Ambiguity in a real option game

In this paper we study a two-player investment game with a first mover advantage in continuous time with stochastic payoffs, driven by a geometric Brownian motion. One of the players is assumed to be ambiguous with maxmin preferences over a strongly rectangular set of priors. We develop a strategy and equilibrium concept allowing for ambiguity and show that equilibira can be preemptive (a player invests at a point where investment is Pareto dominated by waiting) or sequential (one player invests as if she were the exogenously appointed leader). Following the standard literature, the worst case prior for the ambiguous player if she is the second mover is obtained by setting the lowest possible trend in the set of priors. However, if the ambiguous player is the first mover, then the worst case prior can be given by either the lowest or the highest trend in the set of priors. This novel result shows that "worst case prior" in a setting with geometric Brownian motion and kappa-ambiguity does not equate to "lowest trend".