Bertrand competition under uncertainty

We look at a Bertrand model in which each firm may be inactive with a known probability, so the number of active firms is uncertain. The model has a mixed-strategy equilibrium, in which industry profits are positive and decline with the number of firms, the same features which make the Cournot model attractive. Unlike those in a Cournot model with similar uncertainty, Bertrand profits always increase in the probability that firms are inactive. Profits decline more sharply than in the Cournot model, the pattern found empirically in Bresnahan and Reiss [1991].