Subjective expected utility

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.^[1] 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).^[2]

SEU is a different approach from 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.^[3]

The main contribution to formalizing SEU was done by L. J. Savage in 1954 (see Savage's axioms),^[4]^[5] following previous work by Ramsey^[6] and von Neumann.^[7]^{[nb 1]} 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:

\mathrm {E} [u(X)]=\sum _{i}\;u(x_{i})\;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)^[8] and Ellsberg (1961).^[9]

Notes

^ 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.

References

^ Kreps, David (1988). Notes on the Theory of Choice. Westview Press. ISBN 978-0813375533.
^ Gilboa, Itzhak (2009). Theory of Decision under Uncertainty. New York: Cambridge University Press. ISBN 978-0521741231.
^ Machina, Mark J.; Schmeidler, David (1992). "A More Robust Definition of Subjective Probability". Econometrica. 60 (4): 745–780. doi:10.2307/2951565.
^ 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 <http://www.dictionaryofeconomics.com/article?id=pde2008_S000479> doi:10.1057/9780230226203.1474
^ Ramsey, Frank (1931). "Chapter 4: Truth and Probability". In Braithwaite, R. B. (ed.). The Foundations of Mathematics and Other Logical Essays. London: Kegan Paul, Trench, Trubner, & Co.
^ von Neumann, John; Morgenstern, Oskar (1944). Theory of Games and Economic Behavior. Princeton University Press. ISBN 978-0691130613. {{cite book}}: ISBN / Date incompatibility (help)
^ Allais, M. (1953). "Le Comportement de l'Homme Rationnel Devant Le Risque: Critique des Postulats et Axiomes de L'Ecole Americaine". Econometrica. 21 (4): 503–546. doi:10.2307/1907921. JSTOR 1907921.
^ Ellsberg, Daniel (1961). "Risk, Ambiguity and Savage Axioms" (PDF). Quarterly Journal of Economics. 75 (4): 643–79. doi:10.2307/1884324. JSTOR 1884324.

Charles Sanders Peirce and Joseph Jastrow (1885). "On Small Differences in Sensation". Memoirs of the National Academy of Sciences. 3: 73–83. http://psychclassics.yorku.ca/Peirce/small-diffs.htm
Ramsey, Frank Plumpton; “Truth and Probability” (PDF), Chapter VII in The Foundations of Mathematics and other Logical Essays (1931).
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.
de Finetti, Bruno. 1937, “La Prévision: ses lois logiques, ses sources subjectives,” Annales de l'Institut Henri Poincaré,

de Finetti, Bruno. "Foresight: its Logical Laws, Its Subjective Sources," (translation of the 1937 article in French) in H. E. Kyburg and H. E. Smokler (eds), Studies in Subjective Probability, New York: Wiley, 1964.

de Finetti, Bruno. Theory of Probability, (translation by AFM Smith of 1970 book) 2 volumes, New York: Wiley, 1974–5.
Donald Davidson, Patrick Suppes and Sidney Siegel (1957). Decision-Making: An Experimental Approach. Stanford University Press.
Pfanzagl, J (1967). "Subjective Probability Derived from the Morgenstern–von Neumann Utility Theory". In Martin Shubik (ed.). Essays in Mathematical Economics In Honor of Oskar Morgenstern. Princeton University Press. pp. 237–251.
Morgenstern, Oskar (1976). "Some Reflections on Utility". In Andrew Schotter (ed.). Selected Economic Writings of Oskar Morgenstern. New York University Press. pp. 65–70. ISBN 0-8147-7771-6.