Bayesian beliefs with stochastic monotonicity: An extension of Machina and Schmeidler

Machina and Schmeidler show that the probabilistic sophistication can be obtained in an Anscombe-Aumann setting without imposing expected utility by maintaining stochastic monotonicity and adding a new axiom loosely analogous to Savage's P4. This analogous axiom, however, is very strong. In this note, we obtain probabilistic sophistication using a weaker (and more natural) analog of Savage's P4. Stochastic monotonicity is sufficient to bridge the gap, where Anscombe and Aumman use independence twice, we use stochastic monotonicity twice.