As mentioned in Section 1.1, if the preference over Y is clear and complete enough to identify a certain y* ∈ Y as the best choice, then x* ∈ f −1(y*) will be a final choice of the decision problem. If one can specify a υalue function υ(y): Y→ R 1, so that υ(y 1) >υ(y 2) implies that y 1 is prefered to y 2, then the decision problem reduces to max υ(f(x)) over x ∈ X. Unfortunately, such a value function proves difficult to obtain in practice. Before we make strong assumptions to construct a “reasonable” value function (Chapter 5), it may be more important for us to understand what “preference” is and what its basic characteristics are. We shall study these in terms of binary relations that encompass real-valued functions as special cases.
KeywordsPartial Order Binary Relation Preference Information Weak Order Transitivity Property
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