Facing an Uncertain Future
This paper, which was published as Technical Report No. 359 from the Institute of Mathematical Studies in the Social Sciences, Stanford University (1982), analyses uncertainty in an intertemporal context, and is designed to use to the fullest extent possible the overlapping theorem presented in ’The structure of utility functions’ (Ch. 12). At each point of time, a set of states of the world is possible, presenting the agent with a utility tree; the agent has a complete set of preferences over all possible states of the world at all periods of time; one of these states occurs, and in the next period, he or she faces some smaller portion of the tree. Assumption 2 requires the states that follow any branch of the tree to be separable from those events that can no longer happen––conditional on the actual history followed; Gorman calls this the ’very weak independence axiom’; the only other assumption used here is that the agent only examines the future closely for the next two periods, and, for the rest of the future, is content with a summary statistic. Hence Assumption 3 requires the future from t + 2 to the horizon to be separable from its complement at each t. These two assumptions are enough to generate ’Bentham and Bernoulli at a stroke’.
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