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inverse-S: they find it for risk, and more pronounced for uncertainty; latter also concerns: uncertainty amplifies risk
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inverse-S: they find it for risk, and more pronounced for uncertainty; latter also concerns: uncertainty amplifies risk linear utility for small stakes: they assume linear utility. real incentives: random incentive system They say that the probability transformation function can depend on the source of uncertainty. This is an unfortunate terminology because the probability weighting function w(p) depends only on p under common terminology, and it is ten logically impossible that it would depend on a source or whatever else other than p. If I may be allowed to write about own work, in the three-stage decomposition W(A) = w((P(A))), proposed in Footnote 2 of Wakker (2004, Psychological Review, p. 239), (and not w) can depend on the source, and this is what may be happening here. In the source method of Abdellaoui et al. (2011) a composition wS(P(A)) is considered with P additive and wS depending on the souce, but wS is not called probability transformation but source function. Their idea to have risk (rather than ambiguity) attitude depend on source is so confusing that I usually avoid citing this paper, although otherwise it has many great ideas. This terminology is just too confusing. I was the AE editor handling this paper for MS, and did everything allowed within the boundaries for editors to make the authors change terminology, but did not succeed. Find that pessimism decreases for more familiar sources (competence effect). %} Kilka, Michael & Martin Weber (2001) “What Determines the Shape of the Probability Weighting Function under Uncertainty,” Management Science 47, 1712–1726. {% foundations of statistics proposes as index a probability of replicating an effect. Has several references to discussions. Several discussions in December Issue of 2005. %} Killeen, Peter R. (2005) “An Alternative to Null-Hypothesis Significance Tests,” Psychological Science 16, 345–353. {% foundations of statistics; reply to Wagenmakers & Grünwald (2006) %} Killeen, Peter R. (2005) “The Problem with Bayes,” Psychological Science 17, 643–644. {% DC = stationarity: p. 603 bottom of 2nd column, and p. 604 1st column l. 8. This paper considers receipt of one nonzero outcome at some time point. It proposes not to use a multiplicative model to integrate utility and discounting, but an additive model (Eq. 6). Puts this forward as its central contribution (p. 605 directly following Eq. 6). Although it also argues at length that we should look at utilities of outcomes and not at outcomes and puts this also forward as a similarly central contribution (p. 606 last para of column 1). One difficulty I have with the additive-multiplicative is that this form, in the absence of other nonzero outcomes, is purely ordinal and we can just apply the exponential function to get back the multiplicative form after all, after which the only point at which this model generalizes classical exponential discounting is that a power transformation of time is added. But it still is multiplicative then. Another difficulty is that there is a time point at which the value of a positive outcome becomes 0. The author view this point from its sunny side, with a numerical example that $250 in 21 years from now should have value 0 (p. 605 middle of 2nd column). These insights are extremely new to anyone who has worked on intertemporal choice so far. P. 611 has another extremely interesting move: the author proposes to use his additive instead of multiplicative model also for risky choice, and sees sunny sides here too. The factual observation that he puts forward on p. 611, 2nd colum, 2nd para: “Consumers do not multiply the payoff by its probability; they sum utility functions on magnitude and probability” of course provides strong evidence supporting his insight. So then we get to deal with models where people have a strict preference for increasing an outcome obtained with 0 probability, but the author has his defense in place: “it is a mark of humanity that some individuals can always be found who will take that foolhardy gamble.” (p. 611 2nd column 2nd para) So, again, extremely new insights, be it now for all working on risky choice. I was surprised on p. 602 to find that the derivative of discounting (rather than utility) is taken to be Bernoulli’s utility idea. P. 604 top of 2nd column tells us, citing Luce, that power utility satisfies all empirical and theoretical desiderata for utility. With this publication the top journal Psychological Review gives us many ideas that we would never have dreamed of otherwise. %} Killeen, Peter R. (2009) “An Additive-Utility Model of Delay Discounting,” Psychological Review 116, 602–619. {% Table 2: Kahneman & Tversky (1979) is 2nd most cited paper in the economic literature between 1970 and 2005. %} Kim, E. Han, Adair Morse, & Luigi Zingales (2006) “What Has Mattered to Economics since 1970,” Journal of Economic Perspectives 20, 189–202. {% revealed preference %} Kim, Taesung (1987) “Intransitive Indifference and Revealed Preference,” Econometrica 55, 95–115. {% revealed preference %} Kim, Taesung & Marcel K. Richter (1986) “Nontransitive-Nontotal Consumer Theory,” Journal of Economic Theory 38, 324–363. {% intuitive versus analytical decisions; Reflective equilibrium: utility elicitation; compares utility assessment methods, implemented on the computer, regarding acceptance by participants if recommended choice is contrary to intuitive choice. Their “UF” program had an interactive resolution of inconsistencies built in. This worked well and clients had more confidence in this program than in programs that did not consider inconsistencies. %} Kimbrough, Steven O. & Martin Weber (1994) “An Empirical Comparison of Utility Assessment Programs,” European Journal of Operational Research 75, 617–633. {% The authors examine intertemporal discounting, distinguising between the delay effect and the interval effect. Probably the former refers to discounting with the immediacy effect included and the latter without. But I did not read the paper long enough to be able to figure out what exactly the authors mean. DC = stationarity: several places suggest that the authors equate them (abstract, p. 88 ll. 3-5, p 88 footnote 1) but never clearly. Maybe (I do not know) their distinction between delay and interval refers to the distinction between stopwatch time and calendar time and then it would mean that they do distinguish. %} Kinari, Yusuke, Fumio Ohtake, & Yoshiro Tsutsui (2009) “Time Discounting: Declining Impatience and Interval Effect,” Journal of Risk and Uncertainty 39, 87–112. {% Seems to be useful in showing that pointwise continuity implies countable additivity. %} Kindler, Jürgen. (1983) “A Simple Proof of the Daniel-Stone Representation Theorem,” American Mathematical Monthly 90, 396–397. {% Nice verbal, but superficial, exposition of Bayesian Testing; nice annoted literature %} King, Raymond O. & Terrence B. O’Keefe (1989) “Belief Revision from Hypothesis Testing,” Journal of Accounting Literature 8, 1–24. {% real incentives/hypothetical choice: for time preferences: used real incentives; Seems to assume Mazur’s discounting function, linear utility, dynamic inconsistency. Experiment 1: fitting at individual level; 4 out of 24 participants had discount functions with “unusual shape” and were neither exponential nor hyperbolic; 14% unusually shaped discount curves Experiment 2: fitting at individual level; 1 out of 28 had increasing impatience; 3% unusually shaped discount curves Experiment 3: fitting at individual level; 1 out of 20 had increasing impatience %} Kirby, Kris N. (1997) “Bidding on the Future: Evidence against Normative Discounting of Delayed Rewards,” Journal of Experimental Psychology: General 126, 54–70. {% real incentives/hypothetical choice: for time preferences: used RIS. Delays considered were some weeks. Results are as the title says, where additivity refers to intertemporal addition. So the study both confirms intertemporal additivity and linearity of utility. %} Kirby, Kris N. (2006) “The Present Values of Delayed Rewards are Approximately Additive,” Behavioural Processes 72, 273–282. {% decreasing ARA/increasing RRA: paper tests constant relative and constant absolute risk aversion (although the author does not know these terms or concepts) and finds them all violated, arguing that we have to search for different utility families. Exp. 1 uses matching to infer indifferences, and (p. 465) uses BDM (Becker-DeGroot-Marschak), but nicely follows it up with a choice question to verify, although the latter was not really incentivized. Then he tests constant relative risk aversion, by testing whether or not in indifferences (⅓: 3x, ⅓: x, ⅓:0) ~ (⅓: 2x, ⅓: y(x), ⅓:0) y is a linear function of x or not, finding it falsified. Thus he rejects power utility. The experiments all have groups of about N=20. P. 466 3rd para: BDM is hard for subjects. Experiment 2 uses choice lists. P. 4665th para: those take more time. Now uses indifferences 3x½x ~ 2x½y(x) to test constant relative risk aversion. P. 466 penultimate para: strangely enough, does not allow for convex-utility answers. Exp. 3 considers indifferences (⅓: 5x, ⅓: 3x, ⅓:x) ~ (⅓: 3.25x, ⅓: 2.75x, ⅓:y(x)) to test constant relative risk aversion, and (⅓: (x+24), ⅓: (x+12), ⅓:x) ~ (⅓: (x+13.50), ⅓: (x+10.50), ⅓:y(x)) to test constant relative risk aversion. Again, strangely enough, he only allows for concave utility by only considering negative exponential utility. Experiment 4 considers what I call logarithmic utility, ln(hx+1) with h the free parameter, for which he cites Rachlin (1992) but it dates back from long ago in economics. %} Kirby, Kris N. (2011) “An Empirical Assessment of the Form of Utility Functions,” Journal of Experimental Psychology: Learning, Memory and Cognition 37, 461–476. {% Seems that: real incentives/hypothetical choice: for time preferences; more discounting for hypothetical than for real; DC = stationarity; Assume linear utility throughout. Mazur discounting. Kept delayed reward constant, varied delay, asked for reward today that yields indifference (matching). Repeated this for several delayed rewards. Delays were from 3 to 29 days. Rewards ranged from $14.75 to $28.50. Real rewards in experiment 1 through an auction (nice). Repeated the study in experiment 2 with hypothetical rewards. Find that hyperbolic discounting fits better than exponential discounting. Discount rates were lower for hypothetical rewards than for real ones. No evidence for reward-size-dependent discounting, so no magnitude effect. Fitting of data at individual level; “the most curious result of these experiments was the failure to find reliable decreases in discounting rates as delayed reward size increased.” (The decrease was very small). %} Kirby, Kris N. & Nino.N. Marakovic (1995) “Modeling Myopic Decisions: Evidence for Hyperbolic Delay-Discounting with Subjects and Amounts,” Organizational Behavior and Human Decision Processes 64, 22–30. {% Seems that: real incentives/hypothetical choice: for time preferences DC = stationarity; Claim that “most arguments against exponential discounting have tacitly assumed that the discounting rate parameter is independent of amount.” Real rewards Choice between amount tonight and other amount after delay. Varied delay, amount tonight and amount after delay. Since it was “tonight” they did not start with t=0 (=immediately). Choice task instead of matching. Delays ranged from 10 days to 75 days. Delayed rewards ranged from $30 to $85. Immediate rewards ranged from $15 to $83. Discount rate decreased as reward increased. %} Kirby, Kris N., & Nino N. Maraković (1996) “Delay-Discounting Probabilistic Rewards: Rates decrease as Amounts Increase,” Psychonomic Bulletin and Review 3, 100–104. {% Seems that: real incentives/hypothetical choice: for time preferences Delays are in days. Choice based task: choice between smaller, immediate reward and larger, delayed reward. Rewards were below $100 and delays were below 186 days. Participants had a 1 in 6 chance of receiving the reward of one of the choices. Authors use questionnaires for impulsiveness (nice!) and it turned out that the answers to the questionnaires were correlated with discount rates. Real rewards. Higher rewards were discounted less than small rewards. Heroin patients discounted more than the control group. Difficult to determine whether results could be explained by utility actually being convex or concave. %} Kirby, Kris N., Nancy M. Petry, & Warren K. Bickel (1999) “Heroin Addicts Have Higher Discount Rates for Delayed Rewards than Non-Drug-Using Controls,” Journal of Experimental Psychology: General 128, 78–87. {% real incentives/hypothetical choice: for time preferences: seems to be %} Kirby, Kris N. & Mariana Santiesteban (2003) “Concave Utility, Transaction Costs, and Risk in Measuring Discounting of Delayed Rewards,” Journal of Experimental Psychology: Learning, Memory and Cognition 29, 66–79. {% Big study on decisions with and without time pressure. 1700 subjects from Sweden, Austria, US. Time pressure increases the reflection effect of PT. No effect on loss aversion, but little data on it; for it they assume that “risk aversion” is the same for gains and losses (p. 55), which I do not understand. More noise under time pressure. They elicit only one certainty equivalent under gains and one under losses, so that they cannot measure insensitivity. cognitive ability related to risk/ambiguity aversion: all their results agree with time pressure increasing the role of system 1 (intuitive decision making) versus system 2 (deliberate/rational decision making). P. 57: “One interpretation of the current findings is that time pressure decreases System 2 processing compared to time delay and thus increases the reflection effect. Following this logic, and as pointed out by Kahneman (2011), the S-shaped value function of Prospect Theory may primarily be a result of System 1 processing.” P. 57 has the common sentence: “Our results are potentially important for real-world decision making since most everyday decisions entail some degree of risk.” %} Kirchler, Michael, David Andersson, Caroline Bonn, Magnus Johannesson, Erik Ø. Sørensen, Matthias Stefan, Gustav Tinghög, & Daniel Västfjäll (2017) “The Effect of Fast and Slow Decisions on Risk Taking,” Journal of Risk and Uncertainty 54, 37–59. {% %} Kiresuk, Thomas J. & Robert E. Sherman (1968) “Goal Attainment Scaling: A General Method for Evaluating Comprehensive Community Mental Health Programs,” Community Mental Health Journal 4, 443–453. {% ratio bias: find it. Participants find 1:20 less likely than 10:200. Experiments show that people judge a probability n/7 to be smaller than 10n/100: the ratio bias. The authors suggest that we have two different systems of probabilistic assessments. There is the rational one, making us be consciously aware of numerical probabilities that we can tell to other people. There is, however, also the experiental one, that makes us automatically act right in many situations but that we are not aware of and cannot express numerically. %} Kirkpatrick, Lee A. & Seymour Epstein (1992) “Cognitive-Experiential Self-Theory and Subjective Probability: Further Evidence for Two Conceptual Systems,” Journal of Personality and Social Psychology 63, 534–544. {% %} Kirkwood, Craig W. (1993) “An Algebraic Approach to Formulating and Solving Large Models for Sequential Decisions under Uncertainty,” Management Science 39, 900–913. {% %} Kirkwood, Craig W. & Rakesh K. Sarin (1980) “Preference Conditions for Multiattribute Value Functions,” Operations Research 28, 225–232. {% Seems to argue against representative agent. P. 119 seems to write: “… it is clear that the “representative” agent deserves a decent burial, as an approach to economics analysis that is not only primitive, but fundamentally erroneous.” %} Kirman, Alan P. (1992) “Whom or What Does the Representative Individual Represent?,” Journal of Economic Perspectives 6, 117–136. {% survey on nonEU %} Kischka, Peter & Clemens Puppe (1992) “Decisions under Risk and Uncertainty: A Survey of Recent Developments,” Methods and Models of Operations Research 36, 125–147. {% %} Kitayama, Shinobu, Alana Conner Snibble, Hazel Rose Markus, & Tomoko Suzuki (2004) “Is there Any “Free” Choice,” Psychological Science 15, 527–533. {% %} Klayman, Joshua (1995) “Varieties of Confirmation Bias,” Psychology of Learning 32, 385–418. {% Nice references to early literature on multiattribute value theory (is MAUT without risk involved). Develop interpretations and vocabulary to better communicate in qualitative terms than the standard analytical representation. %} Klein, David A., Martin Weber, & Edward H. Shortliffe (1992) “A Framework for Computer-Based Explanation of Multiattribute Decisions in Expert Systems.” In: Ambrose Goicoechea, Lucien Duckstein, & Stanley Zionts (eds.) IX-th International Conference on Multiple Criteria Decision Making, 159–171, Springer Verlag, Berlin. {% %} Klein, Gary A. (1993) “A Recognition-Primed Decision (RPD) Model of Rapid Decision Making.” In Gary A. Klein (ed.), Decision Making in Action: Models and Methods, 138–147, Ablex Pub Norwood, NJ. {% %} Kleindorfer, Paul R., Howard C. Kunreuther, & Paul J.H. Schoemaker (1993) “Decision Sciences. An Integrative Perspective.” Cambridge University Press, Cambridge. {% intuitive versus analytical decisions; criticize Dawes, Faust, & Meehl (1989) for being too narrow. %} Kleinmuntz, Benjamin, David Faust, Paul E. Meehl, & Robyn M. Dawes (1990) “Clinical and Actuarial Judgment,” Science 247 (Jan. 12) 146–147. {% Seems to argue on pp. 113-114 for a design of assessment where biases cancel each other out, something applied by Bleichrodt (2002). %} Kleinmuntz, Don N. (1990) “Decomposition and the Control of Error in Decision Analytic Models.” In Robin M. Hogarth (eds.) Insights in Decision Making: A Tribute to Hillel J. Einhorn, 107–126, University of Chicago Press, Chicago. {% %} Kleinmuntz, Don N. (1991) “Decision Making for Professional Decision Makers,” Psychological Science 2, 135–141. {% %} Klement, Erich Peter & Dan Ralescu (1983) “Nonlinearity of the Fuzzy Integral,” Fuzzy Sets and Systems 11, 309–315. {% dynamic consistency; axiomatizes, in AA framework (so EU for given probabilities in a second stage) with uncertainty aversion (quasi-concavity in posterior probability mixing à la Gilboa & Schmeidler, 1989), the Epstein & Wang 94 model for dynamic consistency; is intertemporal with payment at each time point and also a future opportunity set to reckon with at each time point. That leads to state dependence (I haven’t studied it enough to understand in detail). He assumes equivalence of outcomes over different states, and points out that this restricts his model for regular state dependence but is reasonable in his model where state dependence results from the opportunity sets. In view of outcomes at each time point, intertemporal substitution is relevant. %} Klibanoff, Peter (1995) “Dynamic Choice with Uncertainty Aversion,” Northwestern University, Evanstone, IL. {% Assumes Anscombe-Aumann setup. For two acts there does not exist a CEU (Choquet expected utility) model showing a violation of betweenness iff either one act dominates the other or they are comonotonic. %} Klibanoff, Peter (2001) “Characterizing Uncertainty Aversion through Preference for Mixtures,” Social Choice and Welfare 18, 289–301. {% %} Klibanoff, Peter (2001) “Stochastically Independent Randomization and Uncertainty Aversion,” Economic Theory 18, 605–620. {% An accessible account of this model, describing its underlying assumptions, is in Marinacci (2015 §4). Kahneman & Tversky (1975 p. 30 ff) have the smooth model for ambiguity for two outcomes. event/utility driven ambiguity model: utility-driven. source-dependent utility: the essence of their approach, although interpretations may differ. The authors (KMM) consider a two-stage-expectation representation as in Kreps & Porteus (1978), i.e., EXP[(EXPS[U(f(s))d])d], where 1. EXPS[…] denotes expectation over S. S is a Savagean (1954) state space, f is an act, U is a usual utility function to be used in regular expected utility, and is a subjective probability measure over S à la Savage. 2. KMM assume that there is ambiguity about what the proper is. This is reflected by a second-order probability measure over the set of all first-order probability measures over S. This reflects subjective perception. (Thus this paper calls the last stage, to the right in the tree, first-order, and the first stage, to the left of the tree, second-order.) ------------------------------------------------ At each stage KMM assume EU but can be nonlinear and, hence, it is not EU overall. It means that they do commit to the backward-induction version of dynamic nonEU (formally stated in their Assumption 3, p. 1857), giving up RCLA. They also assume that S has an Anscombe-Aumann-like decomposition. In other words, they assume that objective probabilities are given in S about which there is no ambiguity, so that all ’s considered (in the support of ) agree there with those objective probabilities. They use these to derive U and, later, to define ambiguity. A recursive EU-type two-stage model as above (for simplicity we follow the authors in not counting the AA part as an extra stage) has been considered before by Kreps & Porteus (1978), who interpreted it as an intertemporal model with a nonlinear modeling attitudes towards the timing of the resolution of risk. Reinterpreting such a two-stage Kreps-Porteus setup for ambiguity where the two stages reflect resolutions of uncertainty of a different level of ambiguity, was considered simultaneously and independently by Nau (2006) and Ergin & Gul (2009), and before by Neilson (1993, published 2010) as I learned from KMM’s citations. The authors cite Segal for the general use of 2nd-order probabilities to model ambiguity (but without a recursive EU), but this has been done in many papers before 1990 (Gärdenfors 1979; Gärdenfors & Sahlin 1983; Kahneman & Tversky 1975 p. 30 ff.; Larson 1980; Yates & Zukowski 1976). As do the aforementioned studies, KMM assume that acts, called second-order acts, are available whose outcomes are contingent on the second-order uncertainty resolution; i.e., on which subjective probability measure on S applies. An example of such a second-order act is if you get $100 if the over S is 1 and $200 if it is 2; and so on. The big difference of the present paper (KMM) with preceding ones is that KMM allow the two-stage decomposition to be endogenous, so that possibly it cannot be related to exogenous criteria. So, KMM consider choices between bets (their second-order acts) such as: We are going to derive from your preferences what your subjective probability of rain tomorrow is. If we discover that you consider rain at least as likely as 0.45, you receive million dollar. If we discover that you consider rain less likely than 0.45, then you receive 0 dollar. Would you rather have that gamble or 200,000 dollar for sure? It also involves the decision maker determining subjective probabilities about endogenous aspects of own preferences, such as me gambling with probability 0.6 that I prefer apple to pear, without this preference being related to any exogenous objective verifiable event such as needed when considering conditioning. Thus, KMM consider bets with payments contingent on endogenous aspects of preference. Such bets do exist in the special case where the events pertaining to are exogenous and physically definable, e.g. when referring to the unknown composition of an urn (then however the ’s are only objective), or maybe to an unknown parameter in statistics. (These, however, while outcome-relevant, are usually not treated as observable in the sense that we can construct any bet on them; Bayesian statisticians who assume priors implicitly assume such bets to be available but are, I suspect, usually not well aware of the problematic observable status of bets on parameter-values.) Such cases are in the domain of Kreps & Porteus, Nau, and others, which includes all examples of this kind put forward in the KMM paper. The generalization added here of allowing the outcome-relevant events for second-order acts to also be endogenous greatly enhances the scope of applicability of the theory, but along with it brings in this observability problem, and tractability problems. KMM discuss the pros and cons on p. 1856. With all events regarding assumed observable etc. via second-order acts, KMM can separate ambiguity-beliefs (this is how above is interpreted) and ambiguity-attitudes (this is how above is interpreted). KMM characterize concavity of as follows: they take utilities U(f(s)) as observable outcomes, which is plausible if we interpret them as standard gamble probabilities: U(f(s)) = p can be taken as a Mpm lottery with M big outcome with U(M) = 1 and m small outcome with U(m) = 0. Then is concave if and only if every act f is less preferred than its -expectation U(f(s)). So, this is the usual definition of weak risk aversion. A difficulty of this condition is that the -expectation is not directly observable because is a subjective probability, only inferrable through elaborate elicitations of preferences over second-order act s (derived concepts in pref. axioms; that subjective probabilities are indeed subjective and cannot be direct inputs is argued for instance by Budescu & Wallsten (1987, p. 68). Things are doable from the observability perspective if there exists a subset of with probability 0.5 because this is easy to infer from choice and using only this event is enough to characterize concavity of . It also implies that two persons can be compared regarding ambiguity aversion only if they have the same risk preferences. A drawback that all the approaches mentioned, including Kreps & Porteus, have and share with for instance Chew’s (1983) weighted utility (sum pif(xi)U(xi)/sumpif(xi) for DUR) is that all extra mileage is obtained from a function that, like U, applies to outcomes ( indirectly via utility U). Thus, not only the risk-attitude-like-EU behavior, but also the ambiguity attitude, is driven entirely by the outcome domain we are facing, and not by the uncertainty-domain we are facing. This is apparent from Corollary 3 (p. 1865) with ambiguity attitude described by the Pratt-Arrow measure of at an outcome, and Assumption 5.ii (p. 1869) with ambiguity attitude specified through the interval of outcomes. The approach of this paper, like most others, cannot separate absence of ambiguity from ambiguity neutrality. P. 1870 is remarkable in having ambiguity defined through relating it to known exogenous probabilities—which I like. The definition of ambiguity is inextricably linked with ambiguity aversion or seeking. Likelihood sensitivity, with a symmetric capacity, is taken here as unambiguous (Proposition 5). It means that KMM only consider source preference and not source sensitivity. For example, the extreme case of likelihood insensitivity (source insensitivity), with weight 0 for empty event and weight 1 for universal event, and weight 0.5 for all other events, according to the authors' definition means that there be no ambiguity. This is not correct. Note that a decision maker can be more ambiguity averse towards source1 of events that towards source2 in two ways: either by either taking more concave, or by taking the endogenous two-stage decomposition more dispersed. In KMM’s interpretation it should only be the second way. should be a stable within-person property independent of source. A person’s ambiguity aversion should be independent of the source! I expect that most people applying KMM will not work this way, but will vary concavity of within a person as in Chew et al. (2008). For descriptive purposes, if we find ways to identify and from data, then it can become an empirical question. P.1859 end of §2, Corollary 1, states that on S the authors need not commit to EU, but could also handle nonEU models, where the authors consider Quiggin’s RDU. In the more problematic second stage, where ambiguity is handled, the authors do need EU. For the axiomatization, however, EU on S is used. I summarize what I consider to be drawbacks of the KMM approach in my comments to Epstein (2010, Econometrica). Download 7.23 Mb. Share with your friends:
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