Irreversible, Unobservable, Costly Investment in the Presence of Rivals

I identify circumstances in which an agent wants to make a costly but unobservable irreversible investment that affects the subsequent noisy economic environment. In equilibrium, rivals may eventually infer that the agent is strong even though it initially appeared weak, so long as enough "strength" is seen subsequently. Comparative statics reveal that the higher is the rivals' opportunity cost, the more likely is the agent not to make the costly investment in equilibrium. In addition, as the amount of noise decreases, the probability that the agent invests increases and the probability that a challenge occurs decreases.