The standard framework of decision theory suffers from the fact that ambiguity (non-stochastic uncertainty, indeterminacy) can not be taken properly into account. In most applications, neither the maximin paradigm (relying on complete ignorance on the states of nature) nor the classical Bayesian paradigm (assuming perfect probabilistic information on the states of nature) adequately reflects the situation under consideration. This paper extends classical decision theory to situations where prior and/or sampling information is ambiguous. It gives a framework to generalize expected utility theory to interval probability and imprecise probabilities. Firstly, a general algorithm will be presented to calculate optimal actions under ambiguous prior information. Then it is shown how to incorporate ambiguous sampling information. As a by-product, the results also lead to a question concerning coherent inference with imprecise probabilities and robust Bayesian inference. It will be shown that, under ambiguity, inference based on the imprecise posterior distribution may lead to suboptimal actions. Under ambiguity, the posterior distribution does no longer contain all the relevant information.
Keywords. Decision Making, Generalized Expected Utility, Ambiguity, Linear Programming Chooquet expected utility, $\Gamma$-minimax principle, critic of robust Bayesian analysis
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Authors addresses:
SFB 386, Department of Statistics,
University of Munich,
Akademiestr.1,
D 80799 Munich, Germany
E-mail addresses:
| Thomas Augustin | thomas@stat.uni-muenchen.de |