Dynamic programming for deterministic discrete-time systems with uncertain gain
International Journal of Approximate Reasoning
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This paper deals with the impact of information on the decisions of an agent whose beliefs are imprecise and whose preferences are not in accordance with the Subjective Expected Utility (SEU) model. We assume that his one-shot preferences are representable by a Hurwicz criterion with pessimism-optimism index @a. We moreover assume that in a sequential decision making situation the decision maker acts according to the root dictatorship version of McClennen's Resolute Choice model: he evaluates strategies at the root of the decision tree by the Hurwicz criterion and enforces the best strategy, thus behaving in a dynamically consistent manner. The use of Resolute Choice in an imprecise probability environment raises a general question: is information processed correctly in this model? To show that this question can be given a positive answer in standard cases (and also motivated by the accident-no accident variable in an automobile insurance context), we study the basic situation in which data are provided by the random sampling of a binary variable, and find the influence of the pessimism-optimism index on the optimal decisions to be decreasing with the sample size, the optimal decision rule only depending, asymptotically, on the relative frequencies observed. Then, we turn to the second question raised by the well known feature of Resolute Choice: non-consequentialism. Does the fact that the optimal decision rule may depend on unrealized outcomes necessarily lead to criticisable choices? We study a two-period insurance problem in which an individual has to choose his coverage at period two after observing the period one outcome (loss or no loss). It turns out that in the case where no loss happened, a seemingly irrelevant data - the first period deductible level - may influence the decision maker's second period insurance choice. We analyse this result in relation with the existence and value of the pessimism-optimism degree.