Risk, Ambiguity, and the Rank-Dependence Axioms

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1 American Economic Review 2009, 99:1, Risk, Ambiguity, and the Rank-Dependence Axioms By Mark J. Machina* Choice problems in the spirit o Ellsberg (1961) suggest that rank-dependent ( Choquet expected utility ) preerences over subjective gambles might be subject to the same diiculties that Ellsberg s earlier examples posed or subjective expected utility. These diiculties stem rom event-separability properties that rank-dependent preerences partially retain rom expected utility, and suggest that nonseparable models o preerences might be better at capturing eatures o behavior that lead to these paradoxes. (JEL D81) Consider an urn holding 101 balls, each marked with a number rom 1 through 4. You don t know the number o balls o each type, but you do know that exactly 50 are marked with either a 1 or a 2, and 51 are marked with either a 3 or a 4. This is a variation on the classic urns o Daniel Ellsberg (1961). Given its inormation structure, you don t know the probability o any given number being drawn, but you do know there s an exact 50/101 chance o it being either a 1 or a 2, and a 51/101 chance o it being either a 3 or a 4. Say you were oered the ollowing pair o bets on this urn. Which one would you choose? (O course, you could be indierent between the two bets.) Table Example 1First pair o bets2 50 balls 51 balls E 1 E 2 E 3 E $8,000 $8,000 $4,000 $4, $8,000 $4,000 $8,000 $4,000 Say you were instead oered the ollowing bets. In this case, which would you choose? Table Example 1Second pair o bets2 50 balls 51 balls E 1 E 2 E 3 E $12,000 $8,000 $4,000 $ $12,000 $4,000 $8,000 $0 Call this urn and the bets above the 50:51 example. * Department o Economics, University o Caliornia, San Diego, 9500 Gilman Drive, La Jolla, Caliornia ( mmachina@ucsd.edu). I am indebted to Michael Birnbaum, Larry Epstein, Itzhak Gilboa, Edi Karni, Peter Klibano, Ehud Lehrer, Olivier L Haridon, Duncan Luce, Anthony Marley, David Schmeidler, Uzi Segal, Marciano Siniscalchi, Joel Sobel, Peter Wakker, Jiankang Zhang, and especially Robert Nau and Jacob Sagi or helpul comments and suggestions on this material. All errors and opinions are my own. 385

2 386 THE AMERICAN ECONOMIC REVIEW march 2009 The purpose o this paper is to explore how this and similar examples may pose diiculties or the well-known rank-dependent or Choquet expected utility model o choice under subjective uncertainty, similar to those posed by Ellsberg s original counterexamples to the classical subjective expected utility hypothesis. The ollowing section recalls some o Ellsberg s examples and the approach taken by the Choquet model to address them. Section II shows how (typical?) choices in the above and similar problems may pose Ellsberg-type diiculties or the Choquet model. Section III discusses the sources o these diiculties and their implications or modeling choice under subjective uncertainty. I. Savage, Ellsberg, and Choquet The classic subjective expected utility (SEU) model o choice under uncertainty, as axiomatized by Leonard Savage (1954), involves bets o the orm x 1 on E 1 ; ; x n on E n 4 or some mutually exclusive and exhaustive collection o events 5E 1,,E n 6 and (not necessarily distinct) outcomes x 1,, x n. Savage s axioms imply the existence o a cardinal unction U1 2 over outcomes and a subjective probability measure m1 2 over events, such that the individual evaluates such bets according to an ordinal preerence unction o the orm W SEU K W SEU 1x 1 on E 1 ; ; x n on E n 2 K g n i51 U1x i2m1e i 2. This model o risk preerences and belies has seen widespread application in the economics and decision theory literature. A key aspect o the model, which can be seen rom the additive structure o W SEU 1 2, is that preerences are separable across mutually exclusive events. 1 The classic counterexample to the SEU model is the well-known Ellsberg Paradox (Daniel Ellsberg 1961), which involves the ollowing pairs o bets on a 90-ball urn: Table 3 Three-Color Ellsberg Paradox 30 balls 60 balls red black yellow 1* 1 2 $100 $0 $0 2* 1 2 $0 $100 $0 3* 1 2 $100 $0 $100 4* 1 2 $0 $100 $100 s s When aced with these choices, most individuals express a strict preerence or 1* 1 2 over 2* 1 2 and or 4* 1 2 over 3* 1 2, as indicated to the right o the table. However, these preerences violate the SEU unctional orm g n i51 U1x i2 m1e i 2, since they imply the inconsistent inequalities U($100) m(red) 1 U($0)m(black). U($0)m(red) 1 U($100)m(black) and U($100)m(red) 1 U($0) m(black), U($0)m(red) 1 U($100)m(black). More speciically, they violate event-separability, since individuals ranking o the subacts [$100 on red; $0 on black] versus [$0 on red; $100 on black] depends upon whether the mutually exclusive event yellow yields a payo o $0 or $100. The intuition behind these choices is clear: 1* 1 2 oers the $100 prize on an objective 1/3- probability event, whereas 2* 1 2 oers it on one element o an inormationally symmetric but subjective partition ({black, yellow}) o a 2 /3-probability event. Similarly, 4* 1 2 oers the prize on an objective 2 /3- probability event, whereas 3* 1 2 oers it on the union o a 1 /3- probability event 1 Axiomatically, event-separability ollows rom Savage s Axiom P2 (1954, 23), termed the Sure-Thing Principle.

3 VOL. 99 NO. 1 machina: Risk, Ambiguity, and the Rank-Dependence Axioms 387 and the other element o that subjective partition. In each case, the purely objective bet is preerred to its inormationally equivalent 2 subjective counterpart. Though the urn described above is the most well known o Ellsberg s examples, it is not the only one. Another example (1961, 654, n. 4), suggested to Ellsberg by Kenneth Arrow, involves bets on the ollowing our-color urn. Again, the objective bets and are preerred to their inormationally equivalent subjective counterparts and Table 4 Four-Color Ellsberg Paradox 100 balls 50 balls 50 balls red black green yellow ƒ $100 $100 $0 $0 ƒ $100 $0 $100 $0 ƒ $0 $100 $0 $100 ƒ $0 $0 $100 $100 s s These examples and others (e.g., Ellsberg 1961, 2001) suggest that individuals preerences depart rom the classic SEU model by exhibiting a systematic preerence or objective over subjective bets, a phenomenon known as ambiguity aversion. Such examples have spurred the development o alternatives to subjective expected utility, most notably the expected utility with rank-dependent subjective probabilities or Choquet expected utility model o preerences over subjective bets. Axiomatized by Itzhak Gilboa (1987) and David Schmeidler (1989), it posits a preerence unction W CEU 1 2 o the orm 3 (1) W CEU K W CEU 1x 1 on E 1 ; ; x n on E n 2 K g n U1x i51 i 23C 1< j51 i E j2 C 1< j5 1 E j24 or some cardinal utility unction U1 2 and nonadditive measure or capacity C 1 2, deined over 1 2 s decumulative events < j51 i E j and satisying C 1[2 5 0 and C 1< j51 n E j2 5 1, where the outcomes x 1,, x n must be labeled in order o weakly decreasing preerence 1outcomes labeled in this manner are denoted by the notation x 1,, x n 2. Since it replaces subjective probabilities m1e i 2 by weights o the orm 3C 1< j51 i E j2 C 1< j5 1 E j24, W CEU 1 2 no longer exhibits event- separability. However the telescoping nature o these weights ensures that, like subjective probabilities, they sum to unity. 4 There are several alternative axiomatic derivations o this model, but each involves some orm o the ollowing property: 5 2 I {E 1,, E n } is an inormationally symmetric partition o an objective event E with probability p, we will say that any k-element union o its events is inormationally equivalent to any objective event with probability pk/n. Acts are said to be inormationally equivalent i they assign their respective payos to inormationally equivalent events. 3 In the ollowing ormulas, unions o the orm < 0 i 1 j51 are taken to equal [, and sums gj5 i are taken to equal zero. 4 The property o summing to unity prevents the nonmonotonicity exhibited by preerence unctions o the orm W U1x 1 2C 1E U1x n 2C 1En 2. Since the telescoping property also implies U1x23C 1< j51e i j 2 C 1< E j51 j24 1 U1x23C 1< i 11 E j51 j2 C 1< j51e i j 24 5 U1x23C 1< i 11 E j51 j2 C 1< E j51 j24 5 U(x)[C 11< E j51 j2<(e i <E i11 22 C 1< E j51 j24, equal outcomes x i 5 x i11 and their events can be combined, and an individual outcome s event can be split. 5 This property is similar, though not quite equivalent, to P2* o Gilboa (1987) and Axiom (ii) o Schmeidler (1989). See also the treatments o Yutaka Nakamura (1990), Rakesh Sarin and Peter P. Wakker (1992), Soo Hong Chew and Edi Karni (1994), Chew and Wakker (1996), Wakker (1996), Mohammed Abdellaoui and Wakker (2005), and the review o Veronika Köbberling and Wakker (2003).

4 388 THE AMERICAN ECONOMIC REVIEW march 2009 Comonotonic Sure-Thing Principle: I 3x 1 on E 1 ; ; x n on E n 4 3y 1 on E 1 ; ; y n on E n 4, i x i 5 y i and xˆ i 5 ŷ i and i x i 5 xˆ i and y i 5 ŷ i or all i [ I # 51,, n6, or all i o I, then 3xˆ 1 on E 1 ; ; xˆ n on E n 4 3ŷ 1 on E 1 ; ; ŷ n on E n 4. That is to say, i two acts have the same set o decumulative events E 1, E 1 <E 2, E 1 <E 2 <E 3, and common outcomes over some subamily o events 5E i Z i [ I 6, then the outcomes they actually have over that subamily should not aect their ranking. This implies a partial orm o event- separability, which we term tail-separability, and which is apparent rom the ollowing decomposition o W CEU 1 2 into terms involving its upper-tail, middle, and lower-tail decumulative events: (2) W CEU 1x 1 on E 1 ; ; x n on E n 2 K g i9 i51u1x i 2 3C 1< j51 i E j2 C 1< j5 1 E j24 1 g i0 i5i911u1x i 2 3C 1< j51 i E j2 C 1< j5 1 E j24 1 g n i5i011u1x i 2 3C 1< j51 i E j2 C 1< j5 1 E j24. This model can represent the ambiguity-averse rankings o 1* 1 2 s 2* 1 2 and 3* 1 2 a 4* 1 2 in the three-color Ellsberg example, since any capacity satisying C 1red2 5 1 /3. C 1black2 and C 1red<yellow2, 2 /3 5 C1black<yellow2 will imply W CEU 1 1* /3 3 U1$ /3 3 U1$02. W CEU 1 2* 1 22 and W CEU 1 3* 1 22, 2 /3 3 U1$ /3 3 U1$02 5 W CEU 1 4* The modal preerences in the our-color Ellsberg urn can be modeled in a similar manner. As shown by Gilboa, Schmeidler, and others, Choquet expected utility preerences retain much o the structure and predictive power o classical subjective expected utility, but are suiciently more general to be able to accommodate the various Ellsberg Paradoxes and other systematic departures rom expected utility and probabilistic belies. II. An Ellsberg Paradox or Choquet Expected Utility? The Choquet expected utility model o subjective act preerences successully captures the type o ambiguity aversion displayed in Ellsberg s examples, and has received widespread attention in the literature. It is not clear, however, how it perorms in other cases, such as the 50:51 example above. From the tables, it is clear that acts and are obtained rom and by a pair o common-outcome tail shits, namely $8,000 up to $12,000 in event E 1 and $4,000 down to $0 in E 4. By tail-separability, a Choquet expected utility maximizer would preer to i and only i he or she preers to A pair o reversed rankings, such as s and a 4 1 2, would contradict tail-separability, and hence contradict Choquet. Whether individuals (including the reader) actually exhibit such reversed rankings is o course an empirical matter. But there is a strong Ellsberg-like argument why they might. Though neither pair consists o inormationally equivalent acts as in Ellsberg s examples, each involves a tradeo between objective and subjective uncertainty. On the one hand, the lower act in each pair diers rom the upper act only in its placement o the outcomes $4,000 and $8,000 between the events E 2 and E 3. Since the urn has a 50:51 ball count, in each pair the lower act has a slight objective advantage over the upper act. On the other hand, the lower act in each pair is more ambiguous than the upper act, though the extent o the ambiguity dierence is not the same or the two pairs. In the irst pair, ƒ is distinctly more ambiguous than 1 1 2: whereas s payos are perectly corrected with the objec-

5 VOL. 99 NO. 1 machina: Risk, Ambiguity, and the Rank-Dependence Axioms 389 tive partition 5E 1 <E 2, E 3 <E 4 6, s payos are not correlated at all. Thus an ambiguity-averse individual may well eel that s ambiguity more than osets its slight objective advantage, and choose But in the second pair, is only somewhat more ambiguous than 3 1 2: while s outcomes are somewhat less correlated with the partition 5E 1 <E 2, E 3 <E 4 6 than are s, the dierence in these correlations is less stark than between and The individual may eel that this smaller ambiguity dierence does not oset s objective advantage, and choose III. Ambiguity Aversion as Event-Nonseparability How is it that rank-dependent preerences, which successully capture ambiguity-averse behavior in Ellsberg s original examples, are violated by what would seem to be similar behavior in our modiied version o these examples? The answer lies in a property that rank-dependent preerences retain rom classical expected utility preerences, namely, tail-separability. In the three-color Ellsberg urn, the acts 3* 1 2 and 4* 1 2 are obtained rom 1* 1 2 and 2* 1 2 by a common-outcome shit o $0 up to $100 in the event yellow. By ull event-separability, this should not alter the preerence ranking o the upper versus lower act within each pair. But this shit reverses the ambiguity properties o the upper versus lower act, rom 1* 1 2 being ully objective and 2* 1 2 subjective, to the other way around or 3* 1 2 and 4* 1 2. Thus, an ambiguity-averse individual would exhibit the reversed rankings 1* 1 2 s 2* 1 2 and 3* 1 2 a 4* 1 2. In other words, models that exhibit ull event-separability such as subjective expected utility are incompatible with cross-event eects due to attitudes toward ambiguity. Models that drop ull event-separability but retain tail-separability such as rank-dependent subjective expected utility retain this type o incompatibility. In the 50:51 example, the common-outcome tail shits that convert acts and to acts and have similar crossevent eects on the ambiguity properties o the upper versus lower acts. These shits do not reverse the ambiguity properties as they do or Ellsberg. They still aect them, however, by reducing the extent o the ambiguity dierence between the upper and lower act. I ambiguity matters to the decision maker, the magnitude o this ambiguity dierence can determine whether it does or does not oset some other dierence between the acts, such as an objective probability dierence (as in our example) or, alternatively, an outcome dierence. 6 Common-outcome tail shits do not aect only ambiguity properties. Consider a choice between and below. It is not clear which o these acts is more ambiguous: has a payo dierence o $8,000 riding on the subjective subpartition 5E 3,E 4 6, whereas divides this stake, with $4,000 riding on each o the subpartitions 5E 1,E 2 6 and 5E 3,E 4 6. Thus, gives a 100 percent chance o $4,000 riding on the unknown composition o the urn, whereas gives a 50 percent chance o $8,000 riding on this subjective uncertainty. 7 As in the 50:51 example, an ambiguity averter s choice may well depend on their trade-o rate between this objective and subjective uncertainty, and their attitudes toward the dierent payo levels involved. I their personal trade-o rate exactly matched that o the problem, they could be indierent. Say the individual does have a strict preerence, s Since and dier rom and by an ordered sequence o common-outcome tail shits, 8 tail-separability implies that this ranking should extend to s But as seen in Table 5, is an inormationally symmetric let-right relection o 5 1 2, and is a let-right relection o Surely, anyone 6 To create an example o the latter, change the ball count to 50:50, and change the E 3 payo in acts and ƒ rom $4,000 up to $4, Neither act can be said to have a greater correlation o its outcomes with the objective partition {E 1 <E 2,E 3 <E 4 }. 8 First, shit $0 up to $4,000 in event E 4, then shit $4,000 down to $0 in E 1.

6 390 THE AMERICAN ECONOMIC REVIEW march 2009 Table 5 Relection Example 50 balls 50 balls E 1 E 2 E 3 E $4,000 $8,000 $4,000 $ $4,000 $4,000 $8,000 $ $0 $8,000 $4,000 $4, $0 $4,000 $8,000 $4,000 with the ranking s ƒ should have the relected ranking a Besides aecting the relative ambiguity properties o acts, tail-shits can also aect their symmetry properties, again leading to Ellsberg-like diiculties or the Choquet model. Call this urn and bets the relection example. O course, there is no reason why an individual need have a strict preerence in either pair o acts, in which case the relections cause no problem. Indeed, one might argue that since the decumulative events o and are either identical or inormationally symmetric to each other, a Choquet individual must be indierent between them, and similarly between and 8 1 2, so there is no relection paradox. This is not a counterargument to the point being made here, so much as its contrapositive: i no Choquet individual would have a strict preerence in either pair, then anyone who does have a strict preerence cannot be Choquet. It is much like arguing that Ellsberg s three-color urn does not violate SEU, since no SEU individual would ever exhibit the reversed rankings 1* 1 2 s 2* 1 2 and 3* 1 2 a 4* 1 2. In all such choice problems, the issue is not how individuals ought to choose, but rather, how they do choose. In recent experimental tests, Olivier L Haridon and Lætitia Placido (orthcoming) ound that over 90 percent o subjects expressed strict preerence in choice problems with the above structure, and that roughly 70 percent violated tail-separability by exhibiting reversed rankings o the orm s and a 8 1 2, or the orm a and s I there is a general lesson to be learned rom Ellsberg s examples and the examples here, it is that the phenomenon o ambiguity aversion is intrinsically one o nonseparable preerences across mutually exclusive events, and that models that exhibit ull or even partial event-separability cannot capture all aspects o this phenomenon. This suggests that the study o choice under uncertainty should proceed in a manner similar to standard consumer theory, which develops theoretical results like the Slutsky equation which do not require separability across individual commodities and models empirical phenomena like inerior goods which are not compatible with this property. 9 9 Ehud Lehrer (2008) demonstrates that the partially speciied probabilities model o Lehrer (2007) and the concave integral preerence unction o Lehrer (orthcoming), neither o which are event-separable or tail-separable, are each consistent with the rankings s and a in the 50:51 example, as well as with s and a in the relection example. Jacob Sagi (private correspondence) has shown that the nonseparable models o Peter Klibano, Massimo Marinacci, and Sujoy Mukerji (2005) and Chew and Sagi (2008) are similarly consistent with s and a Marciano Siniscalchi (2008a, b) shows that the vector expected utility model o Siniscalchi (2008a) is consistent with the plausible preerence pattern in the relection example, and explores urther implications or the notion o ambiguity aversion. Aurélien Baillon, L Haridon, and Placido (2008) explore conditions under which multiple-priors models such as the maxmin expected utility model o Gilboa and Schmeidler (1989) are/are not consistent with the 50:51 example and the relection example, and Kin Chung Lo (2007) has shown how a modiied version o the relection example can be used to test the model o Klibano (2001).

7 VOL. 99 NO. 1 machina: Risk, Ambiguity, and the Rank-Dependence Axioms 391 The arguments and examples o this paper have involved theoretical properties o models and hypothesized properties o preerences. As with any such questions, the issues must ultimately be decided by the data. 10 REFERENCES Abdellaoui, Mohammed, and Peter P. Wakker The Likelihood Method or Decision under Uncertainty. Theory and Decision, 58(1): Baillon, Aurélien, Olivier L Haridon, and Lætitia Placido Risk, Ambiguity and the Rank- Dependence Axioms: Comment. Unpublished. Birnbaum, Michael H Tests o Rank-Dependent Utility and Cumulative Prospect Theory in Gambles Represented by Natural Frequencies: Eects o Format, Event Framing, and Branch Splitting. Organizational Behavior and Human Decision Processes, 95(1): Birnbaum, Michael H New Paradoxes o Risky Decision Making. Psychological Review, 115(2): Birnbaum, Michael H., and Alredo Chavez Tests o Theories o Decision Making: Violations o Branch Independence and Distribution Independence. Organizational Behavior and Human Decision Processes, 71(2): Birnbaum, Michael H., and Juan B. Navarrete Testing Descriptive Utility Theories: Violations o Stochastic Dominance and Cumulative Independence. Journal o Risk and Uncertainty, 17(1): Birnbaum, Michael H., Jamie N. Patton, and Melissa K. Lott Evidence against Rank-Dependent Utility Theories: Violations o Cumulative Independence, Interval Independence, Stochastic Dominance, and Transitivity. Organizational Behavior and Human Decision Processes, 77(1): Camerer, Colin, and Martin Weber Recent Developments in Modeling Preerences: Uncertainty and Ambiguity. Journal o Risk and Uncertainty, 5(4): Carbone, Enrica, and John D. Hey A Comparison o the Estimates o EU and Non-EU Preerence Functionals Using Data rom Pairwise Choice and Complete Ranking Experiments. Geneva Papers on Risk and Insurance Theory, 20(1): Chateauneu, Alain On the Use o Capacities in Modeling Uncertainty Aversion and Risk Aversion. Journal o Mathematical Economics, 20(4): Chew, Soo Hong, and Edi Karni Choquet Expected Utility with a Finite State Space: Commutativity and Act-Independence. Journal o Economic Theory, 62(2): Chew, Soo Hong, and Jacob S. Sagi Small Worlds: Modeling Attitudes toward Sources o Uncertainty. Journal o Economic Theory, 139(1): Chew, Soo Hong, and Peter Wakker The Comonotonic Sure-Thing Principle. Journal o Risk and Uncertainty, 12(1): Ellsberg, Daniel Risk, Ambiguity, and the Savage Axioms. Quarterly Journal o Economics, 75(4): Ellsberg, Daniel Risk, Ambiguity and Decision. New York: Garland. Epstein, Larry G A Deinition o Uncertainty Aversion. Review o Economic Studies, 66(3): Fennema, Hein, and Peter P. Wakker A Test o Rank-Dependent Utility in the Context o Ambiguity. Journal o Risk and Uncertainty, 13(1): Gilboa, Itzhak Expected Utility with Purely Subjective Non-Additive Probabilities. Journal o Mathematical Economics, 16(1): Gilboa, Itzhak, and David Schmeidler Maxmin Expected Utility with Non-Unique Prior. Journal o Mathematical Economics, 18(2): Empirical studies o rank-dependent preerences and the Comonotonic Sure-Thing Principle include John Hey and Chris Orme (1994), Lukas Mangelsdor and Martin Weber (1994), Wakker, Ido Erev, and Elke Weber (1994), George Wu (1994), Enrica Carbone and Hey (1995), Hein Fennema and Wakker (1996), Michael Birnbaum and Alredo Chavez (1997), Birnbaum and Juan Navarrete (1998), Birnbaum, Jamie Patton, and Melissa Lott (1999), Birnbaum (2004), and Hey, Gianna Lotito, and Anna Maioletti (2008). See also the theoretical and empirical discussions o Uzi Segal (1987), Alain Chateauneu (1991), Colin F. Camerer and Martin Weber (1992), David Kelsey and John Quiggin (1992), Peter Kischka and Clemens Puppe (1992), Larry Epstein (1999), Wakker (2001), Anthony Marley and R. Duncan Luce (2005), Birnbaum (2008), and L Haridon and Placido (2008).

8 392 THE AMERICAN ECONOMIC REVIEW march 2009 Hey, John D., Gianna Lotito, and Anna Maioletti The Descriptive and Predictive Adequacy o Theories o Decision Making under Uncertainty/Ambiguity. University o York Discussion Paper in Economics 2008/04. Hey, John D., and Chris Orme Investigating Generalizations o Expected Utility Theory Using Experimental Data. Econometrica, 62(6): Kelsey, David, and John Quiggin Theories o Choice under Ignorance and Uncertainty. Journal o Economic Surveys, 6(2): Kischka, Peter, and Clemens Puppe Decisions Under Risk and Uncertainty: A Survey o Recent Developments. Mathematical Methods o Operations Research, 36(2): Klibano, Peter Stochastically Independent Randomization and Uncertainty Aversion. Economic Theory, 18(3): Klibano, Peter, Massimo Marinacci, and Sujoy Mukerji A Smooth Model o Decision Making under Ambiguity. Econometrica, 73(6): Köbberling, Veronika, and Peter P. Wakker Preerence Foundations or Nonexpected Utility: A Generalized and Simpliied Technique. Mathematics o Operations Research, 28(3): L Haridon, Olivier, and Lætitia Placido An Allais Paradox or Generalized Expected Utility Theories? Economics Bulletin, (4)19: 1 6. L Haridon, Olivier, and Lætitia Placido. Forthcoming. Betting on Machina s Relection Example: An Experiment on Ambiguity. Theory and Decision. Lehrer, Ehud Partially-Speciied Probabilities: Decisions and Games. Unpublished. Lehrer, Ehud A Comment on Two Examples by Machina. Unpublished. Lehrer, Ehud. Forthcoming. A New Integral or Capacities. Economic Theory. Lo, Kin Chung Risk, Ambiguity, and the Klibano Axioms. Unpublished. Mangelsdor, Lukas, and Martin Weber Testing Choquet Expected Utility. Journal o Economic Behavior and Organization, 25(3): Marley, A. A. J., and R. Duncan Luce Independence Properties vis-à-vis Several Utility Representations. Theory and Decision, 58(1): Nakamura, Yutaka Subjective Expected Utility with Non-Additive Probabilities on Finite State Spaces. Journal o Economic Theory, 51(2): Sarin, Rakesh K., and Peter Wakker A Simple Axiomatization o Nonadditive Expected Utility. Econometrica, 60(6): Savage, Leonard The Foundations o Statistics. New York: John Wiley and Sons. Revised and Enlarged Edition, New York: Dover Publications, Schmeidler, David Subjective Probability and Expected Utility without Additivity. Econometrica, 57(3): Segal, Uzi The Ellsberg Paradox and Risk Aversion: An Anticipated Utility Approach. International Economic Review, 28(1): Siniscalchi, Marciano. 2008a. Vector Expected Utility and Attitudes Toward Variation. Northwestern University CMS EMS Discussion Paper Siniscalchi, Marciano. 2008b. Machina s Relection Example and VEU Preerences: A Very Short Note. Unpublished. Wakker, Peter P The Sure-Thing Principle and the Comonotonic Sure-Thing Principle: An Axiomatic Analysis. Journal o Mathematical Economics, 25(2): Wakker, Peter P Testing and Characterizing Properties o Nonadditive Measures through Violations o the Sure-Thing Principle. Econometrica, 69(4): Wakker, Peter P., Ido Erev, and Elke U. Weber Comonotonic Independence: The Critical Test between Classical and Rank-Dependent Utility Theories. Journal o Risk and Uncertainty, 9(3): Wu, George An Empirical Test o Ordinal Independence. Journal o Risk and Uncertainty, 9(1):

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