Fig. 1: Task and conditions sketch.

a Gamified dual-decision paradigm. On each trial, participants viewed and made right/left decisions about two consecutive dot motion stimuli (lead and target), which they were told represented flocks of sheep. We explicitly informed participants that if they were correct about the first decision, then the target stimulus would be going to the right, and if incorrect, then it would be going to the left. This meant that, in an optimal observer, the prior for a rightward target stimulus should be equal to the lead decision certainty. They also rated their confidence in the target decision. In Experiment 2, the paradigm was the same except there was a 2-s delay after viewing the target stimulus, before participants were allowed to make the target decision. b Conditions. We manipulated the coherence of the lead and target stimuli (each of which could have L: low, M: medium, or H: high coherence), here depicted with the circle transparency, to create two conditions that had matched available posterior information but differed in whether the lead or target stimulus was stronger. This was tested at three overall posterior levels - low posterior information (L + M), medium posterior information (L + H), and high posterior information (M + H). c Sketch of the Stronger-Lead vs Stronger-Target manipulation. The posterior percept (orange) should optimally be the precision-weighted integration of the prior (yellow), which in our task was always rightward and could span from 50–100%, and likelihood (red). Both conditions led to the same available amount of posterior information, which (in the optimal case) leads to the same probability of a correct choice as well as confidence. Hence, the target accuracy and confidence will only differ between conditions if the two sources of information are not integrated optimally in the decisions and/or confidence. Note that this is only a sketch aimed at conveying the intuition of how the conditions were matched in terms of posterior information. The prior for a rightward target stimulus is captured more accurately by a step function, shown in Fig. 8a.