Bibliography Peter P. Wakker March 16, 2011 This file in pdf format If some symbols don’t show up on your screen properly. Elucidation on keywords and annotations:  Annotations are between signs {% and %} above the references.  Use of key words (are bold printed): By means of the FIND function, you can use the keywords below to find related references in this file. For example, if you use the key word ambiguity seeking for losses and search then you will find 28 references on this topic. key words: ambiguity seeking; ambiguity seeking for losses; ambiguity seeking for unlikely; anonymity protection; Best core theory depends on error theory; binary prospects identify U and W; binary RDU; binary RDU violated; bisection > matching; cancellation axioms; CBDT; CE bias towards EV; PT, applications; confirmatory bias; completenesscriticisms, applications; coalescing; collapse: see coalescing; conditional prob.; Concave utility for gains, convex utility for losses (see also “Risk averse for gains, risk seeking for losses”, and please don’t confuse risk aversion with concave utility etc. unless expected utility is the explicit working hypothesis!); consequentialism/pragmatism; conservation of influence (see preference for flexibility for future influence); crowdingout; simple decision analysis cases using EU; decreasing absolute/increasing RRA; decreasing/increasing impatience; derived concepts in pref. axioms; discounting normative; dominance violation by pref. for increasing income (see also: preferring streams of increasing income); Dutch book (see also “ordered vector space” or “referencedependence test”); dynamic consistency; DC = stationarity; equilibrium under nonEU; equilibriumdiscussion; equityversusefficiency; error theory for risky choice; EU+a*sup+b*inf; eventsplitting: see coalescing; finite additivity; folding back/normal form, descriptive; formula Bayes intuitively; foundations of probability; foundations of quantum mechanics; foundations of statistics; freewill/determinism; game theory for nonexpected utility; Harsanyi’s aggregation; homebias; information aversion (see also “value of information”); intertemporal separability criticized; intuitive versus analytical decisions (see also “Reflective equilibrium”); inverseS; Jeffrey, R.C.; just noticeable difference (other terms are minimally perceptible threshold/difference or just noticeable increment); Kirsten&I; law and decision theory; linear utility for small stakes; loss aversion without mixed prospects; losses from prior endowment; marginal utility is diminishing; measure of similarity; Maths for econ students; Nash equilibrium discussion; Newcomb’s paradox; normal/extensive form; onedimensional utility; ordered vector space; ordering of subsets (see also preference for flexibility); ill overestimate QALY; partwhole bias (special case for uncertainty: coalescing); parametric fitting depends on families chosen; paternalism/Humeanviewofpreference; preference for flexibility; preferring streams of increasing income (see also: dominance violation by pref. for increasing income); presentvalue; Principle of Complete Ignorance; probability elicitation (see also “proper scoring rules”); probabilitycommunication; probability triangle; producing random numbers (people are not able to produce really random numbers); proper scoring rules (see also “probability elicitation”); proper scoring rulescorrection; qualitative probability: see ordering of subsets; PT falsified; quasiconcave so deliberate mixing; questionnaire for measuring risk aversion; questionnaire versus choice utility, see also “utility = representational?”; random incentive system; betweenrandom incentive system (paying only some subjects); ranking economists; ratio bias; ratiodifference principle (see also ratio bias); revealed preference; RCLA (= reduction of compound lotteries assumption); real incentives/hypothetical choice (see also “crowdingout” and “losses from prior endowment”); real incentives/hypothetical choice: for time preferences; restricting representations to subsets; referencedependence test (= assetintegration test; see also losses from prior endowment); relative curvature; Risk averse for gains, risk seeking for losses (see also “Concave utility for gains, convex utility for losses”); risk seeking for (symmetric) fiftyfifty gambles; risku = strength.pr.v; risku = transform of strength.pr.v; risku = transform of strength.pr.v, latter doesn’t exist; risk seeking for smallprob. gains; Savage: DUU = DUR; secondorder probabilities; SEU = SEU; SG doesn’t do well; SG higher than CE (see also “SG higher than others” and “CE bias towards EV”); SG higher than others (see also “SG higher than CE”) ; SIIA/IIIA; small probabilities; small worlds; social sciences cannot measure; sophisticated choice; standardsequence invariance (see also TO method); statedependent utility; strengthofpreference representation; substitutionderivation of EU; survey on nonEU; time preference; threeprisoners problem (also known as Monthy Hall’s three doors); TO method; TO method’s error propagation; Total utility theory; uncertainty amplifies risk; updating; probability intervals; utility concave near ruin; utility depends on prob.; utility elicitation; utility families parametric; utility measurement: correct for prob. distortion; Utility of gambling; utility = representational?; Valueinduced beliefs; value of information (see also “information aversion”); Ver&I.l.av; violation of the certainty effect (see also “risk seeking for (symmetric) fiftyfifty gambles”); Z&Z; sleaping key words: AHP; chained utility elicitation; CPT: data on probability weighting; Christiane, Ver.&I; common knowledge; decision under stress; Games with incomplete information; HYE; Methoden & Technieken; Nash bargaining solution; preference reversal; Reflective equilibrium; SG gold standard; statistics for C/E; Abbreviations: ARA: absolute risk aversion AHP = analytical hierarchy process BDM: BeckerDeGrootMarschak C/E = costeffectiveness CE = certainty equivalent CEU = Choquet expected utility CPT = cumulative prospect theory DC = dynamic consistency DUR = decision under risk DUU = decision under uncertainty EU = expected utility EV = expected value HYE = healthy years equivalent IIA = independence of irrelevant alternatives inverseS: inverseS shaped probability transformation nonEU = nonexpected utility QALY = quality adjusted life years RA: risk aversion RCLA: Reduction of compound lotteries RDU: rankdependent utility RRA: relative risk aversion RS: risk seeking SEU = subjective expected utility SG: standard gamble (used as in medical decision making, designating the probability equivalent method and not the certainty equivalent method) TO method = tradeoff method TTO = time tradeoff method WTA: willingness to accept WTP: willingness to pay REFERENCES {% freewill/determinism %} Aarts, Henk (2006) “Onbewust Doelgericht Gedrag en de Corrosie van de Ijzeren Wil,” inaugurale rede, Department of Social Psychology, Utrecht University, Utrecht, the Netherlands. {% equityversusefficiency; after this paper follows a discussion%} Abasolo, Ignacio & Aki Tsuchiya (2004) “Exploring Social Welfare Functions and Violation of Monotonicity: An Example from Inequalities in Health,” ^ 23, 313–329. {% onedimensional utility; Analyzes the case where expectedutility, multiattributeutility, etc., preferences remain unaffected after transformations of the arguments. Does this as a general principle, with constant absolute risk aversion and constant relative risk aversion as two special cases. %} Abbas, Ali E. (2005) “Invariant Utility Functions and Certain Equivalent Transformations,” working paper. {% %} Abbas, Ali E. (2005) “Maximum Entropy Utility,” ^ 54, 277–290. {% CPT: data on probability weighting; finds that prob. transformation for gains for losses;%} Abdellaoui, Mohammed (1995) “Comportements Individuels devant le Risque et Transformation des Probabilités,” Revue d'Économie Politique 105, 157–178. {% %} Abdellaoui, Mohammed (1996) “Is the Estimated Probability Transformation Function Sensitive to the Magnitude of Losses,” Note de Recherche GRID 96–03. {% %} Abdellaoui, Mohammed (1999) “Les Nouveaux Fondements de la Rationalité devant le Risque: Théories et Expériences,” GRIDCNRS, ENS, Cachan, France. {% ^ ; utility elicitation; TO method: First, the TO method is used to elicit utility. Then these are used to elicit the probability weighing function. More precisely, first a sequence x_{0}, ..., x_{6} is elicited that is equally spaced in utility units. Then equivalences x_{i} ~ (p_{i},x_{6}; 1p_{i},x_{0}) elicit p_{i} = w^{}^{1}(i/6) and, thus, the weighting function. Concave utility for gains, convex utility for losses: P. 1506 Finds concave utility for gains (power 0.89), convex for losses (power 0.92). P. 1508 finds more pronounced deviation from linearity of probability weighting for gains than for losses. inverseS: this is indeed found for 62.5%. 30% had convex prob transformation, rest linear. P. 1507: bounded SA is confirmed p. 1510: Finds nonlinearity for moderate probabilities, so not just at the boundaries. p. 1502: Uses real incentives for gains but not for losses. p. 1504: Finds 19% inconsistencies, which is less than usual, but this may be because the consistency questions were asked shortly after the corresponding experimental questions. p. 1506: Fitting power utilities gives median 0.89 for gains and 0.92 for losses. p. 1510: no reflection, w^{+} (for gains) is different (less elevated) from w^{} for losses, also different than dual, so CPT is better than RDU. This goes against complete reflection. %} Abdellaoui, Mohammed (2000) “ParameterFree Elicitation of Utility and Probability Weighting Functions,” Management Science 46, 1497–1512. {% TO method: Is applied theoretically in a dual manner, on prob. transformation; %} Abdellaoui, Mohammed (2002) “A Genuine RankDependent Generalization of the von NeumannMorgenstern Expected Utility Theorem,” Econometrica 70, 717–736. {% hypothetical choice was used, and discussed on pp. 851 & 862. TO method: use it in intertemporal context. Now not subjective probabilities, but discount weights, drop from the equations. P. 847: The asymmetry found between discounting for gains and for losses may have been generated by the common assumption then of linear utility, which works out differently for gains (where utility is concave) than for losses (where utility is close to linear and even some convex). This paper corrects for utility but does still find asymmetry (p. 859) They find, though not very clear, that discounting is less for losses than for gains, but deviation from constant discounting is the same. risku = strength.pr.v: measure intertemporal utility, and find that it agrees well with utility as commonly measured under risk (p. 860). p. 855: convex utility for losses: Do it in an intertemporal context. With nonparametric analysis, they find linear utility for losses (slightly more convex but insignificant), and concave utility for losses. With parametric analyses, they have no significant deviations from linearity although it is in direction of concavity for gains and convexity for losses. There it agrees with utility as commonly measured under risk. P. 857: For gains 55 had decreasing impatience and 12 had increasing. For losses, 47 decr, 18 incr., and 2 constant. They find almost no evidence for the immediacy effect, which drives quasihyperbolic discounting. P. 860: if not correcting for utility curvature, then overly strong discounting, but the deviation is not big at the aggregate level. %} Abdellaoui, Mohammed, Arthur E. Attema, & Han Bleichrodt (2010) “Intertemporal Tradeoffs for Gains and Losses: An Experimental Measurement of Discounted Utility,” ^ 120, 595–611. 845 866 {% probability elicitation ; inverseS; ambiguity seeking for unlikely %} Abdellaoui, Mohammed, Aurélien Baillon, Laetitia Placido, & Peter P. Wakker (2010) “The Rich Domain of Uncertainty: Source Functions and Their Experimental Implementation,” ^ forthcoming. {% TO method; SG higher than CE; typo on p. 363 (def. of expopower): z should be x. %} Abdellaoui, Mohammed, Carolina Barrios, & Peter P. Wakker (2007) “Reconciling Introspective Utility with Revealed Preference: Experimental Arguments Based on Prospect Theory,” Journal of Econometrics 138, 336–378. Link to paper {% %} Abdellaoui, Mohammed & Han Bleichrodt (2007) “Eliciting Gul’s Theory of Disappointment Aversion by the Tradeoff Method,” Journal of Economic Psychology 28, 631–645. {% %} Abdellaoui, Mohammed, Han Bleichrodt, & Hilda Kammoun (2010) “Are Financial Professionals Really Loss Averse?,” mimeo. {% Discuss pros and cons of parametric fitting. real incentives: seems like kind of random incentive system. They used very large outcomes, such as 10,000 euros, in the experiment, but for real incentives scaled down by a factor 10. For losses they found slightly concave utility, but yet risk seeking. ^ : find concave utility for gains, and slightly concave utility for losses. Risk averse for gains, risk seeking for losses: they find this. The finding of concave utility for losses, but risk seeking, is a nice empirical counterpart to Chateauneuf & Cohen (1994). inverseS: find it, both for gains and losses, fully in agreement with the predictions of PT. Use a measurement method where utility is measured through parametric fitting, assuming power utility. %} Abdellaoui, Mohammed, Han Bleichrodt, & Olivier L'Haridon (2008) “A Tractable Method to Measure Utility and Loss Aversion under Prospect Theory,” ^ 36, 245–266. {% Concave utility for gains, convex utility for losses: find concave utility for gains, convex for losses Table 1 gives a nice summary of the various definitions of loss aversion used in the literature. The first measure some utilities for gains and losses through the tradeoff method, getting some utility midpoints. Using that, the measure w^{}^{1}(0.5) for both gains and losses. Then they know so much that from indifferences between mixed prospects they can measure loss aversion efficiently. %} Abdellaoui, Mohammed, Han Bleichrodt, & Corina Paraschiv (2007) “Loss Aversion under Prospect Theory: A ParameterFree Measurement,” ^ 53, 1659–1674. {% %} Abdellaoui, Mohammed, Olivier L'Haridon, & Corina Paraschiv (2009) “ExperiencedBased vs. DescribedBased Uncertainty: Do We Need Two Different Prospect Theory Specifications?,” working paper. {% Propose a parametric probability weighting function family of the form w(p) = ^{1}^{}p^{} if 0 p and w(p) = 1 (1)^{1}^{}(1p)^{} if p > with 0 1, 0 < . The function is inverseS, has many nice properties, is given preference foundation, and fits data well. reflects elevation (antiindex of pessimism) and reflects sensitivity (curvature; antiindex of inverseS). Abdellaoui, Mohammed, Olivier L'Haridon, & Horst Zank (2010) “Separating Curvature and Elevation: A Parametric Probability Weighting Function,” ^ 41, 39–65. {% %} Abdellaoui, Mohammed & Bertrand R. Munier (1994) “The ClosingIn Method: An Experimental Tool to Investigate Individual Choice Patterns under Risk.” In Bertrand R. Munier & Mark J. Machina (eds.) Models and Experiments in Risk and Rationality, Kluwer Academic Publishers, Dordrecht. {% %} Abdellaoui, Mohammed & Bertrand R. Munier (1996) “Utilité Dépendant des Rangs et Utilité Espérée: Une Étude Expérimentale Comparative,” ^ 47, 567–576. {% %} Abdellaoui, Mohammed & Bertrand R. Munier (1997) “Experimental Determination of Preferences under Risk: The Case of very Low Probability Radiation,” Ciência et Tecnologia dos Materiais 9, Lisboa. {% describes how different heuristics apply to different regions of the prob. triangle.%} Abdellaoui, Mohammed & Bertrand R. Munier (1998) “The RiskStructure Dependence Effect: Experimenting with an Eye to DecisionAiding,” ^ 80, 237–252. {% TO method: test it when formulated dually, i.e. directly on probability weighting. Find that rankdependence does sometimes provide a useful generalization of EU. A more detailed test than Abdellaoui & Munier (1999, in Machina & Munier, eds), which preceded this one.%} Abdellaoui, Mohammed & Bertrand R. Munier (1998) “Testing Consistency of Probability Tradeoffs in Individual DecisionMaking under Risk,” GRID, Cachan, France. {% ^ : test it when formulated dually, i.e. directly on probability weighting. Reports an indirect test in probability triangles whose consequences are a standard sequences (u(x_{3}) u(x_{2}) = u(x_{2}) u(x_{1})). With this at hand probability tradeoff consistency can be tested across triangles. %} Abdellaoui, Mohammed & Bertrand R. Munier (1999) “How Consistent Are Probability Tradeoffs in Individual Preferences under Risk?” ^ Mark J. Machina & Bertrand R. Munier (eds.) Beliefs, Interactions and Preferences in DecisionMaking, 285–295, Kluwer Academic Publishers, Dordrecht. {% %} Abdellaoui, Mohammed & Bertrand R. Munier (2000) “Substitutions Probabilistiques et Décision Individuelle devant le Risque: Expériences de Laboratoire,” ^ 111, 29–39. {% real incentives/hypothetical choice: used flat payment and hypothetical choice, because utility measurement is only interesting for large amounts that cannot easily be implemented. inverseS & uncertainty amplifies risk: confirm less sensitivity to uncertainty than to risk. This implies: ambiguity seeking for unlikely TO method to elicit utility, (Concave utility for gains, convex utility for losses:) gives concave utility for gains (powerfitting gives power of about 0.88 on average) and some convex, but close to linear, utility for losses. They use mixed prospects, so that they can let the standard sequence start at 0 and get utility over a domain [0,x_{6}], including 0 (see just before §3.1, p. 1387). They use an uncertain event E, not given probability, to measure the standard sequence. They measure matching probabilities, x_{p}0 ~ x_{E}0. Test twostage model of PT with W(E) = w(B(E)). Here W is measured from PT by first measuring utility using the tradeoff method (§3.1), and then extending Abdellaoui's (2000) and Bleichrodt & Pinto's (2000) method for measuring probability weighting to uncertainty: 1_{E}0 ~ x then W(E) = U(x), assuming U(0) = 0 and U(1) = 1 (§3.2). B, called choicebased probability by the authors, is measured through matching probabilities: 1_{E}0 ~ 1_{p}0 then B(E) = p (§3.3). They then derive w as w(p) = W(B^{ }^{1}(p)). W satisfies bounded SA (= inverseS extended to uncertainty) for almost all subjects. Bounded SA is similar for gains and losses, but elevation is larger for losses. Bounded SA also holds for the factor B (p. 1395 bottom of first column), and for w, so that all common hypotheses of diminishing sensitivity of Fox & Tversky (1998), Tversky & Fox (1995), Wakker (2004), and others are confirmed. One small deviation is that for losses they find overweighting of unlikely events but no significant underweighting of likely events (§5.4, p. 1394). P. 1398: "The similarity of the properties of judged probabilities and choicebased probabilities comes as good news for the link between the psychological concept of judged probabilities and the more standard economic concept of choicebased probabilities." Pp. 13981399 top has nice texts on status of source preference, as comparative phenomenon that may not be part of transitive individual choice. ambiguity seeking for unlikely gains and ambiguity seeking for losses are confirmed by bounded SA ^ : do so on p. 1394, §5.3 end. %} Abdellaoui, Mohammed, Frank Vossmann, & Martin Weber (2005) “ChoiceBased Elicitation and Decomposition of Decision Weights for Gains and Losses under Uncertainty,” Management Science 51, 1384–1399. {% TO method. This is the best paper I ever coauthored. Unfortunately, the journal printed all its papers very inefficiently in those days, taking twice as many pages as other journals. Whereas in any other journal the paper would have taken 36 pages, in this journal it takes 73. %} Abdellaoui, Mohammed & Peter P. Wakker (2005) “The Likelihood Method for Decision under Uncertainty,” Theory and Decision 58, 3–76. Link to paper Link to comments {% about associativityfunctional equation%} Abel, Niels H. (1826) “Untersuchungen der Functionen Zweier Unabhängigen Veränderlichen Grössen x and y, wie f(x,y), Welche die Eigenschaft Haben, dass f[z,f(x,y)] eine Symmetrische Function von x,y und z ist,” Journal für die Reine und Angewandte Mathematik 1, 1–15, Academic Press, New york. Reproduced in Oevres Completes de Niels Hendrik Abel, Vol. I, 61–65. Grondahl & Son, Christiani, 1881, Ch.4. {% ^ : surely not if evaluated using EU; Typical of decision analysis is that simple choices are used to (derive utilities and other subjective paramters and then) predict more complex decisions. This paper performs this task in an exemplary explicit manner. The authors first use simple choice questions (SG with risk for chronic health states and TTO with time tradeoffs for chronic health states) to get basic utility assessments. For SG they calculate utility both assuming EU and assuming PT. Then they use the findings to predict preferences between more complex risky prospects (involving no real intertemporal tradeoffs), and between more complex (nonchronic) health profiles (involving no real risk). For decisions under risk, PT better predicts future choices than EU. It does so both when SGPT utilities are used as inputs, and when TTObased (riskless!) utility measurements are used as inputs. Bleichrodt (08Jan10, personal communication) told that TTO utility inputs and then PT work as well as SG inputs (no significant differences), which supports risku = strength.pr.v with intertemporal utility iso strength of pr. Butif I understand well, for intertemporal decisions TTO utilities did somewhat better than SG utilities, although with one exception the differences were not significant. %} AbellanPerpiñan, Jose Maria, Han Bleichrodt, & José Luis PintoPrades (2009) “The Predictive Validity of Prospect Theory versus Expected Utility in Health Utility Measurement,” Journal of Health Economics 28, 1039–1047. {% Find that power utility fits best for EQ5D, better than linear or exponential. That is, they take model QT^{r} with Q quality of life and T duration for chronic health states. They also consider nonchronic health profiles. Optimal fitting r is r = 0.65. Impressive sample of about N = 1300 (see p. 668), representative of Spanish population. %} Abellán, José M., José L. Pinto, Ildefonso Méndez, & Xabier Badía (2006) “Towards a Better QALY Model,” Health Economics 15, 665–676. {% %} Abouda, Moez & Alain Chateauneuf (2002) “Characterization of Symmetrical Monotone Risk Aversion in the RDEU Model,” Mathematical Social Sciences 44, 1–15. {% %} Abouda, Moez & Alain Chateauneuf (2002) “Positivity of BidAsk Spreads and Symmetrical Monotone Risk Aversion,” Theory and Decision 52, 149–170. {% foundations of quantum mechanics%} Accardi, Luigi (1986) “NonKolmogorovian Probabilistic Models and Quantum Theory,” text of Invited talk at 45th ISI session, Amsterdam, the Netherlands. {% The funny popular paradoxes such as the threedoor problem, the waitingtime paradox, etc. %} Aczel, Amir D. (2004) “Chance. A Guide to Gambling, Love, The Stock Market and just about Everything Else.” Thunder’s Mouth Press, New York. {% Theorem 2.1.1.1 (on p. 34) and top of p. 35: Cauchy equation implies that f is linear as soon as f is continuous at one point or bounded from one side on a set of positive measure.%} Aczél, János, (1966) “Lectures on Functional Equations and Their Applications.” Academic Press, New York. (This book seems to be a translation and updating of a 1961 German edn.) {% %} Aczél, János, (1987) “^ Kluwer, Dordrecht. {% Aczél’s citation on Catalonian oath of allegiance to Aragonese kings (15^{th} century); I got it in 1992: We, who are as good as you, swear to you, who are not better than us, that we do accept you as our king and sovereign lord, provided that you do observe all our liberties and laws—but if you don’t, then we won’t.%} {% restricting representations to subsets%} Aczél, János (2006) “Utility of Extension of Functional Equations—when Possible,” ^ 49, 445–449. {% %} Aczél, János & Claudi Alsina (1984) “Characterizations of Some Classes of Quasilinear Functions with Applications to Triangular Norms and to Synthesizing Judgements,” Methods of Operations Research 48, 3–22. {% Functional equations (interval scale differentiable equation), when crossing boundaries x_{1}= x_{2}, “shift.” %} Aczél, János, Detlof Gronau, & Jens Schwaiger (1994) “Increasing Solutions of the Homogeneity Equation and of Similar Equations,” Journal of Mathematical Analysis and Applications 182, 436–464. {% A psychophysical application is given where w(1) = 1 is not necessary. %} Aczél, János & R. Duncan Luce (2007) “A Behavioral Condition for Prelec’s Weighting Function on the Positive Line without Assuming W(1) = 1, ^ 51, 126–129. {% %} Adams, David R. (1981) “Lectures on L^{p}Potential Theory,” University of Umea, Department of Mathematics, Umea, Sweden. {% Maybe he showed that Savage’s finitely additive probability measures lead to violations of strict pointwise monotonicity and other things?%} Adams, Ernest W. (1962) “On Rational Betting Systems,” Archiv für Mathematische Logik und Grundlagenforschung 6, 7–18 and 112–128. {% %} Adams, Ernest W. (1966) “On the Nature and Purpose of Measurement,” Synthese 16, 125–169. {% %} Adams, Ernest W. & Robert F. Fagot (1959) “A Model of Riskless Choice,” Behavioral Science 4, 1–10. {% %} Adams, Ernest W., Robert F. Fagot, & Richard E. Robinson (1970) “On the Empirical Status of Axioms in Theories of Fundamental Measurement,” Journal of Mathematical Psychology 7, 379–409. {% Individual decisions versus group decisions with many factors analyzed and referenced that amplify or moderate extreme decisions. They study a large data set of people who betted on ice breakups in Alaska. Obviously, there are selectio effects with more than average risk seeking, for instance, as the authors point out. %} Adams, Renée & Daniel Ferreira (2010) “Moderation in Groups: Evidence from Betting on Ice Breakups in Alaska,” ^ 77, 882–913. {% %} Adamski, Wolfgang (1977) “Capacitylike Set Functions and Upper Envelopes of Measures,” Mathematische Annalen 229, 237–244. {% Investigate how receipt of new info affects risk attitude, i.e. how people change consumption of beef after info on mad cow disease. %} Adda, Jérôme (2007) “Behavior towards Health Risks: An Empirical Study Using the “Mad Cow” Crisis as an Experiment,” Journal of Risk and Uncertainty 35, 285–305. {% reformulate Popper’s claims about inductive probability probabilistically%} Agassi, Joseph (1990) “Induction and Stochastic Independence,” ^ 41, 141–142. {% time preference; some nice results, in particular Theorem 11: not! DC = stationarity; they very carefully distinguish%} Ahlbrecht, Martin & Martin Weber (1995) “Hyperbolic Discounting Models in Prescriptive Theory of Intertemporal Choice,” Zeitschrift für Wirtschafts und Sozialwissenschaften 115, 535–566. {% %} Ahlbrecht, Martin & Martin Weber (1996) “The Resolution of Uncertainty: An Experimental Study,” Journal of Institutional and Theoretical Economics 152, 593–607. {% time preference %} Ahlbrecht, Martin & Martin Weber (1997) “An Empirical Study on Intertemporal Decision Making under Risk,” Management Science 43, 813–826. {% Empirically test Kreps & Porteus model, whose predictions are rejected. §1 gives elementary description of the KP model. %} Ahlbrecht, Martin & Martin Weber (1997) “Preference for Gradual Resolution of Uncertainty,” Theory and Decision 43, 167–185. {% Extends Mertens & Zamir to multiple priors. %} Ahn, David S. (2007) “Hierarchies of Ambiguous Beliefs,” ^ 136, 286–301. {% Jeffrey, R.C.; ordering of subsets: This paper axiomatizes a model of maximization of average expected utility over sets, similar to Jeffrey (1965). The objects are interpreted as probability distributions over outcomes where the set reflects ambiguity over which is the right probability distribution. In this axiomatization, both probability and utility u are subjective/endogenous, so that the model is essentially the same in a mathematical sense as Jeffrey (1965) and Bolker (1966, 1967). There are some technical differences regarding continuity and Ahn’s model having singletons present in the domain and JBB not. The model can be considered a modification of multiple priors or it’s maxmin generalization. The usual ArrowPratt characterization of * being more concave than is given in Proposition 4 and is now taken as more ambiguity averse. %} Ahn, David S. (2008) “Ambiguity Without a State Space,” ^ 75, 3–28. {% Consider three states of nature denoted x, y, z. The subjects are told that y has probability 1/3, and are told that x and z have unknown probability. Subjects were not told more. In reality, x and z also have objective probability 1/3. (The authors generated event x by first letting a number p_{x} be selected at random (uniform distribution) from [0,2/3], and then let x be chosen with probability p_{x}, and z with probability 2/3 p_{x}. This is, however, only a roundabout manner for generating probability 1/3. Given that this was not told to the subjects, so does not matter for them, and given that any researcher who knows probability calculus knows that it is objective probability 1/3, no use doing the twostage procedure.) Let subjects choose prospects organized similarly as budget sets. The axiom of revealed preference is reasonably well satisfied. Consider the following models: (1) “Kinked,” being RDU with fixed decision weight 1/3 for state y (amounting to EU for known probabilities). Thus RDU for the remaining states is like binary RDU, and comprises most other models such as Gilboa & Schmeidler’s (1989) maxmin EU, Schmeidler’s (1989) RDU, maxmin, and Gajdos et al.’s (2008) contraction expected utility. (2) Recursive EU, where as secondorder distribution they take the uniform prior over [0,2/3], and where the two utility functions are exponential with possibly different exponents. It is useful to note that the rho parameter of utility for risk can be identified from bets on s_{2}, and then the parameter for ambiguity can be identified from bets on s_{1} and s_{3} while keeping the payment under s_{2} equal 0. §6: They do leastsquares data fitting without probabilistic error theory. The find that RDU (“kinked) fits better than recursive. %} Ahn, David S., Syngjoo Choi, Douglas Gale, & Shachar Kariv (2009) “Estimating Ambiguity Aversion in a Portfolio Choice Experiment,” Department of Economics, UCLA, Los Angeles. {% Their model is called partitiondependent SEU. Consider decision under uncertainty in an AnscombeAumann model, with partitiondependent SEU, as follows. They do not take an act as a function from S to outcomes, as Savage did, but, as Luce did, as a 2ntuple, so that the act and its preference value can depend on the partition chosen. Thus they can accommodate event splitting and so on. In their model there exists a utility function u and a nonadditive measure . For a partition (E_{1},…E_{n}) of S, SEU is maximized wrt u and P(E_{j}) = (E_{j})/((E_{1}) + ^{...} + (E_{n})), so with for single events but normalized. They presents axiomatizations. First, they, obviousy, assume usual axioms giving SEU within each partition. They use AnscombeAumann axioms. (I would have preferred tradeoff consistency; oh well …) This withinpartition representation does not yet relate betweenpartition representations in any sense. A monotonicity condition implies the same u for all partitions. For the rest (for the role of ), they consider two special cases: CASE 1. The collection of partitions considered is nested: for all two partitions, one is a refinement of the other. Then an extra surething principle characterizes the model with : If acts f and g agree on event E, then the preference between f and g is not changed if the common outcomes on E are replaced by other common outcomes, but also not if the partition outside of E is changed (so refined or coarsened). This axiom ensures the consistent conditioning in P(E_{j}) = (E_{j})/((E_{1}) + ^{...} + (E_{n})), from always the same . CASE 2. The collection of partitions considered is the collection of all partitions. Then besides the version of the s.th.pr. of Case 1, also an acyclicity axiom is imposed. P. 656: To the autjors' knowledge, they are the first to incorporate framing and partitiondependence in a formal model. However, Luce preceded here. An accessible account of his ideas is in Luce (1990, Psychological Science 1). A complete account is in the book Luce (2000). Luce also worked on such models in the 1970s, such as in Ch. 8 of Krantz et al. (1971). Luce uses the term experiment instead of the term partition, and the elements of Luce's experiment need not always give the same union (so they are conditional on their union) Ahn & Ergin always have S as the union. The topic of partition dependence is even more central in Birnbaum's work. He does write formal models but does not do formal work with them such as axiomatizations (although he does give derivations of logical relations between preference conditions). He does very comprehensive empirical work, testing every empirical detail of framing. Birnbaum, Michael H. (2008, Psychological Review 115, 463–501) provides a comprehensive summary. He usually (always?) assumes known probabilities. There is also much empirical evidence on event splitting by Loomes, Sugden, Humphrey, and others. The authors relate their work to support theory. is indeed an analog of the support function. A difference pointed out by the authors is that support theory focuses on probability judgment (Tversky and I started working on a decision theory but he died too soon) whereas they have preferences between acts. A difference not pointed out by the authors is that in support theory there are not only the (partitions of) hypotheses but also there is another layer, of events, and there is a distinction between implicit and explicit unions. Mainly this distinction between hypotheses and events drives why support theory deviates from classical models. Thus I disagree with the claim on p 663 that this paper provide an extension of support theory to decision theory, or that they provide a decision foundation. P. 657: The authors relate their model to unforeseen contingencies. A big difference is that in this paper the union of events in a partition is always S, whereas with unforeseen contingencies there are typically events outside of S. A topic for future research is to what extent the particular partitiondependence proposed here, with consistent conditioning on one nonadditive measure, is of interest empirically or normatively. The EU assumed within given partitions of course meets empirical violations of EU, although there is empirical evidence that using the same partition for describing all acts greatly reduces the violations. The model of this paper is also reminiscent of the source method by Abdellaoui, Baillon, Placido, & Wakker (AER), where different sources are different partition. One difference is that the source method does not give up extensionality, and acts are functions from states to outcomes. Another is that the source method allows for violations of EU throughout, also within a source/partition. In the source method, there can be subjective probabilities within each source but they can be transformed differently for different sources. %} Ahn, David & Haluk Ergin (2010) “Framing Contingencies,” Econometrica 78, 655–695. {% good reference for Möbius function and Möbius transform %} Aigner, Martin (1979) “Combinatorial Theory.” ^ Wiss. 234, Springer, Berlin. {% May have introduced hyperbolic discounting; or was it Chung & Herrnstein (1967)? %} Ainslie, George (1975) “Specious Reward: A Behavioral Theory of Impulsiveness and Impulse Control,” Psychological Bulletin 82, 463–496. {% %} Ainslie, George (1986) “Beyond Microeconomics. Conflict among Interests in a Multiple Self as a Determinant of Value.” ^ John Elster (ed.) The Multiple Self, 133–175, Cambridge University Press, New York. {% dynamic consistency%} Ainslie, George W. (1992) “Picoeconomics.” Cambridge University Press, Cambridge. {% Seems to argue that we are more insensitive with respect to the time dimension that to many pother dimensions. %} Ainslie, George W. (2001) “^ Cambridge University Press, Cambridge. {% real incentives/hypothetical choice: for time preferences: seems to be %} Ainslie, George W. & Vardim Haendel (1983) “The Motives of Will.” In Edward Gottheil, Keith A. Druley, Thomas E. Skolda & Howard M. Waxman (eds.) Etiologic Aspects of Alcohol and Drug Abuse. Charles C. Thomas, Springfield, IL. {% discounting normative: p. 63, 2^{nd} paragraph suggests that (steep) discounting would not be selected in evolution%} Ainslie, George W. & Nick Haslam (1992) “Hyperbolic Discounting.” ^ George F. Loewenstein and John Elster (1992), Choice over Time, 57–92, Russell Sage Foundation, New York. {% %} Airoldi, Mara, Daniel Read, & Shane Frederick (2005) “Longitudinal Dynamic Inconsistency,” in preparation. {% P. 27: “It is wellknown that Constant Relative Risk Aversion (CRRA) preferences sustain the BlackScholes model in equilibrium …” and then it gives many references. P. 38 points out that CRRA does not fit data well. %} AïtSahalia, Yacine & Andrew W. Lo (2000) “Nonparametric Risk Management and Implied Risk Aversion,” ^ 94, 9–51. {% Measure of fit is 2LlnL + 2k where L designates likelihood and k the number of parameters. %} Akaike, Hirotugu (1973) “Information Theory and an Extension of the Maximum Likelihood Principle.” In B.N. Petrox & F. Caski (eds.) Second International Symposium on Information Theory, 267–281, Akademiae Kiado, Budapest. {% %} Akerlof, George A. (1970) “The Market for ‘Lemons’: Quality Uncertainty and the Market Mechanism,” ^ 84, 488–500. {% Gives many examples of procrastination etc., phenomena where a small initial expense is used day after day to postpone something that on the long run brings way higher expenses. Obedience can be similar such as in Milgram’s famous experiment. Reminds me of the “frog effect” (when heating water at a sufficiently slow speed a frog never jumps and gets boiled so dies). P. 2: “Individuals whose behavior reveals the various pathologies I shall model are not maximizing their 'true' utility.” §1 describes how salient information has more effect on decisions than equivalent nonsalient information. Several places (e.g. §III.a p. 5) express disagreement with Becker et al’s rational addiction. %} Akerlof, George A. (1991) “Procrastination and Obedience,” ^ 81, 1–19. {% %} Akerlof, George A. (2002) “Behavioral Macroeconomics and Macroeconomic Behavior,” American Economic Review 92, 411–433. {% crowdingout: Their model seems to imply that severe punishment of crime may increase crime, because of the crowdingout effect. %} Akerlof, George A. & William T. Dickens (1982) “The Economic Consequences of Cognitive Dissonance,” ^ 72, 307–319. {% In Amer. J. Agr. Econ. 91 p. 1175, Akerlof (2009) writes: “ … Shiller and I … challenge the economic wisdom that got us into this mess …and put forward a bold new vision and policies that will transform economics and restore wold prosperity.” There is no limit or concession to nuances in the author’s enthusiasm about his own work! The authors argue, in this book written for popular reading, that animal spirits should get a bigger role in economics. They consider 5 psychological facts in particular: overconfidence, fairness, corruption and bad faith, money illusion, and stories (a catchall category). On p. 3 they cite Keynes (1921): “they are not, as rational economic theory would dictate, “the outcome of a weighted average of quantitative benefits multiplied by quantitative probabilities.” “ [Italics from original] %} Akerlof, George A. & Robert J. Shiller (2009) “^ .” Princeton University Press, Princeton, NJ. {% Russian, writes usually in Russian, about web theory %} Akivis, Maks A. {% %} AlAwadhi, Shafeeqah A., & Paul H. Garthwaite(1998) “An Elicitation Method for Multivariate Normal Distributions,” Communications in Statistics—Theory Meth. 27, 1123–1142. {% §3.4 correctly cites de Finetti on his arguments against countable additivity. Unfortunately, it also suggests that Savage disliked countable additivity but Savage (1954, §3.4) did not have such an opinion. For Savage it was not central and only a pragmatic matter of convenience. He used all subsets of the state space and not a sigmaalgebra only for expositional purposes, actually preferring sigmaalgebra other than for exposition. Savage did express a slight preference for not committing to countable additivity but, again, not out of principle but only pragmatically, and not committing clearly. (Probably to quite some extent so as not to get in conflict with de Finetti who was in a less refined league than Savage.) The paper considers to what extent infinitely many observations necessarily lead to unique probabilities of all events through the law of large numbers. If the set of events considered is very complex and large, and way more so than the nr. of observations, and if probability is finitely additive, then probabilities may not get uniquely determined. This is of course a mathematical result in the sense that it really builds on finite additivity and complexity degrees of infinity. §4: This paper derives a set of priors from learning, and only then derives decisions from that. %} AlNajjar, Nabil I. (2009) “Decision Makers as Statisticians: Diversity, Ambiguity, and Learning,” Econometrica 77, 1370–1401. {% Establish a model of undescribable events where the best coinsurance is no coinsurance. Assume that any finite description can be given, but complete outcomerelevant description should be infinite. Although the basic point is technical, the authors eloquently give many nice examples. %} AlNajjal, Nabir I., Luca Andelini, & Leonardo Felli (2006) “Undescribable Events,” ^ 73, 849–869. {% Something different than bounded rationality. Gives precise formal definitions from logic it seems. %} AlNajjar, Nabil I., Ramon CasadesusMasanell, & Emre Ozdenoren (2003) “Probabilistic Representation of Complexity,” Journal of Economic Theory 111, 49–87. {% proper scoring rules; problem that calibration tests can be passed by charlatans disappears if there are more than one expert. %} AlNajjar, Nabil I., & Jonathan Weinstein (2008) “Comparative Testing of Experts,” Econometrica 76, 541–559. {% This paper criticizes the normatively motivated modern ambiguity aversion literature. I, as Bayesian, only and purely study ambiguity for descriptive reasons, and fully agree that the nonEU models (including ambiguity) are not rational. Empirically, though, there is considerable ambiguity seeking. The paper, appropriately, writes on p. 252 2nd para that its arguments have been known before by specialists. The paper is written with enthusiasm of a kind that will especially appeal to young readers, but it is informal and not very sophisticated. I disagree with many nuances. Central to the paper are the rationality problems of ambiguity models in dynamic decision making and updating. These are, however, general problems of nonexpected utility and not particularly of ambiguity. Because the paper assumes expected utility for risk (and then can assume payment in utils so that it is risk neutrality), a debate of ambiguity (which is about differences between unknown and known probabilities) the debate about ambiguity is the same as the debate about nonexpected utility. It has been widely known since Hammond (1988), and was explained more clearly before in the impressive Burks (1977, Ch. 5), that nonEU violates convincing principles in dynamic decision making. The best paper to start on this debate is Machina (1989). Ghirardato (2002) is also good. He appropriately used the term folk theorems for the results, because they were widely known. I wrote Wakker (1999), http://people.few.eur.nl/wakker/pdf/alias.pdf The debates are often hard to pin down because the relevant assumptions discussed are so selfevident (surely I as Bayesian think so) that people often assume some of those critical conditions implicitly, and verbal descriptions often can equally well refer to one condition as to the other. In the resolute choice approach one gives up what Machina (1989) called consequentialism so as to maintain dynamic consistency. Then one’s decisions depend on risks borne in the past, i.e. on events that could have happened at some stage in the past but are now known to be counterfactual and nonexistent. In Wakker (1999) I described this as believing in ghosts. This was Machina’s preferred way to go, and also McClennen’s who coined the term resolute for it, and also Jaffray’s. In sophisticated choice one gives up dynamic consistency, so as to maintain consequentialism. Then prior and posterior preferences are not the same, and from a prior perspective one may violate dominance (one is willing to pay for precommitment). This was preferred by Karni & Safra and is the least unconvincing for nonEU in my opinion. In Wakker (1999) I called this split personality. A third approach is to give up reduction of compound lotteries, which for uncertainty is something like event invariance. These are models about not being indifferent to the timing of the resolution of uncertainty. I will not discuss them further. Footnote 1, p. 250 suggests that probabilistic sophistication (Machina & Schmeidler’s P4*) is a special case of the surething principle but this is not so. P4* implies Savage’s P4 which is logically and conceptually different from the surething principle (Savage’s P2). P. 251 ll. 12: ”The allconsuming concern of the ambiguity aversion literature is the Ellsberg “paradox.” “ erxpresses well my impression: the field is too much focused on the Ellsberg paradox. P.l; 254 and elsewhere: it is not true that capacities (weighting functions) are interpreted as indexes of belief in nonEU. Some people, especially novices, do so, but experienced people know that this need not be. Abdellaoui et al. (2011 AER) wrote, where source functions capture the nonadditivity of capacities/weighting functions: “Source functions reflect interactions between beliefs and tastes that are typical of nonexpected utility and that are deemed irrational in the Bayesian normative approach.” They reference preceding contributions by Winkler, Vernon Smith, and others. Wakker (2004, Psychological Review) suggested that inverseS/sourcesensitivity could be a belief component but pessimism/sourcepreference/ambiguityaversion not so. Also in multiple priors many are aware of the difference. It is very explicit in contraction expected utility by Gajdos, Hayashi, Tallon, & Vergnaud (2008, JET), for instance. KMM’s smooth model also has it very explicitly. The paper then assumes risk neutrality, or, in other words, EU plus payment in utils. P. 259 discusses what the authors call irrelevance of sunk costs but what amounts to the additivity axiom (discussed in Wakker, 2010, Ch. 1) restricted to constant acts in combination with some updating. It is well known that nonEU can depend on counterfactual risks and costs (see above on resolute choice). What the authors call factbased on p. 261 is like sophisticated choice. The informal presentation does not allow for an exact pinning down. P. 267, on dynamic inconsistency à la Strotz, takes it purely as externallyimposed (say ingrained in your genes) and not as decision based, thus ducking the central questions there. The dynamic inconsistency resulting under ambiguity is not taken that way in this paper. Hence the difference ... P. 275 criticizes multiple priors for the concept of unknown true probability, with which I agree. They then refer to previous work by themselves with limiting theorems on identifying betterknowing experts versus pretendingphonyexperts. §5 (announced before on p. 255) argues that ambiguity aversion may be a misapplied social instinct. In some places it is suggested that it then could be rational, but misapplications do not seem to be rational I would think. This instinctmisapplicationinterpretation does not invalidate attempts to model using ambiguity models. Note also that the considerable ambiguity seeking found empirically shows that more is going on. Another problem in this explanation is that most interactions with other human beings can be expected to be favorable rather than unfavorable, because human beings have more common interests than conflicting interests. So I think that the misapplied social instincts should generate more ambiguity seeking than ambiguity aversion. In the conclusion section, pp. 280281, the authors will argue that their misapplied heuristics model is descriptively superior to existing models. Such a claim, with almost no knowledge of the empirical literature, based mostly on theoretical examples on updating (see their first problem there), is naïve. The second problem on p. 281 has a strange and incomprehensible mix of rational and descriptive requirements. The third problem seems to be unaware that descriptively working people know well that not only fit but also parsimony are important, a standard fact in statistics in all empirical fields. %} AlNajjar, Nabil I. & Jonathan Weinstein (2009), “The Ambiguity Aversion Literature: A Critical Assessment,” ^ 25, 249–284. {% DC = stationarity on p. 100 top; Seems to correct a number of mathematical problems of LoewensteinPrelec (1992). %} AlNowaihi, Ali & Sanjit Dhami (2006) “A Note on the LoewensteinPrelec Theory of Intertemporal Choice,” Mathematical Social Sciences 52, 99–108. {% Critical condition assumes multistage prospects with folding back and then varies upon Luce’s condition by taking only two outcomes but three stages. %} AlNowaihi, Ali & Sanjit Dhami (2006) “A Simple Derivation of Prelec's Probability Weighting Function,” ^ 50, 521–524. {% inverseS: Seems to provide counterevidence. Propose that w for choice between (p,x) and (q,y) should depend on both p and q. Can explain anomalies such as preference reversals but is hard to assess. Some properties of weighting functions are derived from stylized choices from the literature. Only one nonzero outcome is considered, and, hence, the power is undetermined.%} Alarie, Yves & Georges Dionne (2001) “Lottery Decisions and Probability Weighting Function,” ^ 22, 21–33. {% Consider twooutcome prospects, and partition the probabilityoutcome combinations into subsets with particular “qualities,” which are used to accommodate all kinds of empirical findings. %} Alarie, Yves & Georges Dionne (2006) “Lottery Qualities,” Journal of Risk and Uncertainty 32, 195–216. {% %} Albers, Wulf, Robin Pope, Reinhard Selten, & Bodo Vogt (2000) “Experimental Evidence for Attractions to Chance,” German Economic Review 1, 113–130. {% revealed preference %} Alcantud, José C.R. (2002) “Revealed Indifference and Models of Choice Behavior,” Journal of Mathematical Psychology 46, 418–430. {% revealed preference %} Alcantud, José Carlos R. (2008) “Mixed Choice Structures, with Applications to Binary and NonBinary Optimization,” Journal of Mathematical Economics 44, 242–250. {% ordering of subsets: Additive representations for finite subsets, with a simple set of sufficient conditions. %} Alcantud, José C.R. & Ritxar Arlegi (2008) “Ranking Sets Additively in Decisional Contexts: An Axiomatic Characterization,” ^ 64, 147–171. {% risku = transform of strength.pr.v, latter doesn’t exist: Writes on p. 50: “In effect the utility whose measurement is discussed in this paper has literally nothing to do with individual, social or group welfare, whatever the latter may be supposed to mean.” Paper gives nice account, didactical with numerical examples etc., of the difference between ordinal utility and cardinal vNM utility. Nice for students with little mathematical background. P. 31: “Whether or not utility is some kind of glow or warmth, or happiness, is here irrelevant;” Footnote 4 on that page is pessimistic about the step, called psychological, philosophical, of relating utility to satisfaction, happiness, etc. P. 34 ll. 23 does the naive “expected utilitycism” of saying that all of life is decision under uncertainty. P. 37 2^{nd} para gives the nice argument for vNM independence that goods contingent upon exclusive events are never consumed jointly, that also convinced Samuelson. P. 37 last para states that different ways of generating same probability distribution should be equivalent. Paper makes very clear that whether a function is ordinal/cardinal etc. depends on what we want the function to do, such as on p. 40 middle. P. 43 bottom states the ^ . P. 42 already has the probability triangle. P. 44 very clearly states the prospect theory/Markowitz idea that outcomes are taken as changes with respect to a reference point, and not as final wealth. He later refers to Markowitz for it. P. 45 shows this weird past convention of calling convex what is nowadays called concave. P. 46: On difficult observable status of reference point theories in absence of theory about location of reference point: “Markowitz recognizes that until an unambiguous procedure is discovered for determining when and to what extent current income deviates from customary income, the hypothesis will remain essentially nonverifiable because it is not capable of denying any observable behavior.” %} Alchian, Armen A. (1953) “The Meaning of Utility Measurement,” ^ 43, 26–50. {% %} Alessie, Rob J. M., Stefan Hochguertel, & Arthur van Soest (2002) “Household Portfolios in the Netherlands.” In Luigi Guiso, Michael Haliassos, & Tullio Jappelli (eds.) Household Portfolios, The MIT Press, Cambridge, MA. {% Nice empirical study on asymmetric loss functions. The idea was central in Birnbaum, Coffey, Mellers, & Weiss (1992), p.325 and Elke Weber (1994), two studies not cited. %} Alexander, Marcus & Nicholas A. Christakis (2008) “Bias and Asymmetric Loss in Expert Forecasts: A Study of Physician Prognostic Behavior with Respect to Patient Survival,” ^ 27, 1095–1108. {% inverseS is found. Bettor’s subjective probs are estimated from portion of money bet on a horse. Objective probs are estimated from percentage of times that some horse (say favorite, or no. 5favorite, etc.) wins. Thus, bettors overestimate small probs of winning and understimate large probs. of winning. Uses power family to estimate utility and find that bettors are risk seeking (P.s.: no wonder, for horse race bettors!%} Ali, Mukhtar M. (1977) “Probability and Utility Estimates for Racetrack Betting,” ^ 85, 803–815. {% %} Ali, Iqbal, Wade D. Cook, & Moshe Kress (1986) “On the Minimum Violations Ranking of a Tournament,” Management Science 32, 660–672. {% Maths for econ students. %} Aliprantis, Charalambos D. & Kim C. Border (1999) “^ ’s Guide.” Springer, Berlin. {% Hammond (1976): Says that this book was the first to consider endogenously changing tastes: consumer regretting his earlier choice; explicitly restricted attention to the case where no changing or inconsistent choice occurs.%} Allais, Maurice (1947) “^ Imprimerie Nationale, Paris. {% Used just noticeable difference for cardinal utility.
