# Two-Stage Lotteries without the Reduction Axiom

https://doi.org/0012-9682(199003)58:2<349:TLWTRA>2.0.CO;2-7
p. 349-377

Uzi Segal

This paper analyzes preference relations over two-stage lotteries, i.e., lotteries having as outcomes tickets for other, simple, lotteries. Empirical evidence indicates that decision makers do not always behave in accordance with the reduction of compound lotteries axiom, but it seems that they satisfy a compound independence axiom (also known as the certainty equivalent mechanism). It turns out that although the reduction and the compound independence axioms together with continuity imply expected utility theory, each of them by itself is compatible with all possible preference relations over simple lotteries. By using these axioms I analyze three different versions of expected utility for two-stage lotteries. The second part of the paper is devoted to possible replacements of the reduction axiom. For this I suggest several different compound dominance axioms. These axioms compare two-stage lotteries by the probability they assign to the upper and lower sets of all simple lotteries \$X\$. (For a simple lottery \$X\$, its upper (lower) set is the set of lotteries that dominate (are dominated by) \$X\$ by first order stochastic dominance.) It turns out that these axioms are all strictly weaker than the reduction of compound lotteries axiom. The main theoretical results of this part are: (1) an axiomatic basis for expected utility theory that does not require the reduction axiom and (2) a new axiomatization of the anticipated utility model (also known as expected utility with rank-dependent probabilities). These representation theorems indicate that to a certain extent the rank dependent probabilities model is a natural extension of expected utility theory. Finally, I show that some paradoxes in expected utility theory can be explained, provided one is willing to use the compound independence rather than the reduction axiom.