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A Continuous-time Tractable Model for Present-biased Agents
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Proposes a continuous-time variational model of progress-based tasks for present-biased agents, giving closed-form trajectories under exponential and hyperbolic discounting and deriving optimal goal-setting and reward-scheduling interventions.
Problem
Existing present-bias models of goal achievement are defined on discretized time with a step size and only support quasi-hyperbolic discounting, limiting tractability and applicability to exponential and hyperbolic discount functions.
Approach
Model agents as repeatedly choosing the minimum-perceived-cost future progress trajectory in continuous time via the variational principle; the cost is convex for power-law cost functions, so the Euler-Lagrange equation yields a global optimum and closed-form trajectories for exponential and hyperbolic discounting.
Key results
- Under exponential discounting, the optimal goal theta* = [((alpha-1)/k)(1-exp(-kT/(alpha-1)))]^((alpha-1)/alpha) R^(1/alpha), unchanged whether exploitative rewards are allowed.
- For fixed N reward divisions, equal period and reward splits are optimal under both discount functions, and the objective f(N) increases monotonically toward the limit T^((alpha-1)/alpha) R^(1/alpha) (or T^(1/2)R^(1/2) for hyperbolic, alpha=2).
- With hyperbolic discounting (alpha=2), the agent may abandon a task midway when T > (1+sqrt3)/k, indicating time-inconsistent behavior, while exponential agents never abandon once started.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Allow exploitative rewards in optimal goal setting | Continuous-time model with exponential discounting · Optimal goal setting | maximum final progress | No clear effect |
| Allow exploitative rewards in optimal goal setting | Continuous-time model with hyperbolic discounting (α=2) · Optimal goal setting | maximum final progress | No clear effect |
| Increase the number of reward-schedule divisions | Continuous-time model with exponential discounting · Optimal reward scheduling | total progress | Helped |
| Increase the number of reward-schedule divisions | Continuous-time model with hyperbolic discounting (α=2) · Optimal reward scheduling | total progress | Helped |
| Use equal-length periods and equal rewards across stages | Continuous-time model with exponential discounting · Optimal reward scheduling | total progress | Helped |
| Use equal-length periods and equal rewards across stages | Continuous-time model with hyperbolic discounting (α=2) · Optimal reward scheduling | total progress | Helped |
| Use exponential rather than hyperbolic discounting | Continuous-time present-biased agent · Task abandonment | time consistency | Helped |
| Use hyperbolic discounting rather than exponential discounting | Continuous-time present-biased agent · Task abandonment | time consistency | Hurt |
| Use sufficiently fine reward divisions | Continuous-time model with exponential or hyperbolic discounting · Optimal reward scheduling | final progress | Helped |
Surprising findings
- Exploitative rewards have no effect on the optimal goal under exponential or hyperbolic discounting, in contrast to quasi-hyperbolic discrete-time models.
- Unlike discrete quasi-hyperbolic models, finer reward division is always optimal in continuous time regardless of discount parameters.
Limitations
- Closed-form trajectory requires the zeta(t) non-increasing assumption; cases with repeated abandonment and resumption are left for future work.
- For hyperbolic discounting with alpha different from 2, the trajectory requires numerical integration.
- Purely theoretical analysis with no empirical validation against human behavior data.
Experimental setup
Baselines: quasi-hyperbolic discounting discrete-time models
Builds on: Kleinberg and Oren graph model Albers and Kraft progress-based model quasi-hyperbolic discounting
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