Comparison of multiplicative and additive hyperbolic and hyperboloid discounting models in delayed lotteries involving gains and losses

Model-based experimental manipulation of probabilistic behavior in interpretable behavioral latent variable models

Introduction

Interpretable latent variable models that probabilistically link behavioral observations to an underlying latent process have increasingly been used to draw inferences on cognition from observed behavior. The latent process usually connects experimental variables to cognitive computation. While such models provide important insights into the latent processes generating behavior, one important aspect has often been overlooked. They may also be used to generate precise and falsifiable behavioral predictions as a function of the modeled experimental variables. In doing so, they pinpoint how experimental conditions must be designed to elicit desired behavior and generate adaptive experiments.

Methods

These ideas are exemplified on the process of delay discounting (DD). After inferring DD models from behavior on a typical DD task, the models are leveraged to generate a second adaptive DD task. Experimental trials in this task are designed to elicit 9 graded behavioral discounting probabilities across participants. Models are then validated and contrasted to competing models in the field by assessing the ouf-of-sample prediction error.

Results

The proposed framework induces discounting probabilities on nine levels. In contrast to several alternative models, the applied model exhibits high validity as indicated by a comparably low prediction error. We also report evidence for inter-individual differences with respect to the most suitable models underlying behavior. Finally, we outline how to adapt the proposed method to the investigation of other cognitive processes including reinforcement learning.

Discussion

Inducing graded behavioral frequencies with the proposed framework may help to highly resolve the underlying cognitive construct and associated neuronal substrates.

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.

Everything Comes at a Price: Considerations in Modeling Effort-Based Choice

When observing human behavior, one of the key factors determining choice is effort. It is often assumed that people prefer an easier course of action when the alternative yields the same benefits. However, recent research demonstrates that this is not always the case: effort is not always costly and can also add value. A promising avenue to study effort-based choice is to utilize formal decision models that enable quantitative modeling. In this paper, we aim to present an overview of the current approaches to modeling effort-based choice and discuss some considerations that stem from theoretical and practical issues (present and previous) in studies on the role of effort, focusing on the connections and discrepancies between formal models and the findings from the body of empirical research. Considering that effort can, in some circumstances, act as a cost and as a benefit, reconciling these discrepancies is a practical and theoretical challenge that can ultimately lead to better predictions and increased model validity. Our review identifies and discusses these discrepancies providing direction for future empirical research.

A Model Guided Approach to Evoke Homogeneous Behavior During Temporal Reward and Loss Discounting

Background

The tendency to devaluate future options as a function of time, known as delay discounting, is associated with various factors such as psychiatric illness and personality. Under identical experimental conditions, individuals may therefore strongly differ in the degree to which they discount future options. In delay discounting tasks, this inter-individual variability inevitably results in an unequal number of discounted trials per subject, generating difficulties in linking delay discounting to psychophysiological and neural correlates. Many studies have therefore focused on assessing delay discounting adaptively. Here, we extend these approaches by developing an adaptive paradigm which aims at inducing more comparable and homogeneous discounting frequencies across participants on a dimensional scale.

Method

The proposed approach probabilistically links a (common) discounting function to behavior to obtain a probabilistic model, and then exploits the model to obtain a formal condition which defines how to construe experimental trials so as to induce any desired discounting probability. We first infer subject-level models on behavior on a non-adaptive delay discounting task and then use these models to generate adaptive trials designed to evoke graded relative discounting frequencies of 0.3, 0.5, and 0.7 in each participant. We further compare and evaluate common models in the field through out-of-sample prediction error estimates, to iteratively improve the trial-generating model and paradigm.

Results

The developed paradigm successfully increases discounting behavior during both reward and loss discounting. Moreover, it evokes graded relative choice frequencies in line with model-based expectations (i.e., 0.3, 0.5, and 0.7) suggesting that we can successfully homogenize behavior. Our model comparison analyses indicate that hyperboloid models are superior in predicting unseen discounting behavior to more conventional hyperbolic and exponential models. We report out-of-sample error estimates as well as commonalities and differences between reward and loss discounting, demonstrating for instance lower discounting rates, as well as differences in delay perception in loss discounting.

Conclusion

The present work proposes a model-based framework to evoke graded responses linked to cognitive function at a single subject level. Such a framework may be used in the future to measure cognitive functions on a dimensional rather than dichotomous scale.