseconditionsaresufficientconditionsforsatisfyingthepostulates.

WiththehelpofTable 1,wearefullyinformedonwhatpostulatesorArrow’sconditionsoneshouldgiveupduringthe definitionofan aggregationoperator. Thetableshowsthatreliabilityaggregationisdifferentfrompreferenceaggregation: thepostulatearenotcompatiblewithArrow’sconditions.Italsomeansthatsomeunderlyingpriorityshouldbeconsidered inthedefinitionoftheoperator:ifwegiveprioritytothefulfillmentofthepostulatesthenfulfillmentofArrow’sconditions isnotthatimportant.Inthenextsection,we illustratethisissuebyconsideringasimpleaggregationproceduresatisfying allthepostulatesbutabandoningoneArrow’scondition.

6.3. Lexicographic-basedaggregation

Theproposedaggregationprocedureisalexicographicprocedure[14],widelyusedinpreferenceaggregationandbelief merging.In[2],theauthorsshowhowpreferencerelationscanbeaggregated.Firstconsiderahierarchyamongn preorders

(with no looseof generality,

≤

1 is the most importantwhile

≤

n is the leastimportant). According to thishierarchy, if

all k first preorders statea

=

kb andatk

+

1, a

<

k+1b thena will be considered aslessreliablethan b in theresulting

preorder.Nodivergenceentailsthata andb areconsideredasequalintheresultingpreorder.Let

⊕

L _{denote thereliability}

aggregationoperatordefinedasfollows:

Definition12.If

≤

1

,

...

≤

n aren reliabilitypreorders on a setofagents A and if

≤

L denotes thepreorder

⊕

L

(

≤

1

...

≤

n

)

,

then

≤

L _{isdefinedby:}

•

a

<

L_{b iff}

_∃

_k

₌

₁

_...

_n

_∀

_j

_{∈ {}

₁

_...

_k

₋

₁

_}

_a

₌

jb anda

<

kb.

•

a

=

L_belse.

Example5.Assumethefollowingtotalpreordersdefinedoverthesetofagents A

= {

a

,

b

,

c

,

d

,

e

}

. a

=

1b

<

1c

<

1d

=

1e

.

a

<

2b

=

2c

<

2d

=

2e

.

e

=

3b

<

c

=

3d

=

3a

.

Weobtainthefollowingaggregatedpreordera

<

L_bL_c

_<

L_e

_<

L_{d.Forinstance,a}

_<

L_{b holdsbecausea}

₌

1b anda

<

2b.

ItiswellknownthatamongthefourArrow’sconditions, Non-dictatorship doesnotholdforLexicographicaggregation (see[2]).I.e.,

Theorem3.[14]

⊕

L_{satisfiesUniversality,Unanimity,IndependenceofIrrelevantAlternativesanddoesnotsatisfy} Non-dictator-ship.

Corollary1.Ifallpreorders

<

isatisfyPostulatesP1–P7,thenPreorder

<

LalsosatisfiesPostulatesP1–P7.

Itmeansthatthisreliabilityaggregationoperatorisagoodcandidatetoaggregatereliabilitypreordersaslongasthese individualpreordersalsosatisfy P1–P7.

Example6.Consideragenta whoseaimistoassesstherelativereliabilityofthreecommunicatingagentsb

,

c andd.With thespecificpair

<

measure

,

contribution

>

itchose,a getsthepreorder:b

≺

sc

=

sd,i.e.,b hastobeconsideredasstrictly

lessreliablethanc andd whohavetobeconsideredasequallyreliable.Assumethatbeforeapplyingthismethod,a hadan aprioriaboutthethreeagentsandthoughtthatd wasmorereliablethanthetwootherswhichwereequallyreliable,i.e.,

b

=

ac

≺

ad (where

≺

a denotestheaprioripreorder).Thisaprioriinformationcanbeusedbya todecideweather c ord

istheleastreliable.Moreprecisely,byaggregating thetwopreordersandbygivingpriorityto

≺

s,a gets:b

≺

Lc

≺

Ld. 6.4. Avoidingdictatorshipcondition

Ifwewanttoconsidereachinconsistencymeasureinanequalway,weshouldavoidthedictatorshipconditionand in-steadgotowardsvotingrulesforthedefinitionofanon-dictator reliabilityaggregationoperator.Animmediateconsequence is thatpostulate P7 will notbe satisfiedinthe aggregatedpreorder. Anotherconsequenceisthat Arrow’stheoremforces usto give up an other condition sincewe wantthe non-dictatorship condition tobe satisfied.It meansthat some other postulateswillalsohavetobeabandoneddependingontheconditionthatwillbegivenup(IIA or Unanimity).

Let usconsider theclassical preferential voting rules such asCondorcet, Borda andCopeland[13],which enforce the non-dictatorship condition. Allthese rules usually considerstrict and totalpreorders as input andprovidea preorder as output. In our context,we have to considera variant whereinput is a set ofpreorders that may not be strict. We also consider that the aggregation method should take into account the relative position of each agent (w.r.t. other agents) inallpreorders:agenta maybe poorlyrankedaccordingtosome inconsistencymeasurewhileitmayobtainagoodrank accordingtosomeotherinconsistencymeasure.Hencetherankofanagentshouldbebalancedbyitspositionintheoverall preorder.Itmeansthatacountingbasedprocedureismoreadaptedthanapairwisecomparisonmethod,whicharguesfor aBordabasedaggregationprocedureinsteadofanaggregationprocedurebasedonCondorcetorCopeland.

The ruleproposed hereafter rephrases theBorda-basedrule initiallyproposed in [13,Chapter 13]whichhandles total preorders (that maybestrict).As mentioned,we want totake intoaccount therelative positionin thepreorder (‘’good” or‘’poor”position).Consequently,foranyagenta andpreorder

≤

i,therulecomputesascorewhichtakesintoaccountits