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risky utility u = transform of strength of preference v

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Risk averse for gains, risk seeking for losses

risky utility u = transform of strength of preference v: stated in Ch. 12 pp. 166-168. %}

Yates, J. Frank (1990) “Judgment and Decision Making.” Prentice Hall, London.

Yates, J. Frank, Paul C. Price, Ju-Whei Lee, & James Ramirez (1996) “Good Probabilistic Forecasters: The “Consumer’s” Perspective,” International Journal of Forecasting 12, 41–56.
{% Seems to find negative discounting for losses. %}

Yates J. Frank & Royce A. Watts (1975) “Preferences for Deferred Losses,” Organizational Behavior and Human Performance 13, 294–306.

{% Real incentives are implemented.
suspicion under ambiguity: done by letting subjects choose the winning color (“designation of valuable chip”)
second-order probabilities to model ambiguity: two-color Ellsberg urns. (Actually bags with 10 chips.) Game G is risk. Game G' is second-order probability, very clearly generated by the subjects themselves. Game G'' is just unknown probability. Find G ~G'  G''. So no aversion to 2^nd order probability, but aversion to pure ambiguity. So there is more to ambiguity aversion than second-order probabilities. %}

Yates, J. Frank & Lisa G. Zukowski (1976) “Characterization of Ambiguity in Decision Making,” Behavioral Science 21, 19–25.

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Yearsley, James M. (2017) “Advanced Tools and Concepts for Quantum Cognition: A Tutorial,” Journal of Mathematical Psychology 78, 24–39.

{% dynamic consistency: %}

Yechiam, Eldad & Jerome R. Busemeyer (2006) “The Effect of Foregone Payoffs on Underweighting Small Probability Events,” Journal of Behavioral Decision Making 19, 1–16.

{% Present a model and evidence that loss aversion is driven more by overattention to losses than by extremer utility (for which the authors use the term weight) for losses.
They also show that losses take more attention and, thus, lead to better decisions. For instance, subjects can choose between 35 for sure or 200_0.5X where X = 1 or X = 1. It is reasonable to take the risky choice as rational. Paradoxically, with X = 1 subjects more often chose risky than with X = 1. This indirect violation of montonicity is comparable to the zero-outcome effect paradox of Slovic-Birnbaum ((.95, $96; .05, $24) receives lower CE than (.95, $96; .05, $0); Birnbaum, Coffey, Mellers, & Weiss (1992)) but now without outcome 0 involved. %}

Yechiam, Eldad & Guy Hochman (2013) “Losses as Modulators of Attention: Review and Analysis of the Unique Effects of Losses over Gains,” Psychological Bulletin 139, 497–518.

{% They add results to Yechiam & Hochman (2013) on the Slovic-Birnbaum paradox but with no 0 outcome involved. Here, for instance, subjects can choose between 50 for sure or 200_0.5X where X = 1 or X = 1, with again, paradoxically, with X = 1 subjects more often chose risky than with X = 1. %}

Yechiam, Eldad, Matan Retzer, Ariel Telpaz, & Guy Hochman (2015) “Losses as Ecological Guides: Minor Losses Lead to Maximization and not to Avoidance,” Cognition 139, 10–17.

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Yechiam, Eldad, Julie C. Stout, Jerome R. Busemeyer, Stephanie L. Rock, & Peter R. Finn (2005) “Individual Differences in the Response to Forgone Payoffs: An Examination of High Functioning Drug Abusers,” Journal of Behavioral Decision Making 18, 97–110.

{% reflection at individual level for risk: correlation between risk aversion for gains and losses seem to be positive. %}

Yechiam, Eldad & Eyal Ert (2011) “Risk Attitude in Decision Making: In Search of Trait-Like Constructs,” Topics in Cognitive Science 3, 166–186.

{% Several studies have found that choices under losses are more difficult and, hence, noisier than choices under gains (de Lara Resende, Guilherme, & Wu 2010 p. 129; Gonzalez, Dana, Koshino, & Just 2005 JE; Lopes 1987). Somewhat different in spirit but not contradictory is that rewarding in terms of imposing losses to punish mistakes can work more effectively than imposing gains for good acts in making people make right choices. The presence of losses can make people pay more attention, improving decision quality.
PT falsified: this paper has an interesting experiment: people can choose between safe 35 and risky 200_0.51, and also between safe 35 and risky 200_0.5(1). (Unit of outcome is points converted into small money amounts at the end of the experiment, with repeated payments so income effects.) They more often choose risky in the second case, amounting to a violation of transitivity or stochastic dominance! The explanation is that the loss makes people pay more attention and, thus, they more rationally choose the highest expected value. This goes against the spirit of loss aversion. Interesting finding. They show that it is increased attention rather than contrast effect, because if the risky option has lower expected value then the loss makes people more often choose against the, now inferior, risky prospect. (cognitive ability related to risk/ambiguity aversion) Note that, in general, loss aversion can be generated by increased attention for losses (rather than losses having lower utility), but the above increased attention is of a different kind.
They also find Slovic/Birnbaum-type paradoxes where changing a zero outcome into a loss increases evaluation, which is one of these weird zero-outcome paradoxes.
The conclusion writes: “losses may be treated as signals of attention and not only as signals of avoidance. … Our findings demonstrate that the attentional effect of losses is indeed distinct from loss aversion,” %}

Yechiam, Eldad & Guy Hochman (2013) “Loss-Aversion or Loss-Attention: The Impact of Losses on Cognitive Performance,” Cognitive Psychology 66, 212–231.

{% dynamic consistency: nice empirical test of forgone-event independence %}

{% decreasing ARA/increasing RRA: they find it.
Present 50-50 risky choices, framed as good/bad harvest, to N=262 farmer households in Ethiopia, 6 gain choices and 6 mixed choices, using the Binswanger (1981) method to measure in each of those 12 choices. Real incentive for each of the gain choices (with stakes some days of salary), so that income effects do arise. For losses only real incentives if first gained enough in gains (which is a mild form of deception regarding the gains) (deception when implementing real incentives) and only if they accept to participate, which only 76 of the 226 offered did. They only had to pay losses if not exceeding a threshold. This all gives huge biases as the authors properly point out on p. 1026 and defend given the limitations of the setting. More risk aversion they find for mixed than for pure-gain. %}

Yesuf, Mahmud & Randall A. Bluffstone (2009) “Poverty, Risk Aversion, and Path Dependence in Low-Income Countries: Experimental Evidence from Ethiopia,” American Journal of Agricultural Economics 91, 1022–1037.

{% Comparative statics for the smooth ambiguity model. %}

Yi-Chieh Huang, Larry Y. Tzeng, Lin Zhao (2015) “Comparative Ambiguity Aversion and Downside Ambiguity Aversion,” Insurance: Mathematics and Economics 62 257–269.

{% Investigate in a simple setup with hypothetical data how time and risk interact when one fixed positive amount is involved. They do it for one small and one big amount. A central point in their writing is that probability and delay can be combined into a single metric. Find that hyperbolic discounting fits well. Because only one positive gain, utility of outcomes plays no role. %}

Yi, Richard, Xochitl de la Piedad, & Warren K. Bickel (2006) “The Combined Effects of Delay and Probability in Discounting,” Behavioural Processes 73, 149–155.

{% dynamic consistency Seems to show that, under some natural dynamic conditions on multistage CEU (Choquet expected utility), it can only be SEU. %}

Yoo, Keuk-Ryoul (1991) “The Iterative Law of Expectation and Non-Additive Probability Measure,” Economics Letters 37, 145–149.

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Yoo, Keuk-Ryoul (1991) “Steady-State Probabilities under Non-Additivity,” Dept. of Business Administration, Dongduck Women’s University, Seoul, Korea.

{% They seem to show that any finitely additive measure  can be decomposed uniquely as  = ₁ + ₂ with ₁ countable additive and ₂ “pure,” that is any countable additive measure between zero and ₂ must be zero. (There is a sequence of events, all with measure 1, but converging to the empty set.)

. Seem to show it for Borel sigma-algebras on Hausdoff topological spaces. Aliprantis & Border (1999) have more. %}

Yosida, Kosaka & Edwin Hewitt (1952) “Finitely Additive Measures,” Transactions of the American Mathematical Society 72, 46–66.
{% Consider decision under pure risk with decision where the uncertain events are partly influenced by the decision maker (cf. Drèze 1959). In the latter case, they ask the decision maker for probability estimates for the latter events. They then fit PT. That way they get probability weighting for the two kinds of events (source functions!?). There then is source preference for the events under own control. %}

Young, Diana L., Adam S. Goodie, & Daniel B. Hall (2011) “Modeling the Impact of Control on the Attractiveness of Risk in a Prospect Theory Framework,” Journal of Behavioral Decision Making 24, 47–70.

{% Risk averse for gains, risk seeking for losses: they find this.
In two risky choice experiments with gains, and PT of Tversky & Kahneman (1992) data fitting, they find that time pressure increases risk seeking, but the effects on utility and probability weighting alone are not clear. In a similar experiment with losses, time pressure increases likelihood insensitivity, but does not affect risk aversion or risk seeking.
They asked almanac questions about sizes of states in the US, and asked to express j  25% confidence levels. How these were used for risky questions, and whether the expressed confidence levels were used as probabilities, was not clear to me. They asked for direct assessments of certainty equivalents, but how these were incentivized was not clear to me either. P. 181 2^nd column 2^nd para writes that they used RIS in the gains-choices of experiment 1. P. 182 1^st column 3^rd para suggests that it was incentive compatible. %}

Young, Diana L., Adam S. Goodie, Daniel B. Hall, & Eric Wu (2012) “Decision Making under Time Pressure, Modeled in a Prospect Theory Framework,” Organizational Behavior and Human Decision Processes 118, 179–188.

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Young, H. Peyton (1975) “Social Choice Scoring Functions,” SIAM Journal of Applied Mathematics 28, 824–838.

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Young, H. Peyton (1987) “Progressive Taxation and the Equal Sacrifice Principle,” Journal of Public Economics 32, 203–214.

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Young, H. Peyton (1987) “On Dividing an Amount According to Individual Claims or Liabilities,” Mathematics of Operations Research 12, 398–414.

{% risky utility u = strength of preference v (or other riskless cardinal utility, often called value): considers tax schedules in some countries, such as US. Assumes equal sacrifice principle of John Stuart Mill: all people paying tax should lose the same amount of utility (leading to flat tax rate under logarithmic utility). Then from the amounts that the authorities let be paid by the various levels of income, we can derive the marginal utility that the authorities assume there, and then the cardinal utility. For 1957 US tax data, CRRA 1.61 fits the data well. This could be interpreted as cardinal welfare utility and be left as that. The author, however, does not shy away from relating this to utility measured from risky choice. On p. 255 2nd column the author very explicitly relates the utility found to risky utility, writing for instance: "The equal sacrifice hypothesis will be plausible if: (i) the estimated utility function is reasonably consistent with utility theory; … In the modern theory of risk bearing, …" %}

Young, H. Peyton (1990) “Progressive Taxation and Equal Sacrifice,” American Economic Review 80, 253–266.

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Younger, Daniel H. (1963) “Minimum Feedback Arc Sets for a Directed Graph,” IEEE Transactions on Circuit Theory 10, 238–245.

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Zabell, Sandy L. (1982) “W.E. Johnson’s “Sufficientness” Postulate,” Annals of Statistics 10, 1091–1099.

{% foundations of statistics: history %}

Zabell, Sandy L. (1989) “R.A. Fisher on the History of Inverse Probability,” Statistical Science 4, 247-263.

{% foundations of statistics: history. Laplace’s rule of succession: if on n trials we see m successes, then then next trial has success probability (m+1)/(n+1). (The rule I use privately lifelong.) %}

Zabell, Sandy L. (1989) “The Rule of Succession,” Erkenntnis 31, 283-321.

{% foundations of statistics: history %}

Zabell, Sandy L. (1992) “R.A. Fisher and the Fiducial Argument,” Statistical Science 7, 369–387.

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Zabell, Sandy L. (2011) “Carnap and the Logic of Inductive Inference.” In Dov M. Gabbay, John Woods, & Stephan Hartmann (Eds.), Handbook of the History of Logic Vol. 10., 265–309.

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Zachow, Ernst-Wilhelm (1979) “Expected Utility in Two-Person Games,” Mathematics of Operations Research 4, 186–195.

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Zadeh, Lofti A. (1965) “Fuzzy Sets,” Information and Control 8, 338–353.

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Zadeh, Lofti A. (1973) “Outline of a New Approach to the Analysis of Complex Systems and Decision Processes,” IEEE Transactions on Systems, Man and Cybernetics 3, 28–44.

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Zadeh, Lofti A. (1975) “Calculus of Fuzzy Restrictions.” In Lofti A. Zadeh, King-Sun Fu, Kazu Tanaka, & Masamichi Shimura (eds.) Fuzzy Sets and their Applications to Cognitive and Decision Processes, 1–39, Academic Press, New York.

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Zadeh, Lofti A. (1975) “Fuzzy Logic and Approximate Reasoning,” Synthese 30, 407–428.

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Zadeh, Lofti A. (1978) “Fuzzy Sets as a Basis for a Theory of Possibility,” Fuzzy Sets and Systems 1, 3–28.

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Zak, Paul J., Robert Kurzban, & William T. Matzner (2004) “The Neurobiology of Trust,” Annals of the New York Academy of Sciences 102, 224–227.

{% Seems to show that people are insensitive to the time dimension. %}

Zakay, Dan (1998) “Attention Allocation Policy Influences Prospective Timing,” Psychonomic Bulletin and Review 5, 114–118.

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Zakay, Dan (1985) “Post-Decisional Confidence and Conflict Experienced in a Choice Process,” Acta Psychologica 58, 75–80.

{% If you observe one CE (certainty equivalent) of a risk averse EU maximizer, you can derive inequalities for the subjective probabilities. %}

Zambrano, Eduardo (2008) “Expected Utility Inequalities: Theory and Applications,” Economic Theory 36, 147–158.

{% Argues for paternalism that just seeks for efficiency. %}

Zamir, Eyal (1998) “The Efficiency of Paternalism,” Virginia Law Review 84, 229–286.

{% Risk averse for gains, risk seeking for losses
Much risk aversion for mixed. The authors find in an experiment that mostly loss aversion drives clients’ preferences for contingent-fee arrangements regarding attorney’s fees, rather than other components of risk aversion. Experiment 1 did hypothetical legal situations. Experiment 2 (N = 27) did real incentives, with the real payments a proportion of the amounts mentioned in the legal story. Four more experiments were done. Probabilities were always given. %}

Zamir, Eyal & Ilana Ritov (2010) “Revisiting the Debate over Attorneys’ Contingent Fees: A Behavioral Analysis,” Journal of Legal Studies 39, 245–288.

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Zang, Lian-Wen (1986) “Weights of Evidence and Internal Conflict for Support Functions,” Information Sciences 38, 205–212.

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Zank, Horst (1999) “Risk and Uncertainty: Classical and Modern Models for Individual Decision Making,” Ph.D. dissertation, Dept. of Economics, Maastricht University, Maastricht, the Netherlands.

{% Characterizes PT for parametric utility which simplifies the derivation of the underlying PT, essentially generalizing Wakker & Zank (2002) from RDU to PT. Does a similar thing but now with multiattribute outcomes, and utility independence type conditions similarly simplifying the underlying PT derivation. Nice thing here is that just tail independence (or, similarly, the stronger comonotonic independence) already give a kind of state-dependent-utility generalization of RDU and PT, so that the axioms for parametric utility or utility independence need to be imposed only on gains and losses separately. %}

Zank, Horst (2001) “Cumulative Prospect Theory for Parametric and Multiattribute Utilities,” Mathematics of Operations Research 26, 67–81.

{% Characterizes PT in the context of welfare. Uses conditions to characterize particular forms of utility, to simplify the underlying derivation of PT, generalizing Wakker & Zank (2002) from RDU to PT. Shows that concavity at reference point is a kind of loss aversion. %}

Zank, Horst (2007) “Social Welfare Functions with a Reference Income,” Social Choice and Welfare 28, 609–636.

{% An axiomatization of RDU for risk that is alternative to Abdellaoui (2002). The paper used the same notation with cumulative probabilities. It weakens Abdellaoui’s main axiom in the same appealing manner as Chateauneuf (1999) weakened the tradeoff consistency for outcomes of Wakker (1989, 2010), using a midpoint version rather than a general tradeoff version. %}

Zank, Horst (2010) “Consistent Probability Attitudes,” Economic Theory 44, 167–185.

{% Discusses definitions of loss aversion, and proposes a new one that also has implications for probability weighting. The new proposal is:
0  (p:x, 12p:0, p:x) for all x > 0 and p  ½. Holds under PT iff w⁺(p)U(x)  \w^(p)U(x). %}

Zank, Horst (2010) “On Probabilities and Loss Aversion,” Theory and Decision 68, 243–261.

Zank, Horst (2010) “Consistent Probability Attitudes,” Economic Theory 44, 167–185.

{% Proposes, for a prospect x, a representation PT*(p: PT(x+), q:PT(x-)) where: PT* may be an entirely different PT functional than PT; p is the total probability of x yielding a gain (outcome > 0); q is the total probability of x yielding a loss (outcome < 0), 1-p-q is the probability of getting 0; x+ is the CONDITIONAL probability distribution of x given that it is a gain; x- is the CONDITIONAL probability distribution of x given that it is a loss. %}

Zank, Horst (2016) “A General Measure for Loss Attitude,” working paper.

{% doi: http://dx.doi.org/10.1509/jmkr.46.4.543
nonconstant discount = nonlinear time perception: not fully that point, but nonlinear perception of time is central in their paper.
Decompose discounting into subjective time perception and then weighting of that, and cite many preceding works on the idea of subjective time perception. When reading the first pages of the paper, I never saw the mystery revealed of how will they measure subjective time perception? P. 546 shows how psychologists can do this: they asked subjects to indicate on a line “how long” various periods of time were. Oh well.
Seem to find that perception of time is more labile than perception of money.
Köbberling, Schwieren, & Wakker (2007, Theory and Decision) used the introduction of the Euro to separate what they called numerical perception out of the utility of money based on revealed-preference. %}

Zauberman, Gal, B. Kyu kim, Selin A. Malkoc, & James R. Bettman (2009) “Discounting Time and Time Discounting: Subjective Time Perception and Intertemporal Preferences,” Journal of Marketing Research 66, 543–556.

{% Seems to have been the first to formally model moral hazard. %}

Zeckhauser, Richard J. (1970) “Medical Insurance: A Case Study of the Tradeoff between Risk Spreading and Appropriate Incentives,” Journal of Economic Theory 2, 10–26.

{% suspicion under ambiguity: p. S445 points out that suspicion can drive Ellsberg paradox. %}

Zeckhauser, Richard J. (1986) “Comments: Behavioral versus Rational Economics: What You See Is What You Conquer,” Journal of Business 59, S435–S449.

{% Many examples and lessons about good investments when probabilities could not be known. Ricardo gained a fortune buying English bonds 4 days before the battle of Waterloo.
P. 14: “Prospect theory, the most important single contribution to behavioral decision theory to date, …”
P. 15 has nice experiment. Ambiguous event is that 10,000-ton asteroid passed within 40,000 miles of earth during last decade. To get anchor probability, asked a random sample of people to guess probability until a distance was found where the median estimated probability was 0.03. Took that as anchor probability for measuring ambiguity attitude. Nice! However, seems to assume that for such small likelihood one will find ambiguity aversion still, contrary to many empirical findings.
P. 34, §V: Buffett made much money reinsuring earth quakes in California. His capital was so big that he could still be risk neutral (if we can say so for unknown probabilities) for such high amounts.
P. 36, about ambiguity aversion: ”Maxim G: discounting for ambiguity is a natural tendency that should be overcome, just as should be overeating.” %}

Zeckhauser, Richard J. (2006) “Investing in the Unknown and Unknowable,” Capitalism and Society 1, Article 5, 1–39.

{% inverse-S: the authors several times emphasize that small probabilities are overweighted. P. 559 2nd column l. -15: individuals have great difficulties comprehending extremely low-probability events. (Suggests it’s cognitive; cognitive ability related to likelihood insensitivity (= inverse-S)) P. 560 l. 3 suggests inverse-S in probability estimation.
P. 5672 5th para nicely points out that in environments with learning possibilities we should prefer unknown probabilities (ambiguity seeking). %}

Zeckhauser, Richard & Kip W. Viscusi (1990) “Risk within Reason,” Science 248 no. 4955, 559–564.

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Zeelenberg, Marcel (1999) “Anticipated Regret, Expected Feedback and Behavioral Decision Making,” Journal of Behavioral Decision Making 12, 93–106.

{% Contains much of the  literature up to 2007. %}

Zeelenberg, Marcel & Rik Pieters (2007) “A Theory of Regret Regulation 1.0,” Journal of Consumer Psychology 17, 3–18.

{% DFE where subjects quickly receive much feedback from normal distributions. The authors present an RDU model for sequential sampling showing that in one task participants weighted larger payoffs more. %}

Zeigenfuse, Matthew D., Timothy J. Pleskac, & Taosheng Liu (2014) “Rapid Decisions from Experience,” Cognition 131, 181–194.

{% Section 2 nicely reviews stability across domains, tasks, and time.
Fit the same parametric family as T&K 92 to CE (certainty equivalent) measurements. Do measurements month apart, to test time stability. If I remember right, Cohen, Jaffray, & Said (1987) did two measurements a week apart.

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