In his classical contribution Szpilrajn (1930) shows that any quasi-ordering (a reflexive and transitive binary relation) can be extended to an ordering (a complete quasi-ordering). The purpose of this paper is to provide an analogous extension result for non-transitive preferences. The case of quasi-transitivity (transitivity of the strict preference) is trivial since it suffices to complete' the original preference by putting indifference for all pairs of alternatives that are non-comparable. The focus of our analysis is a different notion of non-transitivity called $\sigma$-transitivity and put forward bySen (1970).Our main result is an extension theorem which shows that any $\sigma$-transitive preference can be extended to a complete preference preserving$\sigma$-transitivity. We also provide the dual result in terms of a strict preference approach.Finally, we show an application to the theory of choice by providing a new characterization of the Weak Axiom of Revealed Preference.

Brighi, Luigi. "An Extension Theorem for Non-Transitive Preferences" Working paper, MATERIALI DI DISCUSSIONE, Dipartimento di Economia Politica - Università di Modena e Reggio Emilia, 2005. https://doi.org/10.25431/11380_634622

### An Extension Theorem for Non-Transitive Preferences

#### Abstract

In his classical contribution Szpilrajn (1930) shows that any quasi-ordering (a reflexive and transitive binary relation) can be extended to an ordering (a complete quasi-ordering). The purpose of this paper is to provide an analogous extension result for non-transitive preferences. The case of quasi-transitivity (transitivity of the strict preference) is trivial since it suffices to complete' the original preference by putting indifference for all pairs of alternatives that are non-comparable. The focus of our analysis is a different notion of non-transitivity called $\sigma$-transitivity and put forward bySen (1970).Our main result is an extension theorem which shows that any $\sigma$-transitive preference can be extended to a complete preference preserving$\sigma$-transitivity. We also provide the dual result in terms of a strict preference approach.Finally, we show an application to the theory of choice by providing a new characterization of the Weak Axiom of Revealed Preference.
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2005
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Brighi, Luigi
Brighi, Luigi. "An Extension Theorem for Non-Transitive Preferences" Working paper, MATERIALI DI DISCUSSIONE, Dipartimento di Economia Politica - Università di Modena e Reggio Emilia, 2005. https://doi.org/10.25431/11380_634622
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11380/634622
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