A Theory of Quantifiable Beliefs

Building upon the works of Anscombe and Aumann (1963) and Karni and Schmeidler (1981), we develop a general axiomatic theory of quantifiable beliefs - a form of probabilistic sophistication that does not preclude state-dependent preferences and does not require the reduction of compound lotteries. The theory includes the state-dependent expected utility model of Karni and Schmeidler (1981) and the state-independent non-expected utility model of Machina and Schmeidler (1995) as special cases. The theory is flexible enough to admit recursivity in the decision-making process. One specific example of this recursive class is shown to be compatible with a quantifiable beliefs version of Schmeidler's (1989) Choquet expected utility maximizing model and thus capable of rationalizing Ellsberg-paradox type behavior.

[1]  Colin Camerer.  Individual Decision Making , 2020, The Handbook of Experimental Economics.

[2]  D. Schmeidler,et al.  A More Robust Definition of Subjective Probability , 1992 .

[3]  Edi Karni,et al.  Probabilities and beliefs , 1996 .

[4]  David Schmeidleis.  SUBJECTIVE PROBABILITY AND EXPECTED UTILITY WITHOUT ADDITIVITY , 1989 .

[5]  D. Schmeidler,et al.  Bayes without Bernoulli: Simple Conditions for Probabilistically Sophisticated Choice , 1995 .

[6]  Peter C. Fishburn,et al.  A Mixture-Set Axiomatization of Conditional Subjective Expected Utility , 1973 .

[7]  S. Chew.  A Generalization of the Quasilinear Mean with Applications to the Measurement of Income Inequality and Decision Theory Resolving the Allais Paradox , 1983 .

[8]  F. J. Anscombe,et al.  A Definition of Subjective Probability , 1963 .

[9]  Peter C. Fishburn,et al.  Utility theory for decision making , 1970 .

[10]  Nils-Eric Sahlin,et al.  Decision making with unreliable probabilities , 1983 .

[11]  D. Ellsberg.  Decision, probability, and utility: Risk, ambiguity, and the Savage axioms , 1961 .

[12]  M. Schervish,et al.  State-Dependent Utilities , 1990 .

[13]  Larry G. Epstein.  Are Probabilities Used in Markets ? , 2000, J. Econ. Theory.

[14]  Simon Grant,et al.  Subjective probability without monotonicity: or how Machina's mom may also be probabilistically sophisticated , 1995 .

[15]  Larry G. Epstein,et al.  Dynamically Consistent Beliefs Must Be Bayesian , 1993 .

[16]  Edi Karni,et al.  On the Determination of Subjective Probability by Choices , 2000 .