Therefore, this framework will pave the way for the integration of more prefer- ence reasoning engines and allow the user to parameterize the conversational RSs by specifying how the user model is represented (e.g., weights vector, cp-tree), which can also adapt automatically to the user and problem in context.
Two instances of this framework are developed and are presented in this chap- ter. The first, described in Section 4.4, is based on a simple quantitative preferences formalism, involving a sum of weights (one for each feature of the recommended products), with an associated language of linear inequalities. This is a very com- monly used model for preference representation, specifically, in MAUT (Dyer 2005) described in Section 2.5.1 of Chapter 2. The second instance of the framework (Sec- tion 4.5) is a qualitative preference formalism, where models adopt a kind of gener- alised lexicographic order, and constraints are expressed as comparative preference statements in a language generalising CP-nets (Boutilier et al. 2004a).
4.4
Sum of weights-Model Approach
Additive models allow compact representation of a utility function where the degree of desirability of the user is expressed separately for each attribute by a real value (Balabanovic 1998, Shen 2007, Kim et al. 2011, Shen et al. 2005). For the reasons described in (Keeney & Raiffa 1993, Stewart 1996, Tsoukiàs 2008, Caballero et al. 2010), the additive model is usually used in many real decision-making problems.
4.4.1
Models
The set of models contains all vectors of weights w = (w1, . . . , wn), where wi is a
non-negative real number. wi is the weight assigned to feature Fi. Given a weights
vector w, the overall value w(α) of a configuration α is the sum of weights of the fea- tures included in α, i.e., w(α) = X
i:α(i)=1
wi, which also can be written as
X
i
wiα(i).
This is used to define the ordering on configurations. We define the preference re- lation <w for model w by α <w β if and only if w(α) ≥ w(β), i.e., if and only if
X
i
wi α(i) − β(i) ≥ 0. Thus <w is a total pre-order on configurations.
4.4.2
Constraint language
Constraints on the user’s preferences are expressed as inequalities between sums of weights. The weights summed are related to the features included in the configura- tions that are shown to the user. For instance, when the user selects a configuration α from a set of three configurations proposed by the system (i.e., α, β, γ), the system
infers that the sum of weights of features in α (denoted by w(α)) is greater or equal than the two sums of weights w(β) and w(γ).
The set of constraints will, at a later stage, determine whether a configuration α dominates another configuration β. In fact, during dominance computation the advisor is interested in all models that are compatible with these inequalities.
4.4.3
Dominance relation
Given a set of constraints on preferences (or preference statements) Φ, and two con- figurations α and β, we say α <Φ β if and only if α <w β for all weight vectors w
that comply with all preference statements in Φ.
Example. Let V = {F1, F2, F3 be the set of features and let the user be repre-
sented as a vector of weights w = (w1, w2, w3). The weights w1, w2 and w3 cor-
respond to F1, F2 and F3 respectively. Let Φ be the set of preference statements:
f1f¯2f¯3 ≥ ¯f1f¯2f3, and f1f¯2f3 ≥ f1f2f¯3. Let α be the configuration f1f¯2f¯3, and let β be
the configuration ¯f1f2f¯3. α dominates β if and only if all weights vectors w that sat-
isfy the user’s preferences also agree that the weight of α is greater than the weight of β. Weights vector w complies with the preference statement f1f¯2f¯3 ≥ ¯f1f¯2f3 if
and only if w(f1f¯2f¯3) ≥ w( ¯f1f¯2f3), i.e., w1 ≥ w3. w complies with the preference
statement f1f¯2f3 ≥ f1f2f¯3 if and only if w(f1f¯2f3) ≥ w(f1f2f¯3), i.e., w3 ≥ w2.
Therefore, w satisfies Φ if and only if w1 ≥ w3 and w3 ≥ w2. On the other hand,
w satisfies α <w β if and only if w1 ≥ w2. the statement w1 ≥ w2 can be easily
induced from w1 ≥ w3and w3 ≥ w2. Therefore, any weight vector w that satisfies Φ
necessarily satisfies α<w β. The system will then conclude that α dominates β (i.e.,
α <Φ β).
4.4.4
Dominance computation
One assumption is that including a feature is always at least as good as not including that feature. This assumption is translated in this instance by the set of constraints wi ≥ 0, for i = 1, . . . , n, which says that all feature weights are non negative. Thus,
these statements are included in Φ as constraints on preferences at the beginning of the dialogue. While we go forward in the dialogue, additional constraints on preferences are induced and added to Φ. Given two configurations α and β, to show α <Φ β, the advisor needs to check whether Φ entails α ≥ β with real-valued
variables wi. Φ entails α ≥ β is equivalent to saying that all weights vectors that
agree or satisfy all constraints in Φ, also agree with α being preferred over β, i.e., P
i α(i) − β(i)wi ≥ 0. The lack of expressiveness of this quantitative approach is
4.4. Sum of weights-Model Approach
A standard form of linear programming can answer the question above. This is immediate since each constraint in Φ is represented as a linear inequality between linear combinations of weights (e.g., w1 + w2 ≥ w1 + w3). For instance, we can
have the two following linear inequalities: c1 : w1 + w3 ≥ w2 + w3 and c2 : w1 ≥
w2 + w3. These inequalities might be induced when the user makes her choices
among the queries that were suggested by the system. α ≥ β is equivalent to the following inequality: g : w1 ≥ w2 if α has only feature F1 (i.e., f1f¯2f¯3) and β has
only feature F2 (i.e., ¯f1f2f¯3). A linear programming solver, which uses the Simplex
algorithm (Brearley et al. 1975), can then check whether Φ entailsP α(i) − β(i)w
i.
Technically, the solver checks whether or not c1 and c2 imply w1 ≥ w2. In other
words, the solver can check whether all weights vectors w whose values comply with c1 and c2 also comply with w1 ≥ w2. An implementation of this using a linear
programming solver is to express it as a linear optimisation problem. Define gminto
be the minimum value of P
i α(i) − β(i)wi ≥ 0 subject to constraints Φ. It can
be shown that α <Φ β if and only Φ entails
P
i α(i) − β(i)wi ≥ 0 if and only if
gmin ≥ 0.
Example 1. Let Φ be the pair of statements: f1f¯2f3 ≥ ¯f1f2f3, and f1f2f¯3 ≥ ¯f1f2f3. Let α be the configuration f1f¯2f¯3, and let β be the configuration ¯f1f2f3. Weights vec-
tor w satisfies the constraint f1f¯2f3 ≥ ¯f1f2f3 if and only if w(f1f¯2f3) ≥ w( ¯f1f2f3),
i.e., w1+ w3 ≥ w2+ w3, which holds if and only if w1 ≥ w2. By similar reasoning, w
satisfies Φ if and only if w1 ≥ w2and w1 ≥ w3. Also, w satisfies α ≥ β if and only if
w1 ≥ w2+ w3. Thus Φ does not entail α ≥ β, so we do not have α<swΦ β, since, for
example, weights vector w with w1 = 4, w2 = 2 and w3 = 3 satisfies Φ but does not
satisfy α ≥ β. 2
Example 2. Suppose now that there are four features, and let Ψ be the pair of state- ments f1f¯2f3f4 ≥ ¯f1f2f3f4 and f1f2f¯3f¯4 ≥ f1f¯2f3f¯4. With the sum of weights se-
mantics this implies f1f2f¯3f4 ≥ ¯f1f2f3f4, since the first statement implies w1 ≥ w2,
and the second statement implies w2 ≥ w3, which implies w1 ≥ w3. Thus, we con-
clude that the third statement is satisfied. We therefore have f1f2f¯3f4 <Φ f¯1f2f3f4.
In fact, we have strict dominance: f1f2f¯3f4 Φ f¯1f2f3f4 since we do not have
¯