Summary
A semiorder can be thought of as a binary relationP for which there is a utilityu representing it in the following sense:xPy iffu(x) −u(y) > 1. We argue that weak orders (for which indifference is transitive) can not be considered a successful approximation of semiorders; for instance, a utility function representing a semiorder in the manner mentioned above is almost unique, i.e. cardinal and not only ordinal. In this paper we deal with semiorders on a product space and their relation to given semiorders on the original spaces. Following the intuition of Rubinstein we find surprising results: with the appropriate framework, it turns out that a Savage-type expected utility requires significantly weaker axioms than it does in the context of weak orders.
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We wish to thank Tatsuro Ichiishi, Jorge Nieto, Ariel Rubinstein, Efraim Sadka and especially David Schmeidler and anonymous referees for stimulating discussions and comments. I. Gilboa received partial financial support from NSF grants nos. IRI-8814672 and SES-9113108, as well as from the Alfred P. Sloan Foundation.
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Gilboa, I., Lapson, R. Aggregation of semiorders: intransitive indifference makes a difference. Econ Theory 5, 109–126 (1995). https://doi.org/10.1007/BF01213647
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DOI: https://doi.org/10.1007/BF01213647