My question is a bit general, but it is more common in solving dynamic optimization. I think dynamic optimization is more like a way of solving problems. We first need to divide the problem into several stages. My question is
How can we know or prove the answer is independent of the way we divide the stages?
To be specific, when solving a resource distribution problem. Like allocating 5 scientists to 3 countries($A, B, C$), How can I know that it does not matter whether we choose $A$ country to be the first stage or $B$ be the first or $C$?
Suppose you solve your problem twice, once using the ordering A-B-C and again using the ordering B-A-C, and suppose that the solution to the A-B-C version has a better objective value than the solution to the B-A-C version. Assuming that the problem is amenable to dynamic programming (not every problem is), you should be able to demonstrate that the solution to the A-B-C version is also a feasible solution to the B-A-C version, with the same objective value, contradicting the assumption that the optimal solution to B-A-C is inferior to the optimal solution to A-B-C.