Calculating closed‐form solutions of general Cournot models where firms have private information about costs is very cumbersome. Most authors therefore consider linear demands and constant marginal costs. However, within this framework, the non‐negativity constraint on prices (and quantities) has not been properly dealt with and the correct calculation of all Bayesian Nash equilibria is more complicated than expected. Moreover, multiple symmetric and interior Bayesian equilibria may exist for an open set of parameters. The reason for this is that linear inverse demand is not really linear, since there is a kink at zero price: P(Q)=max{a−bQ,0} rather than P(Q)=a−bQ.