b1 = C * 1/2
b2 = C * 1/4
b3 = C * 1/8
.
.
.
where C is some arbitrary constant and bn is the additive benefit provided by a given infinite research level, such that the total benefit accruing to the sum of the researches winds up being the sum of the geometric series:
C * lim n→∞ Σ(1/(2^n))
which, if do research forever, is just "C" because the infinite sum just approaches "1". This is sort of like trying to leave a room by going halfway to the door every 5 seconds. We go slower and slower, decelerating at a rate that ensures perpetual minute progress, but without any hope of an eventual exit.
In my opinion, if I'm right about the above, this kind-of sucks. My understanding is that infinite research has been provided as a way to incentivise extreme late-game "megabase" play, and specifically to address the pre-0.15 problem that mostly all there was to do once you won factorio was to keep launching more rockets, which, aside from glory, provides no benefit to the player (indeed, for the imaginary in-game hero, it makes pretty much no sense at all). I agree with that assessment, and I agree that some kind of infinite research is a fine amelioration of this somewhat unsolvable problem. But a "bounded" infinitude as specified above is very quickly going to become equivalent to the old arrangement; it simply makes it more expensive to reach the same result.
By way of example -- and let me stress that these are made-up, hypothetical numbers, so, caveat emptor -- let's say we're talking about infinite mining productivity research and C is 1000%. Then when we reach the 20th level of research, the benefit we stand to gain is: 1000% / 2^20, approximately 0.001%. That is to say, basically no appreciable benefit at all. And the further we go down this rabbit hole, the more absurdly pathetic this incentive will become. Perhaps, in a sense, this is as it should be, but no matter how hard we try, we'll never break through that 1000% boundary, which is demoralizing.
It means that once we've effectively achieved a benefit that's psychologically equivalent to 1000% (and at level 19 above, I'd say we clearly already have, as net benefit has already exceeded 99.998%), we have no more reason to continue, just like we did with rockets. The only thing left is glory.
I want to argue that there is something you can do to the numbers to fix this problem.
A sixty-three bit integer -- which is what I presume are the type of "benefit" values we are talking about -- can hold something on the order of 10^19 -- something like a thousand times the distance between the earth and the sun in millimeters. So, we have some wiggle room. One solution is to make C very large. But no matter how large we make it, there's some basic psychological problem knowing that beyond a certain readily attainable level, no appreciable benefit is coming.
I'd like to propose an alternative arrangement. Imagine the C * (infinite sum) setup were preserved, but we picked some other infinite series, whose constituent terms still approach zero as n approaches infinity, but which is not bounded in summation. A classic example would be the harmonic series:
xn = 1/n
which is just {1, 1/2, 1/3, 1/4, 1/5,...}
This series converges to zero but its partial summation diverges to infinity. There are, of course, other series one could pick with this quality, but I think it's an OK starting point.
However, the sum benefit would be C*1 at research level one, which is awkward. So instead, I like the idea of starting a few terms into the series, say, at n=4: {1/4, 1/5, 1/6, ...}. If C=1, the research levels now look like this:
research level ---> cumulative benefit as a fraction of one*
1 ---> 0.25
2 ---> 0.45
3 ---> ~0.62
4 ---> ~0.76
9 ---> ~1.00
100 ---> ~3.35
1000 ---> ~5.65
10^5 ---> ~10.26
10^6 ---> ~12.57
10^7 ---> ~14.86
So, as you can see, an absolutely mind-boggling amount of research is going to need to occur before anyone reaches anything like a C*15 benefit level, therefore, it would be possible to put an absolute maximum in, by simply specifying that, just as a way to keep things from breaking, after the 9,999,999th level of research, no further benefit accrues, and testing that C*15 is not a game-breaking level of benefit. Such a restriction should still provide a meaningful very-late-game incentive up to many weeks of gameplay (and one could still accumulate useless "glory" levels up to 2^63 which should not be a constraint for any vanilla campaign).
Let me know your thoughts. Sorry for the tldr and math!
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