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

Białaszek, Wojciech; Marcowski, Przemysław; Cox, David J.

doi:10.1371/journal.pone.0233337

Cited by 7 publications

(12 citation statements)

References 71 publications

(105 reference statements)

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“…Section Behavioral Models and Model Parameters), the obtained values are difficult to compare. However, the results are in line with the few studies who have considered the modified hyperboloid model and have shown its superiority [( 49 , 61 – 63 , 100 ); but see also ( 44 )], and which show that the hyperbolic model is not a comparably good fit ( 61 – 63 , 100 ).…”

Section: Discussionsupporting

confidence: 88%

“…Note though that this interpretation should be evaluated with caution since we observed a moderate correlation between s and κ in the reward condition. While a previous study evaluating the modified hyperboloid model did not find differences in the scaling parameter s between discounted rewards and losses (49), while others did not explicitly compare the parameters between tasks (38,61,62), the obtained results are in line with the frequent observation of lower discounting rates during loss discounting, also termed "sign-effect" [see also (41,111); for an overview see (112)]. This sign-effect was also reflected in the lower frequency of discounted choices for the loss condition as compared to the reward condition observed for all runs and experiments despite explicitly prolonging delays for this condition (see Supplementary Results S2.1-S2.3).…”

Section: Discussionmentioning

confidence: 93%

“…Section Behavioral Models and Model Parameters) and assessed their ability to predict the behavior in the opposing run, that is, inferring parameters on run A and predicting behavior in run B and vice versa. The two experimental runs thus allowed us to assess an estimate of the out-of-sample PE which is less biased and preferred over in-sample estimates ( 70 , 88 ), commonly used in the field [e.g., ( 38 , 61 , 89 )]. Figure 3F shows the model comparison results averaged over predictions in both runs.…”

Section: Resultsmentioning

confidence: 99%

“…As most studies in the field do not report out-of-sample prediction errors ( 9 , 21 , 38 , 41 , 44 , 49 , 61 , 63 , 72 , 74 , 99 – 101 ), or report predicted log-likelihood ( 40 ), or predicted accuracies ( 54 ), which may be far above the predicted probabilities reported here, and since the prediction error depends on trial difficulty (i.e., on how close trials are to the indifference point and therefore on the precise experimental manipulation, cf. Section Behavioral Models and Model Parameters), the obtained values are difficult to compare.…”

Section: Discussionmentioning

confidence: 99%

“…Another caveat of the model-based approaches is that they almost exclusively rely on the hyperbolic discounting model (with one exception (53)), and thus depend on the implicit assumption of this model being 'true' , or at least on it accounting for a substantial proportion of intra-individual variability. However, studies primarily focusing on comparing the goodnessof-fit of different discounting functions have suggested that this may not be the case [ (53); see also (38,49,(61)(62)(63)], such that multiple alternatives to the hyperbolic model have been proposed [ (53); see also (5,27,(64)(65)(66)(67)(68)(69)]. Adaptive design procedures may therefore benefit from taking into account model comparison results.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2022

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BackgroundThe 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.MethodThe 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.ResultsThe 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.ConclusionThe 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.

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Section: Discussionsupporting

confidence: 88%

Section: Discussionmentioning

confidence: 93%

Section: Resultsmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

et al. 2022

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Variations on a theme by Rachlin: Probability discounting

Killeen

2022

J Exper Analysis Behavior

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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.

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