A FRAMEWORK FOR QUALITATIVE MULTI-CRITERIA

PREFERENCES

Wietske Visser, Reyhan Aydo˘gan, Koen V. Hindriks and Catholijn M. Jonker

Man-Machine Interaction Group, Delft University of Technology, Delft, The Netherlands

Keywords:

Qualitative multi-criteria preferences.

Abstract:

A key challenge in the representation of qualitative, multi-criteria preferences is to ﬁnd a compact and ex-

pressive representation. Various frameworks have been introduced, each of which with its own distinguishing

features. In this paper we introduce a new representation framework called qualitative preference systems

(QPS), which combines priority, cardinality and conditional preferences. Moreover, the framework incor-

porates knowledge that serves two purposes: to impose (hard) constraints, but also to deﬁne new (abstract)

concepts. In short, QPS offers a rich and practical representation for qualitative, multi-criteria preferences.

1 INTRODUCTION

A key challenge in the representation of qualitative,

multi-criteria preferences is to ﬁnd a compact and at

the same time expressive representation. A frame-

work for preference representation provides an ad-

equate tool if it is sufﬁciently expressive to com-

pactly represent a broad range of preference order-

ings. To this end, various frameworks have been in-

troduced, each of which with its own distinguishing

features. For example, in lexicographic approaches

(e.g. (Andr´eka et al., 2002)) preference over out-

comes is determined by combining multiple criteria

according to priority. Goal-based approaches (e.g.

(Brewka, 2004)) use cardinality and compare alter-

natives by the number of goals they satisfy. CP-nets

(Boutilier et al., 2004) are well-known for their abil-

ity to represent conditional preferences. A CP-net is

a qualitative graphical representation of preferences

that reﬂects conditional preference statements under

a ceteris paribus (all else being equal) interpretation.

(Brafman and Domshlak, 2002) extend CP-nets to so-

called TCP-nets which also allow for expressing rel-

ative importance of preference variables. In general,

however, (T)CP-nets are not able to represent lexico-

graphic orderings (Wilson, 2004).

In this paper we introduce a rich and practical

new representation framework for qualitative multi-

criteria preferences called qualitative preference sys-

tems (QPS). This framework enables preference rep-

resentation by using priority, cardinality and condi-

tional preferences. Moreover, the framework incor-

porates knowledge that serves two purposes: as usual,

knowledge can be used to impose (hard) constraints,

but also to deﬁne new (abstract) concepts. To il-

lustrate, it can represent facts about the world (e.g.

Barcelona is in Spain), the feasibility of options (e.g.

hotel X is fully booked in July), and deﬁnitions (e.g.

the cost of a holiday is the sum of the costs of the

ﬂight, hotel and food).

QPSs are based on the lexicographic rule studied

in (Andr´eka et al., 2002). This rule is a fundamental

part of the framework presented as it offers a princi-

pled tool for combining basic preferences. We believe

this ability to combine preferences is essential for any

practical approach to representing qualitative prefer-

ences. It is needed in particular for constructing multi-

criteria preferences. It is not sufﬁcient, however,

since more expressivity is needed and useful in prac-

tice. Therefore, QPSs in addition provide a tool for

representing knowledge, for abstraction, for counting,

and provide a layered structure for representing pref-

erence orderings. We show that QPSs are able to rep-

resent various strategies for deﬁning preference order-

ings, and are able to handle conditional preferences.

To be precise, we show in Section 3 that Logical Pref-

erence Description language (LPD; (Brewka, 2004))

can be embedded into the QPS framework and that

there is an order preserving embedding of CP-nets in

the QPS framework. In addition we consider the key

issue of compact preference representation and show

that these embeddings provide a representation that is

just as succinct as the LPD expressions and CP-nets.

243

Visser W., Aydo

gan R., V. Hindriks K. and M. Jonker C..

A FRAMEWORK FOR QUALITATIVE MULTI-CRITERIA PREFERENCES.

DOI: 10.5220/0003718302430248

In Proceedings of the 4th International Conference on Agents and Artiﬁcial Intelligence (ICAART-2012), pages 243-248

ISBN: 978-989-8425-95-9

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

2 QPS

The main aim of a QPS is to determine preferences

between outcomes in a purely qualitative way. An

outcome is an assignment of values to a set of rel-

evant variables. Every variable has its own domain

of possible values. Constraints on the assignments of

values to variables are expressed in a knowledge base.

Outcomes are deﬁned as variable assignments that re-

spect the constraints in the knowledge base.

The preferences between outcomes are based on

multiple criteria. Every criterion can be seen as a rea-

son for preference, or as a preference from one par-

ticular perspective. We distinguish between simple

criteria that are based on a single variable and com-

pound criteria that combine multiple criteria in order

to determine an overall preference.

Deﬁnition 1 (Qualitative Preference System). A

qualitative preference system (QPS) is a tuple

⟨Var, Dom, K, C⟩. Var is a ﬁnite set of variables. Every

variable X ∈ Var has a domain Dom(X) of possible

values. K (a knowledge base) is a set of constraints

on the assignments of values to the variables in Var. A

constraint is an equation of the form X = Expr where

X ∈ Var is a variable and Expr is an algebraic ex-

pression that maps to Dom(X). An outcome

is an

assignment of a value x ∈ Dom(X) to every variable

X ∈ Var, such that no constraints in K are violated.

denotes the value of variable X in outcome

. C

is a ﬁnite rooted tree of criteria, where leaf nodes are

simple criteria and other nodes are compound crite-

ria. Child nodes of a compound criterion are called

its subcriteria. Weak preference between outcomes by

a criterion c is denoted by the relation ⪰

. ≻

denotes

the strict subrelation, ≈

the indifference subrelation.

Example 1. When comparing holidays, some

variables could be Destination, NeverBeenThere

and Cost, with Dom(Destination) = {Barcelona,

Rome, NewYork}, Dom(NeverBeenThere) = {⊺, },

Dom(Cost) = Z

. The deﬁnition of concepts (e.g.

the cost of a holiday is the sum of the costs of

the ﬂight, hotel and food) can be straightforwardly

represented with the following constraint: Cost =

FlightC+HotelC+ FoodC. Equational constraints are

also sufﬁciently expressive to model different kinds

of knowledge. For example, suppose I want to ex-

press that I have never been to Barcelona, i.e. in all

outcomes where Destination = Barcelona, we should

have NeverBeenThere = ⊺. To do this, we ﬁrst intro-

duce an auxiliary variable B with Dom(B) = {⊺, }.

Then we add B = (Destination = Barcelona) and B =

B∧NeverBeenThere to the constraint base K. This en-

sures that there are no outcomes where Destination =

Barcelona and NeverBeenThere = .

2.1 Simple Criteria

A simple criterion speciﬁes a preference ordering on

the values of a single variable. Its preference between

outcomes is based solely on the value of this variable

in the considered outcomes.

Deﬁnition 2 (Simple Criterion). A simple criterion c

is a tuple ⟨X

, u

⟩, where X

∈ Var is a variable, and

, a preference relation on the possible values of X

is a preorder on Dom(X

). ⋗

is the strict subrelation,

≐

is the indifference subrelation. We call c a Boolean

simple criterion if X

is Boolean and ⊺ ⋗

. A sim-

ple criterion c = ⟨X

, u

⟩ weakly prefers an outcome

over an outcome

, denoted

⪰

, iff

Example 2. A criterion ‘economy’ can be deﬁned as

⟨Cost, u⟩ where for all x, x

′

∈ Dom(Cost), x u x

′

iff

x ≤ x

′

. An example of a Boolean criterion is ‘explo-

ration’, deﬁned as ⟨NeverBeenThere, {(⊺, )}⟩. The

criterion ‘economy’ prefers any holiday with a lower

cost over any holiday with a higher cost, irrespective

of the values of other variables.

Observation 1. Let c = ⟨X

, u

⟩ be a simple criterion.

Then ⪰

is a preorder. If u

is total, then so is ⪰

2.2 Compound Criteria

Qualitative preference systems offer two ways in

which to combine multiple criteria: lexicographic cri-

teria and cardinality criteria. In a lexicographic crite-

rion, preference is determined by the subcriteria with

the highest priority; lower priority subcriteria only in-

ﬂuence the preference if the higher priority subcriteria

are indifferent. In a cardinality criterion, all subcri-

teria have the same priority, and preference is deter-

mined by a kind of voting mechanism that counts the

number of subcriteria that support a certain preference

and those that do not.

2.2.1 Lexicographic Criteria

A lexicographic criterion consists of a set of subcri-

teria and an associated priority order (a strict partial

order, which means that no two subcriteria can have

the same priority). It weakly prefers outcome

over

outcome

if for every subcriterion, either this sub-

criterion weakly prefers

over

, or there is another

subcriterion with a higher priority that strictly prefers

over

. This deﬁnition of preference by a lexico-

graphic criterion is equivalent to the priority operator

as deﬁned by (Andr´eka et al., 2002). It generalizes the

familiar rule used for alphabetic ordering of words,

such that the priority can be any partial order and the

combined preference relations can be any preorder.

ICAART 2012 - International Conference on Agents and Artificial Intelligence

244

Deﬁnition 3 (Lexicographic Criterion). A lexico-

graphic criterion c is a tuple ⟨C

, ⊳

⟩, where C

is a

nonempty set of criteria (the subcriteria of c) and ⊳

, a

priority relation among subcriteria, is a strict partial

order (a transitive and asymmetric relation) on C

. A

lexicographic criterion c = ⟨C

, ⊳

⟩ weakly prefers an

outcome

over an outcome

, denoted

⪰

, iff

∀s ∈ C

(

⪰

∨∃s

′

∈ C

(

≻

′

∧s

′

⊳

s)).

Example 3. Consider a lexicographic criterion c =

⟨{s

, s

}, {(s

, s

)}⟩ where s

is the ‘exploration’ cri-

terion and s

the ‘economy’ criterion from Example

2. Consider three outcomes such that

Destination

Rome,

Cost

= 500, and

NeverBeenThere

= ;

Destination

= Barcelona,

Cost

= 350, and

NeverBeenThere

= ⊺; and

Destination

= NewYork,

Cost

= 700, and

NeverBeenThere

= ⊺. Then we have

≻

. Note that even though

is cheaper than

and hence preferred by criterion

, criterion c prefers

because subcriterion s

has higher priority than s

and s

prefers

Proposition 1. Let c = ⟨C

, ⊳

⟩ be a lexicographic cri-

terion. If for all subcriteria s ∈C

, ⪰

is a preorder,then

the relation ⪰

is also a preorder.

Proof. Preservation of reﬂexivity follows directly

from the deﬁnition of ⪰

(if all subcriteria are reﬂex-

ive, then for every outcome

: ∀s ∈ C

(

⪰

) and

hence

⪰

). Preservation of transitivity has been

proven by (Andr´eka et al., 2002).

2.2.2 Cardinality Criteria

Like a lexicographic criterion, a cardinality criterion

combines multiple criteria into one preference order-

ing. Unlike a lexicographiccriterion, priority between

subcriteria is not a strict partial order, but all subcri-

teria have the same priority. A cardinality criterion

weakly prefers an outcome

over an outcome

if it

has at least as many subcriteria that strictly prefer

over

as criteria that do not weakly prefer

over

Deﬁnition 4 (Cardinality Criterion). A cardinality

criterion c is a tuple ⟨C

⟩ where C

is a nonempty set

of criteria (the subcriteria of c). A cardinality crite-

rion c = ⟨C

⟩ weakly prefers an outcome

over an

outcome

, denoted

⪰

, iff ∣{s ∈ C

∣

≻

}∣ ≥

∣{s ∈ C

∣

/⪰

}∣.

Example 4. Consider a cardinality criterion c =

⟨{s

, s

}⟩ where s

is the ‘exploration’ criterion and

the ‘economy’ criterion from Example 2. For

the three outcomes speciﬁed in Example 3, we have

≻

≈

Unfortunately, transitivity of ⪰

is not guaranteed

for just any set of subcriteria. For example, consider

three outcomes

and three subcriteria s

, s

such that

≻

, and

≻

Then

would be strictly preferred over

strictly

preferred over

and

strictly preferred over

, so the

preference would not be transitive. However, there

are some conditions under which transitivity can be

guaranteed. E.g. if every subcriterion is a Boolean

simple criterion s= ⟨X

, u

⟩, they all induce a total pre-

order of preference that stratisﬁes the outcome space

into two levels: the outcomes where X

= ⊺ are more

preferred and the outcomes where X

=  are less pre-

ferred. This also means that

≻

iff

= ⊺ and

= ; and

/⪰

iff

=  and

= ⊺. So in this

case the deﬁnition preference by a priority class com-

pares the number of ‘goals’ X

that

satisﬁes to the

number of goals that

satisﬁes, just as is done by e.g.

the # strategy of (Brewka, 2004).

Proposition 2. Let c = ⟨C

⟩ be a cardinality criterion

such that for all s∈C

, s is a Boolean simple criterion.

Then ⪰

is a preorder.

Proof. Since all subcriteria of c are reﬂexive (Obser-

vation 1), for any outcome

both ∣{s ∈ C

∣

≻

}∣

and ∣{s ∈ C

∣

/⪰

}∣ are 0, so

⪰

, hence ⪰

is re-

ﬂexive. Since all subcriteria are Boolean simple cri-

teria,

⪰

iff ∣{s = ⟨X

, u

⟩ ∈ C

∣

= ⊺}∣ ≥ ∣{s =

⟨X

, u

⟩ ∈ C

∣

= ⊺}∣. This is just a comparison be-

tween two integers, and hence is transitive.

(Andr´eka et al., 2002) showed that the only opera-

tor to combine any arbitrary preference relations that

satisﬁes the desired properties IBUT (independence

of irrelevant alternatives, based on preferences only,

unanimity with abstentions, and preservation of tran-

sitivity) is the priority operator, which assumes that

priority is a partial order. We observe here that if

only Boolean preference relations (such as those re-

sulting from Boolean simple criteria) are combined,

the cardinality-based rule, in which all combined rela-

tions have equal priority, can also be applied. Requir-

ing antisymmetry in this case would unneccessarily

restrict the expressivity.

2.3 Conditional Preferences

A QPS can be used to express conditional prefer-

ences, i.e. preferences between values of one variable

that depend on the values of other variables.

Example 5. If Anne goes on a holiday to Barcelona

(b), she would like to go together with her friend Juan

(j), but if she goes to Rome (r), she prefers to go

with Mario (m). To express this conditional prefer-

ence in a QPS, we use an auxiliary variable L, whose

domain consists of all combinations of the variables

D (destination) and C (company), i.e. Dom(L) =

A FRAMEWORK FOR QUALITATIVE MULTI-CRITERIA PREFERENCES

245

{(b, j), (b, m), (r, j), (r, m)}. To keep the outcomes

consistent, the constraint L = (D, C) is added to the

knowledge base. Finally, the following simple crite-

rion expresses the conditional preference: c = ⟨L, u

⟩

where u

= {((b, j), (b, m)), ((r, m), (r, j))}.

Instead of representing this kind of preference as

conditional preferences on the values of variables, it

would be more natural to model the underlying rea-

son for the conditional preference, as was argued in

(Visser et al., 2011). This is possible in a QPS, but

outside the scope of this paper.

3 COMPARISON WITH OTHER

FRAMEWORKS

3.1 LPD

(Brewka, 2004) presents a rank-based description lan-

guage for qualitative preferences called logical pref-

erence description language (LPD). The basic expres-

sions of LPD are called basic preference descriptions

which are pairs ⟨s, R⟩ with s one of the strategy identi-

ﬁers ⊺,

, ⊆, # and R a ranked knowledge base (RKB).

An RKB is a set F of propositional formulas together

with a total preorder ≥ on F, representing the relative

importance of the formulas. Alternatively, an RKB

can be represented as a set of ranked formulas ( f, i)

where f is a propositional formula and i, the rank

of f , is a non-negative integer such that f

≥ f

iff

rank( f

) ≥ rank( f

The four strategy identiﬁers refer to different

strategies to obtain preferences over outcomes from

an RKB. Outcomes in this context are propositional

models, i.e. the variables used are Boolean. ⊆ prefers

over

if there is a rank where

satisﬁes a superset

of the formulas that

satisﬁes, and

and

satisfy

the same more important formulas. # prefers

over

if there is a rank where

satisﬁes more formulas

than

, and for all more important ranks,

and

satisfy the same number of formulas. Since (Brewka,

2004) shows that basic preference descriptions ⟨⊺, R⟩

and ⟨

, R⟩ can be transformed into equivalent basic

preference descriptions of the form ⟨⊆, R

′

⟩, we do not

discuss these strategies here.

Theorem 1. There is a QPS ⟨Var, Dom, K, C⟩ with a

criterion c ∈ C that corresponds to a basic preference

description ⟨s, R⟩ for s = # or s =⊆ such that

≥

iff

⪰

for arbitrary outcomes

Proof. A basic preference description ⟨s, R⟩ can be

translated into a QPS ⟨Var, Dom, K, C⟩. Let R = ⟨F, ≥⟩

be an RKB. The propositional variables used in F are

collected in Var; moreover, for each formula f ∈ F a

new variable X

is added to Var and X

= f is added

to the knowledge base K. Clearly, Dom(X) = {⊺, }

for all X ∈ Var. For every formula f ∈ F, a Boolean

simple criterion on the associated variable is deﬁned:

= ⟨X

, {(⊺, )}⟩. If s =⊆, preference of ⟨s, R⟩ is cap-

tured by a lexicographiccriterion c= ⟨C

, ⊳

⟩ such that

= {c

∣ f ∈ F} and c

⊳ c

′

iff f > f

′

. Note that

Boolean criteria that correspond to formulas with the

same rank are incomparable according to the criterion

c. This ensures that an outcome

can only be pre-

ferred to an outcome

on some rank, if there is no cri-

terion that strictly prefers

over

, i.e. there is no for-

mula that

satisﬁes but

does not. This means that

satisﬁes a superset of the formulas that

satisﬁes,

which is the deﬁnition of preference by the ⊆ strategy.

If s = #, for every rank i in R, a cardinality criterion is

deﬁned with as subcriteria all simple criteria associ-

ated to a formula of that rank: c

= ⟨{c

∣ ( f, i) ∈ R}⟩.

The preference of ⟨s, R⟩ is captured by a lexicographic

criterion c= ⟨C

, ⊳

⟩ such thatC

= {c

∣ ( f

′

, i) ∈ R} and

⊳ c

′

iff i > i

′

. This way, a subcriterion of c corre-

sponds with a rank in the RKB R. Now note that how

preferences are induced by c and its subcriteria cor-

responds with how the strategy # induces preferences

over outcomes.

Note that it follows from Theorem 1 that the QPS

corresponding to a basic preference description is just

as succinct as this description. That is, the size of the

QPS is comparable to that of the LPD description (the

size differs at most by a constant factor).

In LPD, complex preference descriptions can be

built from basic ones with the connectives ∧, ∨, > and

−. The meaning of a complex description is deﬁned

in terms of the orderings ≥

and ≥

induced by basic

preference descriptions d

and d

. The order denoted

by d

∧d

is the intersection ≥

∩ ≥

(Pareto ordering),

∨ d

denotes the transitive closure of ≥

∪ ≥

, −d

denotes the reversed ordering ≥

, and d

> d

denotes

the lexicographic ordering of ≥

and ≥

where

strictly preferred to

≥

and

We show that complex descriptions can also be

translated into a QPS that is just as succinct. We ﬁrst

introduce the notion of a reversed criterion that in-

duces the reverse of the ordering induced by the orig-

inal criterion. This can be achieved by reversing the

value preferences of all the simple criteria in a QPS.

Deﬁnition 5 (Reverse of a Criterion). The reverse of

a simple criterion c = ⟨X

, u

⟩ is c

−

= ⟨X

, u

−

⟩ with

−

iff

. The reverse of a cardinality crite-

rion c = ⟨C

⟩ is c

−

= ⟨C

−

⟩ where C

−

= {s

−

∣ s

∈ C

The reverse of a lexicographic criterion c= ⟨C

, ⊳

⟩ is

−

= ⟨C

−

, ⊳

−

⟩ whereC

−

= {s

−

∣ s

∈C

} and s

−

⊳

−

whenever s

⊳

ICAART 2012 - International Conference on Agents and Artificial Intelligence

246

Theorem 2. Let c

and c

be any two criteria. The

lexicographic criterion c

1∧2

= ⟨{c

, c

}, ∅⟩ induces

the order ⪰

∩ ⪰

. The lexicographic criterion c

1>2

⟨{c

, c

}, {(c

, c

)}⟩ induces the order

⪰

1>2

iff

≻

⪰

and

≻

. The criterion c

−

in-

duces the order

⪰

−

iff

⪰

Theorem 2 clearly shows the expressive power of

QPSs. It is very easy to represent speciﬁc opera-

tions for combining preference orderings by means

of QPSs such as creating a Pareto order (∧ operator),

reﬁning a preference ordering by means of a second

one (> operator), and reversing an ordering (− oper-

ator). Moreover, Theorem 2 shows this can be done

just as succinctly with QPSs as with RKBs; i.e. the

size needed differs at most with a constant factor.

The only operator that cannot be represented in

a QPS is disjunction (∨). However, it has been ar-

gued convincingly by (Andr´eka et al., 2002) that this

is not a natural operator, since it does not satisfy the

desired properties ‘indifference to irrelevant alterna-

tives’ and ‘unanimity with abstentions’. Indifference

to irrelevant alternatives means that two outcomes can

be compared solely on their own merits; the presence

or absence of other possible outcomes does not inﬂu-

ence the preference. The disjunction operator is not

indifferent to irrelevant alternatives since it considers

the transitive closure of the union of preference rela-

tions. Unanimity with abstentions means that if all

combined preference relations prefer outcome

over

outcome

, except possibly some that are indifferent,

then the overall preference relation also prefers

over

. The disjunction operator would be indifferent as

soon as one of the combined relations is indifferent,

even if all others strictly prefer

over

We have shown that LPD descriptions (except dis-

junction) can be represented by QPSs just as suc-

cinctly. QPSs are more general, however, than LPD

which is based on ranked knowledge bases. Whereas

RKBs require a total preorder on formulas, QPSs al-

low incomparable priority between subcriteria. QPSs

are not restricted to Boolean variables as LPD is.

Apart from propositional formulas, QPSs support the

use of equational constraints over arbitrary domains.

In particular, QPSs provide a deﬁnitional mechanism

in order to introduce new concepts (abstract variables)

and it is possible to deﬁne preferences over such ab-

stract variables. The knowledge that can be captured

in a QPS therefore is more general.

3.2 CP-nets

(Boutilier et al., 2004) introduce CP-nets: qualita-

tive graphical representations of preferences that re-

ﬂect (conditional) preference statements under a ce-

teris paribus (all else being equal) interpretation.

Deﬁnition 6 (CP-net). (Boutilier et al., 2004) A CP-

net N over variables V = {X

, . . . , X

} is a directed

graph G over X

, . . . , X

whose nodes are annotated

with conditional preference tables CPT(X

) for each

∈ V. Each conditional preference table CPT(X

)

associates a total order ⪰

with each instantiation u

of X

’s parents Pa(X

) = U. A preference ranking ⪰

(a total preorder over the set of outcomes) satisﬁes

a CP-net N iff for each variable X

and for each as-

signment u to the variables in U, yxu⪰ yx

′

u whenever

x ⪰

′

– for all assignments y to the set of variables

Y = V−(U∪{X

}) and all x, x

′

∈ Dom(X

). N entails

≻

, written N ⊧

≻

, iff

≻

holds in every

preference ordering that satisﬁes N. (Boutilier et al.,

2004) show that N ⊧

≻

iff there is a sequence of

improving ﬂips from

. An improving ﬂip of out-

come uxy with respect to variable X

is any outcome

′

y such that x

′

≻

Theorem 3. There is a QPS ⟨Var, Dom, K, C⟩ with a

criterion c∈ C that corresponds to an acyclic CP-net N

over variables V = {X

, . . . , X

} such that if N ⊧

≻

then

≻

for arbitrary outcomes

Proof. The CP-net N can be translated to the QPS S as

follows. All variables in the CP-net are also variables

in the QPS: V ⊆ Var. For every variable X

∈ V, a sim-

ple criterion c

is speciﬁed. If X

is conditionally inde-

pendent, c

= ⟨X

, u

⟩ such that x u

′

iff x ≻

′

. If X

is conditionally dependent, an auxiliary variable X

′

added to Var such that Dom(X

′

) =

∏

{Dom(X) ∣ X ∈

∪ Pa(X

)}. The constraint X

′

∏

∪ Pa(X

)) is

added to K. c

= ⟨X

′

, u

⟩ such that xuu

′

u iff x ≻

′

Finally, a lexicographic criterion c = ⟨C

, ⊳

⟩ is de-

ﬁned such that for every simple criterion c

thus gen-

erated from the CP-net, c

∈ C

, and ⊳

is the transitive

closure of ⊳

′

, where c

⊳

′

iff X

∈ Pa(X

) (note that

since N is acyclic, ⊳

is asymmetric).

Suppose that N ⊧

≻

. This means that there is

a sequence of improving ﬂips from

. First con-

sider the case where this sequence has length 1, i.e.

there is a single improving ﬂip w.r.t. some variable X

from

. Since the preference by a simple crite-

rion is taken from the corresponding CPT,

≻

. If

is not a parent of any variable, all other simple cri-

teria are indifferent between

and

(since they do

not involve X

), so

≻

. If X

is a parent of another

variable X

, ﬂipping its value inﬂuences the value of

the auxiliary variable X

′

. However, c

has higher pri-

ority than c

, so again we have

≻

. Since ⪰

transitive, we also have

≻

if the sequence of im-

proving ﬂips from

is longer than 1.

Note that it follows from Theorem 3 that the QPS

corresponding to an acyclic CP-net is just as succinct

A FRAMEWORK FOR QUALITATIVE MULTI-CRITERIA PREFERENCES

247

as this description. That is, the size of the QPS is

comparable to that of the CP-net (the size differs at

most by a constant factor).

There are some things that CP-nets cannot ex-

press, but a QPS can. Most importantly, we are able to

express abstract preferences based on auxiliary vari-

ables whose values are constrained by the knowledge

base. Consider the well-known example from game

theory called the ‘battle of the sexes’: a husband and

wife have to decide whether to go to the theater or to

a football match. The wife prefers the theater and the

husband prefers football, but both would rather go to-

gether than go to different places. If we let A (resp.

B) stand for ‘the wife (resp. the husband) goes to the

theater’ and ¬A (resp. ¬B) for ‘the wife (resp. the hus-

band) goes to the football match’, then the ordering

AB ≻ ¬A¬B ≻ A¬B ≻ ¬AB represents the wife’s pref-

erences. A CP-net cannot express this ordering, since

there is no improving ﬂip between ¬A¬B and AB. In

a QPS, this preference can be easily expressed by in-

troducing an auxiliary variable T (‘together’), whose

values are constrained by T = A ↔ B. A lexicographic

criterion with two Boolean simple subcriteria, based

on T and A respectively, where the one based on T

has higher priority, induces the desired preference or-

dering TAB ≻ T¬A¬B ≻ ¬TA¬B ≻ ¬T¬AB.

Second, we add priority between criteria, which

allows us to express that a good value for one variable

is more important than a good value for another vari-

able. TCP-nets (Brafman and Domshlak, 2002) are an

extension to CP-nets in which some priority between

variables is taken into account, but this is not strong

enough to represent lexicographic preferences (Wil-

son, 2004). (Wilson, 2004)’s own approach can han-

dle such preferences, but does not allow to use auxil-

iary variables and knowledge as described above.

Third, in a CP-net, every variable occurs exactly

once. In a QPS, some variables may not occur in any

criterion, and some variables may occur in multiple

criteria, e.g. if the preference on its values is different

from different perspectives, or if the preferences of

multiple people are combined.

4 CONCLUSIONS

We introduced Qualitative Preference Systems, a

new framework for representing multi-criteria pref-

erences. QPSs combine different features for com-

pactly expressing preferences. These features include

the well-known lexicographic rule which combines

basic preferences over variables, and a cardinality-

based rule which counts criteria that are satisﬁed. In

addition, QPSs provide a tool for expressing feasibil-

ity constraints as well as abstractions (concept deﬁni-

tions). Finally, such systems support a layered struc-

ture for representing preference orderings.

This combination of features provides a very ex-

pressive preference representation framework which

at the same time allows for a compact representation

of preference orderings. We have shown that the Log-

ical Preference Descriptions introduced in (Brewka,

2004) can be embedded in the QPS framework, with

the exception of the disjunction operator which is

not very natural. The ‘logical’ operators of (Brewka,

2004) translate to structural features of QPSs. We

have also shown that QPSs are able to express con-

ditional preferences by providing an order preserv-

ing embedding of acyclic CP-nets into QPSs. Last

but not least, these embeddings are size preserving,

i.e. the resulting QPSs provide a representation that

is as succinct as the LPD or CP-net representation.

This fact indicates that various problems such as dom-

inance testing for QPSs have an associated computa-

tional complexity that is at most as difﬁcult as these

alternative frameworks for preference representation.

ACKNOWLEDGEMENTS

This research is supported by the Dutch Technology

Foundation STW, applied science division of NWO

and the Technology Program of the Ministry of Eco-

nomic Affairs. It is part of the Pocket Negotiator

project with grant number VICI-project 08075.

REFERENCES

Andr´eka, H., Ryan, M., and Schobbens, P.-Y. (2002). Oper-

ators and laws for combining preference relations. J.

Logic Comput., 12(1):13–53.

Boutilier, C., Brafman, R. I., Domshlak, C., Hoos, H. H.,

and Poole, D. (2004). CP-nets: A tool for representing

and reasoning with conditional ceteris paribus prefer-

ence statements. J. Artif. Intell. Res., 21:135–191.

Brafman, R. I. and Domshlak, C. (2002). Introducing vari-

able importance tradeoffs into CP-nets. In Proc. UAI,

pages 69–76.

Brewka, G. (2004). A rank based description language for

qualitative preferences. In Proc. ECAI, pages 303–

307.

Visser, W., Hindriks, K. V., and Jonker, C. M. (2011).

Interest-based preference reasoning. In Proc.

ICAART, pages 79–88.

Wilson, N. (2004). Extending CP-nets with stronger con-

ditional preference statements. In Proc. AAAI, pages

735–741.

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