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.