Here is why the exponential decrease is the only one which can work as an approximation:
All other decreases would mply a time-dependancy, so the same investment would be valued differently, just because it was made earlier or later. Here an example how this would look like, if we would set the value of a vil at the beginning of the game == 0 after 50 minutes gametime with a) an exponential and b) a quadratic decrease.
In the exponential the reduced investment cost would place the new vil right on the payback line of the older one, while the quadratic (or any polynomial except of course e^0) decrease will underestimate the initial investment, so it seems like the new vil would have a much better payback than the old one.
Orange Line: Villager produced at the beginning of the game
Blue Line: Villager produced 150 s into the game
This would lead to a misconception that it woud be better to delay investments like vils, farms or eco upgrades to get the most out of them, just from the mathematical approach.
Because of that a polynomial decrease would also underestimate the value of a farm because farms lifespan is in the period where the polynomial evalueation of income is in average always worse than exponential ones with the same integral.
With polynomial decreases we also have the problem with additional factors which must be chosen by us: We must chose which part of the graph we use, because x^-a (a>0) is undefined at 0 and falls rapidly between 0 and 1, so we must ignore that part at least. Also a itself is hard to approximate because it must be at least k + 2 if k is the highest polynomial payback we can achieve with any strategy in the game, otherwise the value of that strat would divergate. Since we don’t know the value of k yet we also don’t know how big a must be at least.
But for the example of a towncenter, k is 2, so a must be at least 4, but once we build new tcs with the income of the first, that complete strat would be evalueated as diverging, so we can’t compare it with any other strat. Which would be the goal of any evalueation to get comparable values.
So in my conclusion, a exponential decrease is actually the only viable one to approximate different investments in the game, just because it is absolutely impossible to calculate with any other decrease.