Experimental Design.

(a) Schematic of the operant conditioning chamber. Rats performed a self-paced four-option choice task using nose-poke apertures, each corresponding to a distinct option leading to distribution of reward (sucrose pellets) or punishment (time-out delay) at various size and probability. (b) Outcome structure for each option. Rewards and losses follow either convex (green) or concave (red) probability-magnitude functions. Outcomes are grouped into three probability domains: high-probability Normal Events (NE; 20 − 25%, blue), Rare Events (RE; 5%, yellow), and Rare and Extreme Events (REE; < 1%, pink). (c) Functional characterization of the four choice options based on exposure to rare and extreme gains (“Jackpots”) and rare and extreme losses (“Black Swans”). On the horizontal x-axis are reported (in decreasing order moving away from the origin) the ex-ante probabilities that are unknown to the subject, which only observes the outcome that is measured on the vertical y-axis. For convenience, gains appear in the upper-right quadrant while losses appear in the lower-left one. Convex curve are in green while concave are red. Robust: convex losses and concave gains (avoids both REE). Anti-fragile: convex losses and convex gains (avoids Black Swans while retaining access to Jackpots). Fragile: concave losses and concave gains (exposure to Black Swans, avoidance of Jackpots). Vulnerable: concave losses and convex gains (exposed to both Black Swans and Jackpots). (d) Behavioral sensitivity metrics. Total Sensitivity (y-axis) quantifies overall incorporation of REE by summing convex choices in gain and loss domains. One-sided Sensitivity (x-axis) measures asymmetry between REE seeking and REE avoidance. Example profiles illustrate: an Anti-fragile-like pattern (green), a Fragile-like pattern (red), and a mixed Anti-fragile/Robust profile with strong Black Swan avoidance and partial Jackpot seeking (blue). See Material and Methods for computational definitions.

Total sensitivity to Rare and Extreme Events (REE) (y-axis) plotted against one-sided sensitivity to REE (x-axis), where rightward movement indicates REE seeking and leftward movement indicates REE avoidance.

For a detailed description of the modeling and statistical analysis methods used, please refer to the Materials and Methods section. (a) the 820 black dots represent data from 41 sessions conducted with a total of 20 rats. (b) individual rat profiles, with each square corresponding to a single rat. The colored dots within each square denote the performance across the 41 sessions for each rat, categorized into two primary sensitivity profiles: blue dots indicate rats that exhibit strong sensitivity to REE and are classified as “Black Swan avoiders”, while green dots indicate rats that are responsive to REE but maintain a neutral stance, situated near the midline of the graph. Additionally, two outlier data points are highlighted in red and pink. (c) color-coded dots, each representing the average sensitivity measures across all 41 sessions per rat (n=20 rats).

Behavioral Differences Between Phenotypes in Avoidance and Seeking Behaviors.

This figure illustrates the median behavioral measures of Black Swan Avoidance and Jackpot Seeking across the two phenotypic groups, blue and green. The phenotypes are defined by distinct responses to Rare and Extreme Events (REE), with the blue phenotype exhibiting higher sensitivity to REEs and a tendency towards avoiding Black Swans, while the green phenotype shows less bias and remains closer to neutral behavior. Left Panel: This panel depicts the percentage of nosepokes where rats were exposed (black) or not exposed (light gray) to Black Swans. Increased Black Swan Avoidance is indicated by a higher proportion of non-exposure, suggesting a behavioral preference to avoid aversive outcomes among the blue phenotype compared to green. Right Panel: Conversely, this panel shows the percentage of nosepokes where rats were exposed (black) or not exposed (light gray) to Jackpots. Enhanced Jackpot Seeking is indicated by a higher proportion of exposure, suggesting greater motivation towards reward acquisition in the blue phenotype compared to green. Statistical comparisons between the phenotypes were conducted using Wilcoxon Rank Sum tests. Significant differences are denoted with ⋆ (p < 0.05) and ⋆⋆ (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.

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.

The last four columns present the percentage choice of each option.

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 2. 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.