Abstract
This paper presents a preference foundation for a two-parameter family of probability weighting functions. We provide a theoretical link between the well-established notions of probabilistic risk attitudes (i.e., optimism and pessimism) used in economics and the important independent measures for individual behavior used in the psychology literature (i.e., curvature and elevation). One of the parameters in our model measures curvature and represents the diminishing effect of optimism and pessimism when moving away from extreme probabilities 0 and 1. The other parameter measures elevation and represents the relative strength of optimism vs. pessimism. Our empirical analysis indicates that the new weighting function fits elicited probability weights well, and that it can explain differences in the treatment of probabilities for gains compared to that for probabilities of losses.
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Notes
Note that CRS weighting functions require continuity at 0 and 1, and hence, exclude the linear functions that are discontinuous at extreme probabilities.
We have also estimated parameters for the Goldstein and Einhorn (1987), w(p) = δp γ/(δp γ + (1 − p)γ), and the Prelec (1998) weighting function, \(w(p)=\exp (-\delta (-\ln p)\gamma )\). Although the interpretation of curvature and elevation in relation to these parameters is different and the relative sensitivity measures depend on both parameters and, in general, the probability where it is measured, we think that those parameters can be of general interest and we report the corresponding median values here with interquartile ranges in parentheses. For the Goldstein–Einhorn weighting functions we find δ + = 0.70 (0.53–1.00), δ − = 0.78 (0.59–1.09) and γ + = 0.65 (0.45–0.78), γ − = 0.51 (0.73–0.92). For the Prelec weighting functions we find δ + = 1.13 (0.91–1.31), δ − = 1.04 (0.88–1.25) and γ + = 0.64 (0.39–0.80), γ − = 0.72 (0.48–0.88).
The ρ-probability mixture of P with R is the prospect ρP + (1 − ρ)R = (ρp 1 + (1 − ρ)r 1,...,ρp n + (1 − ρ)r n ). Note that this definition is independent of whether probabilities are cumulative or decumulative. In the case of the cumulative probabilities notation we, obviously, have ρp n + (1 − ρ)r n = 1.
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Appendix: proofs
Appendix: proofs
Proof of Theorem 2
That statement (i) implies statement (ii) follows from the specific form of the representing functional. Jensen-continuity, weak order, and comonotonic independence as well as monotonicity follow immediate. Proportional invariance away from δ follows from substitution of the RDU-functional with a generalized CRS weighting function.
Next we prove that statement (ii) implies statement (i). Obviously statement (ii) in Lemma 1 is satisfied, hence, there exists an additively separable functional representing the preference \( \succcurlyeq \). We restrict the attention to the case that p 1 > 0 and p n − 1 < 1 to avoid the problem of dealing with unbounded V 1,V n − 1. To show that our additive functional in fact is a RDU form with a generalized CRS weighting function we use results presented Diecidue et al. (2009). If δ = 0 (or δ = 1), then proportional invariance comes down to Diecidue, et al.’s common ratio invariance for cumulative (or decumulative) probabilities, and we apply their Theorem 1 to obtain RDU with power weighting function, w(p) = p γ (or dual power weighting function, w(p) = 1 − (1 − p)γ).
Next assume that 0 < δ < 1. We apply the results of Diecidue et al. (2009) presented in their Theorem 3. First we observe that proportional invariance implies the common ratio invariance properties used by Diecidue, et al. We thus obtain, from the proof of their Theorem 3, that RDU holds with a weighting function of the form
with c,b,d > 0, and a = 1/δ c − b(1 − δ)d/δ c. Further, applying proportional invariance away from δ gives that c = d = :γ.
Uniqueness results follow from Theorem 3 of Diecidue et al. (2009). This completes the proof of Theorem 2. □
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Abdellaoui, M., L’Haridon, O. & Zank, H. Separating curvature and elevation: A parametric probability weighting function. J Risk Uncertain 41, 39–65 (2010). https://doi.org/10.1007/s11166-010-9097-6
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DOI: https://doi.org/10.1007/s11166-010-9097-6