Taxation above 100 (long)
 

Setting tax rates above 100 can give you an early-game boost at the cost of losing some population. I did an analysis of how much money you stand to make in the short term, as a percentage of how much money you would make from normal rates, assuming you have enough patrollers to keep unrest at zero. (I.e. it's a best-case analysis of "extra" income if you have spare patrollers lying around.)

For each level of taxation and growth scale, I give the number of turns before your income dips below the baseline (tax 100) level--which is the number of turns for which you're at an advantage--as well as how much more money you have when that point comes, and how much your population has suffered relative to baseline. E.g. for a Growth-3 dominion on 110% taxation, a starting population of 30,000 will have grown to 32,291 by turn 28 instead of 35,457 for an similar untaxed population. I report this as 91% population level.

Numbers come from a simulation, not the game, and I assume (can't find it documented either way) that population loss from taxation happens before population growth/death from scales, thus they're multiplicative and not additive. I also assume that population losses from taxation (formula on manual pg. 38) are rounded like most other integer division in the game, so 110% kills 0.3% pop and not 0.333333...%.

Hopefully this will help people make good decisions about their early game. I did this for my own benefit but thought I might as well post the results. It does turn out that the early-patrol strategy is slightly more advantageous for Death dominions than for Growth dominions, but only at high tax rates and only by a couple of percentage points.

Python code attached just in case there's, you know, a mistake or something in my calculations. ; )

-Max

Edit1: added a couple of decimal places to percentages for clarity's sake.

Edit2: myself pointed out to me that patrollers kill population, too. I forgot about it since it's generally small potatoes in comparison to the taxation loss. Now corrected.

Death-3 Tax rate 110% Breakeven turn 32: Population level 89.86%. Accumulated income 105.30%
Death-3 Tax rate 120% Breakeven turn 28: Population level 81.96%. Accumulated income 110.21%
Death-3 Tax rate 130% Breakeven turn 27: Population level 75.35%. Accumulated income 114.99%
Death-3 Tax rate 140% Breakeven turn 25: Population level 69.93%. Accumulated income 119.90%
Death-3 Tax rate 150% Breakeven turn 25: Population level 64.19%. Accumulated income 123.80%
Death-3 Tax rate 160% Breakeven turn 24: Population level 59.48%. Accumulated income 127.93%
Death-3 Tax rate 170% Breakeven turn 23: Population level 56.14%. Accumulated income 132.70%
Death-3 Tax rate 180% Breakeven turn 22: Population level 52.75%. Accumulated income 137.05%
Death-3 Tax rate 190% Breakeven turn 21: Population level 50.47%. Accumulated income 142.13%
Death-3 Tax rate 200% Breakeven turn 21: Population level 46.31%. Accumulated income 144.66%
===============================

Death-2 Tax rate 110% Breakeven turn 31: Population level 90.16%. Accumulated income 105.21%
Death-2 Tax rate 120% Breakeven turn 28: Population level 82.01%. Accumulated income 109.91%
Death-2 Tax rate 130% Breakeven turn 27: Population level 75.38%. Accumulated income 114.63%
Death-2 Tax rate 140% Breakeven turn 25: Population level 69.97%. Accumulated income 119.51%
Death-2 Tax rate 150% Breakeven turn 25: Population level 64.24%. Accumulated income 123.35%
Death-2 Tax rate 160% Breakeven turn 23: Population level 60.87%. Accumulated income 128.68%
Death-2 Tax rate 170% Breakeven turn 23: Population level 56.19%. Accumulated income 132.18%
Death-2 Tax rate 180% Breakeven turn 22: Population level 52.80%. Accumulated income 136.50%
Death-2 Tax rate 190% Breakeven turn 21: Population level 50.54%. Accumulated income 141.57%
Death-2 Tax rate 200% Breakeven turn 21: Population level 46.38%. Accumulated income 144.05%
===============================

Death-1 Tax rate 110% Breakeven turn 31: Population level 90.16%. Accumulated income 104.94%
Death-1 Tax rate 120% Breakeven turn 28: Population level 82.04%. Accumulated income 109.60%
Death-1 Tax rate 130% Breakeven turn 27: Population level 75.42%. Accumulated income 114.27%
Death-1 Tax rate 140% Breakeven turn 25: Population level 70.01%. Accumulated income 119.11%
Death-1 Tax rate 150% Breakeven turn 25: Population level 64.28%. Accumulated income 122.89%
Death-1 Tax rate 160% Breakeven turn 23: Population level 60.93%. Accumulated income 128.21%
Death-1 Tax rate 170% Breakeven turn 23: Population level 56.24%. Accumulated income 131.65%
Death-1 Tax rate 180% Breakeven turn 22: Population level 52.86%. Accumulated income 135.95%
Death-1 Tax rate 190% Breakeven turn 21: Population level 50.58%. Accumulated income 140.97%
Death-1 Tax rate 200% Breakeven turn 21: Population level 46.45%. Accumulated income 143.41%
===============================

Growth-0 Tax rate 110% Breakeven turn 30: Population level 90.42%. Accumulated income 104.82%
Growth-0 Tax rate 120% Breakeven turn 27: Population level 82.61%. Accumulated income 109.64%
Growth-0 Tax rate 130% Breakeven turn 27: Population level 75.41%. Accumulated income 113.89%
Growth-0 Tax rate 140% Breakeven turn 25: Population level 70.03%. Accumulated income 118.69%
Growth-0 Tax rate 150% Breakeven turn 24: Population level 65.44%. Accumulated income 123.43%
Growth-0 Tax rate 160% Breakeven turn 23: Population level 60.95%. Accumulated income 127.71%
Growth-0 Tax rate 170% Breakeven turn 23: Population level 56.27%. Accumulated income 131.11%
Growth-0 Tax rate 180% Breakeven turn 22: Population level 52.91%. Accumulated income 135.38%
Growth-0 Tax rate 190% Breakeven turn 21: Population level 50.62%. Accumulated income 140.37%
Growth-0 Tax rate 200% Breakeven turn 21: Population level 46.50%. Accumulated income 142.77%
===============================

Growth-1 Tax rate 110% Breakeven turn 29: Population level 90.77%. Accumulated income 104.76%
Growth-1 Tax rate 120% Breakeven turn 27: Population level 82.67%. Accumulated income 109.36%
Growth-1 Tax rate 130% Breakeven turn 26: Population level 76.26%. Accumulated income 114.13%
Growth-1 Tax rate 140% Breakeven turn 25: Population level 70.10%. Accumulated income 118.32%
Growth-1 Tax rate 150% Breakeven turn 24: Population level 65.51%. Accumulated income 123.02%
Growth-1 Tax rate 160% Breakeven turn 23: Population level 61.01%. Accumulated income 127.26%
Growth-1 Tax rate 170% Breakeven turn 23: Population level 56.34%. Accumulated income 130.61%
Growth-1 Tax rate 180% Breakeven turn 22: Population level 52.98%. Accumulated income 134.85%
Growth-1 Tax rate 190% Breakeven turn 21: Population level 50.69%. Accumulated income 139.82%
Growth-1 Tax rate 200% Breakeven turn 21: Population level 46.57%. Accumulated income 142.17%
===============================

Growth-2 Tax rate 110% Breakeven turn 29: Population level 90.75%. Accumulated income 104.50%
Growth-2 Tax rate 120% Breakeven turn 27: Population level 82.70%. Accumulated income 109.06%
Growth-2 Tax rate 130% Breakeven turn 26: Population level 76.28%. Accumulated income 113.78%
Growth-2 Tax rate 140% Breakeven turn 25: Population level 70.14%. Accumulated income 117.94%
Growth-2 Tax rate 150% Breakeven turn 24: Population level 65.54%. Accumulated income 122.59%
Growth-2 Tax rate 160% Breakeven turn 23: Population level 61.07%. Accumulated income 126.80%
Growth-2 Tax rate 170% Breakeven turn 23: Population level 56.40%. Accumulated income 130.10%
Growth-2 Tax rate 180% Breakeven turn 22: Population level 53.04%. Accumulated income 134.32%
Growth-2 Tax rate 190% Breakeven turn 21: Population level 50.75%. Accumulated income 139.25%
Growth-2 Tax rate 200% Breakeven turn 21: Population level 46.64%. Accumulated income 141.56%
===============================

Growth-3 Tax rate 110% Breakeven turn 28: Population level 91.07%. Accumulated income 104.42%
Growth-3 Tax rate 120% Breakeven turn 27: Population level 82.72%. Accumulated income 108.75%
Growth-3 Tax rate 130% Breakeven turn 26: Population level 76.29%. Accumulated income 113.42%
Growth-3 Tax rate 140% Breakeven turn 25: Population level 70.17%. Accumulated income 117.54%
Growth-3 Tax rate 150% Breakeven turn 24: Population level 65.56%. Accumulated income 122.15%
Growth-3 Tax rate 160% Breakeven turn 23: Population level 61.10%. Accumulated income 126.33%
Growth-3 Tax rate 170% Breakeven turn 23: Population level 56.43%. Accumulated income 129.58%
Growth-3 Tax rate 180% Breakeven turn 22: Population level 53.08%. Accumulated income 133.76%
Growth-3 Tax rate 190% Breakeven turn 21: Population level 50.79%. Accumulated income 138.67%
Growth-3 Tax rate 200% Breakeven turn 20: Population level 48.45%. Accumulated income 143.32%

Re: Taxation above 100 (long)
 

Note also that this applies to more than just your home province. If you have an army sitting in a chokepoint province, for instance, and one of your commanders is a not a mage, if you expect to hold onto that province for less than 22 turns you might as well set them on patrol and crank that tax rate up as high as your patrollers can handle, with full confidence that it won't cost you a dime.

-Max

Re: Taxation above 100 (long)
 

Patrollers have a gold upkeep... this is not included within your equation. Patrollers can be good for bottleneck locations during the early part of the game, but using units for expansion is better.

Re: Taxation above 100 (long)
 

Well some nations have very good patrollers, for instance late C'tis Desert Rangers with a patrol bonus of 2, making them quite cost effective I believe.

Re: Taxation above 100 (long)
 

Actually, the masters of early game patroling are Pangaea and Caelum. They can pretty much set at 200% from turn 2 on.

Re: Taxation above 100 (long)
 

Very interesting analysis Max!

Having a look to the results from my newbie point of view, I expected better figures for Growth-3 due to a higher population in a long term basis.
In fact, Death-3 is slightly better in terms of accumulated income and breakeven turn, and it only has a 2% population penalty (roughly speaking) versus the Growth-3 case.

In some of my SP games I wrongly set Growth to 3 expecting a better income in the long run, but now I see those gains are just peanuts.
The lesson I've just learned it is not choose Growth-3 for a raw income bonus, and make a better investment for those 120 points.
Nonetheless, Growth-3 is still advisable for Nations with elder mages (Mictlan, Marignon, etc) or with supersized units like Giants and bulimic Demons (Niefelheim, Lanka, etc).
Does it make sense?

Re: Taxation above 100 (long)
 

Yes. That's what I meant about it being a best-case analysis. Some nations have cheap patrollers, and other times you have a chokepoint province that you have to keep troops in anyway for extended periods. Besides, it's trivial to compute the upkeep for your patrolling force if you want to decide whether 200% taxes is a good idea for a particular province.

-Max

Re: Taxation above 100 (long)
 

Just to make sure we're still on the same page: by turn 22, Growth-3 is going to be pulling in much more money than Death-3. The population percentages are relative to the same population, untaxed. That is, in a 30,000 province Growth-3 will be at 32k (110% tax) instead of 37k (100% tax), whereas Death-3 will be at say 26k (110% tax) instead of 28k (100% tax). I made those latter numbers up but I think they convey the point.

Growth can still be a good investment. I play SP, and I enjoy taking Growth scales so time is on my side. That said, toward the end game gem income is more important than gold income anyway so maybe I should take Death instead for extra points.

-Max

Re: Taxation above 100 (long)
 

Max,

Now I understand what's going on; previously I thought the percentages were based upon a fixed population, forgive my bad math skills... and missing your good explanations in the original post.

Thanks also for your Death/Growth scales comment, it seems reasonable to gear late game towards gem income for broken spells instead of pure gold for troops.

Re: Taxation above 100 (long)
 

As a sheer non-numerical analysis, and speaking of SP - I like growth better than death if I can spare the points and get the paths I want. SP games tend to have many more "slow" turns and thus go longer. I've been around year 6 really wishing I didn't have death-3 as I'd like to sit back and hold my borders as I conjured/itemized and knowing I had to keep conquering to overcome upkeep. Huge swing though, even from death-3 to just growth 1