Hyperbolic Discounting — Temporal Preference

Hyperbolic k: 0.10

Exponential δ (daily): 0.95

Reward A (now): $10

Reward B (delayed): $20

Delay B (days): 10

Shift both (days): 0

Hyperbolic discounting: V(t) = A / (1 + k·t) — discount factor decreases rapidly for near delays, slowly for distant ones.
Exponential discounting: V(t) = A · δ^t — consistent preferences, no reversal (classical economics).
Preference reversal: Hyperbolic agents choose SS now over LL later, but if both rewards are shifted into the future equally, they switch to LL (larger, later).
This explains procrastination, addiction, time-inconsistent behavior. β-δ model: V = δ^t for t>0, β·δ^t for present bias (β < 1).