Boundedly rational expected utility theory

We build a satisficing model of choice under risk which embeds Expected Utility Theory (EUT) into a boundedly rational deliberation process. The decision maker accumulates evidence for and against alternative options by repeatedly sampling from her underlying set of EU preferences until the evidence favouring one option satisfies her desired level of confidence. Despite its EUT core, the model produces patterns of behaviour that violate standard EUT axioms, while at the same time capturing systematic relationships between choice probabilities, response times and confidence judgments, which are beyond the scope of theories that do not take deliberation into account. Electronic supplementary material The online version of this article (10.1007/s11166-018-9293-3) contains supplementary material, which is available to authorized users.

2 This appendix contains three theorems relating to the limiting case of d → 0, and their corresponding proofs, that establish general conditions under which BREUT will produce violations of independence and betweenness like the ones shown in Section 3. Theorem 1 applies the limiting case to analyse what happens when the probabilities of the best consequence are scaled down, as in the CR scenario.
Theorem 2 derives more general properties about the shape of the indifference curves for cores of utility functions restricted to be either weakly concave or weakly convex, which has implications for whether or not BREUT satisfies betweenness. By combining insights from Theorem 1 and Theorem 2, Theorem 3 elaborates further on the circumstances under which the CC effect can be obtained.
Theorems 1 and 2 and their proofs are followed by intuitive explanations of their implications.
The proofs below make use of the following observation. Proof. For convenience, define so that , with > 0. Without loss of generality, assume the 's are ordered according to (Note that, while in principle it may be that for some distinct i, j, there are at least one i and one j such that . So, the above ordering can be done by replacing N with the cardinality of the strict ordering, which is of at least 2.) Let denote the probability of occuring. We proceed by contradiction. Suppose that the result does not hold, so that Pr(S ≻ R) → 0.5 and Pr(S' ≻ R') does not tend to 0 as d → 0. First, E(CE S ) = E(CE R ) and hence which implies Second, since it must be that Pr(S' ≻ R') tends to a number greater than 0 as d → 0 (and specifically 0.5 or 1), then E(CE S' ) ≥ E(CE R' ), and hence which implies .
Combining the two, we obtain: where we have divided by on both sides. Noting that , we write , where ϵ (0, 1). We therefore have: . So: .
Factoring out the left-hand side (LHS) and the right-hand side (RHS) of the equation: Canceling out the common terms on the LHS and the RHS side and grouping the terms that include with those that include , we obtain: … … 4 which can be written as: But notice that for any as it is implied by , which is implied by , which is implied by and by , which are both true because Since this is true for each of the comparisons of the matching terms in the LHS and the RHS sums, it follows that: which is a contradiction and completes the proof.
Theorem 1 concerns lotteries of the form typically used in CR scenarios, S = (x m , p; 0, 1p) and R = (x h , q; 0, 1q), with x h > x m > 0. The commonly used case in which p = 1 is also allowed by the theorem.
To illustrate the intuition, suppose that Pr(S ≻ R) → 0.5 as d → 0 (i.e., the decision maker is indifferent between S and R in the limit, so that E[V(S, R)] = 0) for some p ≤ 1. The key result is that, for a core made of CRRA functions, scaling the probabilities of the best outcome of each lottery down by some factor σ always results in the DM having a strict preference for the scaled down risky lottery R'. Although Theorem 1 starts from perfect indifference between S and R in the scaled-up pair, an immediate corollary is that it will always be possible to obtain a strict preference in favour of S by slightly reducing q, the probability of winning x h in R. That is, in the limit, a CR effect can always be obtained in which the DM has a strict preference for the safer lottery in the scaled-up pair and a strict preference for riskier lottery in the scaled-down pair. A preference reversal in the other direction does not occur for any lotteries of these forms.
Theorem 1 holds as long as there are at least two different CRRA utility functions in the core, without any further assumption about the core distribution, such as the degree of risk aversion implied by each function. The case of a continuous distribution (like the ones used in our simulations) can be approximated by taking an N that is sufficiently large. Maintain the assumptions of Theorem 1 (i.e., the core consists of N CRRA functions, at least two of which are distinct). Take any distinct lotteries S = (x h , p 1 ; x m , p 2 ; 0, 1-p 1p 2 ) and R = (x h , q 1 ; x m , q 2 ; 0, 1q 1q 2 ) for which x h > x m > 0, p 1 , p 2 , q 1 , q 2 ϵ [0, 1], p 1 + p 2 < 1, q 1 + q 2 < 1 and for which Pr(S ≻ R) → 0.5 as d → 0.