The Basics

Monte Carlo

Results

Code: Scenario Artillery Leader Lvl Blockade Obsolete Mean Median 95% quantile 1 0 0 -2 0 0 11.5 11. 17 2 0 0 -1 0 0 9.8 10. 15 3 0 0 -3 0 0 13.4 13. 21 4 0 0 -2 -2 0 15.7 15. 25 5 1 0 -2 0 0 9.5 9. 14 6 5 0 -2 0 0 5.2 5. 8 7 5 4 -2 0 0 3.0 3. 6 8 1 0 -4 0 0 12.4 12. 19 9 5 0 -4 0 0 6.4 6. 10 10 0 0 -6 0 0 23.7 21. 44 11 1 0 -6 0 0 16.4 15. 27 12 0 0 -5 0 0 18.9 18. 31 13 5 0 -6 0 0 7.85 8. 12 14 0 0 -8 0 0 43.2 38. 86 15 1 0 -8 0 0 23.2 21. 41 16 5 0 -8 0 0 9.5 9. 14 17 5 0 -10 0 0 11.6 11. 18

Scenario 1: Fort level 2, no bonuses

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For the experts: As you can see, the distribution looks sorta Poisson-like, which is very much to be expected given that sieges are random and independent. For the experts: As you can see, the distribution looks sorta Poisson-like, which is very much to be expected given that sieges are random and independent.

Scenario 2: Fort level 1, no bonuses

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Scenario 3: Fort level 3, no bonuses

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Scenario 4: Fort level 2, -2 for coastal fort

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Scenario 5: Fort level 2, +1 artillery bonus

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Scenario 6: Fort level 2, +5 artillery

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Scenario 7: Fort level 2, +5 artillery, +4 leader bonus

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Scenario 8: Fort level 4, +1 artillery

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Scenario 9: Fort level 4, +5 artillery

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Scenario 10: Fort level 6

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Scenario 11: Fort level 6, +1 artillery

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Scenario 12: Fort level 5

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Scenario 13: Fort level 6, +5 artillery

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Scenario 14: Fort level 8

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Scenario 15: Fort level 8, +1 artillery

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Scenario 16: Fort level 8, +5 artillery

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Scenario 17: Fort level 10, +5 artillery

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Batch Results

Code: MaxProgress Lvl Art Mean Median 95% quantile 12 3 0 5.86 6 9 12 3 1 5.04 5 8 12 3 2 4.31 4 7 12 3 3 3.63 3 6 12 3 4 3.12 3 6 12 3 5 2.68 3 5 12 3 6 2.35 2 5 12 3 7 2.11 2 4 12 3 8 1.88 2 4 12 2 0 6.62 6 10 12 2 1 5.63 5 9 12 2 2 4.84 5 8 12 2 3 4.19 4 7 12 2 4 3.56 3 6 12 2 5 3.03 3 6 12 2 6 2.62 2 5 12 2 7 2.32 2 4 12 2 8 2.06 2 4 12 1 0 7.53 7 11 12 1 1 6.39 6 10 12 1 2 5.52 5 9 12 1 3 4.73 5 8 12 1 4 4.13 4 7 12 1 5 3.48 3 6 12 1 6 3.02 3 5 12 1 7 2.60 2 5 12 1 8 2.26 2 4 12 0 0 8.59 8 13 12 0 1 7.32 7 11 12 0 2 6.28 6 10 12 0 3 5.40 5 8 12 0 4 4.68 5 7 12 0 5 4.10 4 7 12 0 6 3.46 3 6 12 0 7 2.99 3 5 12 0 8 2.60 2 5 12 -1 0 9.92 10 15 12 -1 1 8.33 8 12 12 -1 2 7.10 7 11 12 -1 3 6.16 6 9 12 -1 4 5.27 5 8 12 -1 5 4.57 4 7 12 -1 6 4.02 4 7 12 -1 7 3.42 3 6 12 -1 8 2.96 3 5 12 -2 0 11.6 11 18 12 -2 1 9.56 9 14 12 -2 2 8.12 8 12 12 -2 3 6.88 7 10 12 -2 4 5.95 6 9 12 -2 5 5.17 5 8 12 -2 6 4.49 4 7 12 -2 7 3.92 4 6 12 -2 8 3.36 3 6 12 -3 0 13.5 13 21 12 -3 1 10.9 11 17 12 -3 2 9.12 9 14 12 -3 3 7.69 7 12 12 -3 4 6.67 6 10 12 -3 5 5.79 6 9 12 -3 6 5.01 5 8 12 -3 7 4.41 4 7 12 -3 8 3.86 4 6 12 -4 0 15.8 15 25 12 -4 1 12.4 12 19 12 -4 2 10.2 10 16 12 -4 3 8.63 8 13 12 -4 4 7.39 7 11 12 -4 5 6.40 6 10 12 -4 6 5.60 5 9 12 -4 7 4.87 5 7 12 -4 8 4.25 4 7 12 -5 0 18.8 18 31 12 -5 1 14.2 13 23 12 -5 2 11.5 11 18 12 -5 3 9.61 9 15 12 -5 4 8.19 8 12 12 -5 5 7.07 7 11 12 -5 6 6.16 6 9 12 -5 7 5.41 5 8 12 -5 8 4.73 5 7 12 -6 0 24.0 21 44 12 -6 1 16.4 15 27 12 -6 2 12.9 12 20 12 -6 3 10.7 10 17 12 -6 4 9.04 9 14 12 -6 5 7.83 8 12 12 -6 6 6.80 7 10 12 -6 7 5.97 6 9 12 -6 8 5.24 5 8 12 -7 0 32.0 28 65 12 -7 1 19.2 18 33 12 -7 2 14.5 14 23 12 -7 3 11.9 11 19 12 -7 4 10.1 10 16 12 -7 5 8.62 8 13 12 -7 6 7.50 7 11 12 -7 7 6.48 6 10 12 -7 8 5.73 6 9 12 -8 0 43.7 39 87 12 -8 1 22.9 21 40 12 -8 2 16.9 16 28 12 -8 3 13.3 13 21 12 -8 4 11.1 11 17 12 -8 5 9.46 9 14 12 -8 6 8.24 8 12 12 -8 7 7.17 7 11 12 -8 8 6.29 6 9 12 -9 0 56.9 52 107 12 -9 1 27.8 26 51 12 -9 2 19.3 18 32 12 -9 3 15.2 14 25 12 -9 4 12.4 12 20 12 -9 5 10.5 10 16 12 -9 6 9.04 9 14 12 -9 7 7.85 8 12 12 -9 8 6.90 7 10 12 -10 0 10000 10000 10000 12 -10 1 37.2 34 70 12 -10 2 22.8 21 40 12 -10 3 17.2 16 28 12 -10 4 13.9 13 22 12 -10 5 11.6 11 18 12 -10 6 9.94 10 15 12 -10 7 8.53 8 13 12 -10 8 7.47 7 11 12 -11 0 10000 10000 10000 12 -11 1 10000 10000 10000 12 -11 2 31.2 28 60 12 -11 3 20.7 19 36 12 -11 4 15.7 15 26 12 -11 5 13.1 12 21 12 -11 6 11.0 10 17 12 -11 7 9.44 9 14 12 -11 8 8.14 8 12 --------------------------------------------------------------- 13 3 0 5.86 6 9 13 3 1 5.03 5 8 13 3 2 4.32 4 7 13 3 3 3.63 3 6 13 3 4 3.11 3 6 13 3 5 2.68 2 5 13 3 6 2.38 2 5 13 3 7 2.11 2 4 13 3 8 1.87 2 4 13 2 0 6.61 6 10 13 2 1 5.60 5 9 13 2 2 4.83 5 8 13 2 3 4.22 4 7 13 2 4 3.53 3 6 13 2 5 3.04 3 6 13 2 6 2.63 2 5 13 2 7 2.32 2 4 13 2 8 2.04 2 4 13 1 0 7.50 7 11 13 1 1 6.43 6 10 13 1 2 5.49 5 9 13 1 3 4.75 5 8 13 1 4 4.13 4 7 13 1 5 3.51 3 6 13 1 6 3.01 3 5 13 1 7 2.60 2 5 13 1 8 2.31 2 4 13 0 0 8.58 8 13 13 0 1 7.27 7 11 13 0 2 6.24 6 9 13 0 3 5.38 5 8 13 0 4 4.65 4 7 13 0 5 4.09 4 7 13 0 6 3.47 3 6 13 0 7 2.97 3 5 13 0 8 2.61 2 5 13 -1 0 9.84 10 14 13 -1 1 8.30 8 12 13 -1 2 7.07 7 11 13 -1 3 6.09 6 9 13 -1 4 5.29 5 8 13 -1 5 4.57 4 7 13 -1 6 4.01 4 6 13 -1 7 3.40 3 6 13 -1 8 2.92 3 5 13 -2 0 11.4 11 17 13 -2 1 9.49 9 14 13 -2 2 8.01 8 12 13 -2 3 6.92 7 10 13 -2 4 5.98 6 9 13 -2 5 5.17 5 8 13 -2 6 4.50 4 7 13 -2 7 3.91 4 6 13 -2 8 3.36 3 6 13 -3 0 13.2 13 20 13 -3 1 10.7 10 16 13 -3 2 9.00 9 13 13 -3 3 7.73 8 12 13 -3 4 6.66 7 10 13 -3 5 5.79 6 9 13 -3 6 5.02 5 8 13 -3 7 4.40 4 7 13 -3 8 3.85 4 6 13 -4 0 15.2 15 23 13 -4 1 12.2 12 18 13 -4 2 10.1 10 15 13 -4 3 8.58 8 13 13 -4 4 7.39 7 11 13 -4 5 6.39 6 10 13 -4 6 5.59 5 8 13 -4 7 4.84 5 8 13 -4 8 4.28 4 7 13 -5 0 17.7 17 27 13 -5 1 13.7 13 21 13 -5 2 11.3 11 17 13 -5 3 9.51 9 14 13 -5 4 8.16 8 12 13 -5 5 7.05 7 11 13 -5 6 6.15 6 9 13 -5 7 5.43 5 8 13 -5 8 4.72 5 7 13 -6 0 21.2 20 35 13 -6 1 15.8 15 25 13 -6 2 12.6 12 19 13 -6 3 10.5 10 16 13 -6 4 9.03 9 14 13 -6 5 7.77 8 12 13 -6 6 6.75 7 10 13 -6 7 5.96 6 9 13 -6 8 5.27 5 8 13 -7 0 26.1 24 46 13 -7 1 18.2 17 30 13 -7 2 14.2 14 23 13 -7 3 11.7 11 18 13 -7 4 9.93 10 15 13 -7 5 8.55 8 13 13 -7 6 7.46 7 11 13 -7 7 6.49 6 10 13 -7 8 5.72 6 9 13 -8 0 34.2 30 67 13 -8 1 21.1 20 35 13 -8 2 16.1 15 26 13 -8 3 13.1 13 20 13 -8 4 11.0 11 17 13 -8 5 9.40 9 14 13 -8 6 8.16 8 12 13 -8 7 7.15 7 11 13 -8 8 6.26 6 9 13 -9 0 44.9 40 87 13 -9 1 25.1 23 44 13 -9 2 18.3 17 30 13 -9 3 14.6 14 23 13 -9 4 12.2 12 19 13 -9 5 10.4 10 16 13 -9 6 8.95 9 13 13 -9 7 7.79 8 12 13 -9 8 6.88 7 10 13 -10 0 57.3 52 107 13 -10 1 29.1 27 51 13 -10 2 20.6 19 34 13 -10 3 16.3 15 26 13 -10 4 13.5 13 21 13 -10 5 11.4 11 17 13 -10 6 9.78 9 15 13 -10 7 8.45 8 12 13 -10 8 7.46 7 11 13 -11 0 10000 10000 10000 13 -11 1 38.4 35 70 13 -11 2 24.3 23 41 13 -11 3 18.4 17 29 13 -11 4 15.0 14 24 13 -11 5 12.6 12 19 13 -11 6 10.7 10 16 13 -11 7 9.30 9 14 13 -11 8 8.05 8 12 --------------------------------------------------------------- 14 3 0 5.85 6 9 14 3 1 5.02 5 8 14 3 2 4.31 4 7 14 3 3 3.61 3 6 14 3 4 3.09 3 6 14 3 5 2.70 3 5 14 3 6 2.36 2 5 14 3 7 2.10 2 4 14 3 8 1.89 2 4 14 2 0 6.60 6 10 14 2 1 5.65 6 9 14 2 2 4.87 5 8 14 2 3 4.20 4 7 14 2 4 3.55 3 6 14 2 5 3.02 3 6 14 2 6 2.62 2 5 14 2 7 2.30 2 4 14 2 8 2.05 2 4 14 1 0 7.43 7 11 14 1 1 6.39 6 10 14 1 2 5.51 5 9 14 1 3 4.78 5 8 14 1 4 4.14 4 7 14 1 5 3.48 3 6 14 1 6 3.00 3 5 14 1 7 2.61 2 5 14 1 8 2.30 2 4 14 0 0 8.54 8 12 14 0 1 7.33 7 11 14 0 2 6.27 6 10 14 0 3 5.39 5 8 14 0 4 4.66 5 7 14 0 5 4.11 4 7 14 0 6 3.45 3 6 14 0 7 2.97 3 5 14 0 8 2.60 2 5 14 -1 0 9.77 10 14 14 -1 1 8.29 8 12 14 -1 2 7.09 7 11 14 -1 3 6.16 6 9 14 -1 4 5.28 5 8 14 -1 5 4.58 4 7 14 -1 6 4.03 4 7 14 -1 7 3.38 3 6 14 -1 8 2.93 3 5 14 -2 0 11.3 11 16 14 -2 1 9.44 9 14 14 -2 2 8.01 8 12 14 -2 3 6.88 7 10 14 -2 4 5.97 6 9 14 -2 5 5.15 5 8 14 -2 6 4.51 4 7 14 -2 7 3.94 4 6 14 -2 8 3.37 3 6 14 -3 0 13.0 13 19 14 -3 1 10.6 10 16 14 -3 2 9.03 9 13 14 -3 3 7.69 7 11 14 -3 4 6.65 6 10 14 -3 5 5.77 6 9 14 -3 6 5.01 5 8 14 -3 7 4.40 4 7 14 -3 8 3.87 4 6 14 -4 0 14.9 14 22 14 -4 1 12.1 12 18 14 -4 2 10.0 10 15 14 -4 3 8.59 8 13 14 -4 4 7.35 7 11 14 -4 5 6.42 6 10 14 -4 6 5.59 5 8 14 -4 7 4.83 5 8 14 -4 8 4.27 4 7 14 -5 0 17.3 17 26 14 -5 1 13.6 13 21 14 -5 2 11.2 11 17 14 -5 3 9.58 9 14 14 -5 4 8.18 8 12 14 -5 5 7.07 7 11 14 -5 6 6.18 6 9 14 -5 7 5.43 5 8 14 -5 8 4.71 5 7 14 -6 0 20.2 19 32 14 -6 1 15.4 15 24 14 -6 2 12.5 12 19 14 -6 3 10.5 10 16 14 -6 4 9.01 9 14 14 -6 5 7.77 8 12 14 -6 6 6.75 7 10 14 -6 7 5.95 6 9 14 -6 8 5.24 5 8 14 -7 0 24.0 23 39 14 -7 1 17.6 17 28 14 -7 2 14.0 13 22 14 -7 3 11.7 11 18 14 -7 4 9.87 10 15 14 -7 5 8.56 8 13 14 -7 6 7.41 7 11 14 -7 7 6.49 6 10 14 -7 8 5.72 6 9 14 -8 0 29.7 27 52 14 -8 1 20.1 19 33 14 -8 2 15.7 15 25 14 -8 3 13.0 12 20 14 -8 4 10.9 11 17 14 -8 5 9.33 9 14 14 -8 6 8.12 8 12 14 -8 7 7.11 7 10 14 -8 8 6.22 6 9 14 -9 0 37.9 34 71 14 -9 1 23.7 22 40 14 -9 2 17.8 17 29 14 -9 3 14.4 14 22 14 -9 4 12.0 11 18 14 -9 5 10.3 10 16 14 -9 6 8.93 9 13 14 -9 7 7.77 8 11 14 -9 8 6.89 7 10 14 -10 0 46.2 42 88 14 -10 1 26.5 25 45 14 -10 2 19.8 19 32 14 -10 3 15.9 15 25 14 -10 4 13.2 13 20 14 -10 5 11.2 11 17 14 -10 6 9.67 9 14 14 -10 7 8.41 8 12 14 -10 8 7.46 7 11 14 -11 0 58.6 54 108 14 -11 1 31.3 29 53 14 -11 2 22.3 21 36 14 -11 3 17.8 17 28 14 -11 4 14.6 14 22 14 -11 5 12.3 12 19 14 -11 6 10.6 10 16 14 -11 7 9.22 9 13 14 -11 8 8.04 8 12 --------------------------------------------------------------- 15 3 0 5.86 6 9 15 3 1 5.00 5 8 15 3 2 4.35 4 7 15 3 3 3.64 3 6 15 3 4 3.13 3 6 15 3 5 2.70 3 5 15 3 6 2.34 2 5 15 3 7 2.09 2 4 15 3 8 1.87 2 4 15 2 0 6.62 6 10 15 2 1 5.62 5 9 15 2 2 4.82 5 8 15 2 3 4.16 4 7 15 2 4 3.52 3 6 15 2 5 3.01 3 6 15 2 6 2.62 2 5 15 2 7 2.32 2 4 15 2 8 2.05 2 4 15 1 0 7.42 7 11 15 1 1 6.44 6 10 15 1 2 5.48 5 9 15 1 3 4.76 5 8 15 1 4 4.13 4 7 15 1 5 3.49 3 6 15 1 6 3.00 3 5 15 1 7 2.63 2 5 15 1 8 2.29 2 4 15 0 0 8.53 8 12 15 0 1 7.26 7 11 15 0 2 6.23 6 10 15 0 3 5.38 5 8 15 0 4 4.67 5 7 15 0 5 4.09 4 7 15 0 6 3.42 3 6 15 0 7 2.98 3 5 15 0 8 2.60 2 5 15 -1 0 9.76 10 14 15 -1 1 8.26 8 12 15 -1 2 7.06 7 11 15 -1 3 6.12 6 9 15 -1 4 5.27 5 8 15 -1 5 4.58 4 7 15 -1 6 4.01 4 7 15 -1 7 3.41 3 6 15 -1 8 2.92 3 5 15 -2 0 11.3 11 16 15 -2 1 9.35 9 14 15 -2 2 8.05 8 12 15 -2 3 6.87 7 10 15 -2 4 5.98 6 9 15 -2 5 5.13 5 8 15 -2 6 4.51 4 7 15 -2 7 3.95 4 6 15 -2 8 3.35 3 6 15 -3 0 12.9 13 19 15 -3 1 10.6 10 16 15 -3 2 8.99 9 13 15 -3 3 7.72 8 11 15 -3 4 6.64 6 10 15 -3 5 5.80 6 9 15 -3 6 5.00 5 8 15 -3 7 4.40 4 7 15 -3 8 3.84 4 6 15 -4 0 14.8 14 22 15 -4 1 12.0 12 18 15 -4 2 10.0 10 15 15 -4 3 8.55 8 13 15 -4 4 7.37 7 11 15 -4 5 6.40 6 10 15 -4 6 5.60 5 8 15 -4 7 4.89 5 8 15 -4 8 4.26 4 7 15 -5 0 17.0 17 25 15 -5 1 13.6 13 20 15 -5 2 11.2 11 17 15 -5 3 9.49 9 14 15 -5 4 8.14 8 12 15 -5 5 7.05 7 10 15 -5 6 6.18 6 9 15 -5 7 5.41 5 8 15 -5 8 4.72 5 7 15 -6 0 19.6 19 30 15 -6 1 15.3 15 23 15 -6 2 12.6 12 19 15 -6 3 10.5 10 16 15 -6 4 8.94 9 13 15 -6 5 7.75 8 11 15 -6 6 6.80 7 10 15 -6 7 5.92 6 9 15 -6 8 5.25 5 8 15 -7 0 23.2 22 37 15 -7 1 17.3 17 27 15 -7 2 13.9 13 21 15 -7 3 11.6 11 18 15 -7 4 9.82 10 15 15 -7 5 8.52 8 13 15 -7 6 7.43 7 11 15 -7 7 6.48 6 10 15 -7 8 5.72 6 9 15 -8 0 28.0 26 46 15 -8 1 19.7 19 32 15 -8 2 15.6 15 24 15 -8 3 12.9 12 20 15 -8 4 10.9 11 17 15 -8 5 9.39 9 14 15 -8 6 8.06 8 12 15 -8 7 7.11 7 11 15 -8 8 6.26 6 9 15 -9 0 35.1 32 65 15 -9 1 22.9 21 38 15 -9 2 17.5 17 28 15 -9 3 14.3 14 22 15 -9 4 12.0 11 18 15 -9 5 10.3 10 15 15 -9 6 8.90 9 13 15 -9 7 7.80 8 11 15 -9 8 6.86 7 10 15 -10 0 40.1 37 73 15 -10 1 25.5 24 43 15 -10 2 19.2 18 30 15 -10 3 15.6 15 24 15 -10 4 13.1 13 20 15 -10 5 11.2 11 17 15 -10 6 9.64 9 14 15 -10 7 8.45 8 12 15 -10 8 7.42 7 11 15 -11 0 48.6 44 88 15 -11 1 28.8 27 48 15 -11 2 21.3 20 34 15 -11 3 17.2 16 26 15 -11 4 14.2 14 21 15 -11 5 12.2 12 18 15 -11 6 10.5 10 15 15 -11 7 9.19 9 13 15 -11 8 8.07 8 12

Notebook

So, I wanted to find out how long sieges generally take in the current game version. I believe the table on the wiki is still from the olden days, when you'd always get 1 siege progress, but since then things have changed quite a bit. Since sieges now depend on the siege history, algebraic expressions become pretty much impossible to use since factoring in all the different outcomes is, well, not easily done. But if you are better at combinatorics than I am, please let me know how to do it.You can find info on how sieges work on the wiki . Apart from the mean siege times table, the info seems up-to-date.The general idea of a Monte Carlo simulation is to take a problem that involves any kind of random process and run the problem a sufficiently large number of times, varying the starting parameters randomly or over a given representative sample if needed. Examples for this can be found in nearly any scientific field, for example you could use it to simulate the motion of molecules, the weather, or the spread of the flu. Generally, the questions the MC simulation is supposed to help answer are parameter changes: what happens if people wash hands more often?In the case of sieges in EU4, I used MC simulations to investigate the probability distribution of how many siege phases you need for certain siege scenarios. Some of the questions one can answer with this are: how much does artillery help? How much do siege leaders help? How long does it take, on average, to siege down a fort of a given level? How long for the 19/20 quantile, or 95% probability of having sieged the fort?I simulated the siege process to the best of my knowledge using Wolfram Mathematica and implementing a siege behaviour like that observed in the game. I started each scenario 10 000 times (i.e. ran 10k virtual sieges) and compiled a histogram from the results. 10k simulations seems good enough if you're not too interested in far outliers and can be calculated in a few seconds on my rig; the average value generally fluctuated less than 0.1 siege phases. I can also evaluate average siege progress on completion, shifted distribution after one already is at a certain siege progress, etc. but I'm too lazy to prepare all the data for presentation, and most players probably don't care about it much.All results are given in siege phases. How long these take depends on several values not covered here and not pertinent to the discussion. Since the distributions all look fairly similar (especially to the untrained eye), I put them in spoilersHere is an overview table of the different scenarios mentioned in detailThis is the standard scenario for the early game non-capital forts.Average: 11.5, Median: 11, 19th/20 quantile: 17Capital-only fort. This is equivalent to a level 2 fort besieged by a 1-pip leaderAverage: 9.9, Median: 10, 19th/20 quantile: 15I actually expected more of a difference here, but capital forts aren't that much weaker than normal forts, requiring only 1.6 additional siege phases. In-game the intuitive difference is probably enlarged by capital forts requiring only 5 artillery for a full siege bonus (see below).Capital plus normal fort, or bastion against a leader pip, etcAverage: 13.4, Median: 13, 19th/20 quantile: 21If the fort is coastal, you need to blockade it or suffer an additional -2 modifier to the die roll. This also works for a bastion if you have no artillery.Average: 15.7, Median: 15, 19th/20 quantile: 25This takes quite a bit longer than the normal level 2 fort, so putting some ships there is a really big advantage. This observation corresponds nicely to my in-game experience.This is a common scenario after artillery becomes available, but is not yet useful in battle, and when widespread application is too expensive.Average: 9.5, Median: 9, 19th/20 quantile: 14You may be wondering why this is not the same as a level 1 fort. The reason is that artillery, in addition to providing a bonus, increases your chance for a breach. Without artillery, you get a breach only on a die roll of 14. With a single artillery regiment, you get it on a roll of 13 or 14, so the probability for a breach doubles. Since breaches are high-impact events that provide a full +3 siege progress, and effectivel increase max progress, this is on average 0.4 siege phases shorter than the level 1 fort. Average number of breaches at siege won increases from 0.44 to 0.89 as expected.+1 artillery is, however, equivalent to getting a +1 obsolete fort bonus, so this scenario also works out if you siege without art but when you already discovered bastions.You got the money, you got the siege.Average: 5.2, Median: 5, 19th/20 quantile: 8Turbo siege mode. More than twice as fast as without artilleryJust out of curiosity how far you can push it. This happened to me in a game only once or twice so farAverage: 3.0, Median: 3, 19th/20 quantile: 6For bastions or coastal fortsAverage: 12.4, Median: 12, 19th/20 quantile: 19Artillery is even more useful here, saving you >3 siege phasesProbably the most common bastion scenarioAverage: 6.4, Median: 6, 19th/20 quantile: 10Star forts or bastions with an unchallenged coastal modifierAverage: 23.7, Median: 21, 19th/20 quantile: 44To put it simply: you could be here a while. Even the average is more than 2 years on a 30 day siege timerStar forts with a sprinkle of artilleryAverage: 16.4, Median: 15, 19th/20 quantile: 27Still a while, but more manageable.Technically a bastion with capital fort, or a star fort against a leader pipAverage: 18.9, Median: 18, 19th/20 quantile: 31Almost two siege phases longer than with an artillery unit on a level 6 fort. The breach chance increase really makes itself felt hereStar forts or bastions with an unchallenged coastal modifierAverage: 7.8, Median: 8, 19th/20 quantile: 12Now we're talking. Get artillery, lots of it.Well, this is only for academic purposes, really. Or are you willing to wait 4 years for each siege on average?Average: 43.2, Median: 38, 19th/20 quantile: 86Let's see if adding artillery helpsAverage: 23.2, Median: 21, 19th/20 quantile: 41This is still quite painful...Maybe morey arty?Average: 9.5, Median: 9, 19th/20 quantile: 14Well, I guess this is sort of ok, but still, those high-level forts are a royal pain in the backsideYou sure you don't wanna put a few ships out there?Average: 11.6, Median: 11, 19th/20 quantile: 18I also ran a larger dataset, for which I didn't compile the results manually. To understand this table, you first need to realize that the artillery bonus and fort level are effective values. A larger fort level, blockade and leader bonus all work the same way - except that in the game, you need more troops for the higher fort level, you will get the same result for a level 4 fort and a level 6 fort besieged by a 2 pip general. Soeffective fort level = fort level - leader siege pips + blockadeThat is, a fort level of -6 in the table below can be a level 6 fort, a level 4 fort with no blockade, a level 7 fort with a 1 pip leader, etc.The same reasoning applies to artillery and the obsolete fort bonus,effective artillery bonus = artillery bonus + obsolete bonusTherefore, the effective fort level varies from +3 (siege 4 leader on a capital fort) to -11 (level 8 coastal capital fort without blockade), while the effective artillery bonus varies from 0 to 8 (5 art + 3 levels obsolete)Then there is MaxProgress: as you discover higher-level forts, the game lets you accumulate higher levels of siege progress. It starts at 12, and increases by 1 for every new fort building you discover. So if you are at tech 14, MaxProgress would be 13.Note that a value of 10000 siege phases means the siege will not terminateIf you have Mathematica and would like to play around with this, here is a copy of my notebook: https://dl.dropboxusercontent.com/u/95609184/sieges.nb