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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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