Temporal Discounting Under Uncertainty: How People Discount Delayed Risky Outcomes

Abstract

How do people discount the value of delayed outcomes when those outcomes are uncertain? Four studies (N₁=142, N₂=128, N₃=165, N₄=119) employed choice tasks with monetary gambles to examine whether temporal discounting rates increase when outcome probability is reduced. Participants chose between immediate certain amounts and delayed probabilistic amounts. Linear mixed-effects models showed a significant interaction between delay and probability (β=0.18, SE=0.04, p<.001, Cohen's d=0.81), indicating that uncertainty compounds temporal discounting. This effect persisted after controlling for individual differences in risk attitude and time preference.

Introduction

The discounted utility model (Samuelson, 1937) predicts that the subjective value of a delayed outcome decays exponentially with time. Numerous studies have documented hyperbolic discounting—where value decreases more sharply at shorter delays—across money, health, and life domains (Laibson, 1997). However, most laboratory studies have assumed that delayed outcomes are certain. In naturalistic decision-making, delays are rarely risk-free: a promised future gain may not materialize, or its magnitude may be subject to volatility (Rachlin, Raineri, & Cross, 1991). How uncertainty and delay interact to shape preferences remains underexplored.

A few studies suggest that people discount more steeply under joint uncertainty and delay (e.g., Keren & Roelofsma, 1995), but findings are mixed and sample sizes have been modest (N<100 per condition). Recent developments in delay discounting methodology—including parametric curve fitting (Mazur, 1987) and hierarchical Bayesian approaches—enable more precise quantification of discounting rates. This study applies these tools to examine whether uncertainty amplifies temporal discounting and whether this amplification varies by individual characteristics.

Method

Participants

We recruited 554 participants (Study 1: N=142, M_age=32.5 years, 54% female; Study 2: N=128, M_age=33.1 years, 57% female; Study 3: N=165, M_age=31.9 years, 59% female; Study 4: N=119, M_age=34.7 years, 52% female) via Amazon Mechanical Turk. Inclusion criteria were age 18–70 and fluent English. Participants received USD 3.00 base compensation plus performance-contingent bonuses (up to USD 10.00, paid out weekly via Amazon gift cards).

Procedure

Participants completed a series of choice tasks modelled on the Kirby (2000) protocol. Each trial presented a choice between an immediate certain amount (e.g., USD 54 today) and a delayed uncertain amount (e.g., USD 80 in 1 month, with 50% probability of receipt). We systematically varied four factors: (1) delay (1 week to 1 year), (2) probability (50%, 75%, 90%, 100%), (3) outcome magnitude (USD 20 to USD 100), and (4) delay × probability interactions. Across the four studies, we manipulated the relative weighting of these factors to probe their interaction. Trials were randomized within-subjects. We fit Mazur's (1987) hyperbolic model V=A/(1+kD) to individual responses, separately for each probability condition, and extracted discount rates (k) for analysis.

Results

Across all four studies, discount rates increased significantly when outcome probability was reduced below 100%. In Study 1, mean k-values were lowest for certain outcomes (k̄=0.0041, SD=0.0028) and increased monotonically with decreasing probability: k̄=0.0068 (75% probability), k̄=0.0089 (50% probability), representing a 117% increase from certainty to 50%-probability conditions (paired t₁₄₁=4.23, p<.001). Study 2 and 3 replicated this pattern with comparable magnitudes. Random-intercept mixed-effects analysis across all four studies, with participants and items as random factors, revealed a significant Probability × Delay interaction (F₃,₄₅₀=11.84, p<.001, partial η²=.07). The fixed effect of probability on discount rate was β=0.18 (SE=0.04, p<.001), equivalent to Cohen's d=0.81 under standardized contrasts.

We tested whether individual differences in risk preference modulated this interaction. Participants completed a separate visual analogue scale assessing general risk tolerance. Risk-seeking individuals showed marginally larger probability effects (β_risk=0.06, p=.07), but this did not eliminate the main effect. When we split participants by median risk score and re-ran analyses, both groups showed significant uncertainty amplification of discounting.

Discussion

These results provide robust evidence that uncertainty and delay interact to reduce the subjective value of future outcomes beyond what either temporal discounting or probability weighting alone would predict. The four-study replication with large samples and systematic variation of parameters strengthens confidence in the finding. The effect size (Cohen's d≈0.80) is substantial, suggesting that practitioners designing incentive structures or benefit schedules must account for how uncertainty multiplies temporal devaluation.

The mechanistic explanation remains speculative. One possibility involves dual-system cognition: uncertain delayed outcomes may evoke greater reliance on automatic, affect-based heuristics (Kahneman & Tversky, 1979), resulting in steeper discounting. Alternatively, cognitive load imposed by joint uncertainty and delay assessment might trigger simplified decision rules. Neuroimaging studies examining activation in the ventromedial prefrontal cortex during decisions involving joint uncertainty and delay could clarify these mechanisms. Additionally, longitudinal work examining real-world savings and insurance decisions would ground these laboratory findings in ecologically valid contexts.

References

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