The peak-and-end rule and differential equations with maxima: a view on the unpredictability of happiness

TL;DR

This paper models happiness dynamics using a differential equation incorporating peak and end effects, revealing that such models can exhibit chaotic behavior under certain conditions.

Contribution

It introduces a novel mathematical framework for happiness evolution based on differential equations with maxima, linking psychological theories to dynamical systems analysis.

Findings

Happiness dynamics can be described by a differential equation with maximum terms.

The model can produce chaotic behavior in happiness over time.

A one-dimensional map captures the system's complex dynamics.

Abstract

In the 1990s, after a series of experiments, the behavioral psychologist and economist Daniel Kahneman and his colleagues formulated the following Peak-End evaluation rule: "The remembered utility of pleasant or unpleasant episodes is accurately predicted by averaging the Peak (most intense value) of instant utility (or disutility) recorded during an episode and the instant utility recorded near the end of the experience", (D. Kahneman et al., 1997, QJE, p. 381). Hence, the simplest mathematical model for time evolution of the experienced utility function $u=u(t)$ can be given by the scalar differential equation $u'(t)=a u(t) + b \max \{u(s) : s\in [t-h,t]\}+f(t) \ (*),$ where $f$ represents exogenous stimuli, $h$ is the maximal duration of the experience, and $a, b \in \mathbb R$ are some averaging weights. In this work, we study equation $(*)$ and show that, for a range of parameters $a, b, h$ and a periodic sine-like term $f$, the dynamics of $(*)$ can be completely described in terms of an associated one-dimensional dynamical system generated by a piece-wise continuous map from a finite interval into itself. We illustrate our approach with two examples. In particular, we show that the hedonic utility $u(t)$ (`happiness') can exhibit chaotic behavior.