O'Callaghan, Patrick (2017): Axioms for Measuring without mixing apples and Oranges.

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Abstract
A mixture set is pathconnected via a suitable collection of paths, the most common example being a convex set. Yet in many economic settings, there are pairs of prospects that are not connected by a path of mixtures. Consider the thought experiment of von Neumann and Morgenstern involving a glass of milk, a glass of tea and a cup of coffee: we are often asked to choose between convex combinations of milk and tea, yet the same cannot be said of tea and coffee. We introduce partial mixture sets (which need not be pathconnected) and provide a formal extension of the wellknown axiomatisation of cardinal, linear utility by Herstein and Milnor. We show that partial mixture sets encompass a variety of settings in the literature and present a novel application to cardinal, nonlinear utility on a stochastic process.
Item Type:  MPRA Paper 

Original Title:  Axioms for Measuring without mixing apples and Oranges 
Language:  English 
Keywords:  Utility, Preferences, mixtures 
Subjects:  C  Mathematical and Quantitative Methods > C0  General C  Mathematical and Quantitative Methods > C0  General > C02  Mathematical Methods D  Microeconomics > D0  General D  Microeconomics > D0  General > D01  Microeconomic Behavior: Underlying Principles 
Item ID:  81196 
Depositing User:  Mr Patrick O'Callaghan 
Date Deposited:  07 Sep 2017 11:46 
Last Modified:  07 Oct 2019 06:47 
References:  K. Back. “Insider Trading in Continuous Time”. In: The Review of Financial Studies 5.3 (1992), pp. 387–409. C. A. Ball and W. N. Torous. “Bond Price Dynamics and Options”. In: The Journal of Financial and Quantitative Analysis 18.4 (1983), pp. 517–531. J. Bertoin and J. Pitman. “Path transformations connecting Brownian bridge, excursion and meander”. In: Bulletin des sciences mathématiques 118.2 (1994), pp. 147–166. S. H. Chew, L. G. Epstein, and U. Segal. “Mixture symmetry and quadratic utility”. In: Econometrica: Journal of the Econometric Society (1991), pp. 139–163. S. H. Chew, L. G. Epstein, and U. Segal. “The projective independence axiom”. In: Economic Theory 4.2 (1994), pp. 189–215. P. C. Fishburn. “Axioms for expected utility in nperson games”. In: International Journal of Game Theory 5.2 (June 1976), pp. 137–149. P. C. Fishburn. The foundations of expected utility. Theory & Decision Library V.31. Springer Science and Business Media, BV, 1982. P. C. Fishburn and F. S. Roberts. “Mixture axioms in linear and mul tilinear utility theories”. English. In: Theory and Decision 9.2 (1978), pp. 161–171. I. Gilboa and D. Schmeidler. A theory of casebased decisions. Cambridge University Press, 2001. I. Gilboa and D. Schmeidler. “Inductive Inference: An Axiomatic Approach”. In: Econometrica 71.1 (2003), pp. 1–26. S. Grant et al. “Generalized utilitarianism and Harsanyi’s impartial observer theorem”. In: Econometrica 78.6 (2010), pp. 1939–1971. I. N. Herstein and J. Milnor. “An Axiomatic Approach to Measurable Utility”. English. In: Econometrica 21.2 (1953), pp. 291–297. E. Karni. “A theory of medical decision making under uncertainty”. In: Journal of Risk and Uncertainty 39.1 (2009), pp. 1–16. E. Karni and Z. Safra. “An extension of a theorem of von Neumann and Morgenstern with an application to social choice theory”. In: Journal of Mathematical Economics 34.3 (2000), pp. 315–327. D. H. Krantz et al. Foundations of measurement, Vol.I: Additive and polynomial representations. New York: Academic Press, 1971. D. M. Kreps. Notes on the theory of choice. Underground classics in economics. Westview Press, 1988. P. Mongin. “A note on mixture sets in decision theory”. In: Decisions in Economics and Finance 24.1 (2001), pp. 59–69. J. R. Munkres. Topology. 2nd. Prentice Hall, 2000. P. A. Samuelson. “Probability, utility, and the independence axiom”. In: Econometrica: Journal of the Econometric Society (1952), pp. 670– 678. B. C. Schipper. “Awarenessdependent subjective expected utility”. In: International Journal of Game Theory 42.3 (2013), pp. 725–753. D. Schmeidler. “Subjective probability and expected utility without additivity”. In: Econometrica 57.3 (1989), pp. 571–587. L. A. Steen and J. A. Seebach. Counterexamples in Topology. 2nd ed. SpringerVerlag New York, 1978. J. von Neumann and O. Morgenstern. Theory of games and economic behavior. Sixtieth anniversary. Princeton and Oxford: Princeton University Press, 1944. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/81196 