Abstract
Let \({\mathcal {E}}\) be a class of events. Conditionally Expected Utility decision makers are decision makers whose conditional preferences \(\succsim _{E}\), \(E\in {\mathcal {E}}\), satisfy the axioms of Subjective Expected Utility (SEU) theory. We extend the notion of unconditional preference that is conditionally EU to unconditional preferences that are not necessarily SEU. We study a subclass of these preferences, namely those that satisfy dynamic consistency. We give a representation theorem, and show that these preferences are Invariant Bi-separable in the sense of Ghirardato et al. (Journal of Economic Theory 118:133–173, 2004). We also show that these preferences have only a trivial overlap with the class of Choquet Expected Utility preferences, but there are plenty of preferences of the \(\alpha \)-Maxmin Expected Utility type that satisfy our assumptions. We identify several concrete settings where our results could be applied. Finally, we consider the special case where the unconditional preference is itself SEU, and compare our results with those of Fishburn (Econometrica 41:1–25, 1973).
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Notes
For a simple act \(f:S\longrightarrow C\), let \(\{c_{1},\ldots ,c_{n}\}\) be the set of its values in C. The act is measurable if \(f^{-1}\{c_{i}\}\in \Sigma \) for every i.
In our setting, uniqueness is a consequence of assumption S0.
By a capacity \(\Gamma \) on \(\mathcal {{\tilde{C}}}=\overset{\text {---}}{co}\left\{ P_{E}\right\} _{E\in {\mathcal {E}}}\) we mean a capacity on the Borel \(\sigma \)-algebra generated by the weak*-topology on \(\mathcal {{\tilde{C}}}\).
Dominiak and Lefort (2011) restrict to a finite state space. Their argument, however, applies here without such a restriction.
In general, the preferences of the corollary are not SEU. An exception obtains when \(n=2\) and \(\alpha =1/2\) (see Amarante 2009, Examples 2 and 3).
As we have seen, these two conditions are automatically satisfied in the case of \({\mathcal {D}}\)-preferences.
It is readily seen that Fishburn’s A7 and A8 are necessary conditions for the uniqueness properties above.
The meaning of the qualification “essentially” will be clarified below.
References
Amarante, M. (2009). Foundations of neo-Bayesian statistics. Journal of Economic Theory, 144, 2146–2173.
Amarante, M., & Filiz, E. (2007). Ambiguous events and maxmin expected utility. Journal of Economic Theory, 134, 1–33.
Anscombe, F. J., & Aumann, R. J. (1963). A definition of subjective probability. Annals of Mathematical Statistics, 34, 199–205.
Beissner, P., Lin, Q., & Riedel, F. (2016). Dynamically consistent \(\alpha \)-maxmin expected utility. Bielefeld University, Center for Mathematical Economics Working Paper No. 535.
Bewley, T. F. (1986). Knightian decision theory: part I, Cowles foundation discussion paper 807. Decision in Economics and Finance, 25(2002), 79–110.
Dellacherie, C., & Meyer, P-A. (1975). Probabilités et potentiel. Hermann, Paris.
Dominiak, A., Duersch, P., & Lefort, J.-P. (2012). A dynamic Ellsberg urn experiment. Games and Economic Behavior, 75, 625–638.
Dominiak, A., & Lefort, J.-P. (2011). Unambiguous events and dynamic choquet preferences. Economic Theory, 46, 401–425.
Eichberger, J., Grant, S., Kelsey, D., & Koshevoy, G. (2011). The \(\alpha \) -MEU model: a comment. Journal of Economic Theory, 48, 1684–1698.
Epstein, L., & Schneider, M. (2003). Recursive multiple priors. Journal of Economic Theory, 113, 1–31.
Fishburn, P. C. (1973). A mixture-set axiomatization of conditional subjective expected utility. Econometrica, 41, 1–25.
Ghirardato, P. (2002). Revisiting Savage in a conditional world. Economic Theory, 20, 83–92.
Ghirardato, P., Maccheroni, F., & Marinacci, M. (2004). Differentiating ambiguity and ambiguity attitude. Journal of Economic Theory, 118, 133–173.
Ghirardato, P., Maccheroni, F., Marinacci, M., & Siniscalchi, M. (2003). Subjective foundations for objective randomization: a new spin on roulette wheels. Econometrica, 71, 1897–1908.
Gilboa, I., & Schmeidler, D. (1989). Maxmin expected utility with a non-unique prior. Journal of Mathematical Economics, 18, 141–153.
Hanany, E., & Klibanoff, P. (2009). Updating ambiguity averse preferences. The BE Journal of Theoretical Economics, 9, 407. (article 37).
Luce, R. D., & Krantz, D. H. (1971). Conditional expected utility. Econometrica, 39, 253–272.
Nehring, K. (1999). Capacities and probabilistic beliefs: a precarious coexistence. Mathematical Social Sciences, 38, 197–213.
Phelps, R. R. (1966). Lectures on Choquet theorem. New York City: van Nostrand.
Phelps, R. R. (1960). Uniqueness of Hahn–Banach extension and uniqueness of best approximation. Transactions of the American Mathematical Society, 95, 238–255.
Schmeidler, D. (1989). Subjective probability and expected utility without additivity. Econometrica, 57, 571–587.
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I wish to thank Fabio Maccheroni, Rich McLean and two anonymous referees for useful suggestions. In particular, one referee clarified a great deal about the nature of the preferences studied in this paper as well as about the place that they occupy in the literature. Financial support from the SSHRC of Canada (Grant # 410-2011-2025) is gratefully acknowledged.
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Amarante, M. Conditional expected utility. Theory Decis 83, 175–193 (2017). https://doi.org/10.1007/s11238-017-9597-9
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DOI: https://doi.org/10.1007/s11238-017-9597-9