Hyperbolic discounting curves crossing
Also known as: Crossing of discount curves, Preference reversal over time
Two reward curves that cross over time produce a predictable last-minute change of heart.
What it means
This is the geometric heart of why hyperbolic discounting causes time-inconsistent choices: when the discounted-value curves of a smaller-sooner and a larger-later reward are plotted over time, the steep hyperbolic shape causes them to cross. Far in advance, the larger-later reward has the higher present value and is preferred; but as the smaller-sooner reward approaches, its value spikes upward and overtakes the larger one, flipping the preference. With exponential discounting the curves never cross, so preferences stay consistent — the crossing is the signature of non-constant discounting. Ainslie used this picture to explain impulsive reversals, relapse, and the value of pre-commitment, which works by removing the smaller-sooner option before its curve can spike. The crossing diagram is the canonical visual explanation of weakness of will.
Examples
Weeks before a party you plan to leave early to study, but as the night arrives the immediate fun's value spikes above the distant exam reward, and you stay.
Choosing Wednesday's lunch on Monday you pick the salad; by Wednesday noon the burger's value has spiked above it. Ordering the salad in advance stops the curves ever crossing.
A month out you are certain you will stop at one drink. At the bar, the second drink's immediate value overtakes tomorrow's clear head — which is why leaving the card at home beats resolve.
First described in George Ainslie (1975, 1992).