Subjective expected utility

{{Short description|Concept in decision 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.{{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}} 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).{{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}}

SEU is a different approach from the one put forward by the one put forward by von Neumann and 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.{{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}}

The main contribution to formalizing SEU was done by L. J. Savage in 1954 (see Savage's axioms),Savage, Leonard J. 1954. The Foundations of Statistics. New York, Wiley.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 {{doi|10.1057/9780230226203.1474}} following previous work by Ramsey{{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}} and von Neumann.{{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}}{{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 $u(x\_i)$ combined with the subjective belief that there is a probability of each outcome $P(x\_i).$ The subjective expected utility is the resulting expected value of the utility:

:$\backslash Epsilon[u(X)]\; =\; \backslash sum\_i\; \backslash ;\; u(x\_i)\; \backslash ;\; P(x\_i)\; .$

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 (1953){{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 }}

and Ellsberg (1961).{{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 }}