Explaining versus Describing Human Decisions. Hilbert Space Structures in Decision Theory

Traditional cognitive theories systematically apply classical set-theoretic structures to model human judgements and decisions under uncertainty. This is particularly evident in theories of rational decision-making, like expected utility theory, where Bayesian, or Kolmogorovian [1], models of probability directly follow from axioms on agents' preferences [2, 3]. However, several cognitive puzzles have been discovered in empirical tests, which provide evidence of systematic deviations from Kolmogorovian probability structures (see, e.g., [4]). For example, Kahneman and Tversky identified a conjunction fallacy in human probability judgements, namely, the law of monotonicity of Kolmogorovian probability does not generally hold in this kind of judgements [5]. Also, in human decision-making, Tversky and Shafir proved that the law of total Kolmogorovian probability does not hold in the disjunction effect [6], while Allais and Ellsberg indicated that people do not always choose by maximizing an expected utility with respect to a Kolmogorovian probability measure [7]. As a consequence of the puzzles above, traditional theories using Kolmogorovian structures, though normatively compelling, are descriptively flawed, which led several authors to elaborate alternative proposals able to more efficiently and realistically represent human behaviour. This was the starting point of the bounded rationality research programme, initially proposed by Herbert Simon [8] and systematically applied by Kahneman and Tversky [5, 6] to describe concrete judgements and decisions. Bounded rationality models give good predictions in a variety of circumstances.