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| 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]]). | | The writer is known as Araceli Gulledge. To play croquet is the pastime I will by no means quit doing. Bookkeeping is what I do for a living. Her husband and her selected to reside in Alabama.<br><br>My webpage ... [http://Www.Farebook.us/index.php?do=/profile-7229/info/ http://Www.Farebook.us/index.php?do=/profile-7229/info] |
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| 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.
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| 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 [[transitive relation|transitivity]].
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| 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 ([[John Quiggin|Quiggin]], 1982, 1993).
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| 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).
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| ==Formal representation==
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| As the name implies, the rank-dependent model is applied to the increasing rearrangement <math>\mathbf{y}_{[ \; ]}</math> of <math>\mathbf{y}</math> which satisfies <math>y_{[1]}\leq y_{[2]}\leq ...\leq
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| y_{[S]}</math>.
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| <math>W(\mathbf{y})=\sum_{s\in \Omega }h_{[s]}(\mathbf{\pi })u(y_{[s]}) </math>
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| where <math>\mathbf{\pi }\in \Pi ,u:\mathbb{R} \rightarrow \mathbb{R} ,</math> and <math>h_{[s]}(
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| \mathbf{\pi })</math> is a probability weight such that
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| <math>h_{[s]}(\mathbf{\pi })=q\left( \sum\limits_{t=1}^{s}\pi _{[t]}\right)
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| -q\left( \sum\limits_{t=1}^{s-1}\pi _{[t]}\right) </math>
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| for a transformation function <math>q:[0,1]\rightarrow [0,1]</math> with <math>q(0)=0</math>, <math>
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| q(1)=1 </math>.
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| Note that
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| <math>\sum_{s\in \Omega }h_{[s]}(\mathbf{\pi })=q\left( \sum\limits_{t=1}^{S}\pi
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| _{[t]}\right) =q(1)=1 </math>
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| so that the decision weights sum to 1.
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| ==References==
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| * Kahneman, Daniel and Amos Tversky. Prospect Theory: An Analysis of Decision under Risk, ''Econometrica'', XVLII (1979), 263-291.
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| * Tversky, Amos and Daniel Kahneman. Advances in prospect theory: Cumulative representation of uncertainty. ''Journal of Risk and Uncertainty'', 5:297–323, 1992.
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| * Quiggin, J. (1982), ‘A theory of anticipated utility’, ''Journal of Economic Behavior and Organization'' 3(4), 323–43.
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| * Quiggin, J. ''Generalized Expected Utility Theory. The Rank-Dependent Model''. Boston: Kluwer Academic Publishers, 1993.
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| == See also ==
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| * [[Favourite-longshot bias]]
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| [[Category:Utility]]
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The writer is known as Araceli Gulledge. To play croquet is the pastime I will by no means quit doing. Bookkeeping is what I do for a living. Her husband and her selected to reside in Alabama.
My webpage ... http://Www.Farebook.us/index.php?do=/profile-7229/info