The equivalence between costly and probabilistic voting models

In costly voting models, voters abstain when a stochastic cost of voting exceeds the benefit from voting. In probabilistic voting models, they always vote for a candidate who generates the highest utility, which is subject to random shocks. We prove an equivalence result: In two-candidate elections, given any costly voting model, there exists a probabilistic voting model that generates winning probabilities identical to those in the former model for any policy announcements, and vice versa. Thus many predictions of interest established in one of the models hold in the other as well, providing robustness of the conclusions to model specifications.