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{{Short description|Concept in decision theory}}

'''Subjective expected utility''' is a method in [[decision theory]] in the presence of [[risk]] originally put forward by [[L. J. Savage]] in [[1954]]. It combines two distinct [[subjective]] concepts: a personal [[utility]] function and a personal [[probability]] analysis based on [[Bayesian probability]] theory.

In [[decision theory]], '''subjective expected utility''' (SEU) is a framework for modeling how individuals make choices under uncertainty. In particular, it posits that decision-makers have 1) a subjective probability distribution over uncertain states of the world; and 2) a utility function over consequences such that their choice behavior can be described as maximizing [[expected utility]] over consequences with respect to their subjective probability.<ref name = "kreps1988">{{cite book |last1=Kreps |first1=David | author1-link = David Kreps | title=Notes on the Theory of Choice |date=1988 |publisher=Westview Press |isbn=978-0813375533}}</ref> This way, the theory of subjective expected utility combines two subjective concepts: a personal [[utility]] function, and a personal [[probability]] distribution (usually based on [[Bayesian probability]] theory).<ref>{{cite book |last1=Gilboa |first1=Itzhak |author1-link=Itzhak Gilboa |title=Theory of Decision under Uncertainty |date=2009 |publisher=Cambridge University Press |___location=New York |isbn=978-0521741231}}</ref>

SEU is a different approach from [[Von Neumann–Morgenstern utility theorem |the one]] put forward by [[John von Neumann| von Neumann]] and [[Oskar Morgenstern |Morgenstern]] in that it does not take (objecive) probabilities (i.e., lotteries) as given. Instead, subjective probabilities are used, which are assumed to be consistent with choice behavior.<ref>{{cite journal |last1=Machina |first1=Mark J. |last2=Schmeidler |first2=David |author1-link=Mark J. Machina |author2-link=David Schmeidler |title=A More Robust Definition of Subjective Probability |journal=Econometrica |date=1992 |volume=60 |issue=4 |pages=745-780 |doi=10.2307/2951565}}</ref>

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> and where you believe that the probability of each outcome is <math>P(x_i)</math>, then your subjective expected utility will be

:<math>\sum_i \; u(x_i) \; P(x_i).</math>

You may be able to make a decision which changes the possible outcomes to <math>\{y_j\}</math> in which case your subjective expected utility will become

:<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.

The main contribution to formalizing SEU was done by [[L. J. Savage]] in 1954 (see [[Savage's subjective expected utility model | Savage's axioms]]),<ref>Savage, Leonard J. 1954. ''The Foundations of Statistics''. New York, Wiley.</ref><ref>Karni, Edi. "Savage's subjective expected utility model." The New Palgrave Dictionary of Economics. Second Edition. Eds. Steven N. Durlauf and Lawrence E. Blume. Palgrave Macmillan, 2008. The New Palgrave Dictionary of Economics Online. Palgrave Macmillan. 23 August 2014 <http://www.dictionaryofeconomics.com/article?id=pde2008_S000479> {{doi|10.1057/9780230226203.1474}}</ref> following previous work by [[Frank P. Ramsey|Ramsey]]<ref name = "ramsey1931">{{cite book |last1=Ramsey | first1 = Frank | editor-last = Braithwaite | editor-first = R. B. | author1-link= Frank P. Ramsey| editor-link= R.B. Braithwaite | title= The Foundations of Mathematics and Other Logical Essays |date=1931 |publisher=Kegan Paul, Trench, Trubner, & Co|___location=London |chapter=Chapter 4: Truth and Probability}}</ref> and [[John von Neumann|von Neumann]].<ref>{{cite book |last1=von Neumann |first1=John |last2=Morgenstern |first2=Oskar |author1-link=John von Neumann |author2-link=Oskar Morgenstern |title=Theory of Games and Economic Behavior |date=1944 |publisher=Princeton University Press |isbn=978-0691130613}}</ref>{{r|g=nb|r=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.}} Savage proved that, if the decision-maker preferences over acts satisfy some reasonable axioms, then their choices can be explained as arising from a utility function <math>u(x_i)</math> combined with the subjective belief that there is a probability of each outcome <math>P(x_i).</math> The subjective expected utility is the resulting [[expected value]] of the utility:

Savage assumed that it was possible to take convex combinations 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>.

:<math>\Epsilon[u(X)] = \sum_i \; u(x_i) \; P(x_i) .</math>

Experiments involving offering people [[lottery]] tickets have suggested that many individuals do not seem to have personally consistent utility functions in the face of risk; Savage's response was not that this showed a flaw in his method, but that applying his method allowed individuals to improve their decision taking.

Experiments have shown that many individuals do not behave in a manner consistent with Savage's axioms of 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 | journal = Econometrica | volume = 21 | issue = 4| pages = 503–546 | doi = 10.2307/1907921 | jstor = 1907921 }}</ref>

and [[Ellsberg paradox|Ellsberg]] (1961).<ref>{{cite journal | last1 = Ellsberg | first1 = Daniel | year = 1961 | title = Risk, Ambiguity and Savage Axioms | url =http://www.dklevine.com/archive/refs47605.pdf | journal = Quarterly Journal of Economics | volume = 75 | issue = 4| pages = 643–79 | doi = 10.2307/1884324 | jstor = 1884324 }}</ref>

== See also ==

* [[Savage's subjective expected utility model]]

== Notes ==

{{reflist|group=nb}}

== References ==

{{reflist}}

* {{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” ([https://web.archive.org/web/20080227205205/http://cepa.newschool.edu/het//texts/ramsey/ramsess.pdf PDF]), Chapter VII in ''The Foundations of Mathematics and other Logical Essays'' (1931).

* [[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,

: [[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.

* [[Bruno de Finetti|de Finetti, Bruno]]. ''Theory of Probability'', (translation by [[AFM Smith]] of 1970 book) 2 volumes, New York: Wiley, 1974–5.

* {{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=[https://archive.org/details/essaysinmathemat0000shub/page/237 237–251]|title=Essays in Mathematical Economics In Honor of Oskar Morgenstern|url=https://archive.org/details/essaysinmathemat0000shub|url-access=registration|editor=[[Martin Shubik]] }}

*{{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|author-link=Oskar Morgenstern}}

== External links ==

* {{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|access-date=2009-02-17}}

{{DEFAULTSORT:Subjective Expected Utility}}

{{Decision theory}}

[[Category:Social philosophy]]

[[Category:Expected utility]]