Rationally Inattentive Intertemporal Choice

The paper asks whether temporal discounting can emerge without an intrinsic preference for immediacy, instead arising from internal uncertainty in mentally simulating future rewards. Building on a Bayesian account where noisy prospective value signals are integrated with priors (leading to as-if hyperbolic discounting), the authors propose that the precision of mental simulation is not fixed but adaptively controlled due to information-processing costs (rational inattention). The central hypothesis is that people invest more cognitive effort to simulate larger rewards, reducing simulation noise and thus decreasing apparent discounting (the magnitude effect), and that this adaptive precision control also links reward magnitude to choice stochasticity. This framework aims to explain regularities in intertemporal choice and connect patience, reward magnitude, and mental effort.

Prior work shows that enhancing the vividness or precision of future simulations reduces discounting: asking participants to imagine spending future rewards, to imagine outcomes in greater detail, or providing episodic tags increases patience. A Bayesian model (Gabaix & Laibson) formalizes discounting as optimal inference from noisy future value simulations that become noisier with delay, producing hyperbolic discounting. Psychological evidence indicates that distant future events are imagined with fewer details, that higher reward cues increase vividness and thought generation, and that prompting justification selectively increases patience for smaller magnitudes, consistent with greater cognitive control already deployed at higher magnitudes. Classic studies document the magnitude effect (greater patience for larger rewards) and its robustness. Related economic theories of rational inattention and rate-distortion have been applied to cognition and decision-making, offering normative principles for allocating limited information-processing resources. Neuroscientific findings implicate prefrontal control networks in intertemporal choice and modulation by reward and effort, with evidence that disrupting dorsolateral prefrontal cortex reduces the magnitude effect. Dopamine has been linked to patience and reward sensitivity, suggesting a neurochemical basis for sensitivity to effort-reward trade-offs.

Modeling: The baseline Bayesian model assumes a true reward value u ~ N(μ, σ_u^2) with μ=0 and a noisy mental simulation s ~ N(u, σ_t^2). Bayesian updating yields a posterior mean proportional to s with discount factor D = 1/(1 + k t), where k = σ_u^2 / σ_t^2. Thus, discounting arises from greater reliance on priors when simulation noise grows with delay. Rational inattention extension: Using rate-distortion theory, the agent selects simulation noise to trade off estimation error (distortion) and information rate (effort). A family of channels parameterized by simulation noise variance is optimized under a rate constraint that increases with reward magnitude and decreases with delay. Under assumptions detailed in Methods, the optimal simulation noise scales with reward magnitude and delay via σ^2 ∝ β r_t / ℓ_r, leading to an optimal discount parameter k = 1/(β r_t). Hence, larger rewards reduce k (less discounting), capturing the magnitude effect. The model also links simulation noise to choice stochasticity, predicting less stochasticity for larger magnitudes and specific interactions with reward variance. Choice modeling: Choices are modeled as arising from noisy simulations; marginalized choice probabilities are expressed via a probit with an inverse-temperature term α that depends on the discount factor and simulation noise. When the sooner option is immediate, α simplifies to a function of the discount factor; decreasing simulation noise with magnitude increases α (less stochasticity). Comparative statics derive predictions: (i) discounting magnitude effect increases as β decreases; (ii) choice stochasticity decreases with magnitude; (iii) the choice-stochasticity magnitude effect weakens with higher reward variance; (iv) increasing sensitivity β reduces the discounting magnitude effect but increases the stochasticity magnitude effect. Empirical analyses:

• Re-analysis 1 (Ballard et al. Study 3; n=1382): Hypothetical choices between immediate vs 1-month delayed rewards with between-subject magnitude manipulation ($20–$2000) and a justification vs no-justification manipulation. Predictions about mean discounting and variability across magnitude and justification were tested via regressions with bootstrapped 95% CIs.
• Re-analysis 2 (Chávez et al.; n=1284): Intertemporal choice questionnaire (27 SS vs LL items; SS immediate). Random-effects Bayesian model selection compared rational inattention models (R1, R2) and hyperbolic/quasi-hyperbolic baselines (H0–H3), using BIC-based evidence and protected exceedance probability (PXP). Parameters capturing magnitude effects on discounting and inverse temperature were estimated.
• New experiment (n=221, MTurk): Titration task with 40 blocks of 6 hypothetical SS ($1 always) vs LL choices; LL rewards from a truncated Normal (mean $5, rounded cents) with low (SD≈1) vs high variance (SD≈5) between-subjects, delays 1–6 months. Tested predictions that reward variance attenuates the magnitude effect on choice stochasticity but not on discounting. Same model set as Chávez re-analysis was fit (maximum likelihood), with PXP for model comparison and parameter contrasts by variance condition.

• Ballard et al. re-analysis (n=1382): Five predictions supported. Mean discount factor k larger in no-justification than justification (CI [0.064, 0.155]); justification effect diminishes with magnitude (interaction CI [-0.036, -0.015]). Variability predictions: standard deviation of k higher at small magnitudes (magnitude main-effect CI [-0.016, -0.008]); lower in justification condition (justification main-effect CI [0.203, 0.501]); and the justification-induced reduction in variability diminishes with magnitude (interaction CI [-0.109, -0.045]).
• Chávez et al. re-analysis (n=1284): The full rational inattention model R2 was decisively favored (PXP > 0.99). Among hyperbolic variants, H3 was best and showed the predicted qualitative patterns: magnitude scaling of inverse temperature m_α > 0 (t(1283)=7.47, p<0.0001) indicating reduced choice stochasticity with larger rewards; magnitude scaling of discounting m_k < 0 (t(1283)=15.42, p<0.0001) indicating reduced myopia with larger rewards. The two magnitude effects were negatively correlated (r = -0.21, p<0.0001). Psychometric functions were better captured by R2 than standard hyperbolic H0, despite R2 having fewer parameters.
• New variance experiment (n=221): Replicated magnitude effects: discounting magnitude effect m_k < 0 (t(220)=7.16, p<0.0001) and choice-stochasticity magnitude effect m_α > 0 (t(219)=9.95, p<0.0001) overall. Critically, the choice-stochasticity magnitude effect was smaller in the high-variance condition (t(219)=2.22, p<0.05), while reward variance had no effect on the discounting magnitude effect (p=0.84). A Bayesian t-test yielded posterior probability >0.99 for the null that variance does not modulate the discounting magnitude effect.

Findings support the hypothesis that temporal discounting can emerge from internal uncertainty in simulated future values, with agents adaptively controlling simulation precision according to information-processing costs and benefits. This rational inattention framework explains the magnitude effect as increased cognitive effort invested for larger rewards, which reduces simulation noise and apparent myopia. It also predicts, and data confirm, a direct link between reward magnitude and reduced choice stochasticity, with this link attenuated by higher reward variance. The results align with psychological and neuroscientific evidence that cognitive control and attention are deployed when stakes are higher, and that prefrontal networks and dopamine modulate effort allocation. The negative correlation between magnitude effects on discounting and stochasticity matches model predictions about sensitivity (β) shifting these effects in opposite directions. Overall, the study provides a normative, information-theoretic account complementing mechanistic models of intertemporal choice and clarifies how patience, reward, and mental effort are intertwined.

The paper advances a unified account of intertemporal choice in which discounting arises from Bayesian inference over noisy future-value simulations whose precision is optimally controlled under rate-distortion (rational inattention) constraints. This yields testable predictions: larger magnitudes reduce discounting and choice stochasticity; justification (increasing cognitive effort) reduces both mean discounting and variability; and reward variance selectively attenuates the magnitude effect on stochasticity but not on discounting. Re-analyses of two large datasets and a new experiment support these predictions, and the rational inattention model outperforms standard discounting models in explaining choices. Future work could test incentive-compatible designs at long delays, extend the framework to losses and anticipatory utility (savoring/dread), and probe neurocomputational mechanisms—e.g., how dopamine modulates sensitivity β and the reward–information rate mapping.

• Hypothetical rewards: Many choices were not incentive-compatible due to impractical long delays; although prior work suggests similar patterns for real vs hypothetical rewards, generalizability may be limited.
• Loss domain and anticipatory utility: The current Bayesian discounting account does not capture reversed magnitude effects for losses, savoring, or dread; extensions are needed.
• Assumptions on priors and noise: The model assumes Gaussian priors centered near zero and specific forms of simulation noise and rate constraints; misspecification could affect inferences.
• Indirect measurement: Mental simulation signals are unobserved; using objective rewards as proxies may introduce interpretational errors.
• Between-subject design in some datasets (e.g., Ballard): Limits within-subject quantitative confirmation of parameter relationships.
• External risk not modeled: Results focus on internal uncertainty; interactions with objective uncertainty were not addressed.