=Paper=
{{Paper
|id=None
|storemode=property
|title=Can Adaptive Conjoint Analysis perform in a Preference Logic Framework?
|pdfUrl=https://ceur-ws.org/Vol-949/kese8-03_01.pdf
|volume=Vol-949
|dblpUrl=https://dblp.org/rec/conf/ecai/GiurcaSB12
}}
==Can Adaptive Conjoint Analysis perform in a Preference Logic Framework?==
https://ceur-ws.org/Vol-949/kese8-03_01.pdf
Can Adaptive Conjoint Analysis perform in a
Preference Logic Framework?1
Adrian Giurca, Ingo Schmitt and Daniel Baier
{giurca, schmitt, daniel.baier }@tu-cottbus.de
Abstract. Research on conjoint analysis/preference aggre- The research reported by [46] proposed a mathematical op-
gation/social choice aggregation is performed by more than timization approach by translating ratings into algebraic con-
forty years by various communities. However, many proposed straints, but such solution requires acyclicity and transitivity
mathematical models understand preferences as irreflexive, and not changing preferences. New debates on solution pro-
transitive and statical relations while there is human psy- posed by [46] were reported by [31] in the context of non-
chology research work questioning these properties as being additive utility aggregations such as Choquet integral. How-
not enough motivated. This works propose to position the ever, none of these approaches consider non-transitive and/or
conjoint analysis inside a logical framework allowing for non- cyclic preferences, [48].
transitive and globally inconsistent preferences. Using a pref- [23] introduced a logic-based utility but the approach was
erence logics one can define a logic-based utility allowing to limited by a number of assumptions such as consistency
obtain an aggregate semantics of the collective choice. (acyclic preferences) ignorance (of neutral rated questions),
transitivity and the restriction of using only 2 stimuli choice
pair comparisons. Moreover, while it argued on the logical
1 Introduction and Motivation nature of the users ratings and rankings, it does not consider
Conjoint Analysis (CA) in marketing research was introduced preference change and interview adaptation. Many of these
forty years ago [26] being influenced by economics ([36], [35]) restrictions were introduced by the method of computing the
and mathematical psychology ([39], [40], [7]). While the begin- logic-based utility, basically adaptation of the weighted ma-
ning was devoted mostly to understand how individuals evalu- jority learning algorithm allowing only binary preference as
ate products/services and form preferences (see, [26], [34], [43] input.
and possibly others), in the last thirty years the CA litera- As discussed by [24], computing beliefs from ratings and
ture focused more on predicting behavioral outcomes by using rankings is much close to the mental expectations of respon-
statistical methods and techniques ([8]) and this resulted in dents and identified three kinds of beliefs that can be obtained
a widespread variation in CA practice. Recently, applications from question answers. The proposed framework considers
in innovation market were developed ([9]). consistent respondent belief sets but on belief sets aggrega-
The traditional conjoint task is related to the rational econ- tion there is no need to require consistency: moreover this is
omy model where agents tend to action towards maximizing inline with the Arrow’s impossibility theorem (see [5] and [6]).
their utilities. Although traditional non-adaptive conjoint solutions re-
While traditional models obtain significant results when quire static, non-changing, preferences, when data collection
processing complete, transitive and acyclic (consistent) prefer- is interactive one may experience preference change. More-
ences, many communities mention that such models are quite over, the actual online solutions on data collection show many
far from the real life. When asking people about thing they cases when the data is collected over days and not by a stan-
like, then they may not answer (incompleteness), or they may dard survey in a contiguous manner. As such, respondents
change their initial preferences due to reception of new infor- may remake-up their mind therefore change is frequently ex-
mation (preference change). In addition, while it seems that pected. Also, [24] pointed that may be useful to use weighted
the preference system of one respondent must be non contra- beliefs due to the imprecise nature of the user ratings. In
dictory, when processing preferences from many respondents addition, among other distinctions it was emphasized that
this assumption does not remain valid. Some of our previous while individual beliefs are consistent (no assumption of user
work argued towards a logic-based model for conjoint analy- irrationality), collective beliefs may not be consistent. In ad-
sis. dition, while the AGM model [4] considered consolidation as
a maintenance operation of removing some dispensable be-
1 This research is supported by (1)DFG Project SQ-System: En-
liefs resulting in a consistent knowledge base, we would like
twicklung von Konzepten für ein quantenlogikbasiertes Retrieval-
Datenbank-Anfragesystem: Anfragesprache, interaktive Suchfor- to avoid such approach due to missing of motivated criteria
mulierungsowie effiziente Anfrageauswertung and (2) German with respect of belief elimination.
Federal Ministry of Education and Research, ForMaT project The goal of this paper is to argue on the opportunity to
(Forschung für den Markt im Team), Phase II, Innovationslabor: use a preference logics framework allowing non-transitivity
Multimediale Ähnlichkeitssuche zum Matchen, Typologisieren
und Segmentieren
and inconsistency in preference data.
2 Related Work The main assumptions of this method are: (a) there is pref-
erence data for a set of objects O and (b) the utility function
The classical model of computing an utility function is the ad- is linear. Each preference data (e.g., o1 o2 ) is translated
ditive linear model (see [8] for details). Basically, the overall into an inequality between corresponding utilities of the cor-
utility is an additive linear combination on value scores ad- responding objects (u(o1 ) ≤ u(o2 )). The method then involves
justed with attribute scores and compensated with a constant minimizing the sum of errors for the inequalities and the sum
depending on interview i.e., of of the squares of the weights in the utility function.
ni
N X
As usual, each attribute value aij ∈ dom(Ai ), i = 1, ...n has
X (k)
U (oj ) = µ + βkl · xjkl weight θij . We denote θj the weights vector corresponding
(k)
k=1 l=1 to the k-th object oj . The goal is to estimate the individ-
ual partworths w = (w1 , ..., wn ) considering a linear utility
where
function (e.g., the vector model) U (o) = w · θ for each θ cor-
U (oj ) – is the total score on product profile oj ,
responding to an object o ∈ O.
βkl = Uk (akl ) – is the user preference on value akl of attribute
We encode preference data by respondent interviews: at the
Ak , and (k) (k)
k-th question we show a subset Ok = {o1 , ..., onk } ⊂ O
1, if oj .Ak = akl
xjkl = , asking the respondent to choose one object as ”the most
0, otherwise
liked”. Without loosing the generality (via reordering) we can
µ is a calibration constant (mean preference value across all
assume that the respondent choose first object as the pre-
objects). Usually Uk () is called part-utility function or part-
ferred one. This choice is encoded as the set of constraints,
worth function and its specification depends of the attribute (k) (k)
w(θ1 − θi ) ≤ 0, i = 2, ..., nk , and reduce the conjoint prob-
type (categorical and quantitative).
lem to a classification problem. [16] proposes to train a L2 -soft
In practice a conjoint study may contain both types of
margin classifier only with positive examples obtained from
attributes. Significant examples of categorical attributes are
respondent ratings, using a with a hyperplane through the ori-
brand names or verbal descriptions containing levels such as
gin and modeling the answering noise with dummy variables
”high”, ”medium”, ”low” while quantitative attributes are the (k)
εi . It trains one algorithm per respondent to get individual
ones which are measurable on either an interval scale or a ra-
vector weights w(p) for each respondent p and then to com-
tio scale (e.g., speed of a processor, size of a screen). While
pute individual partworths by calibration with the aggregated
there were proposed many models to encode the part-worth 1 (p) (p) (p)
and then w∗ = w 2+we .
P
functions, two models are representative: partworths i.e. w e = |P| p∈P w
The training conditions are:
1. the vector model, Uk (akl ) = wk θkl , where wk is the weight ( P k (k) 2
M inimize : w2 + C pk ∈P n
P
of attribute Ak , and θkl is the weight of the value akl ∈ i=2 (εi )
(k) (k) (k)
dom(Ak )) and suchthat : w(θ1 − θi ) ≤ 1 − εi
2. the ideal point model, Uk (akl ) = wk (θkl − θk0 )2 , where θk0
is the weight of the ideal value ak0 of attribute Ak . where C is a constant depending on the respondents set.
In overall the standard conjoint problem reduces to find all
βkl and µ by using training data of user-rated utilities for a 2.1.2 Learning from Preferences
training object dataset.
Recall the learning problem similar with most of conjoint
analysis tasks:
2.1 Machine Learning Approaches
Given a (very large) set of objects (each object repre-
During the last thirty years, Machine Learning research devel-
sented as a set of attribute-value pairs), and a set of
oped very similar problems, offering either statistically-based
evaluation instances (each object is evaluated by experts
or logic-based solutions. As in traditional conjoint analysis,
obtaining a score, typically a real number) find a learning
the difficulty relates to the fact that the set of all possible
algorithm being able to evaluate any subset of the initial
behaviors given all possible inputs is too large to be covered
set of objects being compliant with expert evaluations.
by the set of observed examples (training data). Hence the
learner must generalize from the training data. Learning from As learning algorithms use evaluated training data it looks
examples towards forecasting the future behavior is one large straightforward to input the learner with a database of ex-
field of research. amples in which the human expert has entered scores for
each possible choice. However, similar with traditional con-
2.1.1 Support Vector Machines joint analysis, there are two critical issues of this approach:
(a) many domains have very large set of possible objects there-
Support Vector Machines, [10], [47] was proposed as a clas- fore is would be a tremendously time consuming for the ex-
sification methodology by machine learning community. Ba- pert to create the complete evaluation rank. Moreover, the
sically, the standard model takes a set of input data and, training dataset must also contain enough ”bad” alternatives
classify each given input as being part of one of two possi- otherwise the expert will be tempted to produce only high
ble categories (such as ”like” and ”unlike”). There is research scores for everything and as such, to obtain a rank which is
proposing to use this model on conjoint analysis too (e.g., not useful; (b) in many cases experts do not think in terms
[16]). of absolute scoring functions therefore will be very difficult,
sometimes impossible, to create training data containing ab- Let O = {(a1 , ..., an )|ai ∈ dom(Ai )} be the set of all
solute scores. These reasons yields many researchers to con- possible object representations and S ⊆ O a set of train-
sider pair comparisons rather than scoring individual alter- ing objects. denotes the preference relation on train-
natives(there is a large literature concerning the way users
W S. Find a weighted full DNF CQQL formula
ing data
create preferences. The reader may consider [37], [12], [40], U = j wj mj (mj is the j-th minterm and wj ∈ [0, 1]
[17] and probably many other). Preference learning was pio- its weight) such that U best fulfills the user preferences
neered by [53] and continued by [55], [33], [20] and possibly i.e. when CQQL evaluation is performed over objects in
others. Basically, given a set of (partial) profiles and a pref- O then the obtained rank is consistent with user initial
erence function of these profiles we want be able to train a preferences.
computer program to classify new (so far unseen) profiles by
assigning a correct rank to each profile. The ratio of correctly If oi2 oi1 then the following constraint is considered
classified data points is called the accuracy of the system.
As such conjoint analysis is similar with a learning task: evalCQQL (U, oi1 ) − evalCQQL (U, oi2 ) ≥ 0
learning utility functions from respondent preferences. The
Because CQQL evaluation has simple arithmetic rules for
conjoint problem can be seen as learning to rank a set of
formula evaluation, from the computational point of view
objects by combining a given collection of initial rankings
the problem reduces to a linear optimization: M aximize :
or preference functions. In machine learning community this P
oi2 oi1 (evalCQQL (U, oi1 ) − evalCQQL (U, oi2 )) under the
problem of combining preferences arises in several applica-
tions, such as that of combining the results of different search above described constraints. The readers may consider [46] for
engines, or the collaborative filtering problem. During the last details on problem solving strategies (such as simplex com-
20 years a number of algorithm were developed: a pioneering putations, feasible and unfeasible states, solutions to avoid
algorithm is described in [14] and [15] as an extension of the overfitting and more.)
early work reported by [38]. Advances in learning from pref- Automated extraction of rules from evidences was largely
erences were reported by [19], [20], and [30]. As described by discussed by connectionist learning community (early work
[20], the task of learning object preferences is: by [41], pioneered by [21] and subsequently discussed by [51],
[25], [52], [11], [49], and possibly others) under the umbrella
Let O = {(a1 , ..., an )|ai ∈ dom(Ai )} be the set of all pos- of a much general task:
sible product representations and let S = {o1 , ..., on } ⊆
How can we extract models from the training data in an
O be a set of training objects (aka full profiles, product
automated manner and use these models as the basis of
representations). Let P be a set of respondents and
an autonomous rational agent in the given domain.
{PS,p : S × S −→ {0, 1}|p ∈ P} the set of pairwise
preferences on training data. Learn a utility function One of the most important features of such an approach is
U : O −→ R that ranks any subset of O. that it combines the computational advantages of connec-
tionist models with the qualitative knowledge representation
Notable, while conjoint analysis typically assume a linear util-
proposed by the AI community.
ity function (see details by [8]), learning from preferences does
It is obvious that a solution of this problem must consider
not require utility linearity but many strategies on learning
two stages: (1) Learning the model and (2)Performing infer-
from preferences still assume linear combinations as potential
ence using this model. This work follows only the first stage
ranking functions. A significant solution introduced by [14]
of the problem – if there is a learned ruleset then there are
and improved in [15] considers learning a global preference as
many opportunities to perform inference according with var-
a weighted linear combination of all respondent preferences,
ious semantics (crisp, probabilistic, fuzzy and so on) and a
and then derive a final ordering which is maximal consistent
discussion of appropriateness of each of them should be large.
with this preference. Other research ([53], [30]) uses a differ-
Inside a rule framework the conjoint problem is to find out
ent strategy, specifically direct learning of the utility func-
a set of rules that best model the respondent preferences.
tion directly from the respondent preferences. [53] introduces
One can consider learning of various kinds of rules (possibly
a two-state symmetric neural network architecture that can
weighted), each of them supporting various semantics includ-
be trained with representations of states and a training sig-
ing probabilistic models [42], incomplete/imprecise informa-
nal (corresponding to the user preferences) indicated the pre-
tion, [54], plausibility-based models [18], [22] or quantum logic
ferred state. Subsequent works on this solution were reported
semantics [45]:
by [55], [29], [33], and [27].
1. Simple rules (propositional rules):
2.1.3 Logic-based Approaches [(¬)Ai1 ∧ ..., ∧(¬)Aik Aik+1 ]
A logic-based approach was proposed by [46]by replacing the where (¬)A denotes a possibly negated attribute;
utility function with a logical formula best fulfilling a set of al- 2. Positive attribute-value rules:
gebraic constraints derived from preference processing. They
use Commuting Quantum Query Language (CQQL, [45]) a [Ai1 ' vi1 ∧ ..., ∧Aik ' vik Aik+1 ' vik+1 ]
logical language based on combinations between Boolean con-
ditions and proximity/similarity conditions over specialized where vij ∈ dom(Aij ), Aij ' vij means that Aij takes a
variants of logical operators producing weighted formulas. value around vij (The reader should notice that ' includes
The problem is formulated as below: ordinal values, e.g., Aij = vij );
3. Attribute-value rules with negation: lective beliefs and (c) performing rule extraction and expla-
nation and formal interpretation.
[(¬)Ai1 ' vi1 ∧ ..., ∧(¬)Aik ' vik Aik+1 ' vik+1 ]
where ¬Aij = vij means Aij 6= vij ; 3.1 Preference Logics
4. General attribute-value rules: We follow the approach defined by [50] on preference logic
introduced as a special case of logic by defining a preference
[(¬)Ai1 ' vi1 ∧ ..., ∧(¬)Aik ' vik (¬)Aik+1 ' vik+1 ]
relation between the interpretations of the underlining logic as
The first three kinds of rules were largely addressed by we consider this approach being simple and powerful. Below
data mining community when learning association rules. we recall some of the [50] results.
Researchers developed different kinds of association rules: Let L be a standard logic and @ a strict partial order on
Boolean (crisp) association rules, quantitative association interpretations ( we say I2 is preferred to I1 and denote I1 @
rules, fuzzy association rules. Association rules were pioneered I2 ). Then, L@ = (L, @) is a new logic, a preference logic. The
by [44] and then established by [2], and [3]). Standard associ- basic artifacts such as satisfaction, validity and entailment
ation rules consider two measures of interestingness: support are defined by [50]. Recall that while the standard logics are
and confidence although other models may add two more: monotonic2 . Recall the definitions of satisfiability, validity and
lift and conviction or adopt non-standard ones, [32]. Learn- entailment:
ing association rules is usually performed under both a user- Definition 1 ([50])
specified minimum support and a user-specified minimum Let F, G ∈ L. Let I be an interpretation.
confidence requirements. I preferentially satisfies F (denoted I |=@ F ) if I |= F and
There were developed many algorithms starting with the there is no I 0 such that I @ I 0 and I 0 |= F . As usual, I is
most known one, Apriori ([3]) and continuing with many oth- called the model of F .
ers (Eclat, FP-growth and so on.) A significant step is the F preferentially entails G (denoted F |=@ G) if
Assoc algorithm [28] which enables mining for generalized as-
sociation rules (including negation i.e. attribute-value rules ∀I, I |=@ F ⇒ I |=@ G
with negation) and does not restrict for minimum support
and confidence. That is the preferred models of G are also preferred models
However, on our knowledge, none of this research consid- of F .
ering the conjoint analysis task: basically the training data
set for learning association rules does not distinguish vari- As described by [50], L@ is a non-monotonic logic because
ous users. All the data is uniform (mostly, it comes from e- there may be formulas F, G ∈ L@ such that both F |=@ G and
commerce transactions) and it may refer to one user (such as F |=@ ¬G. Moreover, it is not necessary that F is inconsistent,
in recommender systems, [1] ) or to many but not consider- it is just sufficient that F do not have preferred models.
ing distinct training data for each of them, therefore the con- A significant case of preference logics was introduced by [13]
joint task is somehow hidden. In addition the conjoint analysis under the name of choice logic. Basically, choice logic defines
problem in the context of learning association rules does not the ordered disjunction (denoted ×) as a special kind of stan-
directly performs from preferences: using transactional data dard disjunction (∨) as such introducing a preference relation
as input, there should be some algorithm computing binary between the interpretations and models. The ordered disjunc-
preferences. tion has the same models as regular disjunction but there is
The first kind of rules were considered, in context of adap- a preference relation between these models. For example, if
tive conjoint analysis, by [23] in conjunction with weighted A × B is a disjunction between two atoms. Then I1 = {A},
CQQL (see [45] for language description), an extension of the I2 = {A, B} and I3 = {B} are its models. Then I3 @ I2 and
relational calculus using quantum logic paradigm which de- I3 @ I1 meaning that I1 and I2 are preferred models.
fines metric(or similarity) predicates, weighted conjunction Intuitively, as [13] reports, the ordered disjunction means
(∧θ1 ,θ2 ), weighted disjunction (∨θ1 ,θ2 ) and quantum negation. that when F1 × ...Fn we prefer models that first satisfies F1
Clearly (as explained by [25] and [52]) there is a need for and if this is not possible then we prefer models satisfying F2 ,
both a preference measure to rank the rules and a learning and so on. Choice logic defines the degree of satisfaction for
algorithm which uses the preference measure to find the best all logic formulas
k rules. The work reported by [23] describes a heuristic and
learning approach to use the respondent preferences on stimuli Definition 2 ([13])
to compute a rule preference relation (called minterm prefer- The optionality of a formula (the number of choices to satisfy
ence because the rules were learned as weighted minterms of a formula) is opt(A) = 1 if A is an atom.
the CQQL full disjunctive normal form) and then use a learn- opt(¬F ) = 1
ing algorithm to compute a ranking on the minterms set. opt(F1 ∨ F2 ) = max(opt(F1 ), opt(F2 ))
opt(F1 ∧ F2 ) = max(opt(F1 ), opt(F2 ))
opt(F1 × F2 ) = opt(F1 ) + opt(F2 )
3 Conjoint Analysis using Preference
Logics
[13] defines the preference relation (@) between models of logic
This section introduces a logical framework allowing (a) en- formulas and consequently the entailment. It is shown that
coding of preferences as choice formulas, (b) defining a logic-
2 In the sense that if F , F , F ∈ L, if F |= F then F ∧ F |= F .
based utility inside a preference logic to allow creation of col- 1 2 3 1 3 1 2 3
the entailment satisfies cautious monotony and cumulative from Table 1 say that OS(”Android”) ∧ Battery(”12h”)
transitivity: is preferred to OS(”Android) ∧ Battery(”6h”) as well
as OS(”W inP hone”) ∧ Battery(”12h”) is preferred to
Proposition 1 ([13])
OS(”Android) ∧ Battery(”4h”) and so on.
Let S be a set of choice logic formulas and A, B be classical
formulas. Definition 3 (Mapping trade-off matrices)
S |=@ A and S |=@ B ⇒ S ∪ {A} |=@ B Let a trade-off matrix based on predicates A1 and A2 .
S |=@ A and S ∪ {A} |=@ B ⇒ S |=@ B If A1 (u) ∧ A2 (v) is preferred to A1 (u0 ) ∧ A2 (v 0 ) then this pref-
erence is encoded into the choice formula:
From the computational point of view, choice logic can be
A1 (u) ∧ A2 (v) × A1 (u0 ) ∧ A2 (v 0 )
translated to stratified knowledge bases.
that is preferring models that, if possible first satisfy
4 Modeling Conjoint Analysis A1 (u) ∧ A2 (v)3 .
Conjoint analysis collects preferences from user interviews us- Definition 4 (Mapping pair comparisons)
ing a variety of question types but the most used ones are Let q be the pair comparison
trade-off matrices and pair-comparisons. A trade-off matrix q = A(a) and B(b) OR C(c) and D(d).
([34]) asks a respondent to consider a pair of attributes. It If the left side is preferred then this preference is encoded into
displays all combinations of values for those attributes, ask- the choice formula:
ing the respondents to provide a ranking for the combinations.
The Table 1 show an example of a trade-off matrix related to A(a) ∧ B(b) × C(c) ∧ D(d)
attributes OperatingSystem and Battery life. While trade-off
If q is rated neutral then this preference is encoded into the
12 hours 6 hours 4 hours 2 hours formula:
Android 1 2 7 5 A(a) ∧ B(b) ∨ C(c) ∧ D(d)
WinPhone 3 4 6 11
other OS 8 9 10 12 Similarly, if the right side is preferred then this preference is
encoded into the choice formula:
Table 1. A trade-off matrix with respondent ranking
C(c) ∧ D(d) × A(a) ∧ B(b)
4.2 Towards Logic-based Conjoint Analysis
matrix are quite efficient on ranking binary stimuli, trade-off
matrices cannot be used if we consider stimuli with more than Let A = {A1 , ..., An } be a set of unary predicates with
two attributes. A solution to these limitations is to use pair dom(Ai ) the domain of values. Let O = {(a1 , ..., an )|ai ∈
comparisons. Pair comparisons are seen as choice questions dom(Ai )} be the set of all possible product representations.
Left side OR Right side Definition 5 (Normal Form)
Android Windows Phone,... A full ordered disjunctive normal form (ODNF) over choice
AND Left AND logic defined by the language A is a formula
≥ 500EUR, ... ≤ 3.5” screen, ...
≥ 4” screen,... And,... U = ×j (L1 (l1j ) ∧ ... ∧ Ln (lnj ))
AND Neutral AND
Battery life 6h WIFI, ... (
where Lk lkj ) is a literal corresponding to the predicate Ak (ei-
≥ 4” screen,... Battery life 10h,...
AND Left AND ther Ak (lkj ) or ¬Ak (lkj )) and lkj ∈ dom(Ak ).
other OS no WIFI, ...
Let C the set of all choice formulas derived from user prefer-
ences. Then, the generic conjoint analysis task is described as
Table 2. Pair Comparisons and Ratings below:
Find U = ×j (L1 (l1j ) ∧ ... ∧ Ln (lnj )) such that U best
fulfills the user preferences i.e. there is a maximal set of
evaluated by favoring either ”the left side” or ”the right side” constraints C 0 ⊆ C such that U |=@ C for all C ∈ C 0 .
or ”neutral”.
Of course, the economics community does not really need the
complete DNF but, most of the cases only a subset of the
4.1 Preferences as Choice Formulas ODNF (the most important clauses). In addition, sometimes
Let A1 , ..., An be a set of attributes (unary predi- the constraints may come weighted (using some weight w ∈
cates) with dom(Ai ) the domain of values. Let O = (0, 1]) and then the concept of maximal set can be replaced
{(a1 , ..., an )|ai ∈ dom(Ai )} be the set of all possible by a subset of constraints with a sum of weights greater than
product representations. The choice logic ordered disjunc- a specified threshold.
tion operator makes this logic suitable candidate to en- 3 This corresponds completely to the psychological meaning of
code user ratings as choice formulas. The trade-off ma- trade-off matrices where the respondent does not reject any of
trices introduces a rank between choices e.g., the matrix the alternatives
Rule extraction from a computed ODNF (or a subset) is Aggregation Require Require Allow Static
straightforward as the experts like to understand the depen- Models Irreflexive Transitive Indifference Preference
dencies of a specific predicate value with respect of the re- CA (econ.) yes yes yes yes
SVM yes yes no yes
maining predicates. As such rules are obtained by transform- Preference yes yes no yes
ing U to conjunctive normal form (CNF) and then deriving Learning
rules from each clause according with specific predicates as Rule yes yes no yes
conclusions. Learning
Let R be a the derived ruleset as described above. Then, Preference yes no yes no
Logic (belief rev)
all preferred models of R corresponds to preferred objects in
O.
As such we propose an updated process chain of adaptive Table 3. Conjoint Analysis Preference Requirements
logic-based conjoint analysis as depicted by Figure 1.
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