Recursive Utility for Stochastic Trees
Statement: This work is important for several reasons. It is the first contribution to the literature on preference over risky time streams in which the utility structure is linked to specific and frequently used stochastic models (stochastic trees), and in which the utility structure is specifically designed to permit tractable methods of expected utility calculation (dynamic programming recursion). The stochastic tree structure with utility recursion is important because it extends the
... ent graphical features of decision trees to the modeling of risky time streams. Such features include ease and clarity in model formulation and presentation, as well as a visual structure which suggests a computational method of solution (rollback). In the near term, this provides graphically intuitive model construction and evaluation capabilities for users who may be less familiar with stochastic modeling and algorithms, especially including users those in the medical community, for whom this tool was developed. The results of the paper characterize all possible recursive utility functions over stochastic trees, providing in the long term a basis for devising useful utility structures beyond what are discussed in the paper. Abstract 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.