Hyperbolic discounting

The quasi-hyperbolic model is the one that gets fitted to data. Alongside an ordinary discount factor δ, a single parameter β marks down everything that is not happening now, so a user who prefers the larger reward when both are in the future flips to the smaller one once it becomes available immediately. Imai, Rutter and Camerer (2021) pooled 220 estimates of β from 31 studies: about 77% sit below one, the average is between 0.95 and 0.97 and significantly below one, and modest selective reporting towards over-reporting present bias pulls corrected estimates to between 0.90–0.93 and 0.99. The effect is real and it is small. There is no rate to borrow either — Frederick, Loewenstein and O'Donoghue (2002) found implicit annual discount rates from −6 percent to infinity with no sign of convergence, so anyone handing you a number for how steeply your users discount is quoting one study out of dozens that disagree.

The split inside that average is the part that decides a build. Studies that paid in money returned β of 0.9723 (p = .08) and 0.9758 (p = .13), not distinguishable from no present bias at all; studies that made people allocate tedious real work returned 0.8815 (p = .0001). The explanation is arbitrage: money can be moved between dates, so a monetary choice measures the user's bank account as much as their patience. Augenblick, Niederle and Sprenger (2015) isolated the difference by running 102 students through a seven-week experiment with a companion money study: aggregate β was 0.988 for money against 0.900 for effort, 56% of subjects were classified present-biased over work against 33% over money, and in the 75 people who did both the two measures correlated at ρ = −0.05. Present bias is about the work you are asking for, not the price you are charging, and impatience in one domain predicts nothing about the other.

“Now” is a much shorter window than a product roadmap assumes. In Imai and colleagues' meta-regression, studies that handed over the current-period reward by the end of the session found more present bias than studies that delivered it later the same day — a few hours of delay moves the estimate. Woolley and Fishbach (2018) held reward size constant across five studies and varied only timing: a more immediate bonus raised intrinsic motivation for the activity itself, independent of magnitude and running through how closely the activity and the reward were associated. Their earlier work (2016) found immediate rewards predicted persistence in health and academic goals better than the delayed rewards people gave as their reason for starting. That is lab and classroom work with self-report measures, so treat it as a direction: the payoff has to land inside the session to count as a payoff.

The hyperbolic shape itself is disputed, and one of the challenges is a free interface change. Read (2001) found strong subadditive discounting — a delay is discounted more when it is split into subintervals — with no evidence of the declining impatience that hyperbolic discounting exists to explain. Read, Frederick, Orsel and Rahman (2005) showed across four experiments that outcomes are discounted less when a time is given as a calendar date (“on October 17th”) than as a delay (“in 6 months”), and that the hyperbolic shape appears only under delay wording. Bröder and Krauß (2025) located the mechanism in perception: people judge a period described as a date to be shorter than the same period described as a delay. This is hypothetical choice by small student samples, but it is the cheapest change the literature offers and it points the same way every time.

The field evidence everyone cites for present bias in a paid product is about gyms and subscriptions, and it is partly about something else. DellaVigna and Malmendier (2006) matched contract choice to daily attendance for 7,752 health club members: those on a flat monthly fee above $70 attended 4.3 times a month, paying over $17 per expected visit where a ten-visit pass cost $10 each and forgoing $600 over the membership. An average of 2.31 months elapsed between a member's last visit and their cancellation, worth $187 in fees, and the lag ran to at least four months for 20% of them. Einav, Klopack and Mahoney (2025) took the same question to card data covering ten subscription services: cancellations spike in the months when a card is replaced and renewal therefore has to be active, a friction they calculate raises seller revenue by between 14% and more than 200%, roughly doubling it. They model inattention and switching costs, not β — so the money a subscription earns from delay may be plain inattention rather than present bias. The interface fix is the same; the story you tell about it should not be.

The self-control tooling built on this principle has a thin record. The best-known result for imposed deadlines, Ariely and Wertenbroch's 2002 proofreading study, was retracted by Psychological Science on 2 September 2026, and Hyndman and Bisin's (2026) replication of its Study 2 found that changing deadlines had a negligible effect on all three performance metrics and on the survey measures, every F-statistic far below the 5% critical value of 3.07 across 124 participants, while the within-subject relationships you would expect if the task worked were present — so the null is not an artifact of a broken study. Burger, Charness and Lynham (2011) found evenly spaced interim deadlines lowered completion rates, and Bisin and Hyndman (2020) found deadlines did not increase completion and that students set them badly. Commitment contracts do better, for the minority who take one: in Royer, Stehr and Sydnor's (2015) four-week gym incentive with 1,000 employees, weekly attendance among existing members rose from about 60% to 80%, the incentive-only arm was back at control by month 3, and the arm also offered a self-funded contract held roughly 10 percentage points above control in months 2 and 3, converging only at month 11 — on 12% take-up. Augenblick's subjects would not pay either: 59% took commitment when it was free, 91% chose flexibility once it cost $0.25.

Who the pattern hits hardest is the reason to be careful with it. Amlung and colleagues' (2017) meta-analysis found steeper delay discounting robustly associated with addiction severity and with quantity-frequency of use across alcohol, tobacco, cannabis, stimulants, opiates and gambling, at r = 0.14, consistent across substance and assessment type with modest publication bias. The association is small, but it runs one way: the users most moved by “have it now, pay later” are the ones for whom later goes worst. Regulators have reached the same place — since 15 July 2026 the FCA regulates UK buy-now-pay-later as deferred payment credit, requiring authorisation, proportionate affordability checks before lending and compliance with the Consumer Duty.