A mathematical model of delay discounting with bi-faceted impulsivity

Abstract

Existing mathematical models of delay discounting (e. g. exponential model, hyperbolic model, and those derived from nonextensive statistics) consider impulsivity as a single entity. However, the present article derives a novel mathematical model of delay discounting considering impulsivity as a multi-faceted quantity. It involves the bi-faceted characteristic of impulsivity, and considers impulsivity as a variable represented by two positive and fluctuating quantities (e.g. these facets may be trait and state impulsivity). To derive the model, the superstatistics method, which has been used to describe fluctuating physical systems like a thermal plasma, has been adapted. According to the standard practice in behavioural science, we first assume that the total impulsivity is a mere addition of the two facets. However, we also explore the possibility beyond an additive model and conclude that facets of impulsivity may also be combined in a non-additive way. We name this group of models the Extended Effective Exponential Model or $E^3M$. We find a good agreement between our model and experimental data.