When many events contributed to an outcome, people consistently judge some more causal than others, based in part on the prior probabilities of those events. For instance, when a tree bursts into flames, people judge the lightning strike more of a cause than the presence of oxygen in the air-in part because oxygen is so common, and lightning strikes are so rare. These effects, which play a major role in several prominent theories of token causation, have largely been studied through qualitative manipulations of the prior probabilities. Yet, there is good reason to think that people's causal judgments are on a continuum-and relatively little is known about how these judgments vary quantitatively as the prior probabilities change. In this paper, we measure people's causal judgment across parametric manipulations of the prior probabilities of antecedent events. Our experiments replicate previous qualitative findings, and also reveal several novel patterns that are not well-described by existing theories. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. singular outcome, rather than type causation, where people judge the causes of an outcome in general).When attempting to understand why some causes stand out while others recede, a key fact is that this distinction is not dichotomous, but rather graded [10]. Rather than judge one event entirely a cause and the others entirely not a cause, people appear to judge causation on some kind of continuum. For instance, people might say that the arsonist was most causal; the lack of rain somewhat causal; and the arsonist's birth the least causal. Although gradation was commonly assumed in many models of type causation [11], the graded nature of token causation has only more recently attracted attention [3,4,[12][13][14].Despite its importance, however, our empirical understanding of gradation in token causal judgment is limited. Many studies have investigated how qualitative shifts in the parameters of causal systems (e.g. the prior probability of the antecedent events) produce qualitative shifts in causal judgment (see [3] for an overview), but fewer have mapped the quantitative relationship between those parameters and the resulting judgments. (For a notable exception, see [15][16][17]). Here, we contribute to closing this empirical gap. We present people with two basic causal systems, systematically manipulate the prior probabilities of the antecedent events in those systems, and elicit people's causal judgments. We find that, even in the most basic systems, the quantitative form of people's judgments can be complex, and is incompletely described by existing theories. This result highlights the need for future research on the quantitative form of token causal judgments.
Qualitative manipulations of token causationSeveral qualitative manipulations are known to affect which events people consider causal. We do not attempt an exhaustive review; rather, we follow Icard et al. [3] and highlight certain ...