User:Jftsang/Bomber problem

In operations research, the bomber problem asks the following: Suppose that a bomber must travel a further t discrete time steps before reaching its target to drop its bombs. At each step, there is a probability p that it will encounter an enemy fighter. The bomber carries n air-to-air missiles. If it fires k missiles when it encounters a fighter, then it has a probability $$1 - \alpha ^ k$$ of surviving the encounter (each missile has a probability $$\alpha$$ of missing the fighter). If the bomber encounters a fighter at a given state $$(n, t)$$, let $$k(n, t)$$ be the optimal number of missiles to fire in order to maximise the probability $$V(n, t)$$ that the bomber will reach its target (where $$V(n,0) = 1$$).

Conjecture
Richard R. Weber has proved that $$k(n,t)$$ is nonincreasing in n and that $$n - k(n,t)$$ is nondecreasing in n. He has also conjectured and attempted to prove that $$k(n,t)$$ is also nondecreasing in n.