Pleiotropy has been suggested as a novel mechanism for stabilising cooperation in bacteria and other microbes. The hypothesis is that linking cooperation with a trait that provides a personal (private) benefit can outweigh the cost of cooperation in situations when cooperation would not be favoured by mechanisms such as kin selection. We analysed the theoretical plausibility of this hypothesis, with analytical models and individual-based simulations. We found that (1) pleiotropy does not stabilise cooperation, unless the cooperative and private traits are linked via a genetic architecture that cannot evolve (mutational constraint); (2) if the genetic architecture is constrained in this way, then pleiotropy favours any type of trait and not ...