Preference relations (Dushnik & Miller 1941, Luce 1956, Scott & Suppes 1958b) are among the fundamental structures on which decision aiding models rely. They are used to filter out the dominated outcomes (a dominated outcome is any option for
which there is at least one other outcome in the results that is preferable with regards to the relation).
3.2.1
Preference relations properties
A preference relation is a binary relation between possible outcomes.
Definition 7. Given an outcome spaceΩ and two different outcomes o and o0, we say thato < o0if and only ifo is preferred (or indifferent) over o0. We sayo ∼ o0if and only if the decision maker is indifferent between the two outcomes. We call the first relation a preference relation and we call the second relation an indifference relation.
When ordering, we consider a set of objects (in our case, outcomes) ordered through different ordering types. Below, we review a few concepts to provide the appropriate background in understanding the various preference orderings that are representable. Next, we list some standard properties of binary relations that are useful in classifying preference relations (Woronowicz & Zalewska 2004, Roubens 1989, Hansson & Grüne-Yanoff 2011, Joseph et al. 2007). A binary relation R over a set Ω is called:
• reflexive, if, ∀a ∈ Ω, (aRa), • irreflexive, if, ∀a ∈ Ω, ¬(aRa),
• symmetric, if, ∀a, b ∈ Ω, (aRb) =⇒ (bRa), • asymmetric, if, ∀a, b ∈ Ω, (aRb) =⇒ ¬(bRa),
• antisymmetric, if, ∀a, b ∈ Ω, (aRb) ∧ (bRa) =⇒ (a = b), • transitive, if, ∀a, b, c ∈ Ω, (aRb) ∧ (bRc) =⇒ (aRc), • complete, if, ∀a, b ∈ Ω, (aRb) ∨ (bRa),
The above properties are not independent. For instance, asymmetry implies irreflex- ivity, while irreflexivity and transitivity imply asymmetry.
Via the composition of such properties, the relation R induces different ordering classes. We give the definition for some useful kinds of ordering here.
Based on its properties, a preference relation R is characterized as follows: • A binary relation is a pre-order or quasi order, if it is reflexive and transitive. If
in addition, it is antisymmetric then it is a partial order.
• A binary relation is a strict partial order (or irreflexive partial order), if it is irreflexive, asymmetric and transitive.
3.2. Preference Relations
• A binary relation is a total order, if it is a partial order and it is also complete. If a preference relation R is a total order, any two outcomes are mutually com- parable under R.
• A binary relation is a weak order, if it is a complete pre-order.
Given a pre-order , we can define the induced strict order as follows: o o0 if and only if o o0, but o0 6 o. We define relation ∼ by o ∼ o0 if and only if o o0,
and o0 o, that is, o and o0 are equally preferred. Then, is a strict partial order
and ∼ is an equivalence relation, i.e., a reflexive, symmetric and transitive relation.
3.2.2
Preference relations application
Many real-life problems call for identifying the best possible set of solutions. In order to solve such problems, techniques for both quantitative and qualitative preference representation and reasoning over a set of attributes have been extensively studied in the literature, e.g., (Benferhat et al. 2001, Brewka 2002, Brewka, Niemelä & Syrjänen 2002, Brewka et al. 2003, Balduccini & Mellarkod 2003, Boutilier et al. 2004a, Boutilier et al. 2004b, Brewka, Benferhat & Berre 2004, Wilson 2004b, Brafman, Domshlak & Shimony 2006, Kärger et al. 2008, Costantini & Formisano 2009, Bouveret et al. 2009a, Wilson 2011).
The preference relation is used to compare two solutions in terms of preferences over attributes of those solutions. The preference relation is also used by algorithms that identify the set of most preferred solutions. A solution is said to be optimal if and only if there is no other solution in the set of feasible solutions that strictly dominates it, i.e., a solution α strictly dominates β if and only if α dominates β and β does not dominate α. The set of solutions which are non-dominant to each other is called the non-dominated set.
Research has been performed on multi-attribute decision theory; a consider- able part of this research has focused on reasoning with quantitative preferences (Fishburn 1967, Fishburn 1970, Keeney & Raiffa 1993, Fishburn 1999). However, in many applications it is more natural for users to express preferences in qualitative terms (Doyle & Thomason 1999, Doyle & McGeachie 2003, Dubois et al. 2002, San- thanam et al. 2011). We will be talking about dominance relations for qualitative preferences. Preference reasoning engines define their dominance testing strate- gies based on particular semantics. For instance, these semantics may give an in- dication about how relatively weak or strong the dominance might be as we dis- cuss in Section 4.9 in Chapter 4. These strategies are well studied in the AI lit- erature (Boutilier et al. 2004a, Wilson 2004b, Wilson 2006, Brafman, Domshlak & Shimony 2006, Santhanam et al. 2008, Wilson 2009b, Santhanam et al. 2010b, San- thanam et al. 2010a). Dominance properties (e.g., weak and strong dominance)
along with a number of optimality notions in the context of CP-nets were studied in (Brafman & Dimopoulos 2004b).
The preference relation is relevant in many applications where we need to know whether a solution is (among) the best, in particular when there is a large number of outcomes that might be feasible within a problem (Boutilier et al. 1997, Boutilier et al. 2004a, Brafman, Domshlak & Shimony 2006). Modifying the dominance relation helps the number of optimal solutions be controlled in the sense that the size of the optimal set can shrink or expand. In fact, to stimulate the dominance relation to be more or less strict makes it hard or easy to find non-dominated solutions. Thus, dominance relation tuning allows to have better control of the set of solutions.