Auditory gambling task. In each trial auditory instructions ask participants to place a bet whether the second card will be lower or higher than the first one. Five seconds after the response the first card is drawn, 5 s later the second card and further 5 s later participants have to indicate whether they have lost or won. All instructions are given auditorily through speakers, while participants maintain fixation. All details for computation of reward, risk, and risk prediction error as well as the numerical values for all possible combinations of cards are given in the Section

First card. (A) Pupil dilation between first and second card relative to the time of drawing the first card split by level of uncertainty after first card. Pupil dilates more if the outcome is sure (low uncertainty, light gray, card was 1 or 10) than for high uncertainty trials (black, cards 4,5,6,7), while medium uncertainty trials (dark gray, cards 2,3,8,9) fall in between. Thick lines denote means over participants, thin lines SEM for high and low uncertainty trials; shaded area denotes time when high uncertainty significantly differs from low uncertainty at an expected FDR of 5% (p < FDR_0.05 = 0.042). (B) Significance of difference between high and low uncertainty trials as given in (A). Results of point-wise t-tests for equality of means; negative logarithmic scale implies values to the top to be more significant (lower p). Horizontal line denotes expected FDR of 5% (FDR_0.05 = 0.042), times of significant differences fall above. (C) Model: Probability of winning (gray) and risk (black) after the first card is drawn as function of the first card. Expected reward is linear in the probability of winning. Units of $ (reward) and $² (risk) omitted. Note that probability of winning depends on the bet, but risk does not. To pool data over both bets for the analysis of the first card, we exploit symmetry: in case of the bet “second card higher” the number representing the card is replaced by its mirror (1 → 10, 2 → 9,…,10 → 1) and all bets are treated as “second card lower.” Mathematically we denote the actual card as “c” and the bet-corrected card as “c*” [see ]. (D) Points: Pupil dilation [as in (A)] at time of peak significance between high and low uncertainty sorted by card (c*, adjusted for bet); mean and SEM over subjects. The parabola-shape resulting from the quadratic dependence of risk on c* [cf. (C)] is evident. Line: fit of a model including risk, probability of winning, and a constant offset, coefficients u, v, and w, respectively. (E) Evolution of fit parameters [as in (D)] over time. Quickly after the first card, the effect of risk (u) rises, while the effect of reward (and probability of winning, v) shows little systematic change over time. The contribution of the constant (w) reflects the general time course of pupil dilation over the trial, which happens irrespective of the card’s value and thus independent of any decision variables. (F) Correlation of pupil dilation to risk (black) or probability (gray). Top: correlation coefficient, bottom: probability of correlation being different from 0. Horizontal line denotes 5% expected FDR for risk (FDR_0.05 = 0.045).

Individual uncertainty. Pupil dilation split according to high uncertainty (black) and low uncertainty (gray) after first card for each of the 12 individuals, mean and SEM over trials. All observers show the same qualitative behavior as the average effect shown in Figure A.

Second card. (A) Pupil dilation after the second card depending on whether surprise is high (gray) or low (black) after the second card. To include all data, high surprise here is defined to occur when expected reward prior to the second card was positive and the actual outcome is a loss (P₁ > 0 and P₂ = −$1) or when the expected reward was negative and the actual outcome is a win (P₁ < 0 and P₂ = +$1); conversely, if expected reward had been positive and outcome is a win (P₁ > 0 and P₂ = $1) or expected reward had been negative and outcome is a loss (P₁ < 0 and P₂ = −$1), surprise is low. Otherwise notation as in Figure A. (B) Significance of difference between high and low surprise according to (A). Notation as in Figure B. (C) Model: Risk prediction error (RE₂) as measure of surprise after the draw of the second card as function of first card (c*) and actual outcome (P₂: win or loss). Note that there are no data for (c* = 1, win) and for (c* = 10, loss), since c* = 1 implies p_win,1 = 0 and thus certain loss (P₂ = −$1) as well as c* = 10 implies p_win,1 = 1 and thus certain win (P₂ = +$1), (D) Points: Pupil dilation at peak significance of (A,B) (t = 1.12 s after second card) split by first card (c*) and outcome (P₂: win or loss); mean and SEM over participants. Lines: fits of risk prediction error (RE₂) according to (C). (E) Correlation of risk prediction error (RE₂) as quantitative measure of surprise and pupil dilation at time point of peak significance. (F) Time course of correlation in (E) for the period after the second card. Top: correlation coefficient, bottom: probability of correlation being different from 0. Horizontal line: alpha-level for expected FDR of 5% (FDR_0.05 = 0.042).

Individual surprise. Pupil dilation split according to high surprise (gray) and low surprise (black) after second card for each of the 12 individuals, mean and SEM over trials. All observers show the same qualitative behavior as the average effect shown in Figure A.

Alternative measures of surprise. Timecourse of correlation between other measures of surprise (left: squared reward prediction error; right: absolute reward prediction error a.k.a. salience) and pupil dilation. Notation as in top panel of Figure F.