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## Parameter-free elicitation of utility and probability weighting functions. (2000)

Venue: | Manag. Sci. |

Citations: | 182 - 5 self |

### Citations

6309 | Prospect Theory: An Analysis of Decision Under
- Kahneman, Tversky
- 1979
(Show Context)
Citation Context ...it individual utility and probability weighting functions for monetary outcomes in the gain and loss domains. Concave utility functions are obtained for gains and convex utility functions for losses. The elicited weighting functions satisfy upper and lower subadditivity and are consistent with previous parametric estimations. The data also show that the probability weighting function for losses is more ’’elevated’’ than for gains. (Decision Making; Expected Utility; Rank-Dependent Expected Utility; Cumulative Prospect Theory; Probability Weighting Function) 1. Introduction In a seminal paper, Kahneman and Tversky (1979) present experimental evidence that preferences between risky prospects are not linear in probabilities. They propose, as well, a theory of choice under risk, Prospect Theory (PT), suggesting that a probability weighting function (that maps the unit interval into itself with discontinuities at 0 and 1) exhibiting overweighting of small probabilities and underweighting of moderate and high probabilities may explain the observed nonlinearities. Subsequently, modern generalizations of PT were proposed through RankDependent Expected Utility (RDEU) theory (Quiggin 1982, Wakker 1994), and Cumulative... |

1716 | Advances in Prospect Theory: Cumulative Representation of Uncertainty
- Tversky, Kahneman
- 1992
(Show Context)
Citation Context ...preferences between risky prospects are not linear in probabilities. They propose, as well, a theory of choice under risk, Prospect Theory (PT), suggesting that a probability weighting function (that maps the unit interval into itself with discontinuities at 0 and 1) exhibiting overweighting of small probabilities and underweighting of moderate and high probabilities may explain the observed nonlinearities. Subsequently, modern generalizations of PT were proposed through RankDependent Expected Utility (RDEU) theory (Quiggin 1982, Wakker 1994), and Cumulative Prospect Theory (CPT) (Tversky and Kahneman 1992). Their main characteristic, in decision under risk, consists of allowing not only the transformation of outcomes into utilities, but also the transformation of decumulative probabilities to obtain decision weights through a probability weighting function. This innovation, however, has been perceived as a factor complicating utility measurement, and therefore, the elicitation of RDEU and CPT models. A variety of methods have been used to determine the shapes of the utility function and the probability weighting function under RDEU and CPT. The predominant approach prespecifies parametric forms... |

570 |
A theory of Anticipated Utility”,
- Quiggin
- 1982
(Show Context)
Citation Context ...seminal paper, Kahneman and Tversky (1979) present experimental evidence that preferences between risky prospects are not linear in probabilities. They propose, as well, a theory of choice under risk, Prospect Theory (PT), suggesting that a probability weighting function (that maps the unit interval into itself with discontinuities at 0 and 1) exhibiting overweighting of small probabilities and underweighting of moderate and high probabilities may explain the observed nonlinearities. Subsequently, modern generalizations of PT were proposed through RankDependent Expected Utility (RDEU) theory (Quiggin 1982, Wakker 1994), and Cumulative Prospect Theory (CPT) (Tversky and Kahneman 1992). Their main characteristic, in decision under risk, consists of allowing not only the transformation of outcomes into utilities, but also the transformation of decumulative probabilities to obtain decision weights through a probability weighting function. This innovation, however, has been perceived as a factor complicating utility measurement, and therefore, the elicitation of RDEU and CPT models. A variety of methods have been used to determine the shapes of the utility function and the probability weighting fun... |

405 |
Gambling with the House Money and Trying to Break Even:
- Thaler, Johnson
- 1990
(Show Context)
Citation Context ...ficiently pronounced, it is necessary to investigate a sufficiently wide interval of outcomes. For all the prospects used in the present experimental study, the outcomes (gains and losses) were between U.S. $200 and U.S. $4,000. Subjects were informed before the experiment that, for gains and only for gains, one subject would be randomly selected to play for actual money on one of her answers in TO-experiments and one of her answers in PW-experiments. 5 For losses, a similar lottery incentive mechanism is ethically objectionable. Hence, only hypothetical payoffs were used (e.g., Camerer 1989, Thaler and Johnson 1990). It will be seen later that the reliability of subjects in TO-experiments was higher for losses than for gains. The experiments were conducted in small groups of six subjects in the experimental decision laboratory at the Ecole Normale SupIerieure de Cachan, Department of Economics and Management. Subjects were seated in front of personal computers and were told to take their time, and encouraged to go at their own pace. Be4 Six subjects were discarded; due to a network problem, their data were lost. 5 Subjects’ answers were not collected as switching outcomes or probabilities (i.e., through ... |

290 |
The Probability Weighting Function”,
- Prelec
- 1998
(Show Context)
Citation Context ...ere the decision weights for losses −i and the decision weights for gains +j are defined by: Table 1 Some ’’Typical’’ Parametric Specifications of u and w Experiments Models u(x) w(p) 1−e−cp 1−e−c Currim and PT 1−e −cx 1−e−c a+ bp+ cp 2 Sarin (1989) log(a + bec log p) Tversky and CPT { x if x ≥ 0 −( − x) if x¡0 p [p+(1−p) ] 1 Kahneman (1992) Tversky and CPT x p [ p+(1−p) ] (1) Fox (1995) Wu and PT x p [p+(1−p) ]Gonzalez (1996) CPT p [p+(1−p) ] 1 a + bp exp(−(−ln p)!)(2) (1) Used by Goldstein and Einhorn (1987) and Lattimore et al. (1992). (2) Initially proposed by Prelec (1998). −i =w −( ∑i k=1pk) − w−( ∑i−1 k=1 pk) for i ≥ 2 and −1 =w −(p1); +j =w +( ∑n k=j pk) − w+( ∑n k=j+1pk) for j ≤ n − 1 and +n =w+(pn). Note that these weights do not necessarily sum to one, and that RDEU corresponds to the special case where the weighting function for losses is the dual of the weighting function for gains, i.e., w−(p) =1 − w+(1 − p) for all p ∈ (0, 1). Up to now, most experimental studies used parametric specifications to infer the shapes of utility functions and weighting functions from individual choices. Tversky and Kahneman (1992) and Tversky and Fox (1995) gave parame... |

290 | Curvature of the probability weighting function.
- Wu, Gonzalez
- 1996
(Show Context)
Citation Context ...mates them through standard techniques (e.g., Tversky and Kahneman 1992, Camerer and Ho 1994, Hey and Orme 1994, Tversky and Fox 1995). However, assuming specific functional forms for the utility function and the probability weighting function makes inference about the shapes of these functions dependent on the choice of functional forms. Two research strategies can avoid the potential problems of parametric estimation. The first strategy consists of testing simple preference conditions to obtain information about the shape of either the utility function or the probability weighting function. Wu and Gonzalez (1996, 1998) follow this strategy. They find that aggregate behavior is consistent with a 0025-1909/00/4611/1497$05.00 MANAGEMENT SCIENCE ? 2000 INFORMS 1526-5501 electronic ISSN Vol. 46, No. 11, November 2000 pp. 1497--1512 ABDELLAOUI Parameter-Free Elicitation of Utility and Probability Weighting Functions concave-convex shaped weighting function independent of any assumption about the value function. The second strategy consists of eliciting the utility and probability weighting functions at the level of individuals, without any parametric assumption. This approach is more demanding, but, in ret... |

251 |
The utility of wealth.
- Markowitz
- 1952
(Show Context)
Citation Context ...sky and Kahneman (1992). 17 5.3. Probability Weighting Functions For Tversky and Kahneman (1992), the most distinctive characteristic of individual behavior under risk is what they called the fourfold pattern of risk attitude. It claims that individuals exhibit: (i) risk seeking for gains and risk aversion for losses of low probability; (ii) risk aversion for gains and risk seeking for losses of high probability. The authors remark that since the fourfold pattern arises over a wide range of outcomes, it cannot be explained by the utility of money as suggested by Friedman and Savage (1948) and Markowitz (1952). Instead, they infer that this fourfold pattern of risk attitude suggests an inverse S-shaped nonlinear transformation of the probability scale as well as an S-shaped utility function. Tversky and Kahneman (1992) recognize, however, that instead of clarity of the overall pattern, the individual data reveal both noise and individual differences. Probability weighting functions were elicited for 40 subjects both for gains and losses. The assessed values were inverse images (w)−1(p), rather than images w(p) for p =i=6, i =1, : : : , 5 (see Figure 4). Table 5 gives medians, means, and standard de... |

228 |
Investigating generalizations of expected utility theory using experimental data.
- JD, Orme
- 1994
(Show Context)
Citation Context ..., but also the transformation of decumulative probabilities to obtain decision weights through a probability weighting function. This innovation, however, has been perceived as a factor complicating utility measurement, and therefore, the elicitation of RDEU and CPT models. A variety of methods have been used to determine the shapes of the utility function and the probability weighting function under RDEU and CPT. The predominant approach prespecifies parametric forms for these functions and then estimates them through standard techniques (e.g., Tversky and Kahneman 1992, Camerer and Ho 1994, Hey and Orme 1994, Tversky and Fox 1995). However, assuming specific functional forms for the utility function and the probability weighting function makes inference about the shapes of these functions dependent on the choice of functional forms. Two research strategies can avoid the potential problems of parametric estimation. The first strategy consists of testing simple preference conditions to obtain information about the shape of either the utility function or the probability weighting function. Wu and Gonzalez (1996, 1998) follow this strategy. They find that aggregate behavior is consistent with a 0025-... |

212 |
The predictive utility of generalized expected utility theories.
- Harless, Camerer
- 1994
(Show Context)
Citation Context .... This allows a straightforward comparison of the treatment of probabilities for gains and losses at the level of individual subjects. Indeed, the data confirm the existence of a significant difference between the probability weighting function for gains and the probability weighting function for losses. Moreover, the data suggest a descriptive superiority of CPT over RDEU. Finally, the hypothesis of linearity of the probability weighting function for moderate probabilities is investigated. This question has received rather contradictory answers in the experimental literature. Camerer (1992), Harless and Camerer (1994), and Abdellaoui and Munier (1998), for instance, obtained results through nonparametric techniques suggesting a linear weighting function for intermediate probabilities (see also Cohen and Jaffray 1988). On the contrary, Wu and Gonzalez (1996), among others, found support for nonlinearity. This paper confirms the latter (i.e., nonlinearity). The paper proceeds as follows. Section 2 reviews RDEU theory and CPT. Then, some previous experimental studies of parametric estimation of RDEU or CPT preference functionals are presented. Section 3 describes the two-step elicitation method. Section 4 des... |

187 | Weighing risk and uncertainty,
- Tversky, Fox
- 1995
(Show Context)
Citation Context ...sformation of decumulative probabilities to obtain decision weights through a probability weighting function. This innovation, however, has been perceived as a factor complicating utility measurement, and therefore, the elicitation of RDEU and CPT models. A variety of methods have been used to determine the shapes of the utility function and the probability weighting function under RDEU and CPT. The predominant approach prespecifies parametric forms for these functions and then estimates them through standard techniques (e.g., Tversky and Kahneman 1992, Camerer and Ho 1994, Hey and Orme 1994, Tversky and Fox 1995). However, assuming specific functional forms for the utility function and the probability weighting function makes inference about the shapes of these functions dependent on the choice of functional forms. Two research strategies can avoid the potential problems of parametric estimation. The first strategy consists of testing simple preference conditions to obtain information about the shape of either the utility function or the probability weighting function. Wu and Gonzalez (1996, 1998) follow this strategy. They find that aggregate behavior is consistent with a 0025-1909/00/4611/1497$05.00... |

124 |
An experimental test of several generalized utility theories.
- Camerer
- 1989
(Show Context)
Citation Context ...y function sufficiently pronounced, it is necessary to investigate a sufficiently wide interval of outcomes. For all the prospects used in the present experimental study, the outcomes (gains and losses) were between U.S. $200 and U.S. $4,000. Subjects were informed before the experiment that, for gains and only for gains, one subject would be randomly selected to play for actual money on one of her answers in TO-experiments and one of her answers in PW-experiments. 5 For losses, a similar lottery incentive mechanism is ethically objectionable. Hence, only hypothetical payoffs were used (e.g., Camerer 1989, Thaler and Johnson 1990). It will be seen later that the reliability of subjects in TO-experiments was higher for losses than for gains. The experiments were conducted in small groups of six subjects in the experimental decision laboratory at the Ecole Normale SupIerieure de Cachan, Department of Economics and Management. Subjects were seated in front of personal computers and were told to take their time, and encouraged to go at their own pace. Be4 Six subjects were discarded; due to a network problem, their data were lost. 5 Subjects’ answers were not collected as switching outcomes or pro... |

121 |
Additive Representations of Preferences: A New Foundation in Decision Analysis.
- Wakker
- 1989
(Show Context)
Citation Context ...of probabilities in the assessment process. Moreover, it allows constructing standard sequences of outcomes even when probabilities are distorted or unknown. 2 A standard sequence of positive outcomes (i.e., gains), is constructed as follows. An outcome x1 is determined to make the subject indifferent between the prospects (x0, p; R,1−p) and (x1, p; r,1−p), denoted (x0, p; R) and (x1, p; r) respectively, where 0≤ r¡R¡x0¡x1, and p∈ (0,1); r, R and x0 are held fixed. This means that the trade-off of R for r, over the ’’(1−p)-axis,’’ outweighs the trade-off of x0 for x1, over the ’(p)-axis’ (see Wakker 1989, p. 35). Then, an outcome x2 is determined to make the subject indifferent between the prospects (x1, p; R) and (x2, p; r). Thus, under CPT, the two constructed indifferences give the following two equations: w+(p)u(x0) + (1 − w+(p))u(R) = w+(p)u(x1) + (1 − w+(p))u(r), (3) w+(p)u(x1) + (1 − w+(p))u(R) = w+(p)u(x2) + (1 − w+(p))u(r): (4) Together, these equations imply: u(x1) − u(x0) =u(x2) − u(x1): (5) Equation (5) implies that x1 is a midpoint outcome in terms of utility in the sequence x0, x1, x2, i.e., the trade-off of x0 for x1 is considered as equivalent to the trade-off of x1 for x2, th... |

117 |
A parameter-free elicitation of the probability weighting function in medical decision analysis. Working paper,
- Bleichrodt, Pinto
- 1998
(Show Context)
Citation Context ...p.311), consider the estimation of such a complex model as CPT to be problematic; this paper shows that it can be done without much trouble. Many experimental findings made in Tversky and Kahneman (1992) and Wu and Gonzalez (1996) have been confirmed in this paper by means of parameter-free elicitation of the utility and probability weighting functions. To my knowledge, this is the first paper that entirely elicits the CPT model (for both gains and losses) at the level of individual subjects without any parametric assumption. 1 1 A few months after a first version of this paper was completed, Bleichrodt and Pinto (1998) and Gonzalez and Wu (1999) finished two papers proposing two methods to elicit probability weighting functions. The first paper reports experimental results regarding probability weighting in medical decision making (using the tradeoff method). The second investigates individual differences in probability weighting for monetary gains through an alternating least square estimation method. Given that the elicitation of the probability weighting function in this paper needs the construction of the utility function to be carried out first, the first empirical question addressed concerns the shape... |

98 |
Eliciting von NeumannMorgenstern Utilities when Probabilities Are Distorted or Unknown".
- Wakker, Deneffe
- 1996
(Show Context)
Citation Context ...ssumption about the value function. The second strategy consists of eliciting the utility and probability weighting functions at the level of individuals, without any parametric assumption. This approach is more demanding, but, in return, provides direct measurements of both functions. This paper implements the latter strategy, proposing a two-step method to successively elicit the decision-maker’s utility and weighting functions under CPT. The first step consists of constructing a sequence of outcomes equally spaced in utility by means of the trade-off method initially proposed by Wakker and Deneffe (1996). Contrary to the other existing methods of utility elicitation, the trade-off method is ’’robust’’ against probability distortion. The second step uses the sequence of outcomes to obtain a sequence of probabilities equally spaced in terms of probability weighting. These two steps were computerized to allow a comfortable questioning of subjects and to avoid some expected framing effects by means of a special sequencing of choice questions. This proposed parameter-free elicitation method is used in an experimental investigation to study the shapes of the utility function and the probability wei... |

89 | Risk attitudes and decision weights
- Tversky, Wakker
- 1995
(Show Context)
Citation Context ...hus the reliability of the measurement of probability weighting depends also (and mainly) on the reliability of the assessment of the standard sequence of outcomes (see §5.1). To summarize, we may experimentally elicit the weighting function in two successive steps: (i) construction of a standard sequence of outcomes (trade-off experiments: TO-experiments); (ii) determination of a corresponding standard sequence of probabilities (probability weighting experiments: PW-experiments). Preference conditions concerning the shape of the weighting function were formulated by Segal (1987), Tversky and Wakker (1995), Wu and Gonzalez (1996, 1998) and Prelec (1998). Tversky and Wakker (1995) define two properties of the weighting function that are rarely violated, at least in parametric estimation. These properties are called lower subadditivity (LS) and upper Figure 1 Illustration of Subadditivity subadditivity (US). LS implies that a ’’lower interval’’ [0, q] has more impact than a middle interval [p, p + q], and US means that an ’’upper’’ interval [1 − q, 1] has more impact than a middle interval [p, p + q], provided the middle interval is bounded away from the lower end point 0 and upper end point 1. L... |

72 |
A belief-based account of decision under uncertainty.
- Fox, Tversky
- 1998
(Show Context)
Citation Context ...f method). The second investigates individual differences in probability weighting for monetary gains through an alternating least square estimation method. Given that the elicitation of the probability weighting function in this paper needs the construction of the utility function to be carried out first, the first empirical question addressed concerns the shape of the utility function. Its qualitative properties issuing from the psychological principle of diminishing sensitivity are confirmed here, in agreement with other recent findings (Wakker and Deneffe 1996, Fennema and van Assen 1998, Fox and Tversky 1998). Then the question of the shape of the probability weighting function is addressed. The data confirm that individuals transform probabilities consistently with the psychological principle of diminishing sensitivity, with the two end points of the probability interval serving as reference points. Overall, these results are consistent (for gains) with those obtained recently by Tversky and Fox (1995). This paper also elicits probability weighting functions for losses. This allows a straightforward comparison of the treatment of probabilities for gains and losses at the level of individual subje... |

67 |
Expression theory and the preference reversal phenomena.
- Goldstein, Einhorn
- 1987
(Show Context)
Citation Context ... xr ≤ 0≤ xr+1 ≤ · · ·≤ xn is: VCPT(P) = r∑ i=1 −i u(xi) + n∑ j=r+1 +j u(xj) (2) where the decision weights for losses −i and the decision weights for gains +j are defined by: Table 1 Some ’’Typical’’ Parametric Specifications of u and w Experiments Models u(x) w(p) 1−e−cp 1−e−c Currim and PT 1−e −cx 1−e−c a+ bp+ cp 2 Sarin (1989) log(a + bec log p) Tversky and CPT { x if x ≥ 0 −( − x) if x¡0 p [p+(1−p) ] 1 Kahneman (1992) Tversky and CPT x p [ p+(1−p) ] (1) Fox (1995) Wu and PT x p [p+(1−p) ]Gonzalez (1996) CPT p [p+(1−p) ] 1 a + bp exp(−(−ln p)!)(2) (1) Used by Goldstein and Einhorn (1987) and Lattimore et al. (1992). (2) Initially proposed by Prelec (1998). −i =w −( ∑i k=1pk) − w−( ∑i−1 k=1 pk) for i ≥ 2 and −1 =w −(p1); +j =w +( ∑n k=j pk) − w+( ∑n k=j+1pk) for j ≤ n − 1 and +n =w+(pn). Note that these weights do not necessarily sum to one, and that RDEU corresponds to the special case where the weighting function for losses is the dual of the weighting function for gains, i.e., w−(p) =1 − w+(1 − p) for all p ∈ (0, 1). Up to now, most experimental studies used parametric specifications to infer the shapes of utility functions and weighting functions from individual choice... |

63 |
Probability versus certainty equivalence methods in utility measurement: Are they equivalent? Management Sci.
- Hershey, Schoemaker
- 1985
(Show Context)
Citation Context ...bject’s choices in the first step were, for instance, A11, A 1 2, A 1 3, A 1 4, A 1 5, and A 1 ∗, she was then asked to successively choose between the prospects A2i =(xi, 1) and B2i =(x6, 3=4; x0), i =1, : : : , 5, and also between the prospects A2∗ =(x3, 1) and B2∗ =(x4, 3=4; x2). Overall, each subject was confronted with six series of six choice questions. Note that with this sequencing of choice questions, the subject can hardly infer that she is in fact asked to match probability p. This special sequencing of choice questions was designed to avoid the problem of ’’framed probabilities’’ (Hershey and Schoemaker 1985) as well as possible biases due to scale compatibility (Tversky et al. 1988, DelquiIe 1993). Once the probabilities p1, : : : , p5, p′2 were assessed, the computer program displayed successively the pairs of prospects A 4i , B 4 i , i =1, : : : , 5, A 4 ∗ , B 4∗ , and asked the subject to designate the preferred prospect in each pair. The aim of this step was to check the subjects’ reliability in PW-experiments. 5. Results 5.1. Reliability We can test the reliability of subjects’ responses by seeing how often they expressed the same preference for the same prospect. As explained above, after e... |

55 | Measuring the utility of losses by means of the trade-off method. - Fennema, Assen - 1998 |

55 |
What determines the shape of the probability weighting function under uncertainty. Working paper,
- Kilka, Weber
- 1998
(Show Context)
Citation Context ...are very close to those obtained by Tversky and Kahneman (1992) for gains and losses, which were 0.61 and 0.69 respectively. Nevertheless, the Lattimore et al. (1992) function provides a clearer separation between losses and gains. The estimates of show that the probability weighting function for median data exhibit more elevation for losses than for gains (see Figure 5). Such a clear distinction is not allowed by Tversky and Kahneman’s one-parameter specification of w. Note that if (+, +) and (−, −) designate the parameters of w+ and w− respectively, duality (i.e., w+(p) =1−w−(1−p)) 19 Kilka and Weber (1998) extend these concepts to uncertainty. MANAGEMENT SCIENCE/Vol. 46, No. 11, November 2000 1509 ABDELLAOUI Parameter-Free Elicitation of Utility and Probability Weighting Functions Figure 5 Weighting Functions Estimates Based on Median Data implies that + =− and + = 1=− 20 . Parameter estimates for median data do not meet these conditions. At the level of individual subjects, elevation is also more pronounced for losses than for gains (i.e., for losses is greater than for gains). This is true in the present experimental study for 27 subjects out of 40 (p¡0:05, one-tailed sign test). Figu... |

52 |
Violations of the independence axiom in common ratio problems: An experimental test of some competing hypotheses.
- Starmer, Sugden
- 1989
(Show Context)
Citation Context ...pressed the same preference for the same prospect. As explained above, after each standard sequence assessment every subject was confronted, once again, with the pair of prospects previously presented in the fourth iteration of the bisection processes concerning outcomes and probabilities. Overall, 19% of the subjects’ responses (i.e., 182 out of 960 responses) reveal reversals in preferences. This number is lower than those obtained by 1504 MANAGEMENT SCIENCE/Vol. 46, No. 11, November 2000 ABDELLAOUI Parameter-Free Elicitation of Utility and Probability Weighting Functions Camerer (1989) and Starmer and Sugden (1989). For TO-experiments, the number of reversals is more important for grains (17.9%) than for losses (13.7%). For PW-experiments, 19.1% and 25% of subjects’ responses reveal reversals for gains and losses respectively. 10 Because responses are chained in the trade-off method (the elicitation of xi requires the value xi−1 as input), one may expect errors to propagate similarly as in other chained methods such as the bisection version of the CE method (i.e., fractile method). For the impact of error propagation (on the assessment of w) to be small, the subject should be ’’uniformly consistent’’ at... |

46 | Lottery equivalents: Reduction of the certainty effect problem in utility assessment. - McCord, Neufville - 1986 |

42 |
On the shape of the probability weighting function. Cognitive Psych.
- Gonzalez, Wu
- 1999
(Show Context)
Citation Context ...of such a complex model as CPT to be problematic; this paper shows that it can be done without much trouble. Many experimental findings made in Tversky and Kahneman (1992) and Wu and Gonzalez (1996) have been confirmed in this paper by means of parameter-free elicitation of the utility and probability weighting functions. To my knowledge, this is the first paper that entirely elicits the CPT model (for both gains and losses) at the level of individual subjects without any parametric assumption. 1 1 A few months after a first version of this paper was completed, Bleichrodt and Pinto (1998) and Gonzalez and Wu (1999) finished two papers proposing two methods to elicit probability weighting functions. The first paper reports experimental results regarding probability weighting in medical decision making (using the tradeoff method). The second investigates individual differences in probability weighting for monetary gains through an alternating least square estimation method. Given that the elicitation of the probability weighting function in this paper needs the construction of the utility function to be carried out first, the first empirical question addressed concerns the shape of the utility function. I... |

36 |
The influence of probability on risky choice.
- Lattimore, Baker, et al.
- 1992
(Show Context)
Citation Context ...T(P) = r∑ i=1 −i u(xi) + n∑ j=r+1 +j u(xj) (2) where the decision weights for losses −i and the decision weights for gains +j are defined by: Table 1 Some ’’Typical’’ Parametric Specifications of u and w Experiments Models u(x) w(p) 1−e−cp 1−e−c Currim and PT 1−e −cx 1−e−c a+ bp+ cp 2 Sarin (1989) log(a + bec log p) Tversky and CPT { x if x ≥ 0 −( − x) if x¡0 p [p+(1−p) ] 1 Kahneman (1992) Tversky and CPT x p [ p+(1−p) ] (1) Fox (1995) Wu and PT x p [p+(1−p) ]Gonzalez (1996) CPT p [p+(1−p) ] 1 a + bp exp(−(−ln p)!)(2) (1) Used by Goldstein and Einhorn (1987) and Lattimore et al. (1992). (2) Initially proposed by Prelec (1998). −i =w −( ∑i k=1pk) − w−( ∑i−1 k=1 pk) for i ≥ 2 and −1 =w −(p1); +j =w +( ∑n k=j pk) − w+( ∑n k=j+1pk) for j ≤ n − 1 and +n =w+(pn). Note that these weights do not necessarily sum to one, and that RDEU corresponds to the special case where the weighting function for losses is the dual of the weighting function for gains, i.e., w−(p) =1 − w+(1 − p) for all p ∈ (0, 1). Up to now, most experimental studies used parametric specifications to infer the shapes of utility functions and weighting functions from individual choices. Tversky and Kahneman (199... |

33 |
Some remarks on Quiggin’s anticipated utility.
- Segal
- 1987
(Show Context)
Citation Context ...tion) through xi and xn. Thus the reliability of the measurement of probability weighting depends also (and mainly) on the reliability of the assessment of the standard sequence of outcomes (see §5.1). To summarize, we may experimentally elicit the weighting function in two successive steps: (i) construction of a standard sequence of outcomes (trade-off experiments: TO-experiments); (ii) determination of a corresponding standard sequence of probabilities (probability weighting experiments: PW-experiments). Preference conditions concerning the shape of the weighting function were formulated by Segal (1987), Tversky and Wakker (1995), Wu and Gonzalez (1996, 1998) and Prelec (1998). Tversky and Wakker (1995) define two properties of the weighting function that are rarely violated, at least in parametric estimation. These properties are called lower subadditivity (LS) and upper Figure 1 Illustration of Subadditivity subadditivity (US). LS implies that a ’’lower interval’’ [0, q] has more impact than a middle interval [p, p + q], and US means that an ’’upper’’ interval [1 − q, 1] has more impact than a middle interval [p, p + q], provided the middle interval is bounded away from the lower end point... |

25 | Contingent weighting in judgement and choice. - Sattah, Slovic - 1988 |

24 |
Generic utility theory: Measurement foundations and applications in multiattribute utility theory.
- Miyamoto
- 1988
(Show Context)
Citation Context ...e x0, x1, x2, i.e., the trade-off of x0 for x1 is considered as equivalent to the trade-off of x1 for x2, the trade-off of R for r being used as a ’’measuring rod.’’ This conclusion holds also under EU, i.e., w+(p) =p. A standard sequence x0, : : : , xn needs the construction of n indifferences (xi−1, p; R)∼ (xi, p; r), i =1, : : : , n. A similar process applied to the corresponding negative prospects allows us to obtain a standard sequence of losses. 3 Such a standard sequence is increasing for gains (i.e., 0¡x0¡ · · ·¡xn) and decreasing for losses (i.e., xn¡ · · ·¡x0¡0). 2 To our knowledge, Miyamoto (1988) was the first to suggest the idea of using preference conditions in non-EU contexts by restricting oneself to particular subsets of lotteries. 3 In this case, 0≥ r¿R¿x0¿x1. 1500 MANAGEMENT SCIENCE/Vol. 46, No. 11, November 2000 ABDELLAOUI Parameter-Free Elicitation of Utility and Probability Weighting Functions 3.2. Standard Sequences of Probabilities Suppose that x0, : : : , xn is a standard sequence of gains. Consider the probabilities pi, i =1, : : : , n−1, satisfying: (xn, pi; x0)v(xi, 1; x0) ≡ xi, (6) where 0¡x0¡xn. Under CPT, this indifference implies w+(pi) = u(xi) − u(x0) u(xn) − u(x0... |

17 |
Prospect versus utility.
- Currim, Sarin
- 1989
(Show Context)
Citation Context ...ted in this paper locates the main source of nonlinearity in the second half (of the interior) of the unit interval. It must be recognized, however, that a more efficient testing of linearity of w needs the assessment of more points (p, w−1(p)) than has been done in the present paper. The elicited weighting function for gains and the elicited weighting function for losses are systematically different. This confirms some early empirical findings showing that individuals distort probabilities differently when gains are transformed into losses in risky choice situations (e.g., Cohen et al. 1987, Currim and Sarin 1989, Abdellaoui 1995b). In fact, weighting functions for losses exhibit significantly more elevation than weighting functions for gains. The empirical results obtained in this paper also have important implications for RDEU and EU. The introduction of two weighting functions in CPT seems more appropriate than duality (assumed in RDEU) for taking into account the observed tendency of individuals to treat probabilities differently when 21 Bleichrodt and Pinto (1998) and Gonzalez and Wu (1999) obtained similar results. 1510 MANAGEMENT SCIENCE/Vol. 46, No. 11, November 2000 ABDELLAOUI Parameter-Free ... |

16 |
Nonlinear weighting of probabilities and violations of the betweenness axiom.
- Ho
- 1994
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Citation Context ...utilities, but also the transformation of decumulative probabilities to obtain decision weights through a probability weighting function. This innovation, however, has been perceived as a factor complicating utility measurement, and therefore, the elicitation of RDEU and CPT models. A variety of methods have been used to determine the shapes of the utility function and the probability weighting function under RDEU and CPT. The predominant approach prespecifies parametric forms for these functions and then estimates them through standard techniques (e.g., Tversky and Kahneman 1992, Camerer and Ho 1994, Hey and Orme 1994, Tversky and Fox 1995). However, assuming specific functional forms for the utility function and the probability weighting function makes inference about the shapes of these functions dependent on the choice of functional forms. Two research strategies can avoid the potential problems of parametric estimation. The first strategy consists of testing simple preference conditions to obtain information about the shape of either the utility function or the probability weighting function. Wu and Gonzalez (1996, 1998) follow this strategy. They find that aggregate behavior is cons... |

8 |
The effect on the preference reversal phenomenon of using choice indifferences.
- Bostic, Herrnstein, et al.
- 1990
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Citation Context ...e subjects were instructed to designate in each displayed pair of prospects which one they would prefer to be faced with. Figure 2 illustrates the generic display for the trade-off questions. In the experiments involving gains, as in those involving losses, standard sequences of six outcomes: x1, : : : , x6, were constructed. For gains and losses, the outcomes |x0|, |r|, and |R |were fixed at the following levels: FF 1,000 (U.S. $200), 0 and FF 500 (U.S. $100) respectively. Subjects were not asked to find switching outcomes, but they were asked to express outright choices between prospects. 6 Bostic et al. (1990) report experimental evidence that choice is ’’more consistent’’ than matching. Indeed, they find that the bisection method for elicitation greatly reduces choice versus pricing reversals, and that judged CEs and 6 Fennema and van Assen (1998) allowed their subjects to match switching outcomes. 1502 MANAGEMENT SCIENCE/Vol. 46, No. 11, November 2000 ABDELLAOUI Parameter-Free Elicitation of Utility and Probability Weighting Functions Table 2 Assessing x1 and p1 Through Bisection (For Gains) Outcomes (F.F.) Probabilities (×1%) Question Alternatives † x1 ∈ Choice Alternatives p1 ∈ Choice A = (x0, ... |

5 |
The risk-structure dependence effect: Experimenting with an eye to decision-aiding.
- Munier
- 1998
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Citation Context ...le 8 shows that the hypothesis of identity of weighting functions for gains and losses, called reflection, is not rejected near p =0 and p =1, and clearly rejected for moderate probabilities. Furthermore, one-tailed paired t tests show that the weighting function for losses is significantly ’’above’’ the weighting function for gains for moderate probabilities at the 0.025 level. In other words, w− exhibits more elevation than w+. Table 8 shows also that (w+)−1(p) and (w−)−1(p) are more correlated on the first half of the unit interval than on its second half. Camerer (1992) and Abdellaoui and Munier (1998) obtained results favorable to EU and ’’locally’’ favorable to EU respectively, inside the unit triangle. In fact, accepting EU inside the unit triangle implies a linear shape of the weighting function over a subset of the interval of middle probabilities. Nevertheless, Wu and Gonzalez (1996) report (parametric and nonparametric) results indicating that the weighting function presents significant curvature strictly within the boundaries of the unit interval. Their data-fitting MANAGEMENT SCIENCE/Vol. 46, No. 11, November 2000 1507 ABDELLAOUI Parameter-Free Elicitation of Utility and Probabilit... |

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Preponderance of the certainty effect over probability distortion in decision making
- Cohen, Jaffray
- 1988
(Show Context)
Citation Context ...een the probability weighting function for gains and the probability weighting function for losses. Moreover, the data suggest a descriptive superiority of CPT over RDEU. Finally, the hypothesis of linearity of the probability weighting function for moderate probabilities is investigated. This question has received rather contradictory answers in the experimental literature. Camerer (1992), Harless and Camerer (1994), and Abdellaoui and Munier (1998), for instance, obtained results through nonparametric techniques suggesting a linear weighting function for intermediate probabilities (see also Cohen and Jaffray 1988). On the contrary, Wu and Gonzalez (1996), among others, found support for nonlinearity. This paper confirms the latter (i.e., nonlinearity). The paper proceeds as follows. Section 2 reviews RDEU theory and CPT. Then, some previous experimental studies of parametric estimation of RDEU or CPT preference functionals are presented. Section 3 describes the two-step elicitation method. Section 4 describes an experimental elicitation of utility functions and weighting functions for gains and losses. The results of this experiment are given in §5. Finally, §6 1498 MANAGEMENT SCIENCE/Vol. 46, No. 11, ... |

2 | Experimental comparisons of individual behavior under risk and under uncertainty for gains and for losses. Organ. Behavior and Human Decision Process. - Said - 1987 |

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Estimating probability transformation functions under RDEU.
- Abdellaoui
- 1995
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Citation Context ...function for losses is the dual of the weighting function for gains, i.e., w−(p) =1 − w+(1 − p) for all p ∈ (0, 1). Up to now, most experimental studies used parametric specifications to infer the shapes of utility functions and weighting functions from individual choices. Tversky and Kahneman (1992) and Tversky and Fox (1995) gave parametric estimates for the median subject. In Camerer and Ho (1994) all subjects were assumed to have the same underlying preferences. Other studies estimate parametric forms at the level of individual subjects, e.g., Lattimore et al. (1992), Hey and Orme (1994), Abdellaoui (1995a). Table 1 gives some typical parametric specifications of u and w. The results of these experimental studies are mostly consistent with an inverse S-shaped weighting function, concave for small probabilities and convex for moderate and high probabilities. They are also consistent with a concave utility function for gains. MANAGEMENT SCIENCE/Vol. 46, No. 11, November 2000 1499 ABDELLAOUI Parameter-Free Elicitation of Utility and Probability Weighting Functions More recently, Wu and Gonzalez (1996) used a qualitative approach to analyze the shape of the probability weighting function. They pro... |

1 | 1511 ABDELLAOUI Parameter-Free Elicitation of Utility and Probability Weighting Functions Allais, - SCIENCEVol - 2000 |

1 |
Inconsistent trade-offs between attributes: New evidence in preference assessment biases.
- DelquiIe
- 1993
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Citation Context ... then asked to successively choose between the prospects A2i =(xi, 1) and B2i =(x6, 3=4; x0), i =1, : : : , 5, and also between the prospects A2∗ =(x3, 1) and B2∗ =(x4, 3=4; x2). Overall, each subject was confronted with six series of six choice questions. Note that with this sequencing of choice questions, the subject can hardly infer that she is in fact asked to match probability p. This special sequencing of choice questions was designed to avoid the problem of ’’framed probabilities’’ (Hershey and Schoemaker 1985) as well as possible biases due to scale compatibility (Tversky et al. 1988, DelquiIe 1993). Once the probabilities p1, : : : , p5, p′2 were assessed, the computer program displayed successively the pairs of prospects A 4i , B 4 i , i =1, : : : , 5, A 4 ∗ , B 4∗ , and asked the subject to designate the preferred prospect in each pair. The aim of this step was to check the subjects’ reliability in PW-experiments. 5. Results 5.1. Reliability We can test the reliability of subjects’ responses by seeing how often they expressed the same preference for the same prospect. As explained above, after each standard sequence assessment every subject was confronted, once again, with the pair of... |

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The utility analysis of choices involving risk. Political Econom.
- Friedman, Savage
- 1948
(Show Context)
Citation Context ...close to those obtained by Tversky and Kahneman (1992). 17 5.3. Probability Weighting Functions For Tversky and Kahneman (1992), the most distinctive characteristic of individual behavior under risk is what they called the fourfold pattern of risk attitude. It claims that individuals exhibit: (i) risk seeking for gains and risk aversion for losses of low probability; (ii) risk aversion for gains and risk seeking for losses of high probability. The authors remark that since the fourfold pattern arises over a wide range of outcomes, it cannot be explained by the utility of money as suggested by Friedman and Savage (1948) and Markowitz (1952). Instead, they infer that this fourfold pattern of risk attitude suggests an inverse S-shaped nonlinear transformation of the probability scale as well as an S-shaped utility function. Tversky and Kahneman (1992) recognize, however, that instead of clarity of the overall pattern, the individual data reveal both noise and individual differences. Probability weighting functions were elicited for 40 subjects both for gains and losses. The assessed values were inverse images (w)−1(p), rather than images w(p) for p =i=6, i =1, : : : , 5 (see Figure 4). Table 5 gives medians, m... |