Lexicographic additive differences

Abstract

Several authors have identified sets of axioms for a preference relation ≻ on a two-factor set A × X which imply that ≻ can be represented by specific types of numerical structures. Perhaps the two best-known of these are the additive representation, for which there are real valued functions f A on A and f X on X such that ( a, x) ≻ ( b, y) if and only if f A ( a) + f X ( x) > f A ( b) + f X ( y), and the lexicographic representation which, with A as the dominant factor, has ( a, x) ≻ ( b, y) if and only if f A ( a) > f A ( b) or { f A ( a) = f A ( b) and f X ( x) > f X ( y)}. Recently, Duncan Luce has combined the additive and lexicographic notions in a model for which A is the dominant factor if the difference between a and b is sufficiently large but which adheres to the additive representation when the difference between a and b lies within what might be referred to as a lexicographic threshold. The present paper specifies axioms for ≻ which lead to a numerical model which also has a lexicographic component but whose local tradeoff structure is governed by the additive-difference model instead of the additive model. Although the additive-difference model includes the additive model as a special case, the new lexicographic additive-difference model is not more general than Luce's model since the former has a “constant” lexicographic threshold whereas Luce's model has a “variable” lexicographic threshold. Realizations of the new model range from the completely lexicographic representation to the regular additive-difference model with no genuine lexicographic component. Axioms for the latter model are obtained from the general axioms with one slight modification.