Tax income as a function of population and tax rate
Hello community,
what I am presenting you here are my interim results from my ongoing mathematical modelling of the tax system in TATW. While I didn't check it yet, I do believe this is nothing TATW specific, but applies to MII TW in general.
First of, I can tell you this topic is huge. I hit on it a few years ago, but back in that time lacked the math skills to make any further progress on it. I will try to explain a bit what this is about, why it could become important, and how I got to a decent solution.
So in the beginning, I was bothered by a simple question: "Should I set a higher or lower tax rate?", or better: "How long does the low tax rate need to surpass the gold income of a high tax rate through the raised population growth?"
To determine this, I simply assumed that the tax was a per-capita tax, as it didn't seem to depend on any other factors than the tax rate and the population, and simply took the formula: Tax Income = Population * Tax Rate.
So I wanted to know what "low tax rate", "normal tax rate", "high tax rate" and "very high tax rate" actually meant in units of money per capita per round; and started gathering data from in the game.
I always sent the governors out of the settlement to eliminate their traits as a variable, and wrote down the population, tax income and set tax rate in that settlement in that round, for a few settlements. I first caluclated some really odd numbers for the tax rate, so I did it for a few more and found: Oh my god! It's not constant! There is no linear correlation between the population and the tax income.
Many of you probably have noticed this already: if the population is higher, you don't get as much money per capita compared to a lower population, at the same set tax rate. But determinating the actual mathematical function of the tax income isn't that easy, and back then, I had no idea how to do it.
Well, I came back to it a few weeks ago and started gathering larger amounts of data. As it stands now, I took all settlements that Gondor starts with, with the Expanded Map mod enabled - they are 17, 15 town type settlements and 2 castle type settlements. I sent out all the governors and wrote the population and tax income down for every of the four set tax rates in the towns, so I had 62 points of data.
From this on, it goes quite easy. I made a diagram for the tax income as a function of the population, with 4 distinct sets of data (one for each tax rate), each with 15 points of data, using Excel. I made Excel give me the logarithmical Trendlines and their functions for each set of data (logarithmical function fits best by far, you can easily see in the diagram that it's a logarithmical function). I also made it give me a line that smoothly connects the point of data I had, to compare the two graphs. The trendline looks pretty good, so I rearranged the functions of the trendlines to only differ in one value, that I could assume the variable representing the tax rate in that function. The number was still a bit odd, but by further rearranging I could get it into a shape that is quite smooth from it's numbers, and easy to remember. So without further ado, the formula I got is:
I = 400 * ( ln(P) - 4.4 ) / R , R out of { 2 ; 2.5 ; 3 ; 3.75 }
Where I is the Income, P is the population, and R is 2 for very high, 2.5 for high, 3 for normal, and 3.75 for low tax rate.
The formula does not give the exact values, it is approximative. But over all my gathered data, it has a mean realtive error of 1.23% which is practically nothing, and this function does not diverge from the actual values in any direction, so you can use it without hesitation.
I still have to analyze and compare how the comulative tax incomes over many rounds behave with consideration of the population growth. You guys could do this as well. But for now, I think this formula is already a pretty good thing for us.
Regards,
Mario
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But as we all know, it's impossible to get it so far down, so who cares. ^^
Originally Posted by Excellion
