This paper provides axiomatic characterizations of two sorts of recursive stochastic choice rules, where the agent makes his current decisions using a forward-looking value function that takes into account his future randomizations.  Both of the choice rules we examine generalize logistic choice and are equivalent to it in static problems.  The rules differ in how the agent views future choice sets and how he views his future randomizations.  One rule is equivalent to the discounted logit used in applied work, and exhibits a "preference for flexibility;" the other is "error-averse" and penalizes the addition of undesirable choice to a menu.