Bayesian model comparison confirmed that delay-choices were best predicted by a hyperbolic model, with the largest reward devaluations occurring at shorter delays.

There is a high degree of correlation between findings from cross-validation and Bayesian model comparison (see for example [98]).

However, the Bayesian model evidence penalizes models in proportion to how far the posterior is from the prior (as quantified by the KL-divergence).

This property renders the Bayesian model evidence a better model comparison criterion than AIC or BIC [95].

By using a robust Bayesian modeling approach (Experiment 1), and by directly measuring participants’ indifference points for different effort levels (Experiment 2), our results instead indicate that effort discounting is best characterized by a sigmoidal two-parameter model that allows initially concave discounting shapes.

Comparison between the resulting fits was conducted using Bayesian model comparison (see Materials and Methods).

Importantly, in either task version, the hyperbolic model outperformed the sigmoidal model ( Bayesian model comparison for delay task involving words: xp 2 0.98; mp = 0.75), consistent with a large literature on delay-based choices.

Our Bayesian Model Comparison accounts for model complexity by using the Kullback-Leibler divergence between prior and posterior densities over parameters.

AB, Bayesian Model Comparison comparing all four (A) or five (B) behavioral models: in both comparisons, the hyperbolic model provides the best explanation for choices on the delay task.

These values determine the results of the Bayesian model comparison shown in Fig.