Very Low Probabilities in the Loss Domain

Abstract

This experimental study uses a non-parametric method to investigate probability weighting functions for very low probabilities in the loss domain. Probability weights in three loss situations containing small, large and heterogeneous losses composed of both small and large losses are elicited. While most of the probabilities under consideration are significantly overweighted, the probability weighting function exhibits the much replicated inverse S-shaped functions when losses are small. Interestingly, the more common probabilities, 0.1 and 0.01, get underweighted by more than half of the sample in small and heterogeneous loss situations, respectively. Probability underweighting is accompanied by risk-loving behaviour that can have implications for design of contracts and policies designed to control risky behaviours.

Appendices

Appendix A

Trade-off method

The trade-off (TO) method consists of eliciting a sequence of outcomes that are equally spaced in terms of utility. The distance between utilities is fixed and individuals differ by their assigned outcomes to these utilities. When applied to losses by Fennema and Van Assen³², the procedure is the following. Given fixed outcomes $x_{0}$, r and R such that $x_{0}\,< \,0<\,r \,< R$, the decision maker is asked to determine the value of $x_{1}$ that makes her indifferent between $(x_{1}, p; R, 1-p)$ and $(x_{0}, p; r, 1-p)$ with $x_{1} \,<\, x_{0}$, and a fixed p.³³ Then, $x_{1}$ is used as an input, and a similar process allows the determination of the outcome $x_{2}\,<\,x_{1}$ that makes the decision maker indifferent between $(x_{2}, p;R, 1-p)$ and $(x_{1}, p; r, 1-p)$. The procedure is implemented n times in order to obtain a sequence $x_{n},\ldots , x_{i},\ldots , x_{0}$. Under CPT, the indifference between $(x_{1}, p; R, 1- p)$ and $(x_{0}, p; r,1- p)$ implies

$$w(1-p)u(R)+ w(p)u(x_{1})= w(1-p)u(r)+ w(p)u(x_{0})$$

(A.1)

$$w(1-p)[u(R)- u(r)]= w(p)[u(x_{0})- u(x_{1})].$$

(A.2)

Consequently, the second indifference implies

$$w(1-p)[u(R)- u(r)]= w(p)[u(x_{1})- u(x_{2})].$$

(A.3)

Thus, since the LHS of Eqs. (A.1) and (A.2) are equal, then $u(x_{1})-u(x_{2})=u(x_{0})-u(x_{1})$. By normalizing $u(x_{0})=0,$ we get $u(x_{2} )=2u(x_{1})$. A similar argument leads to the determination of $x_{3}$’s utility such that $u(x_{3})=3u(x_{1})$. Hence, we get $u(x_{i})=iu(x_{1})$. Inductively, from the last indifference equality we get $u(x_{n-1})-u(x_{n})=u(x_{n-2})-u(x_{n-1})$. By substituting $u(x_{i})=iu(x_{1})$ and normalizing $u(x_{n})=-1,$ we get $u(x_{1})=-\frac{1}{n}$. Thus, one gets $u(x_{i})=-\frac{i}{n}$. In this way, we can elicit the utility over the unit interval non-parametrically.

The TO method suffers from a number of drawbacks. One of the drawbacks of the TO method is that the experimenter can fix the value of either $x_{0}$ or $x_{n}$. Moreover, as reported by Abdellaoui,²⁴ recent studies show that the TO method needs high outcomes in order to obtain utility functions with pronounced curvature. Finally, Wakker and Deneffe¹⁰ pointed out that the chaining of old responses into new lotteries might lead to error propagation. That is, any error in the early stages of utility elicitation may result in a biased sequence of outcomes. Bleichrodt and Pinto³⁴ and Abdellaoui et al.³⁵ have investigated this issue. Their findings suggest that the impact of error propagation is negligible.

Appendix B

Experiment instruction

Please take your time and read these instructions carefully. This experiment consists of two sections. In the first section, you will be presented with 35 lottery pairs and will be asked to choose which of the two lotteries you would prefer to play. In this section, lotteries are presented as pie charts. Each screen consists of two pie charts and you only need to indicate which one you prefer. Here is an example of what the computer display of such a pair of lotteries will look like.

In the above example, the left lottery indicates either a 33 per cent chance of losing £60 or a 67 per cent chance of winning £350. In the right lottery, there is either a 33 per cent chance of losing £1030 or a 67 per cent chance of winning £1100. As you can see, the outcomes are displayed around the pie charts and each coloured area represents the corresponding probabilities.

Each pair of lotteries is shown on a separate screen on the computer. On each screen, you should indicate which of the lotteries you prefer to play by clicking one of the two buttons beneath the lotteries. You should click the LEFT if you prefer the lottery on the left and the RIGHT if you prefer the lottery on the right.

In the second section, you are asked to choose between a simple lottery and a sure loss. So you only need to indicate whether you prefer to play the lottery or lose the sure amount. The screen in this section looks like the following:

In the above example, lottery A indicates either a one in a hundred thousand chance of losing 60 or a 99,999 in a hundred thousand chance of losing 333. This means that if lottery A played hundred thousand times, it would lead to a loss of £60 once and all other times it would lead to a loss of £333.