Rank-dependent expected utility
Encyclopedia
The rank-dependent expected utility model (originally called anticipated utility) is a generalized expected utility
model of choice under uncertainty
, designed to explain the behaviour observed in the Allais paradox
, as well as for the observation that many people both purchase lottery tickets (implying risk-loving
preferences) and insure against losses (implying risk aversion
).
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 to 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
, 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 overweight 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
(Tversky & Kahneman, 1992).
of 
which satisfies 
.
where
and 
is a probability weight such that
for a transformation function
with 
, 
.
Note that
so that the decision weights sum to 1.
Generalized expected utility
The expected utility model developed by John von Neumann and Oskar Morgenstern dominated decision theory from its formulation in 1944 until the late 1970s, not only as a prescriptive, but also as a descriptive model, despite powerful criticism from Maurice Allais and Daniel Ellsberg who showed...
model of choice under uncertainty
Uncertainty
Uncertainty is a term used in subtly different ways in a number of fields, including physics, philosophy, statistics, economics, finance, insurance, psychology, sociology, engineering, and information science...
, designed to explain the behaviour observed in the Allais paradox
Allais paradox
The Allais paradox is a choice problem designed by Maurice Allais to show an inconsistency of actual observed choices with the predictions of expected utility theory.-Statement of the Problem:...
, as well as for the observation that many people both purchase lottery tickets (implying risk-loving
Risk-loving
In economics and finance, a risk lover is a person who has a preference for risk. While most investors are considered risk averse, one could view casino goers as risk loving...
preferences) and insure against losses (implying risk aversion
Risk aversion
Risk aversion is a concept in psychology, economics, and finance, based on the behavior of humans while exposed to uncertainty....
).
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 to 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
Prospect theory is a theory that describes decisions between alternatives that involve risk i.e. where the probabilities of outcomes are known. The model is descriptive: it tries to model real-life choices, rather than optimal decisions.-Model:...
, presented by Daniel Kahneman
Daniel Kahneman
Daniel Kahneman is an Israeli-American psychologist and Nobel laureate. He is notable for his work on the psychology of judgment and decision-making, behavioral economics and hedonic psychology....
and Amos Tversky
Amos Tversky
Amos Nathan Tversky, was a cognitive and mathematical psychologist, a pioneer of cognitive science, a longtime collaborator of Daniel Kahneman, and a key figure in the discovery of systematic human cognitive bias and handling of risk. Much of his early work concerned the foundations of measurement...
(1979). However, all such models involved violations of first-order stochastic dominance
Stochastic dominance
Stochastic dominance is a form of stochastic ordering. The term is used in decision theory and decision analysis to refer to situations where one gamble can be ranked as superior to another gamble. It is based on preferences regarding outcomes...
. In prospect theory, violations of dominance were avoided by the introduction of an 'editing' operation, but this gave rise to violations of transitivity
Transitivity
-In grammar:* Intransitive verb* Transitive verb, when a verb takes an object* Transitivity -In logic and mathematics:* Arc-transitive graph* Edge-transitive graph* Ergodic theory, a group action that is metrically transitive* Vertex-transitive graph...
.
The crucial idea of rank-dependent expected utility was to overweight 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
John Quiggin
John Quiggin is an Australian economist and professor at the University of Queensland. Quiggin studied at the Australian National University, obtaining bachelor's degrees in Arts and Economics in 1978 and 1980 respectively, and completing a master's degree in Economics in 1984. Quiggin was awarded...
, 1982, 1993).
The central idea of rank-dependent weightings was then incorporated by Daniel Kahneman
Daniel Kahneman
Daniel Kahneman is an Israeli-American psychologist and Nobel laureate. He is notable for his work on the psychology of judgment and decision-making, behavioral economics and hedonic psychology....
and Amos Tversky
Amos Tversky
Amos Nathan Tversky, was a cognitive and mathematical psychologist, a pioneer of cognitive science, a longtime collaborator of Daniel Kahneman, and a key figure in the discovery of systematic human cognitive bias and handling of risk. Much of his early work concerned the foundations of measurement...
into prospect theory, and the resulting model was referred to as cumulative prospect theory
Cumulative prospect theory
Cumulative prospect theory is a model for descriptive decisions under risk which was introduced by Amos Tversky and Daniel Kahneman in 1992 . It is a further development and variant of prospect theory...
(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 thatfor a transformation function
with , .Note that
so that the decision weights sum to 1.