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Decision Making under Interval and Fuzzy Uncertainty: Towards an Operational Approach

Traditional decision theory is based on a simplifying assumption that for each two alternatives, a user can always meaningfully decide which of them is preferable. In reality, often, when the alternatives are close, the user is either completely unable to select one of these alternatives, or selects one of the alternatives only “to some extent”. How can we extend the traditional decision theory to such realistic interval and fuzzy cases? In their previous papers, the first two authors proposed a natural generalization of the usual decision theory axioms to interval and fuzzy cases, and described decision coming from this generalization. In this paper, we make the resulting decisions more intuitive by providing commonsense operational explanation. Introduction. Traditional decision theory is based on a simplifying assumption that for each two alternatives, a user can always meaningfully decide which of them is preferable. In reality, often, when the alternatives are close, the user is either completely unable to select one of these alternatives, or selects one of the alternatives only “to some extent”. How can we extend the traditional decision theory to such realistic interval and fuzzy cases? In their papers [1, 2, 3, 4, 5, 6, 7, 8, 9], summarized in [10], the first two authors proposed a natural generalization of the usual decision theory axioms to interval and fuzzy cases, and described decision coming from this generalization. In this paper, we make the resulting decisions more intuitive by providing commonsense operational explanation. This paper is structured as follows: first, we recall the main ideas and results of the traditional decision theory. We then consider the case when in addition to deciding which of the two alternatives is better, the user can also reply that he/she is unable to decide between the two close alternatives; this leads to interval uncertainty. Finally, we consider the general case when the user makes fuzzy statements about preferences. Traditional decision theory: brief reminder. Following [11, 13, 14], let us describe the main ideas and results of the traditional decision theory. Main assumption behind the traditional decision theory. Let us assume that for every two alternatives A′ and A′′, a user can tell: • whether the first alternative is better for him/her; we will denote this by A′′ < A′;