Irreversible, unobservable, costly investment in the presence of rivals

I identify circumstances in which an agent makes a costly, unobservable, irreversible investment that affects the subsequent noisy economic environment. Rivals infer that the agent is strong, even though it initially appeared weak, if 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. As the amount of noise decreases, the probability that a challenge occurs decreases. In addition, as the amount of noise decreases, the probability that the agent invests increases (decreases) if the rivals' opportunity cost is relatively high (low).