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offered a choice between red and yellow balls and black and yellow balls, people assume that there must be fewer than 30 yellow balls as would be necessary to deceive them. When making the decision, it is quite possible that people simply neglect to consider that the experimenter does not have a chance to modify the contents of the urn in between the draws. In real-life situations, even if the urn is not to be modified, people would be afraid of being deceived on that front as well.
419:. Indeed, the amount of the payoff is likewise irrelevant. Whichever gamble is selected, the prize for winning it is the same, and the cost of losing it is the same (no cost), so ultimately there are only two outcomes: receive a specific amount of money or nothing. Therefore, it is sufficient to assume that the preference is to receive some money to nothing (this assumption is not necessary: in the mathematical treatment above, it was assumed
1807:
1797:
407:. If the participant strictly prefers Gamble A to Gamble B, by utility theory, it is presumed this preference is reflected by the expected utilities of the two gambles. We reach a contradiction in our utility calculations. This contradiction indicates that the participant's preferences are inconsistent with the expected-utility theory.
450:—all gambles involve risk. By choosing Gamble D, the participant has a 1 in 3 chance of receiving nothing, and by choosing Gamble A, a 2 in 3 chance of receiving nothing. If Gamble A was less risky than Gamble B, it would follow that Gamble C was less risky than Gamble D (and vice versa), so the risk is not averted in this way.
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range (the probability of getting a black ball). The average person expects there to be fewer black balls than yellow balls because, in most real-world situations, it would be to the advantage of the experimenter to put fewer black balls in the urn when offering such a gamble. On the other hand, when
376:
Gamble C to Gamble D if and only if the participant believes that drawing a red or yellow ball is more likely than drawing a black or yellow ball. It might seem intuitive that if drawing a red ball is more likely than drawing a black ball, drawing a red or yellow ball is also more likely than drawing
116:
Ellsberg's findings indicate that choices with an underlying level of risk are favored in instances where the likelihood of risk is clear, rather than instances in which the likelihood of risk is unknown. A decision-maker will overwhelmingly favor a choice with a transparent likelihood of risk, even
465:
There have been various attempts to provide decision-theoretic explanations of
Ellsberg's observation. Since the probabilistic information available to the decision-maker is incomplete, these attempts sometimes focus on quantifying the non-probabilistic ambiguity that the decision-maker faces – see
457:, which cannot be accounted for in expected utility theory. It has been demonstrated that this phenomenon occurs only when the choice set permits the comparison of the ambiguous proposition with a less vague proposition (but not when ambiguous propositions are evaluated in isolation).
524:
the expected utility and maximize the robustness against uncertainty in the imprecise probabilities. This robust-satisficing approach can be developed explicitly to show that the choices of decision-makers should display precisely the preference reversal that
Ellsberg observed.
528:
Another possible explanation is that this type of game triggers a deceit aversion mechanism. Many humans naturally assume in real-world situations that if they are not told the probability of a certain event, it is to deceive them. Participants make the same decisions in the
625:
is a widely received alternative to utility maximization, taking into account ambiguity-averse preferences. This model reconciles the notion that intuitive decisions may violate the ambiguity neutrality, established within both the
Ellsberg Paradox and
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containing 90 balls: 30 balls are red, while the remaining 60 balls are either black or yellow in unknown proportions. The balls are well mixed so that each ball is as likely to be drawn as any other. The participants then choose a gambling scenario:
219:
Typically, participants were seen to be indifferent between bet 1A and bet 2A (consistent with expected utility theory) but were seen to strictly prefer Bet 1A to Bet 1B and Bet 2A to 2B. This result is generally interpreted to be a consequence of
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as they would about related but not identical real-life problems where the experimenter would be likely to be a deceiver acting against the subject's interests. When faced with the choice between a red ball and a black ball, the probability of
224:(also known as uncertainty aversion); people intrinsically dislike situations where they cannot attach probabilities to outcomes, in this case favoring the bet in which they know the probability and utility outcome (0.5 and $ 1 respectively).
652:
in 1952, Ellsberg left immediately to serve as a US Marine before coming back to
Harvard in 1957 to complete his post-graduate studies on decision-making under uncertainty. Ellsberg left his graduate studies to join the
481:. The agent is told precise probabilities of some outcomes, though the practical meaning of the probability numbers is not entirely clear. For instance, in the gambles discussed above, the probability of a red ball is
639:
Other alternative explanations include the competence hypothesis and the comparative ignorance hypothesis. Both theories attribute the source of the ambiguity aversion to the participant's pre-existing knowledge.
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the expected utility is also impossible. The info-gap approach supposes that the agent implicitly formulates info-gap models for the subjectively uncertain probabilities. The agent then tries to
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scandal then encircling
Ellsberg's life. The book is considered a highly-influential paper and is still considered influential within economic academia about risk ambiguity and uncertainty.
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a black or yellow ball. So, supposing the participant strictly prefers Gamble A to Gamble B, it follows that he/she will also strictly prefer Gamble C to Gamble D, and similarly conversely.
595:
To describe how an individual would take decisions in a world where uncertainty aversion exists, modifications of the expected utility framework have been proposed. These include:
369:). Also, there would be indifference between the choices if the participant thought that a red ball was as likely as a black ball. Similarly, it follows the participant will
606:
integral used as a way of measuring expected utility in situations with unknown parameters. The mathematical principle is seen as a way in which the contradiction between
1010:
Yasuhiro Sakai, Daniel
Ellsberg on J.M. Keynes and F.H. Knight: risk ambiguity and uncertainty. Evolutionary and Institutional Economics Review. 2018. (16): 1-18
513:. No probability information whatsoever is provided regarding other outcomes, so the participant has very unclear subjective impressions of these probabilities.
453:
However, because the exact chances of winning are known for
Gambles A and D and not known for Gambles B and C, this can be taken as evidence for some sort of
141:
There are two urns each containing 100 balls. It is known that urn A contains 50 red and 50 black, but urn B contains an unknown mix of red and black balls.
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In light of the ambiguity in the probabilities of the outcomes, the agent is unable to evaluate a precise expected utility. Consequently, a choice based on
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as a strategic analyst but continued to do academic work on the side. He presented his breakthrough paper at the
December 1960 meeting of the
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Ellsberg's experimental research involved two separate thought experiments: the 2-urn 2-color scenario and the 1-urn 3-color scenario.
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I. Gilboa and D. Schmeidler. Maxmin expected utility with non-unique prior. Journal of
Mathematical Economics, 18(2):141–153, 1989.
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popularized the paradox in his 1961 paper, "Risk, Ambiguity, and the Savage Axioms". It is generally taken to be evidence of
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However, ambiguity aversion would predict that people would strictly prefer Gamble A to Gamble B, and Gamble D to Gamble C.
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470:. That is, these alternative approaches sometimes suppose that the agent formulates a subjective (though not necessarily
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Heath, Chip; Tversky, Amos (1991). "Preference and Belief: Ambiguity and
Competence in Choice under Uncertainty".
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Additionally, the participant may choose a separate gamble scenario within the same situational parameters:
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Ellsberg's findings violate assumptions made within common Expected Utility Theory, with participants
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Utility theory models the choice by assuming that in choosing between these gambles, people assume a
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The experimental conditions manufactured by Ellsberg serve to rely upon two economic principles:
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they believe that drawing a red ball is more likely than drawing a black ball (according to
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673:. The work was made public in 2001, some 40 years after being published, because of the
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Mathematically, the estimated probabilities of each color ball can be represented as
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1073:(1989). "Subjective Probability and Expected Utility without Additivity".
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in instances where the unknown alternative will likely produce greater
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that the non-red balls are yellow versus black, and then compute the
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Since the prizes are the same, it follows that the participant will
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792:. Experimental Economics Center, Georgia State University. 2006
18:
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Daniel Ellsberg's 1962 paper, "Risk, Ambiguity, and Decision"
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Info-gap Decision Theory: Decisions Under Severe Uncertainty
936:"Rationality Intertwined: Classical vs Institutional View"
862:(1995). "Ambiguity Aversion and Comparative Ignorance".
602:: Created by French mathematician Gustave Choquet was a
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427:($ 0), but a contradiction can still be obtained for
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You receive $ 100 if you draw a black or yellow ball
661:. Ellsberg's work built upon previous works by both
614:, and Ellsberg's seminal findings can be reconciled.
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280:You receive $ 100 if you draw a red or yellow ball
97:in which people's decisions are inconsistent with
786:"Experimental Discussion of the Ellsberg Paradox"
144:The following bets are offered to a participant:
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387:Gamble A to Gamble B and Gamble D to Gamble C.
913:(2nd ed.). Academic Press. section 11.1.
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728:, pp. 75–76, paragraph 315, footnote 2.
446:In addition, the result holds regardless of
105:published a version of the paradox in 1921.
256:You receive $ 100 if you draw a black ball
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1029:Foundations of Rational Choice Under Risk
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69:Learn how and when to remove this message
743:"Risk, Ambiguity, and the Savage Axioms"
617:Maxmin expected utility: Axiomatized by
253:You receive $ 100 if you draw a red ball
32:This article includes a list of general
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474:) probability for possible outcomes.
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591:Decisions under uncertainty aversion
121:. When offered choices with varying
1057:. London: Macmillan. pp. 75–76
934:Lima Filho, Roberto (2 July 2009).
415:The result holds regardless of the
1453:What the Tortoise Said to Achilles
648:Upon graduating in Economics from
303:, of which red balls are drawn at
38:it lacks sufficient corresponding
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351:of the two gambles individually.
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968:Journal of Risk and Uncertainty
865:Quarterly Journal of Economics
751:Quarterly Journal of Economics
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1065:– via Internet Archive.
821:International Economic Review
477:One such attempt is based on
411:The generality of the paradox
339:Utility theory interpretation
1847:Probability theory paradoxes
1842:Paradoxes in utility theory
1033:. Oxford University Press.
702:Subjective expected utility
669:, challenging the dominant
99:subjective expected utility
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16:Paradox in decision theory
1832:Decision-making paradoxes
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1054:A Treatise on Probability
909:Ben-Haim, Yakov (2006).
635:Alternative explanations
600:Choquet expected utility
479:info-gap decision theory
1372:Paradoxes of set theory
612:Expected utility theory
431:($ 100) <
423:($ 100) >
391:Numerical demonstration
367:expected utility theory
53:more precise citations.
697:Experimental economics
671:rational choice theory
608:rational choice theory
1837:Statistical paradoxes
468:Knightian uncertainty
461:Possible explanations
361:Gamble A to Gamble B
294:Knightian uncertainty
129:Experimental research
1738:Kavka's toxin puzzle
1510:Income and fertility
1049:Keynes, John Maynard
1025:Anand, Paul (1993).
439:($ 100) =
1852:Thought experiments
1397:Temperature paradox
1320:Free choice paradox
1184:Fitch's knowability
811:Segal, Uzi (1987).
659:Econometric Society
550:is compared to the
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103:John Maynard Keynes
1773:Prisoner's dilemma
1459:Heat death paradox
1447:Unexpected hanging
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692:Ambiguity aversion
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1763:Preparedness
1707:
1595:Productivity
1575:Mandeville's
1529:
1367:Opposite Day
1295:Burali-Forti
1290:Bhartrhari's
1078:
1075:Econometrica
1074:
1059:. Retrieved
1053:
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929:
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794:. Retrieved
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56:
37:
1693:Condorcet's
1545:Giffen good
1505:Competition
1259:White horse
1234:Omnipotence
1061:17 February
726:Keynes 1921
663:J.M. Keynes
604:subadditive
345:probability
300:probability
204:get $ 1 if
186:get $ 1 if
168:get $ 1 if
150:get $ 1 if
59:August 2015
51:introducing
1826:Categories
1768:Prevention
1758:Parrondo's
1748:Navigation
1733:Inventor's
1728:Hedgehog's
1688:Chainstore
1671:Population
1666:New states
1600:Prosperity
1580:Mayfield's
1422:Entailment
1402:Barbershop
1315:Epimenides
713:References
667:F.H Knight
623:Schmeidler
552:lower part
531:experiment
518:maximizing
34:references
1783:Willpower
1778:Tolerance
1753:Newcomb's
1718:Fredkin's
1605:Scitovsky
1525:Edgeworth
1520:Easterlin
1485:Antitrust
1382:Russell's
1377:Richard's
1350:Pinocchio
1305:Crocodile
1224:Newcomb's
1194:Goodman's
1189:Free will
1174:Epicurean
1145:paradoxes
1083:CiteSeerX
998:146410959
976:CiteSeerX
874:CiteSeerX
522:satisfice
275:Gamble D
248:Gamble B
1811:Category
1708:Ellsberg
1560:Leontief
1540:Gibson's
1535:European
1530:Ellsberg
1500:Braess's
1495:Bertrand
1473:Economic
1407:Catch-22
1387:Socratic
1229:Nihilism
1199:Hedonism
1159:Analysis
1143:Notable
1051:(1921).
974:: 5–28.
790:EconPort
741:(1961).
681:See also
472:Bayesian
443:($ 0)).
371:strictly
356:strictly
272:Gamble C
245:Gamble A
101:theory.
1713:Fenno's
1678:Arrow's
1661:Alabama
1651:Abilene
1630:Tullock
1585:Metzler
1427:Lottery
1417:Drinker
1360:Yablo's
1355:Quine's
1310:Curry's
1273:Logical
1249:Sorites
1239:Preface
1219:Moore's
1204:Liberal
1179:Fiction
1105:1911053
944:2389751
896:2946693
842:2526866
796:May 28,
772:1884324
650:Harvard
584:
572:
568:
556:
554:of the
548:
536:
511:
499:
495:
483:
333:
321:
317:
305:
202:Bet 2B:
184:Bet 1B:
166:Bet 2A:
148:Bet 1A:
119:utility
95:paradox
93:) is a
47:improve
1620:Thrift
1590:Plenty
1565:Lerner
1555:Jevons
1550:Icarus
1480:Allais
1442:Ross's
1280:Barber
1264:Zeno's
1209:Meno's
1103:
1085:
1037:
996:
978:
942:
917:
894:
876:
840:
770:
619:Gilboa
403:, and
374:prefer
359:prefer
85:, the
36:, but
1723:Green
1703:Downs
1635:Value
1570:Lucas
1437:Raven
1345:No-no
1300:Court
1285:Berry
1101:JSTOR
994:S2CID
892:JSTOR
838:JSTOR
816:(PDF)
768:JSTOR
746:(PDF)
207:black
171:black
1801:List
1625:Toil
1340:Card
1335:Liar
1063:2020
1035:ISBN
940:SSRN
915:ISBN
798:2022
665:and
621:and
319:vs.
123:risk
89:(or
1093:doi
986:doi
884:doi
870:110
830:doi
760:doi
234:urn
189:red
153:red
81:In
1828::
1099:.
1091:.
1079:57
1077:.
992:.
984:.
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938:.
890:.
882:.
868:.
850:^
836:.
826:28
824:.
818:.
788:.
766:.
756:75
754:.
748:.
610:,
581:90
575:60
565:90
545:90
539:30
508:91
502:30
492:90
486:30
399:,
335:.
1455:"
1451:"
1136:e
1129:t
1122:v
1107:.
1095::
1043:.
1000:.
988::
972:4
946:.
923:.
898:.
886::
844:.
832::
800:.
774:.
762::
630:.
578:/
570:–
562:/
559:0
542:/
505:/
489:/
441:U
437:U
433:U
429:U
425:U
421:U
405:B
401:Y
397:R
330:3
327:/
324:2
314:3
311:/
308:1
213:B
195:B
177:A
159:A
72:)
66:(
61:)
57:(
43:.
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