Towards a general class of parametric probability weighting functions
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Towards a general class of parametric probability weighting functions József Dombi1
· Tamás Jónás2
© The Author(s) 2020
Abstract In this study, we present a novel methodology that can be used to generate parametric probability weighting functions, which play an important role in behavioral economics, by making use of the Dombi modifier operator of continuous-valued logic. Namely, we will show that the modifier operator satisfies the requirements for a probability weighting function. Next, we will demonstrate that the application of the modifier operator can be treated as a general approach to create parametric probability weighting functions including the most important ones such as the Prelec and the Ostaszewski, Green and Myerson (Lattimore, Baker and Witte) probability weighting function families. Also, we will show that the asymptotic probability weighting function induced by the inverse of the so-called epsilon function is none other than the Prelec probability weighting function. Furthermore, we will prove that, by using the modifier operator, other probability weighting functions can be generated from the dual generator functions and from transformed generator functions. Finally, we will show how the modifier operator can be used to generate strictly convex (or concave) probability weighting functions and introduce a method for fitting a generated probability weighting function to empirical data. Keywords Probability weighting functions · Modifier operator · Continuous-valued logic · Prospect theory
1 Introduction It is common knowledge that the probability weighting functions play an important role in non-expected utility theories, including prospect theory and rank-dependent models. Therefore, there has been a consistent interest in them (see, e.g. Abdellaoui et al. 2008; Kahneman and Tversky 2013; Lattimore et al. 1992; Ostaszewski et al. 1998; Prelec 1998; Tversky and Kahneman 1992; Wakker 2010; Wakker and Yang 2019). These functions describe the phenomenon that people tend to overreact those events that occur with a low probability and underreact to events that have a high probability (see, e.g. Bleichrodt 2001; Camerer 2007; Chateauneuf et al. 2007; Köbberling and Wakker 2003; Communicated by A. Di Nola.
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Koszegi and Rabin 2006; Loomes et al. 2002). Thus, in line with the empirical estimates, the probability weighting functions are regressive (first they have values greater than the identity function, then they have values less than the diagonal), inverse S-shaped (first concave, then convex) and asymmetric (intersecting the diagonal at one third) (see, e.g. Levy 1992; Li et al. 2017; Offerman et al. 2009; Prelec 1998). In this study, we present a novel methodology that can be used to generate parametric probability weighting functions by making use of the Dombi modifier operator of continuousvalued logic (Dombi 2012a). This operator is defined as follows. (λ)
Definition 1 The modifier operator m ν,ν0 : [0, 1] → [0, 1] is given by
Tamás Jónás [email protected] József Dombi dombi@i
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