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'''Subjective [[expected utility]]''' is a method in [[decision theory]] in the presence of [[risk]], promoted by [[L. J. Savage]] in 1954<ref>Savage, Leonard J. 1954. ''The Foundations of Statistics''. New York, Wiley.</ref> following previous work by [[Frank P. Ramsey|Ramsey]] and [[John von Neumann|von Neumann]].<ref>Ramsey says that his essay merely elaborates on the ideas of [[Charles Sanders Peirce]]. [[John von Neumann]] noted the possibility of simultaneous theory of personal probability and utility, but his death left the specification of an axiomatization of subjective expected utility incomplete, until Johann Pfanzagl's work.</ref>  The theory of subjective expected utility combines two subjective concepts: first, a personal [[utility]] function, and second a personal [[probability]] distribution (based on [[Bayesian probability]] theory).
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Savage proved that, if you adhere to axioms of rationality, if you believe an uncertain event has possible outcomes <math>\{x_i\}</math> each with a utility to you of <math>u(x_i)</math> then your choices can be explained as arising from a function in which you believe that there is a subjective probability of each outcome is <math>P(x_i)</math>, and your subjective expected utility is the [[expected value]] of the utility,
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:<math>\sum_i \; u(x_i) \; P(x_i).</math>
 
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:<math>\sum_j \; u(y_j) \; P(y_j).</math>
 
Which decision you prefer depends on which subjective expected utility is higher. Different people may make different decisions because they may have different utility functions or different beliefs about the probabilities of different outcomes.
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Savage assumed that it is possible to take [[convex combination]]s of decisions  and that preferences would be preserved. So if you prefer <math>x(=\{x_i\})</math> to <math>y</math> and <math>s</math> to <math>t</math> then you will prefer <math>\lambda x + (1-\lambda )s</math> to <math>\lambda y + (1-\lambda )t</math>, for <math>0<\lambda<1</math>.
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Experiments have shown that many individuals do not behave in a manner consistent with subjective expected utility, e.g. most prominently [[Allais paradox|Allais]] (1953)<ref>{{cite journal | last1 = Allais | first1 = M. | year = 1953 | title = Le Comportement de l'Homme Rationnel Devant Le Risque: Critique des Postulats et Axiomes de L'Ecole Americaine | url = | journal = Econometrica | volume = 21 | issue = 4| pages = 503–546 | doi = 10.2307/1907921 }}</ref>
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* {{cite journal| author=[[Charles Sanders Peirce]] and [[Joseph Jastrow]]|year=1885|title=On Small Differences in Sensation| journal=Memoirs of the National Academy of Sciences|volume=3|pages=73–83|url=http://psychclassics.yorku.ca/Peirce/small-diffs.htm}} http://psychclassics.yorku.ca/Peirce/small-diffs.htm
 
* [[Frank P. Ramsey|Ramsey, Frank Plumpton]]; “Truth and Probability” ([http://cepa.newschool.edu/het//texts/ramsey/ramsess.pdf  PDF]), Chapter VII in ''The Foundations of Mathematics and other Logical Essays'' (1931).
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* [[Bruno de Finetti|de Finetti, Bruno]]. "Probabilism: A Critical Essay on the Theory of Probability and on the Value of Science," (translation of 1931 article) in ''Erkenntnis,'' volume 31, September 1989.
 
* [[Bruno de Finetti|de Finetti, Bruno]]. 1937, “La Prévision: ses lois logiques, ses sources subjectives,” Annales de l'Institut Henri Poincaré,
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: [[Bruno de Finetti|de Finetti, Bruno]]. "Foresight: its Logical Laws, Its Subjective Sources," (translation of the [http://www.numdam.org/item?id=AIHP_1937__7_1_1_0  1937 article] in French) in H. E. Kyburg and H. E. Smokler (eds), ''Studies in Subjective Probability,'' New York: Wiley, 1964.
 
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* {{cite book| title=Decision-Making: An Experimental Approach| author=[[Donald Davidson (philosopher)|Donald Davidson]], [[Patrick Suppes]] and [[Sidney Siegel]]| publisher=[[Stanford University Press]]|year=1957}}
 
*{{cite book|author=Pfanzagl, J|year=1967|publisher=Princeton University Press|chapter=Subjective Probability Derived from the [[Oskar Morgenstern|Morgenstern]]–[[John von Neumann|von Neumann]] [[Expected utility|Utility Theory]]|pages=237–251|title=Essays in Mathematical Economics In Honor of Oskar Morgenstern|editor=[[Martin Shubik]] }}
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*{{cite book|author=Pfanzagl, J.|year=1968|publisher=Wiley|chapter=Events, Utility and Subjective Probability|pages=195–220|title=Theory of Measurement}}
 
*{{cite book|author=Morgenstern, Oskar|year=1976|publisher=New York University Press|chapter=Some Reflections on [[Expected utility|Utility]]|pages=65–70|title=Selected Economic Writings of Oskar Morgenstern|editor=Andrew Schotter|isbn=0-8147-7771-6|authorlink=Oskar Morgenstern}}
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== External links ==
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* {{cite web|url=http://www.econ.jhu.edu/people/Karni/savageseu.pdf|title=Savages’ Subjective Expected Utility Model|author=Edi Karni, Johns Hopkins University|date=November 9, 2005|accessdate=2009-02-17}}


{{DEFAULTSORT:Subjective Expected Utility}}
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