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§8.8 Problem 2 (p. 247) discusses the modeling of loss aversion through piece-wise linear utility with a kink at 0
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| §8.8 Problem 2 (p. 247) discusses the modeling of loss aversion through piece-wise linear utility with a kink at 0.
criticism of monotonicity in Anscombe-Aumann (1963) for ambiguity: §10.7.3, pp. 301-304. uncertainty amplifies risk: §10.4, p. 292, reviews empirical evidence ambiguity seeking for unlikely: Section 10.4.2 cites evidence on insensitivity, which comprises ambiguity seeking for unlikely. biseparable utility for uncertainty: §10.6, pp. 298-299, presents it criticism of monotonicity in Anscombe-Aumann (1963) for ambiguity: §10.7, p. 302, Figure 10.7.1. Example 11.2.2 (p. 321) illustrates how matching probabilities easily capture ambiguity. P. 354, §12.7, reviews the literature finding ambiguity seeking for losses, confirming reflection. %} Wakker, Peter P. (2010) “Prospect Theory: For Risk and Ambiguity.” Cambridge University Press, Cambridge, UK. Additional material
Wakker, Peter P. (2011) “Jaffray’s Ideas on Ambiguity,” Theory and Decision 71, 11–22. Link to paper
Wakker, Peter P. (2014) “Verliesangst,” NRC Handelsblads (Delta Lloyd Magazine) 27 June, p. 9. Link to paper
Wakker, Peter P. & Daniel Deneffe (1996) “Eliciting von Neumann-Morgenstern Utilities when Probabilities Are Distorted or Unknown,” Management Science 42, 1131–1150. Link to paper
Wakker, Peter P., Ido Erev, & Elke U. Weber (1994) “Comonotonic Independence: The Critical Test between Classical and Rank-Dependent Utility Theories,” Journal of Risk and Uncertainty 9, 195–230. Link to paper Link to typo (Link does not work for some computers. Then can:
Wakker, Peter P., Sylvia J.T. Jansen, & Anne M. Stiggelbout (2004) “Anchor Levels as a New Tool for the Theory and Measurement of Multiattribute Utility,” Decision Analysis 1, 217–234. Link to paper
Wakker, Peter P. & Marc P. Klaassen (1995) “Confidence Intervals for Cost/Effectiveness Ratios,” Health Economics 4, 373–381. Link to paper
Wakker, Peter P., Hans J.M. Peters, & Tom B.P.L. van Riel (1986) “Comparisons of Risk Aversion, with an Application to Bargaining,” Methods of Operations Research 54, 307–320. Link to paper
Wakker, Peter P. & Anne M. Stiggelbout (1995) “Explaining Distortions in Utility Elicitation through the Rank-Dependent Model for Risky Choices,” Medical Decision Making 15, 180–186. Link to paper
Wakker, Peter P., Richard H. Thaler, & Amos Tversky (1997) “Probabilistic Insurance,” Journal of Risk and Uncertainty 15, 7–28. Link to paper
Wakker, Peter P., Daniëlle R.M. Timmermans, & Irma A. Machielse (2007) “The Effects of Statistical Information on Risk and Ambiguity Attitudes, and on Rational Insurance Decisions,” Management Science 53, 1770–1784. Link to paper
Wakker, Peter P. & Amos Tversky (1993) “An Axiomatization of Cumulative Prospect Theory,” Journal of Risk and Uncertainty 7, 147–176. Link to paper Link to typos (Link does not work for some computers. Then can:
Wakker, Peter P. & Horst Zank (1999) “State Dependent Expected Utility for Savage’s State Space; Or: Bayesian Statistics without Prior Probabilities,” Mathematics of Operations Research 24, 8–34. Link to paper
Wakker, Peter P. & Horst Zank (1999) “A Unified Derivation of Classical Subjective Expected Utility Models through Cardinal Utility,” Journal of Mathematical Economics 32, 1–19. Link to paper
Wakker, Peter P. & Horst Zank (2002) “A Simple Preference-Foundation of Cumulative Prospect Theory with Power Utility,” European Economic Review 46, 1253–1271. Link to paper
Walasek, Lukasz & Neil Stewart (2015) “How to Make Loss Aversion Disappear and Reverse: Tests of the Decision by Sampling Origin of Loss Aversion,” Journal of Experimental Psychology: General 144, 7–11. {% P. 302 seems to have written, on loss function having to be determined by extraneous nonstatistical factors and using term weight for loss: “The question as to how the form of the weight function W(,) should be determined is not a mathematical or statistical one. The statistician who wants to test certain hypotheses must first determine the relative importance of all possible errors, which will entirely depend on the special purposes of his investigation.” Seems to have proposed maxmin (minmax in terms of loss function). %} Wald, Abraham (1939) “Contributions to the Theory of Statistical Estimation and Testing Hypotheses,” Annals of Mathematical Statistics 10, 299–326. {% It seems that here he proved his famous result that each undominated choice in decision under uncertainty can be taken as maximizing Bayesian subjective expected utility and even subjective expected value. %} Wald, Abraham (1949) “Statistical Decision Functions,” Annals of Mathematical Statistics 20, 165–205. {% event/utility driven ambiguity model: event-driven: Multiple priors: proposed maxmin EU (minmax in terms of loss function) on pp. 18, 26-27. Dutch book (end of Ch. II) Seems to have shown that for finite state spaces, for a risk set that is bounded and closed from below, the set of Bayesian decision rules is complete. The idea is that we choose a Pareto-optimal option, take the tangential hyperplane (in view of the possibility to take mixes of options, the set is convex), then take the orthogonal probability vector, and then take the option chosen as minimizer of expected loss w.r.t. the probabilities generated. Mathematical generalizations are given. This result has often been used to justify the Bayesian use of subjective probabilities. Seems to take as decision under uncertainty model a more general setup than Savage (1954): there is a state space S and an action space A. The “pre-consequence space” (my term) is the product set A x S. Then there is a function f mapping A x S to a consequence space C. Savage’s 1954 model can be considered to be the special case where acts with same consequences for each s are identified and, next, all maps from S to C are available. Conversely, one can interpret the Wald action space as a subset of the Savage act space. Oh well. biseparable utility %} Wald, Abraham (1950) “Statistical Decision Functions.” Wiley, New York. {% %} Wald, Abraham (1952?? Paris conference comments on independence). {% Seems to have written, on loss aversion: “that loss is a more powerful motivator than gain, or that groups threatened with loss will be more likely to protest than groups that seek proactively to achieve a gain.” %} Walder, Andrew G. (1994) “Implications of Loss Avoidance for Theories of Social Movements,” working paper, Harvard University. {% Discusses empirical studies of which kind of lotteries sell best (e.g., many low prizes or not, etc. %} Walker, Ian & Juliet Young (2001) “An Economist’s Guide to Lottery Design,” Economic Journal 111, F700–F722. {% %} Wall, Dan (2014) “Visualize Prospect Theory.” https://decisionsciences.shinyapps.io/Shiny/pt_qtd_shiny.Rmd
Wallach, Michael A. & Cliff W. Wing (1968) “Is Risk a Value?” Journal of Personality and Social Psychology 9, 101–106. {% Dempster’s conditioning %} Walley, Peter (1987) “Belief Function Representation of Statistical Evidence,” Annals of Statistics 15, 1439–1465. {% completeness-criticisms; updating; three-prisoners problem: p. 279 argues, through three-prisoners problem, that Dempster-Shafer updating rule can lead to accept “sure loss.” The argument does not result, contrary to what some have suggested, from dynamic inconsistency, but is primarily based on a de Finetti-like book making with adding up several accepted bets and requiring linear utility. A summary is in Miranda (2008). %} Walley, Peter (1991) “Statistical Reasoning with Imprecise Probabilities.” Chapman and Hall, London. {% foundations of statistics: Argues for likelihood principle but against Bayesianism. P. 33: “It seems to me that Carnap’s programme was unsuccessful because he insisted on a Bayesian solution and therefore failed to satisfy the RIP.” Here RIP means “Representation Invariance Principle,” i.e., independence of the sample space chosen. %} Walley, Peter (1996) “Inferences from Multinomial Data: Learning about a Bag of Marbles,” Journal of the Royal Statistical Society B 58, 3–57. {% %} Walley, Peter & Terrence L. Fine (1982) “Toward a Frequentist Theory of Upper and Lower Probabilities,” Annals of Statistics 10, 741–761. {% Propose procedures that satisfy the likelihood principle, even stronger than that, treat every two parameters with same likelihood the same (so no role for differentiating priors). Procedures avoid subjective inputs and can also satisfy frequentist criteria. As a price to pay, the procedures are conservative. %} Walley, Peter & Serafin Moral (1999) “Upper Probabilities Based only on the Likelihood Function,” Journal of the Royal Statistical Society B 61, 831–847. {% real incentives/hypothetical choice: seems that they criticized the use of hypothetical choice. %} Wallis, W. Allen & Milton Friedman (1942) “The Empirical Derivation of Indifference Functions.” In Oskar Lange, Francis McIntyre, & Theodore O. Yntema (eds.) Studies in Mathematical Economics and Econometrics in Memory of Henry Schultz, 175–189, University of Chicago Press, Chicago. {% Newcombs paradox is that player is physically second to play but mentally is first. %} Walliser, Bernard (1988) “A Simplified Taxonomy of 22 Games,” Theory and Decision 25, 163–191. {% On updating and revising beliefs %} Walliser, Bernard & Denis Zwirn (2002) “Can Bayes’ Rule Be Justified by Cognitive Rationality Principles?,” Theory and Decision 53, 95–135. {% %} Wallsten, Thomas S. (1971) “Subjective Expected Utility Theory and Subjects’ Probability Estimates: Use of Measurement-Free Techniques,” Journal of Experimental Psychology 88, 31–40. {% probability elicitation; Shows that physicans when giving probability judgment, do not provide objective guidelines for probability, but instead the probabilities that they think best support their recommended treatment. %} Wallsten, Thomas S. (1981) “Physician and Medical Student Bias in Evaluating Diagnostic Information,” Medical Decision Making 1, 145–164. {% Seem to explain that subjective probabilities are theoretical constructs (derived concepts in pref. axioms). %} Wallsten, Thomas S. & David V. Budescu (1983) “Encoding Subjective Probabilities: A Psychological and Psychometric Review,” Management Science 29, 151–173. {% %} Wallsten, Thomas S. & David V. Budescu (1995) “A Review of Human Linguistic Probability Processing: General Principles and Empirical Evidence,” Knowledge Engineering Review 10, 43–62. {% Ague that averages (over different judges) of probability estimates are often way better than any individual judgments. %} Wallsten, Thomas S., David V. Budescu, Ido Erev, & Adele Diederich (1997) “Evaluating and Combining Subjective Probability Estimates,” Journal of Behavioral Decision Making 10, 243–268. {% %} Wallsten, Thomas S., David V. Budescu, Amnon Rapoport, Rami Zwick, & Barbara H. Forsyth (1986) “Measuring the Vague Meanings of Probability Terms,” Journal of Experimental Psychology: General 115, 348–365. {% probability elicitation %} Wallsten, Thomas S., David V. Budescu, & Rami Zwick (1993) “Comparing the Calibration and Coherence of Numerical and Verbal Probability Judgments,” Management Science 39, 176–190. {% %} Wallsten, Thomas S., Ido Erev & David V. Budescu (2000) “The Importance of Theory: Response to Brenner (2000),” Psychological Review 107, 947–949. {% Imprecise probabilities: argue that upper and lower probabilities are more natural than precise probabilities, and give nice refs. %} Wallsten, Thomas S., Barbara H. Forsyth, & David V. Budescu (1983) “Stability and Coherence of Health Experts’ Upper and Lower Subjective Probabilities about Dose-Response Functions,” Organizational Behavior and Human Performance 31, 277–302. {% Seems to be one of the inventors of marginal utility, together with Jevons and Menger. marginal utility is diminishing: according to Larrick (1993) one of the first to suggest diminishing marginal utility. %} Walras, M.E. Léon (1874) “Elements of Pure Economics.” Translated by William Jaffé, Irwin, Homewood IL, 1954. {% P. 98 (according to Georegescu-Roegen 1954 QJE p. 513): “all these successive units have for their possessor an intensity of utility decreasing from the first unit which responds to the most urgent need to the last, after which satiety sets in. Walras, M.E. Léon (1896, 3rd edn.) “Eléments d’Économie Politique Pure.” F. Rouge, Lausanne. {% free-will/determinism: epiphenomenalism means that mental is entirely caused by material things. Willusionism is the view that, because of this, free will is an illusion. %} Walter, Sven (2014) “Willusionism, Epiphenomenalism, and the Feeling of Conscious Will,” Synthese 191 2215–2238. {% Proposes that after receipt of outcome, one feels regret or elation as the outcome is above or below the indifference class of the gamble. Those feeling are, however, only temporary and fade away and then the absolute level of the outcome determines the well-being. The speed of the fading away is determined by a time-preference parameter. The participant optimizes anticipating all that. %} Walther, Herbert (2003) “Normal-Randomness Expected Utility, Time Preference and Emotional Distortions,” Journal of Economic Behavior and Organization 52, 253–266. {% Continues on his 2003 model. Theoretically shows how all kinds of properties in discounting and probability weighting can be captured by different functions, adding evolutionary considerations. %} Walther, Herbert (2010) “Anomalies in Intertemporal Choice, Time-Dependent Uncertainty and Expected Utility—A Common Approach,” Journal of Economic Psychology 31, 114–130. {% Many studies find a negative, rather than the usually assumed positive, relation between risk and returns of stocks. This paper puts reference dependence forward as a promising explanation. %} Wang, Huijun, Jinghua Yan, & Jianfeng Yu (2017) “Reference-Dependent Preferences and the Risk–Return Trade-Off,” Journal of Financial Economics 123, 395–414. {% Analyze the famous RAND (“US”) data set on heath insurance, and a similarly nice data set on health insurance from China. real incentives/hypothetical choice: hypothetical choice error theory for risky choice: the novelty of this study is what they call the “mixture model approach.” That is, they do not assume a universal framing as gains or losses etc., but take as an extra parameter in their study whether the subjects perceive the outcomes as gains or losses, and in that manner derive from data who have a gains- and who a loss frame. They estimate costs-probability distributions. For RAND data, their observable is preferred insurance by subjects, for Chinese data set it is WTP. Risk averse for gains, risk seeking for losses: US respondents: risk averse for gains, and risk neutral or maybe some risk averse for losses. Chinese seemed to be risk neutral for gains and risk seeking for losses. This can be reconciled with the fourfold pattern if we assume that the framing in the context of insurance makes people more risk averse, which is well-known, and that in the Chinese group, who had to do WTP and not choice, WTP had the known biases downward. The authors instead resort to cultural differences. %} Wang, Mei & Paul S. Fischbeck (2004) “Incorporating Framing into Prospect Theory Modeling: A Mixture-Model Approach,” Journal of Risk and Uncertainty 29, 181–197. {% Measure loss aversion in 53 countries around the world, using the data set also used by Rieger, Wang, & Hens (2015), and using Hofstede’s indexes. They, properly, control for other components in loss aversion. They used hypothetical choice. I agree that for losses hypothetical is better than the common prior-endowment-and-then-paying-back procedure. Also, a study at this scale is hard to organize anyhow. Individualism, power distance, and masculinity increase loss aversion. Uncertainty avoidance and macroeconomic variables do not have effect. Footnote 6 thanks anonymous referees for the addition of a comment, and, as usual, one can feel that it is a silly remark that was imposed on the authors because referees have too much power on writing subjective opinions today. %} Wang, Mei, Marc Oliver Rieger, & Thorsten Hens (2017) “The Impact of Culture on Loss Aversion,” Journal of Behavioral Decision Making 30, 270–281. {% %} Wang, Shaun W., Virginia R. Young, & Harry H. Panjer (1997) “Axiomatic Characterization of Insurance Prices,” Insurance: Mathematics and Economics 21, 173–183. {% Note that strategic answering, pointed out by Harrison (1986), is more of a theoretical problem than empirical. %} Wang, Stephanie W., Michelle Filiba, & Colin F. Camerer (2010) “Dynamically Optimized Sequential Experimentation (DOSE) for Estimating Economic Preference Parameters,” working paper, California Institute of Technology. {% updating; dynamic consistency %} Wang, Tan (2003) “Conditional Preferences and Updating,” Journal of Economic Theory 108, 286–321. {% %} Wang, Tong V., Rogier J. D. Potter van Loon, artijn J. van den Assem, & Dennie van Dolder (2016) “Number preferences in lotteries,” Journal of Behavioral Decision Making 11, 243–259. {% Many nice citations on uncertain preferences. Use the modified BDM (Becker-DeGroot-Marschak) procedure of Schade & Kunreuther. They assume that, for WTP, there is an interval in which there is a probability of buying. Below it buying is certain, and above it it is certainly not. The authors ask subjects to develop such an interval with a probability distribution, and then generate buying according to this probability distribution. The authors, however, assume, and I disagree, that it is in the subjects’ interest to generate the probability distribution that agrees with their own distribution. If I face future uncertainties (even if regarding my own future tastes) then I integrate them out, come to one fixed current deterministic indifference price, and buy for all lower prices and do not buy for all higher. I have no interest in getting my future probability distribution reproduced at present. For instance, p. 204 2nd column end of 3rd para assumes that, if my future probability of buying is 10%, then at present my “ideal” probability of buying is 10%. %} Wang, Tuo, Ramaswamy Venkatesh, & Rabikar Chatterjee (2007) “Reservation Price as a Range: An Incentive-Compatible Measurement Approach,” Journal of Marketing Research 64, 200–213. {% updating; updating of Dempster-Shafer belief functions. %} Wang, Ying-Ming, Jian-Bo Yang, Dong-Ling Xu & Kwai-Sang Chin (2007) “On the Combination and Normalization of Interval-Valued Belief Structures,” Information Sciences 177, 1230–1247. {% They further test the violation of internality that Gneezy, List, & Wu (2006) called the uncertainty effect, showing that it easily disappears. %} Wang, Yitong, Tianjun Feng & L. Robin Keller (2013) “A Further Exploration of the Uncertainty Effect,” Journal of Risk and Uncertainty 47, 291–310. {% Argues that for Bentham utility was multi-dimensional without aggregation to one-dimensional, so, without completeness of pref. P. 8 l. 5/6 suggests that Bentham got concept of utility from writings of Hume, Helvétius, and Beccaria. Cites Bentham for anonymity condition: “Everybody to count for one, nobody for more than one.” %} Warke, Tom W. (2000) “Mathematical Fitness in the Evolution of the Utility Concept from Bentham to Jevons to Marshall,” Journal of the History of Economic Thought 22, 3–23. {% %} Warke, Tom W. (2000) “Multi-Dimensional Utility and the Index Number Problem: Jeremy Bentham, J.S. Mill and Qualitative Hedonism,” Utilitas 12, 176–203. {% real incentives/hypothetical choice; time preference: military drawdown program of early 1990s, for 65,000 separatees had choice between annuity and lump-sum payment. So, real incentives, big stakes. They consider discounting of money; i.e., linear utility. Majority took lumpsum implying discount rates over 18%. %} Warner, John T. & Saul Pleeters (2001) “The Personal Discount Rate: Evidence from Military Downsizing Programs,” American Economic Review 91, 33–53. {% Uses CenTER panel. Some simple measures of risk aversion are correlated with financial decisions and other things. %} Warneryd, Karl (1996) “Risk Attitude and Risky Behavior,” Journal of Economic Psychology 17, 749–770. {% Legal controversy between Chichilnisky and Wooders %} Warsh, David (1996) “Economic Principals: A Bitter Battle Illuminates an Esoteric World,” Boston Globe Online Business. {% small probabilities; anonymity protection %} Washington, variety of species {% confirmatory bias: (One of the?) first to find the confirmation bias, through the game where cards with a vowel on one side have an even number on the other. %} Wason, Peter C. (1968) “Reasoning about a Rule,” Quarterly Journal of Experimental Psychology 20, 273–281. {% %} Wasserman, Larry A. (1990) “Prior Envelopes Based on Belief Functions,” Annals of Statistics 18, 454–464. {% updating %} Wasserman, Larry A. & Joseph B. Kadane (1990) “Bayes’ Theorem for Choquet Capacities,” Annals of Statistics 18, 1328–1339. {% %} Wasserman, Larry A. & Joseph B. Kadane (1992) “Symmetric Upper Probabilities,” Annals of Statistics 20, 1720–1736. {% DOI: 10.1080/00031305.2016.1154108 foundations of statistics: first part discusses procedures leading to te statement on p-values and is not interesting for me. Then comes the ASA statement. It is useful in general to warn against problems of p-values. Yet I found it a bit disappointing. It only writes standard generalities such as that one should not go by p-value alone but also by others things such as quality of design. And then always the usual point (their Point 4) that one should report all the tests and analyses ever considered, and the choice of the ones reported. This is indeed necessary by the rules of the game and the definition of p-value, but cannot and is never satisfied in any statistical analysis ever done in the history of mankind. For this discrepancy one cannot criticize the requirement to be incorrect given the def. of p-value, and neither mankind for violating it, but one should criticize p-value for being partly nonsensical concept anyhow. %} Wasserstein, Ronald L. & Nicole A. Lazar (2016) “The ASA's Statement on p-Values: Context, Process, and Purpose,” American Statistician 70, 129-133. {% %} Waters, Leonie K. & Michael Collins (1984) “Effect of Pricing Conditions on Preference Reversals by Business Students and Managers,” Journal of Applied Psychology 69, 346–348. {% time preference: seems to find sign dependence in intertemporal choice, with smaller discounting for losses than for gains (“gain-loss asymmetry”). intertemporal separability criticized: habit formation %} Wathieu, Luc (1997) “Habits and the Anomalies in Intertemporal Choice,” Management Science 43, 1552–1563. {% %} Wathieu, Luc (2004) “Consumer Habituation,” Management Science 50, 587–596. {% Seems to have introduced behaviorism. Schijnt te zeggen dat slechts uiterlijk waarneembaar gedrag onderwerp van een objectieve psychologie kan zijn. %} Watson, John B. (1913) “Psychology as the Behaviorist Views It,” Psychological Review 20, 158–177. {% %} Watson, John B. (1930) “Behaviorism.” Norton. {% Criticizes Rabin & Thaler (2001) “Anomalies: Risk Aversion,” Journal of Economic Perspectives. Argues that reasonable persons should not exhibit the risk aversion assumed by Rabin & Thaler. Rabin & Thaler, in their reply, correctly point out that this is irrelevant because their analysis is descriptive and not normative. Next the author argues that the phenomena assumed by Rabin & Thaler would for some gambles require extremely high indexes of RRA (also argued by Palacios-Huerta & Serrano 2006) and that this is not realistic. Rabin & Thaler, in their reply, correctly point out that they know this, agree with it, and always have done so, and that it is part of their reasoning (see, for example, Rabin (2000, Econometrica), p. 1287 2nd paragraph). The point is that this shows that the relative index of risk aversion is not suited for comparing small-stake gambles to high-stake gambles, or choices at different levels of wealth, the index being so very sensitive to where the origin of the scale is located. I expect that the latter deficiency of constant RRA has been known to many people in the present and past. %} Watt, Richard (2002) “Defending Expected Utility Theory,” Journal of Economic Perspectives 16, 227–228. {% Extends Nahs bargaining and other bargaining solutions from expected utility to biseparable utility. %} Webb, Craig S. (2013) “Bargaining with Subjective Mixtures,” Economic Theory 52, 15–39. {% DOI 10.1007/s00199-015-0871-1 Uses the term ambivalence iso (likelihood) insensitivity. The popular and useful neo-additive weighting functions of Chaeauneuf, Eichberger, & Grant (2007) are discontinuous at 0 and 1, which is crude and can sometimes bring theoretical complications. This paper proposes the simplest continuous extension that one can think of: the weighting function is linear on [0,1-k], [1-k, k], and [k,1]. 1-k is much like the best-rank boundary of Wakker (2010) and k is the worst-rank boundary. The nice thing is that the paper gives a preference foundation, where it is further nice that this is done in the Savage framework with richness of states and not of outcomes. %} Webb, Craig S. (2015) “Piecewise Additivity for Non-Expected Utility,” Economic Theory 60, 371–392. {% Tradeoff method: also used dually, to get probability weighting differences. Piecewise linearity means linearity on [0,p1], [p1,p2], and [p2,1]. It is a continuous variation of neo-additive. %} Webb, Craig S. (2017) “Piecewise Linear Rank-Dependent Utility,” Theory and ecision 82, 403–414. {% Characterizes the variational model, using a two-stage setup with backward induction as do AA, but in the second stage using a subjective SEU model by imposing Savage’s axioms there rather than AA’s objective probabilities and EU for risk. It then enogenizes fifty-fifty mixing, and uses this endogenous operation to do AA type things. The fifty-fifty mixing is as follows: Assume for events A,C, we have A B (revealed preference). If we find a subset E of A, and an event E´ disjoint from B, with E ~ E´, such that A\E ~ CuE´, then these two events are midpoints between A and C, and so are all other events B ~ to them. They are called second-stage averages. They are a kind of 50-50 mixture, and can be used to get 50-50 utility mixtures. With these mixtures, subjective analogs of AA mixing, and theorems, can be obtained. Section 9 discusses pros and cons of different models with different kinds of richness. Instead of Savage’s P6, he uses solvability and an Archimedean axiom. I guess that the two-stage setup here rules out finite equally spaced cases. The set of events is assumed to be a sigma algebra. %} Webb, Craig (2017) “Purely Subjective Variational Preferences,” Economic Theory 64, 121–137. {% proper scoring rules %} Webb, Craig & Horst Zank (2008) “Using Proper Scoring Rules to Measure Loss Aversion,” {% EU+a*sup+b*inf: novelty is that they do it using richness of probabilities and not of outcomes. Nice way to easily measure the jumps at 0 and 1. Propose to take these jumps, divided by 1 minus the jumps, as indexes of optimism and pessimism. That is, in the above a-b notation, a/(1ab) and b/(1ab). Thus, if a and b tend to 0.5, both optimism and pessimism tend to , and optimism is for instance, for constant a, an increasing function of b and pessimism. They assume a finite outcome set and, hence, problems about null sets in the Chateauneuf, Eichberger, & Grant (2007) paper do not arise here. They essentially impose vNM independence ( independence of common probability shifts, which in fact is the sure-thing principle for risk), and consistent optimism- and pessimism attitudes, which can be measured from limiting probability-shift properties and then be required to be consistent. %} Webb, Craig S. & Horst Zank (2011) “Accounting for Optimism and Pessimism in Expected Utility,” Journal of Mathematical Economics 47, 706–717. {% Subjects can trade off time against outcome (wait longer for higher outcome with fixed probability) or against probability (wait longer for higher probability at fixed outcome). They want to wait longer for an increase in probability than for an increase in outcome if both entail the same expected value gain. However, stimuli are not just a money amount received with a probability at some time point, but the students are playing a computer game having to shoot many things and either the success-probability of every shot is increased or the damage of every shot. So it is a complex situation that does not directly speak to usual decision theories. %} Webb, Tara L. & Michael E. Young (2015) “Waiting when Both Certainty and Magnitude Are Increasing: Certainty Overshadows Magnitude,” Journal of Behavioral Decision Making 28, 294–307. {% Consider the interaction of risk and time, in particular regarding the topic mentioned in the title. In direct binary choices, where subjects can by heuristic delete common components, they do not find reductions of the certainty effect if delay is added, or of the immediacy effect if risk is added. Here they do not replicate Keren & Roelofsma (1995). In CE or present value evaluation, they still find no reduction of the certainty effect if adding delay when the certainty effect concerns the common consequence version of the Allais paradox (the authors use the term Allais paradox only for the common consequence version). They do find it for the common ratio version of the Allais paradox. They also find reduction of the immediacy effect if adding risk. Hence, the effects are found a bit but not very clearly. %} Weber, Bethany J. & Gretchen Chapman (2005) “The Combined Effects of Risk and Time on Choice: Does Uncertainty Eliminate the Immediacy Effect? Does Delay Eliminate the Certainty Effect?,” Organizational Behavior and Human Decision Processes 96, 104–118. {% Do 3-color Ellsberg paradox for monetary outcomes and for waiting time (for delivery of a good). Choices are hypothetical. In the waiting time setup subjects seem to choose between sure waiting times and ambiguous waiting times (only specified up to an interval), without very clear rationality/ambiguity-neutrality point, and the results are not easily comparable. %} Weber, Bethany J. & Wah Pheow Tan (2012) “Ambiguity Aversion in a Delay Analogue of the Ellsberg Paradox” Judgment and Decision Making 7, 383–389. {% %} Weber, Elke U. (1984) “Combine and Conquer: A Joint Application of Conjoint and Functional Approaches to the Problem of Risk Measurement,” Journal of Experimental Psychology: Human Perception and Performance 10, 179–194. {% %} Weber, Elke U. (1988) “Expectation and Variance of Item Resemblance Distributions in a Convolution-Correlation Model of Distributed Memory,” Journal of Mathematical Psychology 32, 1–43. {% The main point of this paper, stated immediately in the intro, is that an asymmetric loss function, also studied by Birnbaum, can give a motivational (deliberate, not due to misperceptions/biases) justification for nonlinear decision weights. The idea is that for some internal or external reason a person dislikes more underestimating some probability than overestimating it. It is analogous to statistical estimation theory where not the outcome of the gamble but the error of your estimation (whether too high or too low) matters for you. This internal/external reason may be psychologically plausible but it is not part of the decision model and its outcomes. It is something like “your colleague might blame you or you might feel silly the morning after you received the outcome of the gamble if it was way more than you estimated,” and this approach is not decision-theoretic. Therefore, while psychologically plausible, this main point is not of direct interest to me. This notwithstanding, there are many comments and discussions about decision theory that are subtle and valuable, and the paper is very well written. I therefore read it several times and often cite it. uncertainty amplifies risk: p. 237/238 suggests more deviation (inverse-S) from EU under uncertainty than under risk. P. 237 next-to-last paragraph, on pessimism, cites evidence from “impression formation” where cues receive more attention as they are ranked lower between the other cues. P. 238 last paragraph expresses preference for decision weights depending on outcomes over utilities depending on probabilities/events and, thus, for rank-dependent utility over lottery-dependent utility of Becker & Sarin. Footnote 9 gives several refs on utility depending on probability. P. 239 1st column: two-stage model of, first, estimation of probability and, second, configural weighting. questionnaire versus choice utility: p. 239 2nd column end of first para: “Thus, decision analysts’ dogmatic refusal to consider introspective judgements of perceived probability as valid evidence may one day seem as unnecessary in its self-imposed limitations as a behaviorist approach to, say, language acquisition.” P. 239 2nd column l. l 10/-3: “By separating the utility of the outcome itself from the weight given to the outcome as a function of its relative rank or the nature of the task …, changes in preference as a function of elicitation method can be attributed to changes in configural weighting, while allowing the utility of the outcome to remain invariant.”: this citation expresses what Birnbaum calls scale convergence and what I argued for in my ’94 Theory and Decision paper and used in Wakker & Deneffe (1996). See also discussion of Weber, Anderson, & Birnbaum some lines above.) risky utility u = strength of preference v (or other riskless cardinal utility, often called value): many suggestions on p. 239/240, in particular p. 239 2nd column middle of page. P. 240 discusses, twice, that people may want to change the internal constraints that they are imposing upon themselves, which I interpret as meaning that we shouldn’t take any utility function as normatively acceptable. paternalism/Humean-view-of-preference: last sentence, on use of configural-weighting models (is approximately the same as rank-dependence): “and finally help to provide more accurate and consistent estimates of subjective probabilities and utilities in situations where all parties agree on the appropriateness of the expected-utility framework as the normative model of choice.” %} Weber, Elke U. (1994) “From Subjective Probabilities to Decision Weights: The Effects of Asymmetric Loss Functions on the Evaluation of Uncertain Outcomes and Events,” Psychological Bulletin 115, 228–242. {% Study direct judgments of riskiness versus attractiveness of lotteries. %} Weber, Elke U., Carolyn J. Anderson, & Michael H. Birnbaum (1992) “A Theory of Perceived Risk and Attractiveness,” Organizational Behavior and Human Decision Processes 52, 492–523. {% Seem to have a questionnaire for measuring risk aversion %} Weber, Elke U., Ann-Renée Blais, & Nancy E. Betz (2002) “A Domain-Specific Risk-Attitude Scale: Measuring Risk Perceptions and Risk Behaviors,” Journal of Behavioral Decision Making 15, 263–290. {% Risk averse for gains, risk seeking for losses: Exhibit 8: seem to be risk neutral for losses, risk averse for gains. %} Weber, Elke U. & William P. Bottom (1989) “Axiomatic Measures of Perceived risk: Some Tests and Extensions,” Journal of Behavioral Decision Making 2, 113–131. {% %} Weber, Elke U., Ulf Böckenholt, Dennis J. Hilton, & Brian Wallace (1993) “Determinants of Diagnostic Hypothesis Generation: Effects of Information, Base Rates, and Experience,” Journal of Experimental Psychology: Learning, Memory, and Cognition 19, 1151–1164. {% %} Weber, Elke U. & Dennis J. Hilton (1990) “Contextual Effects in the Interpretations of Probability Words: Perceived Base Rate and Severity of Events,” Journal of Experimental Psychology: Human Perception and Performance 16, 781–789. {% Redoes Wakker, Erev & Weber (1994), with several modifications. Shows that, if you deliberately bring in perceptional framing effects by highlighting, boldprinting, larger-font printing, etc. lowest or highest outcomes, then in that manner you can generate rank-dependence. Similarly, if you deliberately bring in motivational effects by letting lotteries be evaluated as buyer or seller etc., then this can also generate rank-dependence effects. %} Weber, Elke U. & Britt Kirsner (1997) “Reasons for Rank-Dependent Utility Evaluation,” Journal of Risk and Uncertainty 14, 41–61. {% Propose a model of variance divided by expectation to determine if people/animals are risk averse or risk seeking and show that in 20 data sets from other studies with choices between sure and two-outcome prospects their formula performs well. A problem may occur if the expected value in the denominator is zero or negative. real incentives/hypothetical choice: pp. 435-436: real incentives give more risk aversion both for gains and for losses. %} Weber, Elke U., Sharoni Shafir, & Ann-Renee Blais (2004) “Predicting Risk-Sensitivity in Humans and Lower Animals: Risk as Variance or Coefficient of Variation,” Psychological Review 111, 430–445. {% %} Weber, Martin (1983) “An Empirical Investigation on Multi-Attribute Decision Making.” In Pierre Hansen, (ed.) Essays and Surveys on Multiple Criteria Decision Making, 379–388, Springer Verlag, Berlin. {% %} Weber, Martin (1985) “A Method for Multiattribute Decision Making with Incomplete Information,” Management Science 31, 1365–1371. {% %} Weber, Martin (1987) “Decision Making with Incomplete Information,” European Journal of Operational Research 28, 44–57. {% %} Weber, Martin (1998) “Remarks on the Paper “On the Measurement of Preferences in the Analytical Hierarchy Process” by Ahti A. Salo and Raimo P. Hämäläinen,” Journal of Multi-Criteria Analysis 6, 320–321. {% %} Weber, Martin (1998) “Comment on Mayer, C., Financial Systems and Corporate Governance: A Review of the International Evidence,” Journal of Institutional and Theoretical Economics 154, 166–169. {% P. 10 gives nice interpretation on finding that decision weights are more problematic than thought: the finding is bad news for MAUT because they turn out to be more problematic. But it is good news for MAUT because henceforth we can better measure because we now know the errors better. %} Weber, Martin & Katrin Borcherding (1993) “Behavioral Influences on Weight Judgments in Multiattribute Decision Making,” European Journal of Operational Research 67, 1–12. {% survey on nonEU %} Weber, Martin & Colin F. Camerer (1987) “Recent Developments in Modelling Preferences under Risk,” OR Spektrum 9, 129–151. {% Risk averse for gains, risk seeking for losses: they study Shefrin & Statman’s (1985) disposition effect, which suggests risk seeking for losses and risk aversion for gains. %} Weber, Martin & Colin F. Camerer (1998) “The Disposition Effect in Securities Trading: An Experimental Analysis,” Journal of Economic Behavior and Organization 33, 167–184. {% %} Weber, Martin, Franz Eisenführ, & Detlof von Winterfeldt (1987) “Bias in Assessment of Attribute Weights.” In Yoshikazu Sawaragi, Koichi Inoue & Hirotaka Nakayama (eds.) Toward Interactive and Intelligent Decision Support Systems, 309–318, Springer Verlag, Berlin. {% Download 7.23 Mb. Share with your friends:
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