Expected utility theory with probability grids and preference formation

We reformulate expected utility theory, from the viewpoint of bounded rationality, by introducing probability grids and a cognitive bound; we restrict permissible probabilities only to decimal (ℓ-ary in general) fractions of finite depths up to a given cognitive bound. We distinguish between measurements of utilities from pure alternatives and their extensions to lotteries involving more risks. Our theory is constructive from the viewpoint of the decision maker. When a cognitive bound is small, the preference relation involves many incomparabilities, but these diminish as the cognitive bound is relaxed. Similarly, the EU hypothesis would hold more for a larger bound. The main part of the paper is a study of preferences including incomparabilities in cases with finite cognitive bounds; we give representation theorems in terms of a 2-dimensional vector-valued utility functions. We also exemplify the theory with one experimental result reported by Kahneman and Tversky.