50:
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
77:
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
590:
929:
337:
390:
664:
439:
233:
434:
800:
128:
722:
150:
85:
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
792:
757:
240:
342:
70:. In prospect theory, violations of dominance were avoided by the introduction of an 'editing' operation, but this gave rise to violations of
51:
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.
595:
585:{\displaystyle h_{}(\mathbf {\pi } )=q\left(\sum \limits _{t=1}^{s}\pi _{}\right)-q\left(\sum \limits _{t=1}^{s-1}\pi _{}\right)}
54:
A number of attempts were made to model preferences incorporating probability theory, most notably the original version of
28:
155:
86:
924:{\displaystyle \sum _{s\in \Omega }h_{}(\mathbf {\pi } )=q\left(\sum \limits _{t=1}^{S}\pi _{}\right)=q(1)=1}
975:
395:
947:
Tversky, Amos and Daniel
Kahneman. Advances in prospect theory: Cumulative representation of uncertainty.
100:
67:
671:
71:
133:
990:
762:
727:
59:
55:
39:, as well as for the observation that many people both purchase lottery tickets (implying
97:
As the name implies, the rank-dependent model is applied to the increasing rearrangement
940:
Kahneman, Daniel and Amos
Tversky. Prospect Theory: An Analysis of Decision under Risk,
36:
984:
44:
79:
63:
332:{\displaystyle W(\mathbf {y} )=\sum _{s\in \Omega }h_{}(\mathbf {\pi } )u(y_{})}
40:
32:
385:{\displaystyle \mathbf {\pi } \in \Pi ,u:\mathbb {R} \rightarrow \mathbb {R} ,}
66:(1979). However, all such models involved violations of first-order
16:
Generalized expected utility model of choice under uncertainty
963:
Generalized
Expected Utility Theory. The Rank-Dependent Model
659:{\displaystyle h_{}(\mathbf {\pi } )=q\left(\pi _{}\right)}
954:
Quiggin, J. (1982), ‘A theory of anticipated utility’,
803:
765:
730:
674:
598:
442:
398:
345:
243:
158:
136:
103:
35:, designed to explain the behaviour observed in the
923:
786:
751:
716:
658:
584:
428:
384:
331:
227:
144:
122:
43:preferences) and insure against losses (implying
228:{\displaystyle y_{}\leq y_{}\leq ...\leq y_{}}
956:Journal of Economic Behavior and Organization
8:
965:. Boston: Kluwer Academic Publishers, 1993.
114:
883:
873:
862:
839:
824:
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764:
729:
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137:
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110:
105:
102:
931:so that the decision weights sum to 1.
429:{\displaystyle h_{}(\mathbf {\pi } )}
7:
859:
535:
482:
436:is a probability weight such that
815:
354:
272:
14:
251:
138:
106:
89:(Tversky & Kahneman, 1992).
949:Journal of Risk and Uncertainty
123:{\displaystyle \mathbf {y} _{}}
21:rank-dependent expected utility
912:
906:
890:
884:
844:
836:
831:
825:
775:
769:
740:
734:
717:{\displaystyle q:\rightarrow }
711:
699:
696:
693:
681:
668:for a transformation function
647:
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623:
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572:
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1:
145:{\displaystyle \mathbf {y} }
29:generalized expected utility
1007:
87:cumulative prospect theory
23:model (originally called
944:, XVLII (1979), 263-291.
976:Favourite-longshot bias
925:
878:
788:
787:{\displaystyle q(1)=1}
753:
752:{\displaystyle q(0)=0}
718:
660:
586:
560:
501:
430:
386:
333:
229:
146:
124:
31:model of choice under
926:
858:
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661:
587:
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481:
431:
387:
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147:
125:
93:Formal representation
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763:
728:
672:
596:
440:
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241:
156:
134:
101:
68:stochastic dominance
25:anticipated utility
951:, 5:297–323, 1992.
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656:
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329:
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152:which satisfies
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109:
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99:
98:
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82:, 1982, 1993).
60:Daniel Kahneman
58:, presented by
56:prospect theory
17:
12:
11:
5:
1004:
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993:
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920:
917:
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827:
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811:
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745:
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713:
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689:
686:
683:
680:
677:
654:
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646:
643:
639:
635:
631:
628:
625:
621:
617:
612:
609:
606:
602:
580:
574:
571:
568:
564:
558:
555:
552:
547:
544:
541:
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532:
528:
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521:
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512:
509:
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491:
488:
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479:
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472:
469:
465:
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456:
453:
450:
446:
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421:
417:
412:
409:
406:
402:
381:
377:
373:
369:
365:
362:
359:
356:
353:
349:
328:
323:
320:
317:
313:
309:
306:
303:
299:
295:
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287:
284:
280:
274:
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268:
264:
260:
257:
253:
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222:
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208:
205:
202:
199:
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188:
185:
181:
177:
172:
169:
166:
162:
140:
117:
113:
108:
94:
91:
37:Allais paradox
15:
13:
10:
9:
6:
4:
3:
2:
1003:
992:
989:
988:
986:
977:
974:
973:
969:
964:
960:
958:3(4), 323–43.
957:
953:
950:
946:
943:
939:
938:
934:
932:
918:
915:
909:
903:
900:
896:
887:
880:
874:
869:
866:
863:
854:
850:
847:
840:
828:
821:
812:
809:
805:
795:
781:
778:
772:
766:
746:
743:
737:
731:
708:
705:
702:
690:
687:
684:
678:
675:
666:
652:
644:
637:
633:
629:
626:
619:
607:
600:
578:
569:
562:
556:
553:
550:
545:
542:
539:
530:
526:
523:
519:
510:
503:
497:
492:
489:
486:
477:
473:
470:
463:
451:
444:
419:
407:
400:
379:
363:
360:
357:
351:
347:
318:
311:
304:
297:
285:
278:
269:
266:
262:
258:
244:
236:
217:
210:
206:
203:
200:
197:
194:
186:
179:
175:
167:
160:
92:
90:
88:
83:
81:
75:
73:
69:
65:
61:
57:
52:
48:
46:
45:risk aversion
42:
38:
34:
30:
26:
22:
962:
961:Quiggin, J.
955:
948:
942:Econometrica
941:
796:
667:
237:
96:
84:
76:
72:transitivity
64:Amos Tversky
53:
49:
24:
20:
18:
797:Note that
41:risk-loving
33:uncertainty
935:References
881:π
860:∑
841:π
816:Ω
813:∈
806:∑
697:→
638:π
620:π
563:π
554:−
536:∑
524:−
504:π
483:∑
464:π
420:π
372:→
355:Π
352:∈
348:π
298:π
273:Ω
270:∈
263:∑
207:≤
195:≤
176:≤
985:Category
970:See also
991:Utility
80:Quiggin
27:) is a
339:where
724:with
592:and
392:and
62:and
19:The
130:of
47:).
987::
794:.
759:,
235:.
74:.
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916:=
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910:1
907:(
904:q
901:=
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891:]
888:t
885:[
875:S
870:1
867:=
864:t
855:(
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837:(
832:]
829:s
826:[
822:h
810:s
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779:=
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773:1
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738:0
735:(
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700:[
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679::
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645:S
642:[
634:(
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627:=
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616:(
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608:S
605:[
601:h
579:)
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570:t
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543:=
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531:(
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511:t
508:[
498:s
493:1
490:=
487:t
478:(
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471:=
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460:(
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452:s
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445:h
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416:(
411:]
408:s
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364::
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312:y
308:(
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248:(
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198:.
190:]
187:2
184:[
180:y
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78:(
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