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Theory and Decision

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16.03.2016

An expected utility theory for state-dependent preferences

verfasst von: Edi Karni, David Schmeidler

Erschienen in: Theory and Decision | Ausgabe 4/2016

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Abstract

This note is a generalization and improved interpretation of the main result of Karni and Schmeidler (An Expected utility theory for state-dependent preferences. Working paper no. 48-80 of the Foerder Institute for Economic Research, Faculty of Social Sciences, Tel Aviv University, 1980). A decision-maker is supposed to possess a preference relation on acts and another preference relation on state-prize lotteries, both of which are assumed to satisfy the von Neumann–Morgenstern axioms. In addition, the two preference relations restricted to a state of nature are assumed to agree. We show that these axioms are necessary and sufficient for the existence of subjective expected utility over acts with state-dependent utility functions and a subjective probability measure. This subjective probability measure is unique when conditioned on the set of states of nature in which not all the prizes are equally desirable.

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As Karni and Mongin (2000) note “...we conclude that even under state independence, the KS (Karni and Schmeidler) rather than the conventional AA (Anscombe and Aumann) probability should be taken to be the decision maker’s subjective probability.” (p. 234 ).

Savage’s extended model contains more comparisons and it led him to suggest the baffling consequence of “being hanged without damage to his health or reputation”. Drèze (1987, p. 68) . We do not require such consequences.

The same redundancy appears in the statements of the consistency and strong consistency axioms in Karni et al. (1983).

These representations of vNM utility already appear in Fishburn’s textbook (Fishburn 1970). For a web accessible presentation where \(X(s)=X\) for all s, see http://www.tau.ac.il/~schmeid/PDF/Decision_Theory_Technical_Notes.pdf.

Karni’s (1985) statement of the following theorem misstated the uniqueness of the utility function. The same mistake appeared in Karni and Mongin’s Proposition 3 (Karni and Mongin 2000), and is implied in the restatement of the result of Karni and Schmeidler (1980) in Karni et al. (1983).

This also is the trivial direction of the BRT.

Anscombe, F. J., & Aumann, R. J. (1963). A definition of subjective probability. Annals of Mathematical Statistics, 43, 199–205.CrossRef

Baccelli, J. (2016). Do bets reveal beliefs? A unified perspective on state-dependent utility issues. Synthese.

Cook, P. J., & Graham, D. A. (1977). The demand of insurance and protection: the case of irreplaceable commodities. Quarterly Journal of Economics, 91, 143–156.CrossRef

Drèze, J. H. (1987). Decision theory wit moral hazard and state-dependent preferences. In Essays on economic decisions under uncertainty. Cambridge: Cambridge University Press.

Dreze, J. H., & Rustichini, A. (2004). State dependent utility and decision theory. In S. Barbera, P. Hammond, & C. Seidl (Eds.), Handbook of utility theory: volume 2 extensions (pp. 839–892). New York: Springer.CrossRef

Fishburn, P. C. (1970). Utility theory for decision making. New York: Wiley.

Fishburn, P. C. (1973). A mixture-set axiomatization of conditional subjective expected utility. Econometrica, 41, 1–25.CrossRef

Fishburn, P. C. (1974). On the foundations of decision making under uncertainty. In M. S. Balch, D. L. McFadden, & S. Y. Wu (Eds.), Essays on economic behavior under uncertainty. Amsterdam: North-Holland.

Hirshleifer, J., & Riley, J. G. (1979). The analytics of uncertainty and information—an expository survey. Journal of Economic Literature, 17, 1375–1421.

Karni, E. (1985). Decision making under uncertainty: the case of state-dependent preferences. Cambridge: Harvard University Press.CrossRef

Karni, E. (2009). State-dependent utility. In P. Anand, P. K. Pattanaik, & C. Puppe (Eds.), Handbook of rational and social choice. Oxford: Oxford University Press.

Karni, E., & Mongin, P. (2000). On the Determination of subjective probability by choice. Management Science, 46, 233–248.CrossRef

Karni, E., & Safra, Z. (2016). A theory of stochastic choice under uncertianty. Journal of Mathematical Economics (forthcoming).

Karni, E., & Schmeidler, D. (1980). An Expected utility theory for state-dependent preferences. Working paper no. 48–80 of the Foerder Institute for Economic Research, Faculty of Social Sciences, Tel Aviv University.

Karni, E., Schmeidler, D., & Vind, K. (1983). On state dependent preferences and subjective probabilities. Econometrica, 51, 1021–1032.CrossRef

Lu, J. (2015). A Bayesian theory of state-dependent utilities. (unpublished).

Nau, R. (2001). De Finetti was right: probability does not exist. Theory and Decision, pp. 89–124.

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von Neumann, J., & Morgenstern, O. (1947). Theory of games and economic behavior. Princeton: Princeton University Press.

Titel: An expected utility theory for state-dependent preferences
verfasst von: Edi Karni
David Schmeidler
Publikationsdatum: 16.03.2016
Verlag: Springer US
Erschienen in: Theory and Decision / Ausgabe 4/2016
Print ISSN: 0040-5833
Elektronische ISSN: 1573-7187
DOI: https://doi.org/10.1007/s11238-016-9545-0

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