The Repetitions Approach to Characterize Cardinal Utility

Building on previous work of A. Camacho, we give necessary and sufficient conditions for the existence of a cardinal utility function to represent, through summation, a preference relation on sequences of alternatives

Clarification of some mathematical misunderstandings about Savage's foundations of statistics, 1954

This note discusses some mathematical misunderstandings about Savage (1954). It is shown that in his model the probability measure cannot be countably additive, that the set of events must be a σ-algebra and not just an algebra, that Savage did not characterize all atomless finitely additive probability measures, and that the state space in his model, while infinite, does not have to be uncountable

A Graph-Theoretic Approach to Revealed Preference

One of the issues in the impossibility theorem of Arrow is the difference between choice behaviour, as considered by Arrow in most of the illustrations for the conditions in his theorem, and binary relations as dealt with in Arrow's theorem. The relations between choice behaviour and binary relations are studied in revealed preference theory, a theory which originates from consumer demand theory. This paper presents a graph-theoretic approach to revealed preference theory. This is done by considering alternatives as vertices, and choice situations as arcs. By means of this method alternative proofs are obtained for some known results. In particular it is shown that many results from literature can be derived from what may be the main result from revealed preference theory, a theorem of Richter (1966). Next a duality approach is sketched, where vertices and arcs are interchanged as done in dual graph theory. Finally some results are given for non-transitive binary relations. For these there is an increasing interest because of Arrow's theorem

Additive representations on rank-ordered sets. I. The algebraic approach

This paper considers additive conjoint measurement on subsets of Cartesian products containing “rank-ordered‘ n-tuples. Contrary to what has often been thought, additive conjoint measurement on subsets of Cartesian products has characteristics different from additive conjoint measurement on full Cartesian products

Additive representation for equally spaced structures

It is shown that additive conjoint measurement theory can be considerably generalized and simplified in the equally spaced case

Transforming Probabilities without Violating Stochastic Dominance

The idea of expected utility, to transform payments into their utilities before calculating expectation, traces back at least to Bernoulli (1738). It is a very natural idea to transform, analogously, probabilities. This paper gives heuristic visual arguments to show that the, at first sight, natural way to do this, at second thought seems questionable. At second thought a sound and natural way is the way indicated by Quiggin (1982) and Yaari (1987a)

Continuity of Preference Relations for Separable Topologies

In Debreu (1954, 1959) some classical results were provided for consumer theory. Necessary and sufficient conditions were given for the existence of (con- tinuous) utility functions to represent preference relations of consumers. Further results are given in Bowen (1968), Jaffray (1975), Richter (1980), and Chateauneuf (1985)

Unbounded Utility for Savage's "Foundations of Statistics," and Other Models

A general procedure for extending finite-dimensional "additive-like" representations for binary relations to infinite-dimensional "integral-like" representations is developed by means of a condition called truncation-continuity. The restriction of boundedness of utility, met throughout the literature, can now be dispensed with, and for instance normal distributions, or any other distribution with finite first moment, can be incorporated. Classical representation results of expected utility, such as Savage (1954), von Neumann and Morgenstern (1944), Anscombe and Aumann (1963), de Finetti (1937), and many others, can now be extended. The results are generalized to Schmeidler's (1989) approach with nonadditive measures and Choquet integrals, and Quiggin's (1982) rank-dependent utility. The different approaches have been brought together in this paper to bring to the fore the unity in the extension process

Continuity of transformations

Let u be a continuous function from Re to Re. Let f be a transformation (which in our terminology does not have to be bijective) from the range of u to Re. v = f(u(.)) is the composition of f and u. It is well-known that continuity of f implies continuity of v. We consider the reversed question: Does continuity of v imply continuity of f? Elementary as this question may be, we did not find a place in literature where the answer is given. In fact it is our experience that the probability that a mathematician at first sight will gamble on the wrong answer, is an increasing function of his familiarity with elementary analysis, and is always above 1/2. This paper will answer the reversed question above, in a somewhat more general setting, and give applications