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Classically, a rational individual is a maximizer: he chooses the best alternative according to some fixed utility function.

In this Chapter we present the basic models used in individual and social choice theory, microeconomics, decision theory, etc., to describe individual preferences and choices over alternatives, and the links between those models. Section 2.2 bears on the main types of binary relations used to represent preferences, namely, linear orders, weak orders, partial orders and acyclic relations. We also define here the notion of a partition of the set of alternatives and study the relation between the class of ordered partitions and the class of weak orders. Section 2.3 defines the notion of choice rationalizable by a binary relation, also called ‘pair-dominant choice’. Each class of preference relations has a class of pair-dominant choice functions rationalizable by the relations from this class associated with it. Another classic model of choice described in Section 2.4 deals with the selection of the alternatives that are ‘optimal’ according to one or several criteria. The case of one criterion leads to the utility maximization model whereas one of its generalizations is the Paretian multicriterion model. It is shown that the choice functions defined by these utility models are the classes of pair-dominant choice functions rationalizable by linear orders, weak orders and partial orders, respectively. Section 2.5 presents several rationality conditions for choice functions and studies their relations.

In Chapter 2 we have shown that the classic utility maximization model is equivalent to the pair-dominant choice model, with the corresponding binary relation being a weak order. An important property of a weak order is that the indifference relation of a weak order is transitive. But in Chapter 1 we discussed several examples in which the indifference relation associated with a preference relation is not transitive. In these examples the intransitivity of the indifference relation leads to the insensitivity to the utility evaluation. In this Chapter, we show how the classic paradigm of binary comparison and maximization choice can be extended to take into account this insensitivity. In order to do so, we introduce an -wide insensitivity zone for the comparison of the utility values and for the choice. Thus, an alternative is preferred to an alternative only if its utility value exceeds the utility value of from the threshold value (i.e., if () > () + ). Throughout this Chapter the value of threshold is assumed to be either a positive constant or a function () which depends on each alternative (while in Chapters 4 and 5 more general definitions of will be considered). In the case where the threshold function is non-negative, in Section 3.2 this model of utility maximization is called the interval choice model.

In this Chapter we consider the utility maximization models with a threshold function depending on both compared alternatives, i.e., the form (). In Section 4.2 we show that any pair-dominant function can be rationalized by utility maximization with such a threshold function. Then we formulate sufficient and/or necessary conditions on the function () in order that the corresponding binary relation satisfies acyclicity, transitivity, negative transitivity or the strong intervality condition. In Section 4.3 we study the case where the threshold function () additively depends on the thresholds of the separate alternatives, i.e., () = () + (). It is shown that the corresponding binary relation is a biorder. In Section 4.4 the case of a multiplicative threshold function is considered, i.e., the case where () = ()(). Moreover, we assume that the threshold function depends on the utility value of the alternative in a special way: () = (), > 0, ∈ ℝ It is shown that in all cases the corresponding binary relation is an interval order. When ∈ [0, 1] the corresponding relation is a semiorder. The inverse statement, namely, that semiorders can be represented by such a model, is proved only for a special class of semiorders, called regular semiorders. In Section 4.5 we give the properties of the choice functions associated with the utility maximization models considered in this Chapter.

In this Chapter we study choice functions rationalizable by utility maximization with a threshold depending on the feasible set and in some cases on the compared alternatives. The threshold function is considered to be in one of the following forms: = (), = (), = (), or = ().

The analysis in the previous Chapters has considered four main ingredients: This analysis allowed us to isolate several classes of binary relations that are of particular interest because they have a simple numerical representation involving and and generate, via some mechanism (e.g., the pair-dominant one), classes of choice functions with remarkable properties.

Our endeavour to provide a systematic view of the utility within a threshold maximization paradigm and its relations with preference and choice models has been completed. Throughout this study we considered the rational choice paradigm defined by maximization of a utility within a threshold, i.e., presented in the form where the threshold takes one of the following six forms: We would now like to discuss and classify these different forms of our general paradigm.