Recursive utility for stochastic trees

Stochastic trees are semi-Markov processes represented using tree diagrams. Such trees have been found useful for prescriptive modeling of temporal medical treatment choice. We consider utility functions over stochastic trees which permit recursive evaluation In a graphically intuitive manner analogous to decision tree rollback. Such rollback is computationally intractable unless a low-dimensional preference summary exists. We present the most general classes of utility functions having specific tractable preference summaries. We examine three preference summaries - memoryless, Markovian, and semi-Markovian - which promise both computational feasibility and convenience in assessment. Their use is illustrated by application to a previous medical decision analysis of whether to perform carotid endarterectomy.