Large compound lotteries

Extending preferences over simple lotteries to compound (two-stage) lotteries can be done using two different methods: (1) using the Reduction of compound lotteries axiom, under which probabilities of the two stages are multiplied; (2) using the compound independence axiom, under which each second stage lottery is replaced by its certainty equivalent. Except for expected utility preferences, the rankings induced by the two methods are always in disagreement and deciding on which method to use is not straightforward. Moreover, sometimes each of the two methods may seem to violate some kind of monotonicity. In this paper we demonstrate that, under some conditions, the disagreement disappears in the limit and that for (almost) any pair of compound lotteries, the two methods agree if the second stage lotteries are replicated sufficiently many times.