A theory of quantifiable beliefs

Building upon the works of Anscombe and Amuann [Ann. Math. Stat. 34 (1963) 199] and Karni and Schmeidler [An expected utility theory for state-dependent preferences. Working Paper 48-80, Foerder Institute for Economic Research, Tel Aviv University], 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 [An expected utility theory for state-dependent preferences. Working Paper 48-80, Foerder Institute for Economic Research, Tel Aviv University] and the state-independent non-expected utility model of Machina and Schmeidler [J. Econ. Theory 67 (1995) 106] as special cases. The theory is flexible enough to admit recursivity in the decision-making process