Experimental design - panel (a), schematic representation of the rat behavioral operant chamber equipped with four holes, each associated with a specific exposure to gains and losses (Anti-fragile, Robust, Vulnerable and Fragile); panel (b), values and frequencies for convex (green)/concave (red) gains/losses, with the largest probabilities highlighted in blue for Normal Events (20 to 25%, NE domain), Rare Events in yellow (5%, RE domain) and Rare and Extreme Events in pink (ε-probability smaller than 1%, REE domain); panel (c), representation and label of the four possible options, with increasing gains/losses (up/down) against decreasing probabilities (right/left): “Robust” has convex losses (avoidance of negative REE called Black Swan) and concave gains (avoidance of positive REE called Jackpot), “Anti-fragile” has convex losses and gains (Black Swan avoidance and Jackpot seeking), “Fragile” has concave losses and gains (Black Swan seeking and Jackpot avoidance) and “Vulnerable” has concave losses (Black Swan seeking) and convex gains (Jackpot seeking); panel (d), representation of two measures of Sensitivity to REE, i.e. “Total Sensitivity” (y-axis) and “One-sided Sensitivity” (x-axis) and examples, where green dot represents sensitivities associated with an hypothetical near-Anti-fragile profile (close to full Jackpot seeking and complete Black Swan avoidance), red dot represents sensitivities associated with an hypothetical near-Fragile profile (close to full Black Swan seeking and Jackpot avoidance), and blue dot represents sensitivities associated with an hypothetical choice mixing mostly Anti-fragile and Robust (near complete Black Swan avoidance and partial Jackpot seeking) - see the Material and Methods section on Modelling and Statistical Analysis for details

Total Sensitivity to REE (y-axis) against One-sided Sensitivity to REE (x-axis; i.e. REE seeking towards the right and avoiding towards the left) - see Material and Methods section on Modelling and Statistical Analysis - panel (a) black-colored points depict 41 sessions run by 20 rats all together; panel (b) in each square representing a rat, colored points depict 41 sessions for each of the 20 rats; the color code indicates the 2 main categories of profiles observed (blue: the most sensitive to REE and black swan avoiders; green: sensitive to REE but neither biased towards seeking nor avoidance (close to midline on the graph); 2 outliers were identified and color-coded in red and pink; panel (c) color-coded points depict averages over the 41 sessions for each of the 20 rats such points. Most points lie on the right half of the square, that is, on the REE Avoidance side. In addition, within the right half most points are located in the upper quarter, that is, above the Robust-Vulnerable vertex, suggesting moderate to high Total Sensitivity as well as Black Swan Avoidance. In the Material and Methods section on Experimental Method Details, we discuss the contribution of the training sessions to such patterns. These two patterns are better seen if we focus on each individual rat.

Median behavioral measures of Black Swan Avoidance and Jackpot Seeking by phenotype, defined as median shares of nosepokes exposed to REE (in percentage) for blue and green phenotypes; phenotypes are indicated in column with a corresponding colored symbol at the center - left panel depicts the percentage of nosepokes exposed to Black Swans in black and not exposed to Black Swans in light gray - Black Swan Avoidance increases with the share of non exposure to Black Swans; right panel depicts the percentage of nosepokes exposed to Jackpots in black and not exposed to Jackpots in light gray - Jackpot Seeking increases with the share of exposure to Jackpots; blue and green phenotype comparisons are carried out by Wilcoxon Rank Sum tests (⋆ means p < 0.05 and ⋆⋆ means p < 0.01)

Responses of Total Sensitivity to REE and of One-sided Sensitivity to REE, averaged over the 41 final sessions, following Jackpots in panel (a) and Black Swans in panel (b), for each of the 20 rats. Each dot indicates one rat and dots are color coded to indicate the profile of the animal, as in Figure 2. Black dotted lines materialize no change in sensitivities after exposure to the REE compared to before. Black solid lines represent spline estimates - see Material and Methods section on Modelling and Statistical Analysis

Selected augmented Q-Learning models by phenotype

Total Sensitivity to REE (y-axis) against One-sided Sensitivity to REE (x-axis) averages for each rat over the 41 sessions; top panel replicates the right panel in Figure 2: observed sensitivities; bottom left panel: predicted sensitivities derived from selected models; bottom right panel: simulated sensitivities computed from runs of selected models over artificial 41 sessions

Parameters of the selected Q-Learning model, estimated for each rat over the 41 sessions - left panel: decision weights for averages of gains and losses and for REE; right panel: learning and forgetting rates for gains and losses; while individual parameter for pink and red rats are represented as dots, parameters for blue and green rats are represented by box plots with 10th and 90th percentiles, interquartile ranges and median; between phenotype comparisons carried out by Wilcoxon Rank Sum tests (⋆ means p < 0.05) - see Material and Methods subsection Augmented Q-Learning Model Estimation and Simulation for details

Time-line of experiments

Statistical moments and decomposition of Jensen gaps for convex and concave exposures

Before/after differences for Total Sentivity to REE and for One-sided Sensitivity to REE, averaged over the 41 sessions, for each of the 20 rats - see Figure 4. We highlight negative mean differences in red and positive mean differences in yellow.

Variation of choices observed in final experimental sessions compared to that observed in training sessions (step 3 in Material and Methods section on Experimental Method Details); panel (a) shows variations of Total Sensitivity to REE (y-axis) and of One-sided Sensitivity to REE (x-axis), that is, dark color-coded dots replicate averages over the 40 sessions for each of the 20 rats - see panel (c) in Figure 2 - and are paired with averages over training sessions for the same rat, depicted with light color-coded dots connected through lines with corresponding dark color-coded dots; panel (b) shows total variation distance, that is, the proportion of total nose pokes in training sessions that need to be changed to replicate nosepokes in final sessions

Position of REE in the sequence

Sequences and types

Succession of events for each sequence

Behavioral measures for each rat - Total and One-sided Sensitivities to REE - TSREE and OSREE in short; Black Swan Avoidance and Jackpot Seeking in percentage - BSA and JPS in short

BIC values for all augmented Q-Learning models, with selected value highlighted

AIC values for all augmented Q-Learning models, with selected value highlighted

Parameter estimates by rat for all selected Q-Learning models

Mean and median parameter estimates for blue and green phenotypes, and difference tests with significant results highlighted

BIC values for all augmented Q-Learning model with outcome range-adaptation, with selected value highlighted

AIC values for all augmented Q-Learning model with outcome range-adaptation, with selected value highlighted

Ex-post total outcomes for each of the 20 rats over the final 41 sessions - summary statistics. Each row represents one rat and the color codes indicate the profile of the animal, as in Figure 2

Convexity Premiums for each of the 20 rats (row) - see Appendix for details. Each row represents a rat and the color code indicates the profile of the animal, as in Figure Towards the right, the dot materializes how many pellets were obtained relative to the number that would have been obtained if a convex menu had been chosen at each trial. Towards the left, the coloured dot indicates the total seconds of penalty obtained relative to what would have been obtained if a concave menu had been chosen at each trial.