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Analytically Tractable Models for Decision Making under Present Bias

Yasunori Akagi, Naoki Marumo, Takeshi Kurashima

TL;DR

The paper tackles time-inconsistency arising from present bias by introducing an analytically tractable, progress-based framework that mirrors long-horizon tasks. It preserves the core quasi-hyperbolic discounting mechanism while enforcing convex, monotone costs over continuous progress, which yields closed-form trajectories and enables exact analysis of task abandonment, optimal goal setting, and reward scheduling. Key contributions include a precise abandonment threshold $β_0$, asymptotic characterizations as the horizon grows, and regime-dependent results for goal setting and reward scheduling that inform incentive design. These findings offer practical guidance for designing interventions and incentives tailored to individuals' present-bias strength in real-world tasks.

Abstract

Time-inconsistency is a characteristic of human behavior in which people plan for long-term benefits but take actions that differ from the plan due to conflicts with short-term benefits. Such time-inconsistent behavior is believed to be caused by present bias, a tendency to overestimate immediate rewards and underestimate future rewards. It is essential in behavioral economics to investigate the relationship between present bias and time-inconsistency. In this paper, we propose a model for analyzing agent behavior with present bias in tasks to make progress toward a goal over a specific period. Unlike previous models, the state sequence of the agent can be described analytically in our model. Based on this property, we analyze three crucial problems related to agents under present bias: task abandonment, optimal goal setting, and optimal reward scheduling. Extensive analysis reveals how present bias affects the condition under which task abandonment occurs and optimal intervention strategies. Our findings are meaningful for preventing task abandonment and intervening through incentives in the real world.

Analytically Tractable Models for Decision Making under Present Bias

TL;DR

The paper tackles time-inconsistency arising from present bias by introducing an analytically tractable, progress-based framework that mirrors long-horizon tasks. It preserves the core quasi-hyperbolic discounting mechanism while enforcing convex, monotone costs over continuous progress, which yields closed-form trajectories and enables exact analysis of task abandonment, optimal goal setting, and reward scheduling. Key contributions include a precise abandonment threshold , asymptotic characterizations as the horizon grows, and regime-dependent results for goal setting and reward scheduling that inform incentive design. These findings offer practical guidance for designing interventions and incentives tailored to individuals' present-bias strength in real-world tasks.

Abstract

Time-inconsistency is a characteristic of human behavior in which people plan for long-term benefits but take actions that differ from the plan due to conflicts with short-term benefits. Such time-inconsistent behavior is believed to be caused by present bias, a tendency to overestimate immediate rewards and underestimate future rewards. It is essential in behavioral economics to investigate the relationship between present bias and time-inconsistency. In this paper, we propose a model for analyzing agent behavior with present bias in tasks to make progress toward a goal over a specific period. Unlike previous models, the state sequence of the agent can be described analytically in our model. Based on this property, we analyze three crucial problems related to agents under present bias: task abandonment, optimal goal setting, and optimal reward scheduling. Extensive analysis reveals how present bias affects the condition under which task abandonment occurs and optimal intervention strategies. Our findings are meaningful for preventing task abandonment and intervening through incentives in the real world.
Paper Structure (25 sections, 10 theorems, 51 equations, 4 figures)

This paper contains 25 sections, 10 theorems, 51 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Proposed Model
  4. Formulation
  5. Analytical solution
  6. Discussion
  7. Task abandonment and present bias
  8. Condition for task abandonment
  9. Case (a): $\alpha (1 - \gamma) \geq 1$, or equivalently $\beta \leq (1 - \frac{1}{\alpha})^{\alpha - 1}$.
  10. Case (b): $\alpha (1 - \gamma) < 1 < \alpha (1 - \gamma^2)$, or equivalently $(1 - \frac{1}{\alpha})^{\alpha - 1} < \beta < (1 - \frac{1}{\alpha})^{\frac{\alpha - 1}{2}}$.
  11. Case (c): $\alpha (1 - \gamma^2) \leq 1$, or equivalently $\beta \geq (1 - \frac{1}{\alpha})^{\frac{\alpha - 1}{2}}$.
  12. Asymptotic formula for $\beta_0$
  13. Optimal Goal Setting
  14. Non-TAI $\beta$
  15. TAI $\beta$
  16. ...and 10 more sections

Key Result

Lemma 1

The following holds for $t=1, 2, \ldots, T$:

Figures (4)

  • Figure 1: Agents' state sequences $((t, x_t))_{t=1}^T$ for $R = \theta = 1$.
  • Figure 2: The plots of $u_t\ (t=\tilde{t}, \ldots, T)$ defined by \ref{['eq:def_ut']} for various $\alpha$, $\beta$, and $T$ when $R = 1$. We do not include cases where $\alpha = 10$ and $\beta = 0.5$, because $\beta_0 < 0.5$.
  • Figure 3: Examples of reward scheduling.
  • Figure 4: Optimal reward interval with $\alpha = 2$. The markers indicate the maximum and minimum lengths of the periods in the optimal solution $(T_i)_{i=1}^k$ computed with \ref{['eq:reward_scheduling_dp']}. The green line shows the nearly optimal reward interval \ref{['eq:nearly_optimal_reward_interval']} derived from the theoretical analysis.

Theorems & Definitions (22)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Definition 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 2
  • ...and 12 more