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Roubens, Vincke & Pirlot have summarized and extended representation theorems for partial semiorders in the 80s and 90s. See e.g. Roubens, M. & Vincke, P.: Preference Modelling. Springer, 1985. Vincke, P. & Pirlot, M.: Semiorders: Properties, Representations, Applications. Springer, 1997.

Basically, for finite domains, if $P$ is a strict preference and $I$ is a nontransitive indifference relation such that $R=P\cup I$ is an incomplete semiorder, then a utility function $u(.)$ and a constant $\delta$ partially represent the ordering in the following way:

$aPb \Rightarrow u(a)>u(b)+\delta$

and

$aIb \Rightarrow |u(a)-u(b)|\leq \delta$

I'm not an economist but currently work on metaethical problems in philosophy, and was wondering whether any economists or decision theorists have used this type of representation, e.g. for consumer preferences or for some utility-maximizing decision making? In other words, is this type of thing an "old hat" in economics or rather uncommon? Do economists use it?

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