Assessing risky weighting functions for positive and negative binary gambles using the logarithmic derivative function

This paper uses the tool of the logarithmic derivative function (LD) to ascertain the functional form of the risky weighting function for probabilistic outcomes. The LD function of a continuous function g(x) is defined as the ratio of the derivative of a function at x to the function itself. The LD is particularly sensitive to changes in the slope of a function thus making it an effective way of distinguishing functions with similar forms. The present study replicates earlier analyses of the LD candidates for the risky weighting function for positive binary gambles and extends the program of study to include negative binary gambles. Empirical estimates for LD values were elicited in an experiment in which participants matched gambles with positive outcomes or with negative outcomes. The risky weighting function for positive and negative gambles differed significantly only in the low range of probability values where p≤0.15. Several candidate models were shown to be incompatible with the observed LD pattern across both types of gambles. Other candidates had a functional form that was similar to the observed LD pattern, but systematically misfit the observed data in one or more regions of the curve. Of the models that predicted the right shape, only one-the Exponential Odds function ω(p)=exp(−s(1−p)bpa)-showed a random error pattern. The Exponential Odds function was also the only candidate function with best-fitting parameters that differed between positive and negative gambles and thereby picking up on an important difference between the two gamble types.