Betweenness satisfying preferences and dynamic choice

We consider the dynamic choice problem where uncertainty is resolved gradually (i.e., decision trees). We impose consistency on the decision maker by requiring that his behavior in trees conform to preference maximization over lotteries. We formulate and defend three different requirements on dynamic behavior and show that for any decision maker whose behavior is consistent with maximizing a continuous, monotone preference relation over lotteries, each one of these conditions is equivalent to the betweenness property of the underlying preferences.