How to Calculate Present Bias (Beta-Delta Discounting)
Present bias weights a payoff you can take today at 1 and a payoff t periods away at beta × delta^t, so the beta penalty lands once on everything that is not immediate.
The Present Bias formula
Calculator
Enter two payoffs plus beta and delta to see what each is worth today and whether the ranking flips from a distance.
In the first comparison this one is immediate, so it carries a weight of 1.
The reward that pays more but makes the person wait.
How far the larger payoff sits behind the smaller one.
The one-time penalty on anything not immediate. Set it to 1 to switch the bias off.
Ordinary patience, applied once for every period of delay.
For the second comparison, where neither payoff is immediate.
Against $100 available immediately, the delayed payoff is worth $90 today, so the smaller, sooner payoff wins.
- Weight on the delayed payoff
- 0.45
- Value of the sooner payoff from a distance
- $29.52
- Value of the later payoff from a distance
- $53.14
- Preference reversal
- Yes, the ranking flips
beta × delta^t comes to 0.45, so the delayed payoff counts for that share of its face value.
Moved out of the present, the smaller payoff is worth $29.52, because it now carries beta as well.
The larger payoff is worth $53.14 seen from the same distance, so the larger, later payoff wins. beta multiplies both sides here, which is why it cannot decide this comparison.
The person plans to wait while both payoffs are distant, then takes the smaller one once it is available now, which is the pattern commitment devices are built for.
How to calculate Present Bias, step by step
- 1Set beta and delta. beta is the one-time penalty on anything that is not immediate, and delta is the ordinary per-period discount factor. Both sit between 0 and 1.
- 2Give the immediate payoff its full weight. A payoff you can take right now is multiplied by 1, with no beta and no delta applied to it.
- 3Weight the delayed payoff. Raise delta to the number of periods of delay, multiply by beta, then multiply by the size of the payoff.
- 4Compare the two weighted values. The larger weighted value is the option the person takes while the sooner payoff is still available immediately.
- 5Push both options into the future and redo it. When neither payoff is immediate they both carry beta, so beta cancels and only delta separates them. A different winner is a preference reversal.
Worked example: Present Bias
Compare $100 today with $200 one period later, using beta = 0.5 and delta = 0.9. The delayed payoff carries a weight of 0.5 × 0.9 = 0.45, so it is worth 0.45 × 200 = $90 today, less than the $100 on the table right now, and the sooner payoff wins. Now push both options 5 periods into the future. The $100 is 5 periods away and worth 0.5 × 0.9^5 × 100 = $29.52, while the $200 is 6 periods away and worth 0.5 × 0.9^6 × 200 = $53.14. The larger, later payoff wins that comparison, and the flip between the two is the preference reversal present bias predicts.
Present Bias questions
What do beta and delta mean in the present bias formula?
delta is the standard per-period discount factor, applied once for each period of delay, and it works the same way for someone with no present bias at all. beta is an extra one-time penalty applied to any payoff that is not available right now, however far away it sits. Setting beta to 1 removes the bias and leaves ordinary exponential discounting.
Why does beta cancel when both options are in the future?
Both payoffs are delayed in that comparison, so each one gets multiplied by beta, and the shared factor drops out when you rank them. What is left is delta raised to the difference in delay, a constant trade-off that does not change with the calendar. That is why plans made about the future look patient and the same plans look impatient once one option becomes immediate.
How is present bias different from simple impatience?
An impatient person discounts steeply but consistently, so the choice they make in advance is the choice they carry out later. Present bias adds a penalty that applies only to what is not immediate, so the ranking flips as the sooner option arrives. That inconsistency is why present-biased people pay for commitment devices and merely impatient people do not.
What value of beta should you use?
beta is a number between 0 and 1 fitted to observed behavior rather than derived from theory, and experiments generally place it below 1 for most people. A beta close to 1 means almost no bias and almost no reversals, while a beta near 0.5 means an immediate payoff counts for at least twice what the same payoff would count for once it is delayed.
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