An alternative approach to quadratic scoring rules using continuous-valued logic

Abstract

AbstractFollowing tfhe seminal paper of Offerman et al. (2009), in this study, adaptations of constructions of continuous-valued logic to prospect theory are presented. Here, we demonstrate that the so-called kappa function and its special cases are viable alternatives to some elements of quadratic scoring rule prospects theory. After, we present the tau-eta scoring rule prospect and show that it may be treated as a generalization of the quadratic scoring rule prospect. Furthermore, we prove that if this new prospect for an uncertain event is evaluated using specific kappa functions as utility functions, then (1) the weighting measure of the event is a function of the optimal value of its reported probability, (2) the inverse of the latter function, and (3) the (risk-) corrected reported probability of the event, also as a function of the optimal value of its reported probability, all have a common formula. The parameters of the common formula are unambiguously determined by four tuning parameters. Lastly, we show that with our approach, by fitting one of the abovementioned functions to corresponding empirical data, we can immediately obtain the other two functions as well.