Delay discounting

Why would someone take ten pounds today over eleven tomorrow, yet wait a year and a day for the eleven?

Take the two choices as arithmetic before calling anyone impatient. Ten today or eleven tomorrow is a ten per cent gain for one day's wait. Ten in a year, or eleven in a year and a day, is the same gain for the same wait. The only difference is how far away both options sit.

Most people take the ten in the first question and wait for the eleven in the second. That is not impatience — impatience would be consistent and take the sooner option both times. It is stranger: a preference between two fixed things that depends on where the person stands. What valuation could do that?

Discounting the future at all is sensible: a promise a year out carries risk, and money now can be used now. The question is never whether future value is shrunk but by what rule.

Try the rule an accountant would use: shrink value by a constant proportion per unit of time. Exponential discounting. Adding a year in front of both options multiplies both by the same factor, since the extra year is common to both — and the same positive factor cannot change which number is larger. So an exponential discounter who prefers ten today must prefer ten in a year too. Whatever people are doing, they are not doing that.

What shape produces the flip? Something steep near the present and flat further out. Then the one-day gap between today and tomorrow straddles the steep part and costs the delayed option a lot, while the identical gap a year out sits on the flat part and costs almost nothing. The family usually fitted is hyperbolic — value divided by one plus a rate times the delay — which has that shape and fits human and animal choice data considerably better than exponential.

Make it concrete. Suppose value is divided by (1 + kD), with D the delay in days and an illustrative k of 0.15 — chosen to fit this example, not an estimate of anything. Standing today, eleven-tomorrow is worth 11 / 1.15 = 9.57, less than ten, so you take the ten. Now stand thirty days before both, the sooner at 30 days and the later at 31: the sooner is worth 10 / 5.5 = 1.82, the later 11 / 5.65 = 1.95. The order has flipped with nothing changed but the vantage point.

That is the point. The reversal is not an error laid over a consistent system; it is what the arithmetic of that curve does, and two selves who disagree are both computing correctly from where they stand.

Now be sceptical: a fitting curve is not a mechanism. What else could produce the reversal?

Uncertainty could. A reward at one day is nearly certain and one at 366 days is not — but if you are already accepting a year of risk, one more day adds very little. Someone applying a probability rather than a time discount reverses in the same direction with no special near-term impatience, and since delay and risk are confounded in almost every real version of the question, this rival is hard to rule out.

Framing could too, and this constrains the strong reading most. Ask the same questions with calendar dates rather than delays, and the steepness drops and reversals shrink markedly. A stable feature of a person's valuation should not care how the interval was worded; that it does suggests part of what is measured is how a delay is represented. I am recalling that result rather than deriving it, and its size varies across studies, though the direction has held up.

A third crack: if a single rate k characterised a person it should not depend on what is discounted. It does — larger amounts are discounted less steeply, and one individual yields different rates for money, food and cigarettes.

The why-do-we-procrastinate piece in this collection takes this curve as an input and moves on to what makes tasks aversive. My fixed point is upstream: the shape of the curve itself, and how much weight it will bear.

Not much, in one respect. Hyperbolic discounting is an excellent description of choice data and a poor candidate for a mechanism: it says people behave as though value were divided by one plus a rate times delay, not what does the dividing. The quasi-hyperbolic alternative — one extra penalty on everything not immediate, then ordinary exponential discounting beyond — fits nearly as well with a different story, that now is categorically special rather than the curve smoothly bending. Choice data alone struggles to separate them, which should temper how mechanistically anyone reads either.

The trait reading needs care too. Steep discounting correlates reasonably robustly with addiction, obesity and gambling, but the causal direction is not established — heavy use plausibly steepens discounting as much as the reverse, and the rate moves with mood, framing and recent consumption. I quote no effect sizes on purpose: published values range widely with sample, commodity and procedure, and one number would suggest a precision the literature does not have.

What survives is the reversal itself. It is easy to demonstrate, appears in animals as well as people, and rules out exponential discounting decisively — which matters, because exponential is what most reasoning about the future quietly assumes. So a plan is not a preference. It is a preference recorded at a distance, and the self who arrives stands somewhere else and will genuinely disagree — which is why commitment devices work where resolutions do not.

• Whether the quasi-hyperbolic account and the smooth curve can be separated by anything but choice data.
• Why animals discount far more steeply over seconds than people do.

Every term the collection defines is gathered in the glossary.