A mathematical model of delay discounting with bi-faceted impulsivity

TL;DR

Delay discounting has traditionally been modeled with a single impulsivity parameter. This work treats impulsivity as two fluctuating facets and derives a bi-faceted extension, the Extended Effective Exponential Model (E^3M), using a superstatistics framework that yields both additive and nonadditive combinations of the facets. The nonadditive formulation with $k_{\text{R}}=k_{\text{T}}/k_{\text{S}}$ leads to a closed-form discount law involving the confluent hypergeometric function $\mathcal{U}$ and gamma factors, while empirical analysis on eight datasets shows excellent agreement and sensible parameter constraints such as $q_{\text{S}}<2$ and $\langle k_{\text{R}}\rangle<1$. Overall, the approach offers a principled bridge between behavioural science and statistical physics, with potential to generalize to more facets and to other decision-making contexts.

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.

Figures (13)

• Figure 1: Fitting observed data obtained from Ref. mansouridata using the EM and EEM. EM: $\kappa_0 = 0.0044 \pm 0.0015$. EEM: $q=9.067 \pm 0.116,~\kappa_1=0.048 \pm 0.013$.
• Figure 2: Variation $\epsilon_{\text{eff}}^{\text{ext}}$ with $D$ for different $q_{\text{S}}$ values and for a fixed $q_{\text{T}}=3.01$.
• Figure 3: Variation $\epsilon_{\text{eff}}^{\text{ext}}$ with $D$ for different $q_{\text{T}}$ values and for a fixed $q_{\text{S}}=1.01$.
• Figure 4: Fitting longitudinal data obtained from Ref. anokhin for monozygotic twins at age 16 using the E$^3$M: $q_{\text{S}}=1.113 \pm 0.040,~q_{\text{T}}= 6.463\pm 0.816$.
• Figure 5: Fitting longitudinal data obtained from Ref. anokhin for monozygotic twins at age 18 using the E$^3$M: $q_{\text{S}}= 1.134\pm 0.038,~q_{\text{T}}= 7.137\pm 0.705$.