×

Constructive decision theory. (English) Zbl 1471.91092

Summary: In most contemporary approaches to decision making under uncertainty, a decision problem is described by a set of states and set of outcomes, and a rich set of acts, which are functions from states to outcomes over which the decision maker (DM) has preferences. Many interesting decision problems, however, do not come with a state space and an outcome space. Indeed, in complex problems it is often far from clear what the state and outcome spaces would be. We present an alternative foundation for decision making, in which the primitive objects of choice are syntactic programs. A representation theorem is proved in the spirit of standard representation theorems, showing that if the DM’s preference relation on objects of choice satisfies appropriate axioms, then there exist a set \(S\) of states, a set \(O\) of outcomes, a way of interpreting the objects of choice as functions from \(S\) to \(O\), a probability on \(S\), and a utility function on \(O\), such that the DM prefers choice \(a\) to choice \(b\) if and only if the expected utility of \(a\) is higher than that of \(b\). Thus, the state space and outcome space are subjective, just like the probability and utility; they are not part of the description of the problem. In principle, a modeller can test for SEU behaviour without having access to states or outcomes. We illustrate the power of our approach by showing that it can capture decision makers who are subject to framing effects and those who are subject to failures of extensionality.

MSC:

91B06 Decision theory

References:

[1] Ahn, D., Ambiguity without a state space, Rev. Econ. Stud., 75, 1, 3-28 (2008) · Zbl 1141.91380
[2] Ahn, D.; Ergin, H., Framing contingencies, Econometrica, 78, 2, 655-695 (March 2010) · Zbl 1229.91098
[3] Anscombe, F. J.; Aumann, R. J., A definition of subjective utility, Ann. Math. Stat., 34, 199-205 (1963) · Zbl 0114.07204
[4] Aumann, R. J., Utility theory without the completeness axiom, Econometrica, 30, 445-462 (1962) · Zbl 0121.15202
[5] Aumann, R. J., Utility theory without the completeness axiom: a correction, Econometrica, 32, 210-212 (1964) · Zbl 0133.13004
[6] Aumann, R. J., Agreeing to disagree, Ann. Stat., 4, 6, 1236-1239 (1976) · Zbl 0379.62003
[7] Benartzi, S.; Thaler, R. H., Diversification strategies in defined contribution savings plans, Am. Econ. Rev., 91, 1, 79-98 (2001)
[8] Bewley, T., Knightian decision theory, Decis. Econ. Finance, 25, 79-110 (2002) · Zbl 1041.91023
[9] Billot, A.; Vergopoulos, V., Expected utility without parsimony, Math. Soc. Sci., 93, 14-21 (2018) · Zbl 1393.91061
[10] Blume, L.; Easley, D.; Halpern, J. Y., Redoing the foundations of decision theory, (Principles of Knowledge Representation and Reasoning: Proc. Tenth International Conference (KR’06) (2006)), 14-24
[11] Dekel, E.; Lipman, B.; Rustichini, A., Representing preferences with a unique subjective state space, Econometrica, 69, 891-934 (2001) · Zbl 1026.91032
[12] Dubra, J.; Maccheroni, F.; Ok, E., Expected utility theory without the completeness axiom, J. Econ. Theory, 115, 118-133 (2004) · Zbl 1062.91025
[13] Eliaz, K.; Ok, E., Indifference or indecisiveness? Choice-theoretic foundations of incomplete preferences, Games Econ. Behav., 56, 61-86 (2006) · Zbl 1151.91409
[14] Fagin, R.; Halpern, J. Y., Belief, awareness, and limited reasoning, Artif. Intell., 34, 39-76 (1988) · Zbl 0634.03013
[15] Feinberg, Y., Subjective reasoning—games with unawareness (2004), Stanford Graduate School of Business, Tech. Rep. Research Paper Series #1875
[16] Fishburn, P. C., Utility Theory for Decision Making (1970), John Wiley and Sons: John Wiley and Sons New York · Zbl 0213.46202
[17] Fishburn, P. C., Reconsiderations in the foundations of decision under uncertainty, Econ. J., 97, 388, 825-841 (1987)
[18] Galaabaatar, T.; Karni, E., Subjective expected utility with incomplete preferences, Econometrica, 81, 255-284 (2013) · Zbl 1274.91187
[19] Ghirardato, P., Coping with ignorance: unforeseen contingencies and non-additive uncertainty, Econ. Theory, 17, 247-276 (2001) · Zbl 0980.91010
[20] Gilboa, I.; Schmeidler, D., Subjective distributions, Theory Decis., 56, 345-357 (2004) · Zbl 1090.91014
[21] Grabiszewski, K., On the rejectability of the subjective expected utility theory, B. E. J. Theor. Econ., 16, 437-454 (2016) · Zbl 1375.91075
[22] Halpern, J. Y., Alternative semantics for unawareness, Games Econ. Behav., 37, 321-339 (2001) · Zbl 1001.03015
[23] Halpern, J. Y.; Rêgo, L. C., Extensive games with possibly unaware players, Math. Soc. Sci., 70, 42-58 (2014) · Zbl 1294.91023
[24] Heifetz, A.; Meier, M.; Schipper, B., Interactive unawareness, J. Econ. Theory, 130, 78-94 (2006) · Zbl 1141.91598
[25] Johnson, E.; Hershey, J.; Meszaros, J.; Kunreuther, H., Framing, probability distortions, and insurance decisions, J. Risk Uncertain., 7, 35-51 (1993)
[26] Karni, E., Subjective expected utility theory without states of the world, J. Math. Econ., 42, 325-342 (2006) · Zbl 1132.91425
[27] Karni, E.; Vierø, M.-L., “Reverse Bayesianism”: a choice-based theory of growing awareness, Am. Econ. Rev., 103, 7, 2790-2810 (2013)
[28] Krantz, D. H.; Luce, R. D.; Suppes, P.; Tversky, A., Foundations of Measurement, vol. 1: Additive and Polynomial Representations (1971), Academic Press: Academic Press New York · Zbl 0232.02040
[29] Kreps, D., Static choice and unforeseen contingencies, (Dasgupta, P.; Gale, D.; Hart, O., Economic Analysis of Markets and Games: Essays in Honor of Frank Hahn (1992), MIT Press: MIT Press Cambridge, Mass)
[30] Kreps, D. M., A representation theorem for “preference for flexibility”, Econometrica, 47, 3, 565-577 (1979) · Zbl 0423.90005
[31] Kreps, D. M., Notes on the Theory of Choice (1988), Westview Press: Westview Press Boulder, Colo.
[32] Lipman, B., Decision theory without logical omniscience: toward an axiomatic framework for bounded rationality, Rev. Econ. Stud., 66, 339-361 (1999) · Zbl 0962.91016
[33] Luce, R. D., Rational versus plausible accounting equivalences in preference judgments, Psychol. Sci., 1, 225-234 (1990), reprinted with minor changes in: Edwards, Ward (Ed.), Utility Theories: Measurements and Applications. Kluwer, Boston, 1992, pp. 187-206
[34] Machina, M., States of the world and state of decision theory, (Meyer, D., The Economics of Risk (2006), W. E. Upjohn Institute: W. E. Upjohn Institute Kalamazoo, MI)
[35] Mandler, M., A difficult choice in preference theory: rationality implies completeness or transitivity but not both, (Millgram, E., Varieties of Practical Reasoning (2001), MIT Press: MIT Press Cambridge MA)
[36] McNeil, B. J.; Pauker, S. J.; Sox, H. C.; Tversky, A., On the elicitation of preferences for alternative therapies, N. Engl. J. Med., 306, 1259-1262 (1982)
[37] Modica, S.; Rustichini, A., Unawareness and partitional information structures, Games Econ. Behav., 27, 2, 265-298 (1999) · Zbl 0943.91026
[38] Nau, R., The shape of incomplete preferences, Ann. Stat., 34, 2430-2448 (2006) · Zbl 1106.62001
[39] Rantala, V., Impossible worlds semantics and logical omniscience, Acta Philos. Fenn., 35, 18-24 (1982) · Zbl 0519.03002
[40] Rockafellar, R. T., Convex Analysis (1970), Princeton University Press: Princeton University Press Princeton, N.J. · Zbl 0229.90020
[41] Savage, L. J., Foundations of Statistics (1954), Wiley: Wiley New York · Zbl 0055.12604
[42] Scott, D., Measurement structures and linear inequalities, J. Math. Psychol., 1, 233-247 (1964) · Zbl 0129.12102
[43] Thaler, R. H.; Sunstein, C., Nudge: Improving Decisions About Health, Wealth, and Happiness (2009), Penguin Books
[44] Tversky, A.; Koehler, D. J., Support theory: a nonextensional representation of subjective probability, Psychol. Rev., 101, 4, 547-567 (1994)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.