Using logarithmic derivative functions for assessing the risky weighting function for binary gambles

• Generalization of hazard and reverse hazard functions.
• Assessing existing proposal for the risky weighting function.
• Strong experimental support for the exponential class of weighting functions.

A logarithmic derivative (LD) of a continuous function g(x)g(x) is itself a function in the form of g′(x)g(x). Hazard and reverse hazard are examples of LD functions that have proven to be useful for discriminating among similar functions for stochastic systems, and the essential idea of LD functions can be used more generally. In this research, an analysis of the logarithmic derivative was employed to evaluate the various proposals for the risky weighting function w(p)w(p) that have been advanced in the psychological and economic literature. Risky weighting functions are the weighting coefficients of the outcome utility values, i.e., if an outcome has an associated probability pp, then w(p)w(p) is the transform of pp that weights the utility of the outcome. An experiment was done to obtain empirical estimates of the logarithmic derivative of the risky weighting function for individuals by utilizing a novel gamble-matching paradigm with binary gambles. Five models from the research literature did not predict the observed shape for the LD function. Four additional models for the risky weighting function could predict the general profile of the LD function but nonetheless resulted in a nonrandom, systematic pattern for the corresponding model fit residuals. The nonrandom pattern of the fit residuals is taken as evidence against the models. Consequently nine models had problems in accounting for the empirical LD function. However, two risky weighting functions provided an accurate description of the empirical LD function. These risky weighting functions are the Prelec function w(p)=e−s(−lnp)a, with aa and ss as fitting parameters, and a novel model, the Exponential Odds function w(p)=e−s(1−p)bpa with aa, bb and ss as fitting parameters.