Rationally Inattentive Intertemporal Choice

The prevailing interpretation of temporal discounting, the tendency to undervalue future rewards, centers on an inherent preference for immediate gratification. However, accumulating evidence suggests an alternative perspective: this discounting might stem from the inherent uncertainty in our mental representations of future events. We mentally simulate future outcomes to assess their value, aiding in delayed gratification; however, these simulations are inherently noisy. Studies consistently show that interventions enhancing the precision of mental simulation, such as detailed imagining or episodic cues, reduce discounting. This suggests that even with neutral intrinsic time preferences, imprecise future prospection can diminish the perceived value of delayed rewards. This concept is formalized within a Bayesian model of discounting, where noisy simulations of future value are combined with prior beliefs using Bayes' rule. Noisier simulations of more distant future events lead to increased reliance on prior beliefs, resulting in hyperbolic discounting, consistent with observed behavior. However, this model assumes a fixed relationship between delay and simulation noise, a simplification that is addressed in this paper. We propose that simulation noise is not fixed but rather adaptively controlled based on the cost-benefit analysis of mental effort, particularly in relation to the magnitude of the reward. We hypothesize that the magnitude effect – increased patience with higher reward magnitudes – arises from this adaptive allocation of cognitive resources: larger rewards warrant more precise, effortful simulations.

The literature on temporal discounting extensively documents the phenomenon of hyperbolic discounting, where the devaluation of future rewards is disproportionately greater for shorter delays. Several studies demonstrate the magnitude effect, showing that discounting decreases with increasing reward magnitude. This finding challenges simple models of constant discounting. Existing models, such as the Gabaix and Laibson model, offer Bayesian explanations incorporating noise in value representations, but they assume fixed noise levels. The current work leverages rate-distortion theory, an information-theoretic framework for modeling optimal internal uncertainty, to provide a more comprehensive model addressing this limitation. This framework has been successfully applied to diverse cognitive domains, including perceptual judgments and working memory, and its application here promises a more nuanced understanding of the cognitive processes underlying intertemporal choice.

The authors build upon the Bayesian model of discounting, incorporating rational inattention to endogenize simulation noise. The model considers an agent choosing between rewards with varying delays. The agent receives a noisy signal of the future reward value through mental simulation. In the original Gabaix-Laibson model, the noise variance increases linearly with delay. The authors extend this by modeling the agent as a communication channel, with the precision of the internal representation subject to an information rate constraint. Using rate-distortion theory, they derive the optimal simulation noise variance as a function of reward magnitude and a sensitivity parameter (β). This parameter reflects the trade-off between the cognitive cost and the benefit of precise simulations. Larger rewards justify greater effort and lead to reduced noise. The model predicts both discounting and choice stochasticity magnitude effects. The authors test their model using three approaches: first, re-analyzing existing datasets from Ballard et al. (2017) and Chávez et al. (2017), investigating the effects of justification and reward magnitude on discounting and choice stochasticity. Second, they compare their rational inattention model (R2) against several alternative models (including quasi-hyperbolic and various hyperbolic discounting models) using Bayesian model selection to assess quantitative model fit and evaluate the qualitative predictions of their theory. Third, they conduct a new experiment manipulating reward variance to test the model’s predictions regarding how variance interacts with reward magnitude in influencing choice stochasticity and discounting. In this experiment, participants made choices between a smaller, immediate reward and a larger, delayed reward, with the larger rewards drawn from Gaussian distributions with different variances.

The re-analysis of the Ballard et al. (2017) dataset supported several key predictions of the rational inattention model. The average discount factor was higher in the no-justification condition, and the justification effect diminished with increasing magnitude. Furthermore, choice variability was higher for smaller magnitudes and lower in the justification condition, with the justification effect diminishing with magnitude. The re-analysis of the Chávez et al. (2017) dataset provided quantitative support for the rational inattention model, demonstrating superior model fit compared to several alternative models of hyperbolic discounting. The model revealed significant magnitude scaling effects for both discounting and choice stochasticity, and a negative correlation between these effects, consistent with theoretical predictions. The new experiment manipulating reward variance also confirmed the model's predictions. The choice stochasticity magnitude effect was significantly lower in the high-variance condition, while there was no effect of variance on the discounting magnitude effect. This pattern strongly supports the rational inattention account, showing how internal noise adaptively modulates the relationship between reward magnitude, cognitive effort, and intertemporal choice.

The findings demonstrate that temporal discounting can be partially explained by a rationally inattentive process of mental simulation of future outcomes. The model successfully accounts for the magnitude effect, showing that increased patience with larger rewards stems from a cost-benefit analysis of the cognitive resources allocated to simulating the future. The model accurately predicts the relationship between reward magnitude, choice stochasticity, and reward variance, connecting the stochasticity of choices to the magnitude effect. This suggests that the seemingly inconsistent behavior in intertemporal choice is a rational response to the limits of cognitive processing capacity. The model also connects the cognitive control mechanisms to the decision-making processes that lead to the magnitude effect. This provides a high-level perspective that complements more mechanistic analyses of cognition and discounting.

This research offers a novel perspective on temporal discounting, integrating Bayesian inference with rational inattention to explain the magnitude effect. The model's success in accounting for behavioral patterns and its consistency with neuroscientific evidence suggest that limitations in mental simulation capacity, rather than simply intrinsic preferences, significantly contribute to discounting. Future research could explore how other factors such as anticipation and loss aversion might be incorporated into the model to provide a more comprehensive understanding of intertemporal decision-making.

The study relies primarily on hypothetical choices, although the authors address this limitation by citing research suggesting that discount rates are highly correlated between hypothetical and real reward scenarios. Future research using real rewards would further strengthen the conclusions. The model assumes a specific form for the relationship between information rate and reward magnitude, which could be further investigated and refined. The generalizability of the model to diverse populations and contexts also warrants future exploration.