This question has come up so many times I thought it might be a good idea to create a blog entry about this. Hopefully I can clear up any remaining confusion about what the catch-up point is, what it isn't, and how it is calculated.
But this seemingly-simple description makes some very powerful statements, which I will try to describe in more detail below.
Every production building upgrade is beneficial in the long term, for two reasons:
- Higher level production buildings can produce resources faster.
- They can hold more resources internally, and can therefore produce for longer periods of time before needing to be collected.
However, as anyone who has had to wait out a long Dark Elixir Drill upgrade can tell you, your resource production suffers quite dramatically while the upgrade is taking place, as resource production ceases for the entire period. So while the long term benefits of upgrading are clear, it is equally clear that there is a short-term resource penalty for upgrading (beyond the resource cost to actually perform the upgrade). The question remains, at what point do the long-term benefits outweigh the short-term penalties? That is exactly what the catch-up point defines.
Consider the following scenario of upgrading a Gold Mine from level 6 to level 7. Below are the relevant statistics of the two levels:
|Level 6||Level 7|
|Time to Fill||15h 23m 5s||18h 45m|
Obviously the level 7 mine is superior from a statistics point of view, but during the 12 hours it takes to upgrade, the mine will be sitting idle. Had we not upgraded it, it would have been happily producing 1,300 gold every hour for us. Thus, until the newly-upgraded mine makes up for its idle time and produces at least as much as the old mine would have, we are still have not reached the point where the upgrade has been beneficial to us. Once we do, we can say that the mine has reached its 'catch-up point'.
So how do we figure out what that catch-up point is? The way to visualize it is to imagine two different gold mines, one we upgrade to level 7, and one we leave at level 6. At what point does the new level 7 mine's production begin to surpass that of the level 6 mine we did not touch? Below is a chart showing the production of the two mines over time:
For the first twelve hours, the level 7 mine does not produce anything, as it is still upgrading. The level 6 mine, on the other hand, continutes to produce 1,300 gold per hour, reaching a total of 15,600 gold at the end of the upgrade period.
Now that the newly-upgraded mine is producing again, it begins producing gold faster than the old level 6 mine, but the old mine has a 15,600 gold head start. Until the level 7 mine can overcome that head start, the old level 6 mine will still have produced more gold over the entire period. When 64 hours have elapsed since the upgrade was initially begun, the two mines have finally produced the same amount of gold: 83,200. At this point, the level 7 mine has "caught up" to the level 6 mine.
Beyond 64 hours, the level 7 mine will always produce more gold than the level 6 mine, because it has both a higher production rate and a longer time to fill.
The result of the calculation, then, is that it takes a total of 52 hours of production once the upgrade is finished (64 hours total production minus the 12 hours of upgrade time) for the level 7 mine to "catch up" to the level 6 mine. We therefore say that the "catch-up point" for the level 7 mine is 52 hours, or 2 days and 4 hours.
Limitations and Things to Consider
Catch-up points are not cumulative, and you can't "get through them quicker" or "catch up on multiple upgrades at once" by upgrading again while you are still in the catch-up period of the earlier upgrade. But don't take to mean that upgrading your production buildings quickly is a bad idea. On the contrary, as we have shown before, all production building upgrades are always beneficial in the long term, and therefore the earlier you perform them the earlier you reap the long-term benefits. You are forgoing production now for more production later by performing the upgrade, but assuming you play the game longer than the maximum catch-up time, that decision is always a good one.
It becomes a little more murky to try to play the math game of "If I start the upgrade during the catch-up period I can eliminate both at the same time" because all the catch-up point is is a modeling of an individual decision of whether or not to upgrade. You can combine them and make a composite decision tree if you like, but you since you don't actually have to make up 4 days of level 1 drill production, 6 days of level 2 drill production, etc. it's not really directly comparable in that way.
In the bigger picture, don't sweat the catch-up point too much; it is an academic exercise and largely irrelevant. The only time it makes much sense to think about it is if you have a short-term need for resources, because for the duration of the upgrade and the catch-up period you will have less resources available than you would have had you not begun the upgrade. However, as stated before, upgrading your resource collectors is always beneficial in the long term (neglecting things like making yourself a juicier raid target, of course!), so as long as you plan on continuing to play the game, upgrade away. :)