DERIVED STRENGTHS OF PREFERENCE RELATIONS ON COORDINATES

A way is indicated to derive, from a preference relation on a Cartesian product, strength of preference relations on the coordinate sets. These strengths of preference relations are then used to reformulate several well-known properties of preference relations, and make their meaning more transparent. A new result for dynamic contexts is given. This paper examines the existence of several kinds of additively decomposable representing functions for preference relations on Cartesian products. This is done under the assumption that the coordinate sets are connected topological spaces, e.g., R y, or Iw. The representing functions which we shall find will be cardinal. Several authors have dealt with cardinal additively decomposable representing functions under the assumption that the coordinate sets are ‘mixture spaces’, the main intended examples of mixture spaces being sets of probability distributions. See, for instance, Anscombe and Aumann (1963), Fishburn (1965), Polk& (1967), Keeney and Raiffa (1976, Karni, Schmeidler and Vind (1983). Usually the axioms of von Neumann and Morgenstem (1953, ch. 3 and the appendix) are assumed, leading to linear representing functions. Cardinality is easily obtained in the presence of linearity because, if one linear function is a (strictly increasing) transformation of another linear function, then in fact it must be a linear transformation. Hence, if two linear functions induce the same preference relation, then they also induce the same strength of preference relation. Matters are more complicated if the assumption of linearity is weakened to the assumption of continuity. To see this, let f and g be two continuous functions on a connected domain. Let f = cp . g for a transformation cp. For guaranteeing that cp is (positively) linear, the assumption of strict increasingness of cp no longer suffices. It must be strengthened to the more complicated assumption that cp preserves (‘first-order’) differences, as shown in Basu (1982). The characterizing properties for preference relations on connected topological spaces, met in literature, reflect this additional complication. This paper handles the additional complication by indicating a way to derive, from a preference relation on a Cartesian product (‘revealed’), strength of preference relations on the coordinate sets. By means of these we formulate several properties and results, met in literature, for connected topological spaces. All reformulated versions will require that certain strengths of preferences,