Taxation above 100 (long)

[MaxWilson]

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%
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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%
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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%
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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%
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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%
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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%
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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%