Abstract
Mixture sets were introduced by Herstein and Milnor (Econometrica 21:291–297, 1953) to prove a generalised expected utility theorem. Mixture sets provide an axiomatisation of convexity suitable for discrete, as well as continuous, environments (Mongin in Decis Econ Finance 24:59–69, 2001). However, the nature of mixture sets over finite domains has been little studied. In this paper, we provide a complete characterisation. More recently, another abstract convex structure for finite domains, the antimatroid, has appeared in the literature on decision theory and social choice. The relationship between mixture sets and antimatroids has not previously been explored. We show here that neither concept is a special case of the other.
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Thanks to Dmitriy Kvasov for prompting me to think harder about this question and to Hitotsubashi University for its hospitality while writing the paper. I have also benefited from the insightful comments of Suren Basov and an anonymous referee, as well as audiences at the University of Auckland and the 28th Australasian Economic Theory Workshop (University of Melbourne).
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Ryan, M.J. Mixture sets on finite domains. Decisions Econ Finan 33, 139–147 (2010). https://doi.org/10.1007/s10203-010-0103-x
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DOI: https://doi.org/10.1007/s10203-010-0103-x