# Regret

Just after the publication of prospect theory and mixed in with the many new generalisations of expected utility theory that were designed, among other things, to deal with the Allais Paradox, was a theory that was every bit as original as prospect theory but not as eye-catching.This is ‘regret theory’. It was developed simultaneously by Bell (1982) and Loomes & Sugden (1982). One of the puzzles that is usually associated with the Allais Paradox involves asking people to choose first between two alternatives, A and B:

A: \$1,000,000 for sure (probability 1.00); or

B: \$5,000,000 with probability 0.10, \$1,000,000 with probability 0.89, and \$0 with probability' 0.01.

Most people choose A. Following this, they are presented with another problem. Choose between two alternatives, C and D:

C: \$5,000,000 with probability 0.10 and \$0 with probability 0.90; or

D: \$1,000,000 with probability 0.11 and SO with probability 0.89.

In round two, most people choose C but this set of choices, A and C, violates the independence axiom of expected utility theory. The ‘trick’ is that A and D are actually the same and so too are В and C, once the common elements are removed (which according to the independence axiom should make no difference). To see this, we can rewrite the options as:

A: \$1,000,000 with probability of 0.89 and \$1,000,000 with probability

0.11; or

B: \$5,000,000 with probability 0.10, \$1,000,000 with probability 0.89, and \$0 with probability 0.01.

C: \$5,000,000 with probability 0.10, SO with probability 0.89 and \$0 with probability 0.01; or

D: \$1,000,000 with probability 0.11 and SO with probability 0.89.

Let us now strike out the common elements:

A: \$1,000,000 with probability of 0.89 and \$1,000,000 with probability

0.11; or

B: \$5,000,000 with probability 0.10. \$1.000.000 with probability 0.89. and \$0 with probability 0.01.

C: \$5,000,000 with probability 0.10, SO with probability 0.89 and \$0 with probability 0.01; or

D: \$1,000,000 with probability 0.11 and SO with probability 0.89.

If the decision-maker says that she prefers A to B, then she reveals a preference for \$1,000,000 with 0.11 probability over \$5,000,000 with 0.10 probability and \$0 with 0.01 probability'. Then, in round two, if she says she prefers C, she reveals a preference for \$5,000,000 with 0.10 probability and \$0 with 0.01 probability over SI,000,000 with 0.11 probability. A complete preference reversal! While this is not universal behaviour or, indeed, the majority decision of people tested, it is the modal preference of decision-makers who participate in economics and psychology' experiments (Raiffa 1968; Slovic & Tversky 1974). That is, the most likely preference to be observed from a random sample of subjects. While all of the generalisations of expected utility' theory, including prospect theory, allow for this type of behaviour, regret theory provides a clear and straightforward explanation for it.

Bell (1982) and Looms & Sugden (1982) built an alternative to expected utility theory and prospect theory in which the decision-maker can anticipate feeling regret and this anticipation of a negative feeling in the future affects their decisions now. In the decision problem that we just discussed, the decision-maker anticipates the regret she will feel if she opts for В instead of A and the least likely outcome of receiving \$0 eventuates when she could have had \$1,000,000 for sure. In an interesting turning of the tables, it was this work by Bell, Loomes and Sugden, all economists writing for economics journals, which initiated the large research program in regret and disappointment in psychology. Of course, people feel regret after something happens but the idea that they can anticipate this regret during the decision-making process and that a rigorous model of choice could be built to incorporate it was new. In recent times, Bleichrodt & Wakker (2015) have argued that regret theory might just be the best alternative to the alternatives to expected utility theory. Despite the success of prospect theory as a fountainhead for the mainstream of behavioural economics, the richness of regret theory and the large volume of psychological results that have flowed from it mean that regret theory sits, albeit in the shadows, as a potential rival to prospect theory. We have taken some first steps towards showing how regret theory can help us understand terrorist behaviour (Phillips & Pohl 2020).