Utility Maximization Within a Threshold Depending on Both Compared Alternatives.

In this Chapter we consider the utility maximization models with a threshold function depending on both compared alternatives, i.e., the form ε(x, y). 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 ε(x, y) 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 ε(x, y) additively depends on the thresholds of the separate alternatives, i.e., ε(x, y) = δ(x) + δ(y). 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 ε(x, y) = δ(x)δ(y). Moreover, we assume that the threshold function δ depends on the utility value of the alternative in a special way: δ(x) = αu(x)β, α > 0, β ∈ ℝ It is shown that in all cases the corresponding binary relation is an interval order. When β ∈ [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. [ABSTRACT FROM AUTHOR]