Still struggling with BR calculations

I've figured out the relationship between BR and wages at this point, but I'm still getting some wrong values for the BR values themselves, especially with respect to cavalry. I'll list a few of my calculations, to see if anyone (Autarch or otherwise) can spot mistakes I've made.

For a first example, let's try ogres (the light infantry version with clubs and hide).

• AC: 4
• UHP: 17, so UHP/8: 2.13
• Morale: +2, so Multiplier: 1.13
• Speed: 4, so Multiplier: 0.81
• Formation: IF, so Multiplier: 0.675
• Combat Multiplier: 4 (after rounding -- not sure how to hand this, might be the problem!)

Multiplying all this together and dividing by 6 gives me a BR of 3.52, or 0.059 per troop. The table values are "4.0" and "0.077" per troop.

Is this just a difference of rounding?

Even if I round "3.5" up to "4.0" (as seems to have happened for the unit value), I still get 0.067 (not 0.077) on a per troop basis.

Or in reverse, taking 60*0.077 gives 4.62. This seems like it should round up to 5.0, not down to 4.0.

I'm getting even larger errors for calvary types. For example, I'm getting BR 5.0 for Goblin Wolf Riders, not 6.5. The only way I could get them up to 6.5 was by using the infantry formula for the speed multiplier (=(charge/7.5)^0.33) instead of the cavalry formula (=charge/9).

Again, maybe this is due to something about rounding in the attack multiplier formula that I don't understand.

Here's an example of my failed attempt to calculated cavalry BRs.

Goblin wolf riders:

• AC: 3
• UHP: 20
• Morale: +2, so Multiplier: 1.13
• Speed: 7, so Multiplier: 0.78
• Formation: IM, so Multiplier: 0.62
• Combat Multiplier: 8.58 (2 rider attacks, 4 mount charge attacks, +1 for spear, = 7, and then increased another 1.58 by an "unrounding modifier")
• BR: 5.86
• DoW table value: 6.5

I used the cleave factor for the overall HD (1.1), not just the rider, but I don't think that's right. But it was the only way I could get close to the value 6.5, aside from using the foot movement speed multiplier formula (which I'm pretty sure is not right to do here).

For orc boar riders, I can get the correct formula by using the cleave factor "1.1" to calculate the combat multiplier. That's one less than the combined HD cleave factor. But I have no idea if that's correct or why "1.1" is the right cleave factor, or if it's all just a lucky accident that it comes out to 8 because I made other errors that canceled out with one another.

As you can see, I'm really confused by all this!

Another inconsistency I've noticed: To match the book values for BR/wage for spear-armed infantry beastmen, it's necessary to give them an extra point of damage for their "max damage on charge" value. If I remove the extra point of max damage for charging, then orcs drop down to 0.5 (instead of 1.0), gnolls to 2.5 (instead of 3.0), and bugbears to 5.0 (instead of 6.0).

But based on the ways the rules are written, they shouldn't be able to gain extra damage from a charge, since they are IF, not FF (page 20).

So either my calculations for infantry are incorrect in another way, or the table values are too high (because they gave an extra damage point for charging with a spear), or the rule for charges is itself wrong (for saying that only FF can get the extra point of damage).

After looking at the hobgoblin light and medium cavalry BR values, I'm now feeling confident that there must be something wrong with the formula itself. I don't think it could create the values in the table. There must be some additional missing factor (maybe mentioned elsewhere in the rules?)

In this case, both of them have identical attack sequences, uhp, and morale, so the only differences are AC, speed, and formation.

The medium cavalry has the edge on AC, by an extra 33% (4 vs 3). But the light cavalry has the edge on both speed (by 33%) and formation (by 30%). So light cavalry, by the formula in the book, should have a better BR than medium cavalry, by at least 30%. (I say "at least", since the light cavalry also have javelins and should be increasing their combat factor to account for a bonus of 0.2 times their missile value.)

I calculate hobgoblin light cavalry at 7.0 (much higher than the table's 5.0), and medium cavalry at 4.5 (a bit lower than the table's 5.0). I think my calculation feels closer to being right. One extra point of AC is not worth giving up the greater mobility and skirmishing potential of a LM unit. Is there some other benefit of FM units I'm overlooking? In some ways I hope not, since I rather like the way that ACKS gives loose-formation units so much utility.

This analysis seems like it should apply to human light and medium cavalry as well.

I think it's fine from an wages standpoint if the formula isn't meant to match up with the table values (maybe the cost differences reflect scarcity of trainable men, etc), but I feel like it's a bigger problem if using abstracted combat inverts the value of light vs medium cavalry.

Here's my attempt to duplicate the BR calculation example format on page 93, replacing all the Heavy Infantry values with Light Infantry values. This is the simplest unit in the game! If I can't even get this right, I might as well give up now.

Hopefully this will let someone spot exactly which numbers are wrong. (I still think it might be something about the combat multiplier roundings.)

EXAMPLE: Let’s calculate the BR of a unit of 120 Light Infantry A (AC 3, 6 uhp, Movement Rate 2/4/6, Formation
LF, 2 Short Sword & Shield 11+ or 2 Thrown Javelin 11+ with Range 2, ML -1).

The formula is (Unit AC) x (uhp/8) x (Morale Modifier) x (Speed Modifier) x (Formation Modifier) x (Combat Modifier) / 6.

We calculate:
» Armor Class is 3.
» Uhp/8 is (6/8), or 0.75.
» Morale Modifier is [(-1+7)/7]^0.5, or 0.93.
» Speed Modifier is (6 / 7.5)^0.33, or 0.93.
» Formation Modifier is 1.
» Melee Value is equal to the unit’s unrounded number of attacks (2.1) divided by its rounded number of attacks (2) times the maximum damage the unit can deal in a charge (2), or 2.1. As a LF unit, Missile Value is equal to (number of missile attacks) x [walking movement + missile range] / 2. For this unit, that is (2.1) x [2 + 2] /2, or 4.2.
» Since Missle Value of 4.2 is greater than Melee Value of 2.1, Combat Modifier is [4.2 + (0.05 x 2.1)], or 4.31.
» Therefore the unit’s BR is (3 x 0.75 x 0.93 x 0.93 x 1 x 4.31) / 6 = 1.39, which rounds to 1.5.

But this is supposed to be 1.0, not 1.5, based on the RAW table values.

On a per-troop basis, this comes out to a value of 0.012 BR/troop and a wage of 9 gp/troop, instead of the expected value of 0.08 BR per troop and 6 gp/troop.

Meanwhile, here's my largest failure in an attempt to calculate a known BR value: medium cavalry.

EXAMPLE: Let’s calculate the BR of a unit of 60 Medium Cavalry (AC 4, 6 uhp, Movement Rate 3/6/9, Formation
FM, 2 Lance & Shield 11+ (with bonus of 3 Hooves 8+ on a charge), ML 1).

The formula is (Unit AC) x (uhp/8) x (Morale Modifier) x (Speed Modifier) x (Formation Modifier) x (Combat Modifier) / 6.

We calculate:
» Armor Class is 4.
» Uhp/8 is (6/8), or 0.75.
» Morale Modifier is [(1+7)/7]^0.5, or 1.07.
» Speed Modifier is (9 / 9), or 1.
» Formation Modifier is 1.
» Melee Value is equal to the unit’s unrounded number of attacks (1.58) divided by its rounded number of attacks (2) times the maximum damage the unit can deal in a charge (2+3 hooves + 1 lance bonus), or 4.73. Missile value is 0, since this unit has no ranged weapon.
» Combat Modifier is just given by melee value, so this is also 4.73.
» Therefore the unit’s BR is (4 x 0.75 x 1.07 x 1 x 1 x 4.73) / 6 = 2.53, which rounds to 2.5.

This is nowhere close to the table value of 5.0. It's off by a factor of 2.

The value for unrounded melee attacks is given in an example on page 84, so I know I did that part right. And most of the other values are so simple it's hard to see how they could be wrong.

I wonder what it would look like if you calculate the medium cavalry based off its charging number of attacks, you would get:
Charging number of attacks: (60 cavalry * 2 hooves + 0.65 cleave factor * 3.5 average damage)/180 = 3.09

Added to the 1.58 primary unrounded attacks, this would give us unrounded attacks of 4.67 and rounded attacks of 5, unchanged charge damage, for a melee value of 5.606.  This would come out to 2.99, rounding to 3.

In conclusion, this did not help in any significant way.  The only helpful information I have is that some values were adjusted through playtesting (as per page 95 Final Notes), which probably accounts for the light infantry discrepancy but might not cover the medium cavalry one.

I get the impression that there were some adjustments on the order of plus/minus 0.5 or so, which amount to "rounding choices" in my mind.

But the medium cavalry thing is really bugging me, since it doesn't seem to make sense from a balance standpoint. The value of a unit of light cavalry really does seem to be higher than medium cavalry in practice (avoiding damage by falling back is better than a point of AC), and so it doesn't make sense to adjust medium cavalry away from the formula -- except maybe for the indirect purpose of rationalizing the economic system to account for scarcity. ("They should be more expensive due to greater training and expensive equipment, and BR is tied to price/wage.")

[quote="EHamilton"]

After looking at the hobgoblin light and medium cavalry BR values, I'm now feeling confident that there must be something wrong with the formula itself. I don't think it could create the values in the table. There must be some additional missing factor (maybe mentioned elsewhere in the rules?)

In this case, both of them have identical attack sequences, uhp, and morale, so the only differences are AC, speed, and formation.

The medium cavalry has the edge on AC, by an extra 33% (4 vs 3). But the light cavalry has the edge on both speed (by 33%) and formation (by 30%). So light cavalry, by the formula in the book, should have a better BR than medium cavalry, by at least 30%. (I say "at least", since the light cavalry also have javelins and should be increasing their combat factor to account for a bonus of 0.2 times their missile value.)

I calculate hobgoblin light cavalry at 7.0 (much higher than the table's 5.0), and medium cavalry at 4.5 (a bit lower than the table's 5.0). I think my calculation feels closer to being right. One extra point of AC is not worth giving up the greater mobility and skirmishing potential of a LM unit. Is there some other benefit of FM units I'm overlooking? In some ways I hope not, since I rather like the way that ACKS gives loose-formation units so much utility.

This analysis seems like it should apply to human light and medium cavalry as well.

I think it's fine from an wages standpoint if the formula isn't meant to match up with the table values (maybe the cost differences reflect scarcity of trainable men, etc), but I feel like it's a bigger problem if using abstracted combat inverts the value of light vs medium cavalry.

[/quote]

All of the values were created using that formula. I have an enormous spreadsheet with numerous tabs and it took literally months. The formula is quite complex so that's why I did it all for you in advance. Some values for particular units were then rounded based on playtesting when they proved too cheap or too expensive.

To address your point above, Formed Mounted deal damage from their mount's hooves on a charge. Light Mounted do not. You're not gaining one point of AC, you're also gaining three attacks on a charge. [On review, I saw that the Hobgoblin Light Cavalry are noted as LM but with the hoof attacks on charge, which is an error and probably the cause of your dismay here. They are Loose Mounted and do not receive the additional hoof attacks. The error was in the unit entry, not in the spreadsheet or the rule mechanic.]

Unrounded Hobgoblin Light Cavalry BR is .086. 60 x .086 is 5.16, which I rounded to 5. Unrounded Hobgoblin Med Cavalry is .095, yielding BR 5.7; I rounded to 5.5. Unrounded Human Medium Cavalry is .05, yielding BR 3. It was raised to 5 after playtesting. Unrounded Human Light Cavalry is .046 for Type A and .055 for Type B, yielding BR 2.76 to 3.3. I set it to 3.5.

The reason that Medium Cavalry had its BR raised is that in practice, the lower AC relative to the AC of Heavy Cavalry was less disadvantageous than the math suggested. Commanders tended to keep their Formed Mounted units out of range until they had the opportunity to strike, then slammed with vicious charges. Since Medium Cavalry had the same offensive power as Heavy Cavalry, this meant that you could get most of the strengths of Heavy Cavalry for much less. if I recall, the largest adjustments in the game were made to medium cavalry so that might be why you're frustrated here.

All of the beastmen BR are calculated as if they can deal an extra point of damage with spear on charge. In the original edition of the rules, they could do so. The ability for Irregular Foot to deal the extra damage was later removed because beastmen proved to be too powerful relative to their cost. If I had then reduced their BR, their cost would have also dropped and the problem wouldn't be fixed.

(Since heavy irregular beastmen tended to have 3 to 4 attacks at 10+ or better attack throws, they were substantially more likely to be able to deal 3 points of damage in one round than human troops. Since 3 points of damage is what triggers a shock roll against a 6 uhp opponent, that made a huge difference. In other words, there's an invisible "inflection point" that happens to coincide at 3 or more attacks vs. 6 or less hp. Also granting +1 spear damage on ANY successful hit made virtually every human unit break when charged by beastmen.)

There are dozens of such case-by-case adjustments throughout the rosters - it would take me hours to outline every single one. I'm honestly not sure why you are re-doing them all by hand -- that rather defeats all the energy I put into calculating and playtesting them for you! If your heart is set on playing with the raw data, I do offer access to the master spreadsheets to high-level Patreon backers so if that will save you a lot of time it might be worth the expense.

I'm trying to understand the formula for the sake of determining how to use it to determine the BR of any new units, in hopes of keeping them consistent with the old ones. I guess this suggests that there's a lot of post-formula adjustments to be made, which in turn suggests that values can be set by fiat and it's not worth sweating the details too much.

I'm... semi-tempted by the Patreon thing, though probably not with my current month's budget. Curse you and your fiendish extortion schemes!

I continue to enjoy the game, and hope you aren't offended by anything I've said here. I understand the desire to tweak values to get the right "feel".

My recommendation would be to use the formula if you find it valuable, but if the value it spits out doesn't seem to work in play to change it. That's what I do!

I didn't take you as being critical of my choices as a game designer, I just didn't comprehend what you were attempting to do. There's not much there to understand apart from the algebra. I knew approximately what the BRs "had" to be based on known wages of troops and so on. I thought of all the things that could influence a unit's BR and made a rough pass at estimating how much they mattered. It spat out some numbers.. I checked against that. I fought a bit on a battle map to see if they seemed reasonable. I adjusted. I tested. I asked my players how much they'd pay in GP for different units. And so on. It was incrementally developed with lots of back and forth. If you're familiar with how the US Military worked out its formula by armor penetration, they did it by firing a lot of guns at a lot of pieces of metal at different ranges and then writing down what happened. Then they made a formula to fit the data. You can use it to estimate penetration of new weapons versus armor. But there's no physical theory behind it in the sense of Newton's theory. It's just a curve that fits the data in a way that works most of the time. The BR formula is like that.