Inequality Measurement with an Ordinal and Continuous Variable What would be the analogue of the Lorenz quasiordering when the variable of interest is continuous and of a purely ordinal nature? We argue that it is possible to derive such a criterion by substituting for the PigouDalton transfer used in the standard inequality literature what we refer to as a Hammond progressive transfer. According to this criterion, one distribution of utilities is considered to be less unequal than another if it is judged better by both the lexicographic extensions of the maximin and the minimax, henceforth referred to as the leximin and the antileximax, respectively. If one imposes in addition that an increase in someoneu0027s utility makes the society better off, then one is left with the leximin, while the requirement that society welfare increases as the result of a decrease of one personu0027s utility results in the antileximax criterion. Incidentally, the paper provides an alternative and simple characterisation of the leximin principle widely used in the social choice and welfare literature.

Generalized transitivity: A systematic comparison of concepts with an application to preferences in the Babington Smith model Abstract   Reciprocal relations are binary relations Q with entries    Q  (  i  ,  j  )  xe2x88x88    0  ,  1     , and such that    Q  (  i  ,  j  )    Q  (  j  ,  i  )    1   . Relations of this kind occur quite naturally in various domains, such as preference modeling and preference learning. For example,      q    i  ,  j      could be the fraction of voters in a population who prefer candidate i to candidate j. In the literature, various attempts have been made at generalizing the notion of transitivity to reciprocal relations. In this paper, we compare three important frameworks of generalized transitivity: gstochastic transitivity, Ttransitivity, and cycletransitivity. To this end, we introduce Etransitivity as an even more general notion. We also use this framework to extend an existing hierarchy of different types of transitivity. As an illustration, we study transitivity properties of probabilities of pairwise preferences, which are induced as marginals of an underlying probability distribution on rankings (strict total orders) of a set of alternatives. In particular, we analyze the interesting case of the socalled Babington Smith model, a parametric family of distributions of that kind.