Comparison of probabilistic choice models in humans

From data through discount rates to the area under the curve

The rate of discounting future goods is a crucial factor in intertemporal trade-offs, upon which depends not only individual well-being but also that of our planet: How much privation now for a temperate future for our grandchildren? What is the best way to measure how the value of future goods decreases with its delay? The most accurate discount functions involve several covarying parameters, making interpretation equivocal. A universal and robust measure is the area under the discount curve, the AuC. The AuC of a hyperbolic discount function is a logarithmic function of the discount rate, k. The same integral also approximates the area under a hyperboloid function. A simple technique converts each datum into estimates of the discount rate, eliminating rogue data points in the process. These trimmed estimates are converted into areas and tested against data, where they succeed at predicting the AuC and its relation to log(k).

Variations on a theme by Rachlin: Probability discounting

Rachlin and colleagues laid the groundwork for treating the discounting of probabilistic goods as a variant of the discounting of delayed goods. This approach was seminal for a large body of subsequent research. The present paper finds the original development problematic: In converting probability to delay, the authors incorrectly dropped trial duration. The subsumption of probability by delay is also empirically questionable, as those are different functions of variables such as magnitude of outcome and commodity versus money. A variant of Rachlin's theme treats human discounting studies as psychophysical matching experiments, in which one compound stimulus is adjusted to equal another. It is assumed that a function of amount (its utility) is multiplied by a function of probability (its weight). Conjoint measurement establishes the nature of these functions, yielding a logarithmic transform on amount, and a Prelec function on probability. This model provides a good and parsimonious account of probability discounting in diverse data sets. Variant representations of the data are explored. By inserting the probabilistically discounted utility into the additive utility theory of delay discounting, a general theory of probabilistic intertemporal choice is achieved.

Sleep deprivation is associated with attenuated parametric valuation and control signals in the midbrain during value-based decision making

Sleep deprivation (SD) has detrimental effects on cognition, but the affected psychological processes and underlying neural mechanisms are still essentially unclear. Here we combined functional magnetic resonance imaging and computational modeling to examine how SD alters neural representation of specific choice variables (subjective value and decision conflict) during reward-related decision making. Twenty-two human subjects underwent two functional neuroimaging sessions in counterbalanced order, once during rested wakefulness and once after 24 h of SD. Behaviorally, SD attenuated conflict-dependent slowing of response times, which was reflected in an attenuated conflict-induced decrease in drift rates in the drift diffusion model. Furthermore, SD increased overall choice stochasticity during risky choice. Model-based functional neuroimaging revealed attenuated parametric subjective value signals in the midbrain, parietal cortex, and ventromedial prefrontal cortex after SD. Conflict-related midbrain signals showed a similar downregulation. Findings are discussed with respect to changes in dopaminergic signaling associated with the sleep-deprived state.

A hyperbolic decay of subjective probability of obtaining delayed rewards

Background

Hyperbolic discounting of delayed and probabilistic outcomes has drawn attention in psychopharmacology and neuroeconomics. Sozou's evolutionary theory proposed that hyperbolic delay discounting may be totally attributable to aversion to a decrease in subjective probability of obtaining delayed rewards (SP) which follows a hyperbolic decay function. However, to date, no empirical study examined the hypothesis, although this investigation is important for elucidating the roles of impatience, precaution, and uncertainty aversion in delay discounting processes.

Methods

In order to (i) determine the functional form of the relation between delay until receipt and SP, and (ii) examine whether delay discounting is attributable to a decrease in SP, we assessed the subjects' SP and their delay and probability discounting. We examined the fitness of hyperbolic and exponential functions to the assessed SP, and relations between the SP, and delay/probability discounting, and subjective-probability discounting for delayed rewards.

Results

The results demonstrated (a) SP decayed hyperbolically as delay increases, (b) a decay of SP was associated with delay discounting, and (c) subjective-probability discounting did not significantly correlate with delay discounting.

Conclusion

Our results demonstrated (i) hyperbolic decay of SP is related to delay discounting, and (ii) delay discounting is, however, not attributable to precautious foresight in intertemporal choice. Further, a novel parameter of pure time preference is proposed.