To preempt or not to preempt? Clustering vs. single entry in an oligopoly

In this work, we develop a real option model in continuous time in which firms compete perfectly on market entry but compete oligopolistically after having entered. We analytically show that in equilibrium, entry may be clustered and that firms tend to enter in weakly larger clusters as the market becomes more crowded. This equilibrium is unique up to a permutation of players. We also show that with moderate market risk, higher volatility delays entry and leads to larger groups.
We explore the following special cases: 1) In Cournot competition with a linear demand function, we show that firms always enter individually. 2) In Bertrand competition, a firm enters only for finite market levels. 3) Firm profits follow an inverse exponential function of the number of firms, and we find a closed-form solution in which the cluster size remains constant over time and increases with risk.