Modelling Probability Judgements that Violate Binary Complementarity

Individuals make biased and variable probability judgements. Recent models such as the Bayesian Sampler and Probability Theory Plus Noise capture these effects by assuming people randomly sample events but are biased towards indifference (i.e., 0.5). However there is a bias they do not capture: systematic violations of binary complementarity, i.e., violations of the simple constraint that judgments of P(A) and P(not A) should sum to 1. Until now, this bias was only captured by the sampling process of the Quantum Sequential Sampler. Here we develop straightforward generalisations of the Bayesian Sampler, by introducing an asymmetric prior, and Probability Theory Plus Noise, by introducing asymmetric noise, that can generate violations of binary complementarity. We next show that these three models make distinct predictions for the mean-variance relationship in repeated judgments. Finally, we investigate violations of binary complementarity in five experiments, where participants judged the probabilities of dice rolls. Participants consistently violated binary complementarity, independent of whether they were in a high or low probability environment or how the alternative options are partitioned. Crucially, participants showed the highest variability for probability judgements below 0.5, an effect captured by an asymmetric prior in the generalised Bayesian Sampler, but not by the biasing mechanisms in the other models.