Delta Matters: An Analytically Tractable Model for $β$-$δ$ Discounting Agents

Humans exhibit time-inconsistent behavior, in which planned actions diverge from executed actions. Understanding time inconsistency and designing appropriate interventions is a key research challenge in computer science and behavioral economics. Previous work focuses on progress-based tasks and derives a closed-form description of agent behavior, from which they obtain optimal intervention strategies. They model time-inconsistency using the $\beta$-$\delta$ discounting (quasi-hyperbolic discounting), but the analysis is limited to the case $\delta = 1$. In this paper, we relax that constraint and show that a closed-form description of agent behavior remains possible for the general case $0 < \delta \le 1$. Based on this result, we derive the conditions under which agents abandon tasks and develop efficient methods for computing optimal interventions. Our analysis reveals that agent behavior and optimal interventions depend critically on the value of $\delta$, suggesting that fixing $\delta = 1$ in many prior studies may unduly simplify real-world decision-making processes.