Hmm, I looked into it again, but I find some of your calculations/assumptions very arbitrary. But it is interesting.
By the way, this can also be done with 0% arbitrarity.
Here we go:
Since this thread is "Swordfig
hters vs. Pikemen", these two units will be the only ones taken in consideration.
By the
law of large numbers, we can assert that the average number of wounds inflicted by 100 arrows equals Bowmen's hit chance.
Thus, the
average number of wounds inflicted by 100 arrows is
15 against Sword Fighters
20 against Pikemen
Since both unit types have 3 Life Points, they both need 3 wounds to die.
Thus, we can say that the
average number of kills per 100 arrows is
5 against Sword Fighters
6,67 against Pikemen
Please note that this does not take in account that arrows may not hit always the same units, which spreads out the damage and lessens the losses; assuming the damage doesn't spread out
biases this comparison in Sword Fighter's favor; this means that, if Pikemen end up resulting better, the result will be certain, while if they don't then the result will have a slight chance of being such due to this bias.
The
percentage of soldiers lost purely due to Bowmen's intervention equals "average number of kills" times "
X" (times 100, since it's a percentage and not a ratio) divided by "number of warriors of the types in the comparison", where X is a coefficient accounting for those factors that apply to both sides, such as the duration of the fight or the actual number of Bowmen and thrown arrows.
The
number of warriors of the types in the comparison equals "Y" divided by "units' economic requirements", where Y is the economical output of their hometowns, which must be assumed equal for both sides for this comparison to be as unbiased as possible.
Now, the
units' economic requirements are:
1 Recruit, 2 Iron Defences, 1 Iron Weapon for Sword Fighters
1 Recruit, 1 Iron Defence, 1 Iron Weapon for Pikemen
Iron Weapons and Iron Defences come from the same production chains up to the very last step, which means that their
relative worth is easily calculated by taking in account the different production times.
The difference in the last step's buildings' size will be ignored in order to make the comparison simpler and because it is very rare that the available space is so limited that this becomes an impactful factor; anyway, when it does come into play, it favors Pikemen, which means that ignoring it yet again
slightly biases this comparison in Sword Fighter's favor.
Thus,
1 Iron Protection is worth 1,24 Iron Weapons.
Recruits are instead from a completely different production chain; what we know for sure is that they are worth less than Iron Weapons, since Schools work very fast and Gold is usually more available than Iron and can be produced much faster and in higher quantities.
Thus, we'll work with the bounds of certainty of their value:
0 ≤ Recruit ≤ 1 Iron Weapon
With these values, we can say that
1 Sword Fighter's worth is between 3,48 and 4,48 Iron Weapons
1 Pikeman's worth is between 2,24 and 3,24 Iron Weapons
Thus, the ratio of their economic worths is between 0,64 and 0,72 (Pikeman/Sword Fighter)
The
ratio of the number of warriors of the types in the comparison thus equals the "inverse of the ratio of their worth" (this being a ratio, Y gets factored out for being both at the numerator and at the denominator).
Thus, the
total number of Pikemen is between 1,38 and 1,55 times that of Sword Fighters; for simplicity the total number of Sword Fighters will be called "
N".
Thus, the
ratio of soldiers lost purely due to Bowmen's intervention for sure
is 5·X/N for Sword Fighters
lies between 3,22·X/N and 3,62·X/N for Pikemen
In percentages:
500·X/N % of the Sword Fighters
something between 322·X/N % and 362·X/N % of the Pikemen
500·X/N is
always higher than both 322·X/N and 362·X/N , for any N ≥ 1 and X > 0 (which are their limit: N can't be less than 1 or else there are no warriors in the fight, X can't be 0 or less or else it means that no arrows have been shot during the battle).
Thus, we can say with certainty that Pikemen as a whole are more durable than Sword Fighters against Bowmen.
Which they aren't supposed to be, as far as I know.
EDIT: Fixed a few mistypings.
Just when you think you know something, you have to look at it in another way, even though it may seem silly or wrong. You must try! - John Keating, "Dead Poets Society"
circa of the fights with the setup from the last "test 1" - except that there were no Bowmen), better than they do when few Bowmen take part in the fight.
