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Analytically Tractable Models for Decision Making under Present Bias
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Proposes an analytically tractable variant of the Kleinberg–Oren present-bias model for progress-accumulation tasks, deriving closed-form agent trajectories and optimal goal-setting and reward-scheduling results.
Problem
Present-bias models of time-inconsistent goal pursuit (e.g., the Kleinberg–Oren graph model) have no closed-form agent behavior and optimal interventions are NP-hard, limiting analysis of task abandonment and incentive design.
Approach
Restricts tasks to monotone accumulation of a real-valued progress over a fixed period with a power cost c(Δ)=Δ^α, under quasi-hyperbolic discounting; this yields an analytic state recursion, an explicit abandonment threshold β0, and tractable optimal goal and reward-scheduling solutions.
Key results
- Task abandonment occurs only if the present-bias parameter β is below a threshold β0 in (1/e, 1) that converges to (1-1/α)^(α-1) as T grows, with β0 − (1−1/α)^(α−1) = Θ(1/log T).
- For α=2 and β ≤ √2−1, offering the reward in T unit-length installments yields progress √(βRT), whereas for β ≥ 1/2 (non-TAI) a single lump-sum reward is optimal.
- Exploitative rewards (rewards never claimed) can raise final progress for agents with strong present bias, and their benefit grows with α, decreasing β, and T.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use frequent intermediate rewards instead of a lump-sum reward | Proposed model with strong present bias · Reward scheduling | final progress | Helped |
| Use a lump-sum reward instead of frequent intermediate rewards | Proposed model with weak present bias · Reward scheduling | final progress | Helped |
| Add an exploitative reward | Proposed model with strong present bias · Optimal goal setting | final progress | Helped |
| Lower the present-bias parameter below the task-abandonment threshold | Proposed model · Task completion | task abandonment | Helped |
Surprising findings
- Exploitative rewards that can never be claimed can still increase progress of agents with strong present bias.
- Frequent intermediate rewards are beneficial for strongly present-biased agents, while a lump-sum reward is optimal for weakly present-biased ones.
Limitations
- Restricted to monotone progress-accumulation tasks; progress cannot decrease.
- Assumes naive agents and a specific power cost function c(Δ)=Δ^α; excludes sunk-cost, sophistication and other biases.
Experimental setup
Metrics: final progress abandonment threshold β0
Baselines: Kleinberg–Oren model
Builds on: Kleinberg–Oren model quasi-hyperbolic discounting
Closest papers by meaning
- A Continuous-time Tractable Model for Present-biased Agents (AAAI 2025 · similarity 0.90)
- Present-Biased Optimization (AAAI 2021 · similarity 0.90)
- Delta Matters: An Analytically Tractable Model for beta–delta Discounting Agents (AAAI 2026 · similarity 0.89)
- Structural Approach to Guiding a Present-Biased Agent (AAAI 2026 · similarity 0.88)
- Expectation Alignment: Handling Reward Misspecification in the Presence of Expectation Mismatch (NeurIPS 2024 · similarity 0.82)
- The Hidden Cost of Waiting for Accurate Predictions (ICLR 2025 · similarity 0.82)
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