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The non-existence of a utility function and the structure of non-representable preference relations

2002; Elsevier BV; Volume: 37; Issue: 1 Linguagem: Inglês

10.1016/s0304-4068(02)00003-4

1873-1538

Alan F. Beardon, Juan C. Candeal, Gerhard Herden, Esteban Induráin, Ghanshyam B. Mehta,

Advanced Algebra and Logic

In this paper we investigate the structure of non-representable preference relations. While there is a vast literature on different kinds of preference relations that can be represented by a real-valued utility function, very little is known or understood about preference relations that cannot be represented by a real-valued utility function. There has been no systematic analysis of the non-representation problem. In this paper we give a complete description of non-representable preference relations which are total preorders or chains. We introduce and study the properties of four classes of non-representable chains: long chains, planar chains, Aronszajn-like chains and Souslin chains. In the main theorem of the paper we prove that a chain is non-representable if and only it is a long chain, a planar chain, an Aronszajn-like chain or a Souslin chain.