 A substantial proportion of the responses to Allais-type phenomena have involved generalisations of SEU that remain conservative enough to preserve the requirement of what Machina &amp; Schmeidler (1992) call “probabilistic sophistication”: that preferences over acts reduce to preferences over lotteries and that these in turn obey Mixture Weak Order, Mixture Continuity and Stochastic Dominance, if not  Independence.[6]  Machina &amp; Schmeidler offer an axiomatic characterisation of probabilistically sophisticated preferences that gives up Savage’s Sure-Thing condition, which plays a critical role in the derivation of Independence, and retains the remainder of his conditions. Since the Sure-Thing principle, however, also plays an important role in ensuring the existence of a suitable probability distribution over the set of events, they strengthen the Weak Comparative Probability condition to the following: