The rank-dependent expected utility model (originally called anticipated utility) is a
generalized expected utility Generalized expected utility is a decision-making metric based on any of a variety of theories that attempt to resolve some discrepancies between expected utility theory and empirical observations, concerning choice under risky (probabilistic) c ...
model of choice under
uncertainty
Uncertainty refers to epistemic situations involving imperfect or unknown information. It applies to predictions of future events, to physical measurements that are already made, or to the unknown. Uncertainty arises in partially observable ...
, designed to explain the behaviour observed in the
Allais paradox
The Allais paradox is a choice problem designed by to show an inconsistency of actual observed choices with the predictions of expected utility theory.
Statement of the problem
The Allais paradox arises when comparing participants' choices in two ...
, as well as for the observation that many people both purchase lottery tickets (implying
risk-loving preferences) and insure against losses (implying
risk aversion
In economics and finance, risk aversion is the tendency of people to prefer outcomes with low uncertainty to those outcomes with high uncertainty, even if the average outcome of the latter is equal to or higher in monetary value than the more c ...
).
A natural explanation of these observations is that individuals overweight low-probability events such as winning the lottery, or suffering a disastrous insurable loss. In the Allais paradox, individuals appear to forgo the chance of a very large gain to avoid a one per cent chance of missing out on an otherwise certain large gain, but are less risk averse when offered the chance of reducing an 11 per cent chance of loss to 10 per cent.
A number of attempts were made to model preferences incorporating probability theory, most notably the original version of
prospect theory
Prospect theory is a theory of behavioral economics and behavioral finance that was developed by Daniel Kahneman and Amos Tversky in 1979. The theory was cited in the decision to award Kahneman the 2002 Nobel Memorial Prize in Economics.
Based ...
, presented by
Daniel Kahneman and
Amos Tversky (1979). However, all such models involved violations of first-order
stochastic dominance. In prospect theory, violations of dominance were avoided by the introduction of an 'editing' operation, but this gave rise to violations of
transitivity.
The crucial idea of rank-dependent expected utility was to overweigh only unlikely extreme outcomes, rather than all unlikely events. Formalising this insight required transformations to be applied to the cumulative probability distribution function, rather than to individual probabilities (
Quiggin, 1982, 1993).
The central idea of rank-dependent weightings was then incorporated by
Daniel Kahneman and
Amos Tversky into prospect theory, and the resulting model was referred to as
cumulative prospect theory
Cumulative prospect theory (CPT) is a model for descriptive decisions under risk and uncertainty which was introduced by Amos Tversky and Daniel Kahneman in 1992 (Tversky, Kahneman, 1992). It is a further development and variant of prospect theor ...
(Tversky & Kahneman, 1992).
Formal representation
As the name implies, the rank-dependent model is applied to the increasing rearrangement
of
which satisfies
.
where
and
is a probability weight such that
and
for a transformation function