July 2017. Revised July 2018.We investigate the entry timing and location decisions under market size uncertainty with Brownian motions in a continuous-time spatial duopoly competition. We show the following results. The entry threshold of the follower non-monotonically increases in volatility, implying that the leader's monopoly periods get longer with volatility. However, the leader is more likely to increase the degree of product differentiation as the volatility rises. A larger entry cost asymmetry between the firms places the leader closer to the edge in a preemption equilibrium although such an asymmetry places the leader closer to the center in a sequential equilibrium
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