nach oben

Theory and Decision

Erschienen in:

11.02.2015

Purely subjective extended Bayesian models with Knightian unambiguity

verfasst von: Xiangyu Qu

Erschienen in: Theory and Decision | Ausgabe 4/2015

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

This paper provides a model of belief representation in which ambiguity and unambiguity are endogenously distinguished in a purely subjective setting where objects of choices are, as usual, maps from states to consequences. Specifically, I first extend the maxmin expected utility theory and get a representation of beliefs such that the probabilistic beliefs over each ambiguous event are represented by a non-degenerate interval, while the ones over each unambiguous event are represented by a number. I then consider a class of the biseparable preferences. Two representation results are achieved and can be used to identify the unambiguity in the context of the biseparable preferences. Finally a subjective definition of ambiguity is suggested. It provides a choice theoretic foundation for the Knightian distinction between ambiguity and unambiguity.

Nächster Artikel Detecting heterogeneous risk attitudes with mixed gambles

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Nur mit Berechtigung zugänglich

This distinction goes by many names: risk versus uncertainty (Knight 1921); unambiguity versus ambiguity (Ellsberg 1961); precise versus vague probability (Savage 1972), and so forth.

See, for example Halevy (2007), Machina (2009) and Placido et al. (2011).

A good reference about biseparable preference is Luce (2000).

The general criticism of Epstein–Zhang’s definition is in Wakker (2008).

Kopylov (2007) discusses another possible structure, called mosaic, of collection of unambiguous events.

The comparison of different approach is discussed in Alon and Schmeidler (2014).

SEU representation result under our setting is introduced in Kobberling and Wakker (2003).

The intense discussion appears in Ghirardato and Marinacci (2001).

Note the above theorem will hold if we replace A5\(^*\) by binary comonotonic tradeoff consistency and the A9 by binary tradeoff consistency. This is because as Kobberling and Wakker (2003) show that with weak order, continuity and monotonicity assumptions, A5\(^*\) and A9 imply binary comonotonic tradeoff consistency and binary tradeoff consistency, respectively. We adopt traditional version of Bisymmetry to make the connection with Alon and Schmeidler (2014) and Ghirardato and Marinacci (2001) clear.

Notice that the midpoint operation \(\langle x;y\rangle \sim ^*\langle y;z\rangle \) defined in (4) is also valid here. The technique tool we provide here is consistent with those used in Ghirardato and Marinacci (2001) and Alon and Schmeidler (2014), which will make the connection transparent.

Another view about complementary additivity appears at Wakker (2004). He claims that many departures from ambiguity neutrality are induced by ambiguity insensitivity, not necessarily ambiguity aversion. He further concludes that complementary additivity is actually ambiguous insensitive.

Alon, S., & Schmeidler, D. (2014). Purely subjective maxmin expected utility. Journal of Economic Theory, 152, 382–412.MATHMathSciNetCrossRef

Amarante, M., & Filiz, E. (2007). Ambiguous events and maxmin expected utility. Journal of Economic Theory, 134(1), 1–33.MATHMathSciNetCrossRef

Anscombe, F., & Aumann, R. (1963). A definition of subjective probability. Annals of Mathematical Statistics, 34, 199–205.MATHMathSciNetCrossRef

Casadesus-Masanell, R., Klibanoff, P., & Ozdenoren, E. (2000). Maxmin expected utility over savage acts with a set of priors. Journal of Economic Theory, 92(1), 35–65.MATHMathSciNetCrossRef

Cerreia-Vioglio, S., Ghirardato, P., Maccheroni, F., Marinacci, M., & Siniscalchi, M. (2011). Rational preferences under ambiguity. Economic Theory, 48, 341–375.MATHMathSciNetCrossRef

Chew, S. H., & Sagi, J. S. (2008). Small worlds: Modeling attitudes toward sources of uncertainty. Journal of Economic Theory, 139(1), 1–24.MATHMathSciNetCrossRef

de Finetti, B. (1937). La prevision: Ses lois logiques, ses sources subjectives. Annales de l’ Institute Henri Poincare, 7, 1–68.

Ellsberg, D. (1961). Risk, ambiguity, and the savage axioms. The Quarterly Journal of Economics, 75(4), 643–669. http://www.jstor.org/stable/1884324.

Epstein, L. G., & Zhang, J. (2001). Subjective probabilities on subjectively unambiguous events. Econometrica, 69(2), 265–306.MATHMathSciNetCrossRef

Ghirardato, P., & Marinacci, M. (2001). Risk, ambiguity, and the separation of utility and beliefs. Mathematics of Operations Research, 26(4), 864–890.MATHMathSciNetCrossRef

Ghirardato, P., & Marinacci, M. (2002). Ambiguity made precise: A comparative foundation. Journal of Economic Theory, 102(2), 251–289.MATHMathSciNetCrossRef

Gilboa, I., & Schmeidler, D. (1989). Maxmin expected utility with non-unique prior. Journal of Mathematical Economics, 18(2), 141–153.MATHMathSciNetCrossRef

Halevy, Y. (2007). Ellsberg revisited: An experimental study. Econometrica, 75(2), 503–536.MATHCrossRef

Hong, C. S., & Karni, E. (1994). Choquet expected utility with a finite state space: Commutativity and Act-independence. Journal of Economic Theory, 62(2), 469–479.MATHMathSciNetCrossRef

Keynes, J. M. (1921). A treatise on probability. London: Macmillan.MATH

Klibanoff, P., Marinacci, M., & Mukerji, S. (2005). A smooth model of decision making under ambiguity. Econometrica, 73(6), 1849–1892.MATHMathSciNetCrossRef

Klibanoff, P., Marinacci, M., & Mukerji, S. (2011). Definitions of ambiguous events and the smooth ambiguity model. Economic Theory, 48, 399–424.MATHMathSciNetCrossRef

Knight, F. H. (1921). Risk, uncertainty and profit. Boston: Houghton Mifflin.

Kobberling, V., & Wakker, P. (2003). Preference foundations for nonexpected utility: A generalized and simplified technique. Mathematics of Operations Research, 28(3), 395–423.MathSciNetCrossRef

Kopylov, I. (2007). Subjective probabilities on “small” domains. Journal of Economic Theory, 133(1), 236–265.MATHMathSciNetCrossRef

Luce, D. (2000). Utility of gains and losses: Measurement theoretical and experimental approaches. London: Lawrence Erlbaum Publishers.

Machina, M. J. (2009). Risk, ambiguity, and the rank-dependence axioms. The American Economic Review, 99(1), 385–392.CrossRef

Machina, M. J., & Schmeidler, D. (1992). A more robust definition of subjective probability. Econometrica, 60(4), 745–780.MATHMathSciNetCrossRef

Nakamura, Y. (1990). Subjective expected utility with non-additive probabilities on finite state spaces. Journal of Economic Theory, 51(2), 346–366.MATHMathSciNetCrossRef

Nehring, K. (1999). Capacities and probabilistic beliefs: A precarious coexistence. Mathematical Social Sciences, 38(2), 197–213.MATHMathSciNetCrossRef

Nehring, K. (2006). Is it possible to define subjective probabilities in purely behavioral terms? A comment on Epstein–Zhang (2001). mimeo.

Pfanzagl, J. (1968). Theory of measurement. New York: Wiley.MATH

Placido, L., Mohammed, A., Baillon, A., & Wakker, P. (2011). The rich domain of uncertainty: Source functions and their experimental implementation. American Economic Review, 101(2), 695–723.CrossRef

Qu, X. (2013). Maxmin expected utility with additivity on unambiguous events. Journal of Mathematical Economics, 49(3), 245–249.MATHMathSciNetCrossRef

Ramsey, F. P. (1931). Truth and probability. In The foundations of mathematics and other logical essays. London: K. Paul, Trench, Trubner, and Co. (Reprinted from Ramsey (1978), Kyburg and Smokler (1980), and Gardenfors and Sahlin (1988)).

Sarin, R., & Wakker, P. (1992). A simple axiomatization of nonadditive expected utility. Econometrica, 60(6), 1255–1272.MATHMathSciNetCrossRef

Savage, L. J. (1972). The foundations of statistics (2nd ed.). New York: Dover.MATH

Schmeidler, D. (1986). Integral representation without additivity. Proceedings of the American Mathematical Society, 97(2), 255–261.MATHMathSciNetCrossRef

Schmeidler, D. (1989). Subjective probability and expected utility without additivity. Econometrica, 57(3), 571–587.MATHMathSciNetCrossRef

Wakker, P. (1984). Cardinal coordinate independence for expected utility. Journal of Mathematical Psychology, 28(1), 110–117.MATHMathSciNetCrossRef

Wakker, P. (2004). On the composition of risk preference and beliefs. Psychological Review, 111(1), 236–241.

Wakker, P. (2008). Uncertainty. In L. Blume & S. Durlauf (Eds.), The New Palgrave: A dictionary of economics (Vol. 8). New York: Palgrave Macmillan.

Wakker, P. P. (2010). Prospect theory for risk and ambiguity. Cambridge: Cambridge University Press.MATHCrossRef

Zhang, J. (2002). Subjective ambiguity, expected utility and Choquet expected utility. Economic Theory, 20(1), 159–181.MATHMathSciNetCrossRef

Titel: Purely subjective extended Bayesian models with Knightian unambiguity
verfasst von: Xiangyu Qu
Publikationsdatum: 11.02.2015
Verlag: Springer US
Erschienen in: Theory and Decision / Ausgabe 4/2015
Print ISSN: 0040-5833
Elektronische ISSN: 1573-7187
DOI: https://doi.org/10.1007/s11238-015-9489-9

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Weitere Artikel der Ausgabe 4/2015

An experiment on case-based decision making

Sub-models for interactive unawareness

An experimental study on the effect of ambiguity in a coordination game

An experimental test of a search model under ambiguity

Detecting heterogeneous risk attitudes with mixed gambles

Two preference metrics provide settings for the study of properties of binary relations