Rank dependent expected utility

Prospect theory is most well known for its assumption that gains are treated differently to losses. Another crucial part of the theory, namely that probabilities are weighted, typically attracts much less attention. Recent evidence, however, is suggesting that probability weighting has a crucial role to play in many applied settings. So, what is probability weighting and why does it matter?

The basic idea of probability weighting is that people tend to overestimate the likelihood of events that happen with small probability and underestimate the likelihood of events that happen with medium to large probability. In their famous paper on 'Advances in prospect theory', Amos Tversky and Daniel Kahneman quantified this effect. They fitted experiment data to equation

where γ is a parameter to be estimated. In interpretation, p is the actual probability and π(p) the weighted probability. The figure below summarizes the kind of effect you get. Tversky and Kahneman found that a value of γ around 0.61 best matched the data. This means that something which happens with probability 0.1 gets a decision weight of around 0.2 (overweighting of small probabilities) while something that happens with probability 0.5 gets a decision weight of only around 0.4 (underweighting of medium to large probabilities).  

Why we observe this warping of probabilities is unclear. But the consequences for choice can be important. To see why consider someone deciding whether to take on a gamble. Their choice is either to accept £10 for certain or gamble and have a 10% chance of winning £90 and a 90% chance of winning nothing. The expected value of this gamble is 0.1 x 90 = £9. So, it does not look like a good deal. But, if someone converts a 10% probability into a decision weight of 0.2 we get value 0.2 x 90 = £18. Suddenly the gamble looks great! Which might explain the appeal of lottery tickets.

There is, though, a problem. It is not enough to simple weight all probabilities. This, as I will shortly explain, doesn't work. So, we need some kind of trick. While prospect theory was around in 1979 it was not until the early 1990's that the trick was found. That trick is rank dependent weighting. The gap of over 10 years in finding a way to deal with probabilities may help explain why probability weighting has had to play second fiddle to loss aversion. Lets, though, focus on the technical details.

Consider the example. Here there are no obvious problems if we just weight probabilities. The 10% chance of winning is converted into a 0.2 decision weight while the 90% chance of losing is converted into a 0.7 decision weight. The overall expected value is then 0.2 x £90 = £18. Everything looks fine.

So, consider another example. Suppose that the sure £10 is now a gamble with a 10% chance of winning £10.09, a 10% chance of winning £10.08, a 10% chance of winning £10.07, and so on, down to a 10% chance of winning £10. If we just simply weight all these 10% probabilities as 0.2 then we get expected value of 0.2 x 10.09 + 0.2 x 10.08 + ... + 0.2 x 10 = £20.09. This is absurd. A gamble that essentially gives £10 cannot be worth over £20! You might say that the problem here is we have ended up with a combined weight of 2. If, though, we normalize weights to 1 we will not have captured the over-weighting of small probabilities. So, normalizing is not, of itself, a solution. 

The problem with the preceding approach is that we have weighted everything - good or bad - by the same amount. Rank dependent probability does away with that. Here we rank outcomes from best to worst. The decision weight we place on an outcome is then the weighted probability of the outcome or something better minus the weighted probability of something better. 

In our original gamble the best outcome is £90 and the worst is £0. The weight we put on £90 is around 0.2 because there is 10% chance of £90, no chance of anything better, and a 10% probability is given weight 0.2. The weight we put on £0 is 0.8 because it is the weighted probability of £0 or better, namely 1, minus the weighted probability of £90, namely 0.2. So, not much changes in this example.

In the £10 gamble the best outcome is £10.09, the next best £10.08, and so on. The decision weight we but on £10.09 is around 0.2 because there is a 10% chance of £10.09 and no chance of anything better. Crucially, the weight we put on £10.08 is only around 0.1 because we have the weighted probability of £10.08 or better, a 20% chance that gives weight around 0.3, minus the weighted probability of £10.09, around 0.2. You can verify that the chance of winning £10.07, £10.06 and so on has an even lower decision weight. Indeed, decision weights have to add to 1 and so the high weight on £10.09 is compensated by a lower weight on other outcomes. For completeness the table below gives the exact weights you would get with the Tversky and Kahneman parameters. Given that decision weights have to add to 1 the expected value is going to be around £10. Common sense restored!

Generally speaking, rank dependent weighting means that we capture, and only capture, over-weighting of the extreme outcomes. So, we capture the fact a person may be overly optimistic about winning £90 rather than £0 without picking up the perverse prediction that every unlikely event is over-weighted. The discussion so far has focused on gains but we can do the same thing with losses. Here we want to capture, and only capture, over-weighting of the worst outcomes. 

So why does all this matter? There is mounting evidence that weighting of probabilities can explain a lot of behavior, including the equity premium puzzle, long shot bias in betting and willingness of households to buy insurance at highly unfavorable premiums. For a review of the evidence see the article by Helga Fehr-Duda and Thomas Epper on 'Probability and risk: Foundations and economic implications of probability-dependent risk preferences'. It is easy to see, for instance, why overweighting of small probabilities could have potentially profound implications for someone's view of insurance. A very small probability of loss may be given a much higher decision weight. That makes insurance look like a good deal.