Expected utility theory with probability grids and preference formation
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Expected utility theory with probability grids and preference formation Mamoru Kaneko1 Received: 30 May 2018 / Accepted: 21 August 2019 © The Author(s) 2019
Abstract 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 (-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. Keywords Expected utility · Measurement of utility · Bounded rationality · Probability grids · Cognitive bound · Incomparabilities JEL Classification C72 · C79 · C91
The author thanks J. J. Kline, P. Wakker, M. Lewandowski, R. Ishikawa, S. Shiba, M. Cohen, O. Shulte, and Y. Rebille for helpful comments on earlier versions of this paper. In particular, comments given by the two referees improved the paper significantly and are greatly appreciated. The author is supported by Grant-in-Aids for Scientific Research Nos. 19K21704 and 17H02258, Ministry of Education, Science and Culture.
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Mamoru Kaneko [email protected] Faculty of Political Science and Economics, Waseda University, Shinjuku, Tokyo 169-0850, Japan
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1 Introduction We reconsider EU theory from the viewpoints of bounded rationality and preference formation. We restrict permissible probabilities to decimal (-ary, in general) fractions up to a given cognitive bound ρ; if ρ is a natural number k, the set of permissible k probabilities is given as Πρ = Πk = { 100k , 101k , . . . , 10 }. The decision maker makes 10k preference comparisons step by step using probabilities with small k to those with larger k to obtain accurate comparisons. The derived preference relation is incomplete in general, but the EU hypothesis holds for some lotteries and it would hold more when there is no cognitive bound, i.e., ρ = ∞ and Π∞ = ∪k
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