This article describes a modification of the Allais paradox that induces preferences inconsistent with two conditions weaker than the independence axiom, namely quasi-convacity (a special case of which is the bemeenness axiom), and Hypothesis II of Machina (also called fanning-out).
These violations can be formally derived from prospect theory by invoking a nonlinear transformation of probability into decision weight.
This article describes a variation on the Allais paradox that presents difficulties for two widely discussed restrictions on nonexpected utility models, namely, Machina's Hypothesis II (1982) which states that stochastically dominating shifts in distribution increase local risk aversion, and quasi-convexity, which implies that a mixture of two distributions may not be strictly preferred to both of its component parts. (An important special case of quasi-convexity is the betweenness or dominance axiom, which implies, in addition, that the mixture may not be considered strictly worse than the two original gambles (Chew, 1983;Chew and MacCrimmon, 1979;Dekel, 1986; Fishburn, 1983). In terms of indifference curves in the probability simplex, Hypothesis II is consistent with curves that "fan out," quasi-convexity with curves that are (weakly) concave with respect to the best (top) outcome, while betweenness/dominance generates curves that are straight (though not necessarily parallel) lines (see figure 1).
The list of current theories that assume at least one of these restrictions is long (for excellent surveys, see Machina, 1987; Weber and Camerer, 1987). Betweenness is assumed by the implicit expected utility theory of Chew (1983) and Dekel(1983), Chew and MacCrimmon's (1979) weighted utility theory. Fishburn's (1983) skew-symmetric bilinear (SSB) theory, and Guls's (1988) recent theory of disappointment aversion; Hypothesis II is, of course, advanced in Machina's articles, but it is also included in Chew and Waller's (1986) fight hypothesis, and in a transitive version of Loomes and Sugden's (1987) regret theory; finally, quasi-convexity is needed to allow for risk aversion in Becker *I would like to thank David Bell, Vijay Krishna, John Pratt, and especially Colin Camerer for helpful comments and criticism.