| A. F. Beardon, Juan Carlos Candeal, Universidad de Zaragoza G. Herden, Esteban Indurain, Universidad Publica de Navarra G. B. Mehta, |
| The Non-Existence of a Utility Function and the Structure of Non-Representable Chains |
| Session: C-11-14 Tuesday 15 August 2000 by Candeal, Juan Carlos |
| In this paper we study and explain the problem of the non-representability of a linearly ordered set (chain) by a real-valued function. Basically, we find that the chains for which the representability fails are closely related to the long line or the lexicographic ordered plane or certain chains, which seem to be new in the literature, that we call Aronszajn Chains. The search for topological conditions for a non-representable chain to be a planar chain, i.e. and roughly speaking, to contain a non-representable subset of the lexicographic plane, is related to the Souslin Hypothesis of set theory. We then introduce the concepts of Aronszajn-like Chain and Souslin Chain, respectively. Finally, we obtain the following structure theorem for non-representable chains: Every non-representable chain is long or planar or an Aronszajn-like Chain or a Souslin Chain. Also, some other topics of topological nature are discussed in the paper. Our approach applies obviously to the case of non-representable preference relations. So, the results we obtain are of interest for both utility theory and welfare analysis. |