And now I know that I'm wrong in doing so, so could someone, please, tell me how does the Hit Points number and the Firepower number get into the equations?
What are their effects?

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Scouse Git[1] -- git1@scousers.net

unit strength = attack (or defense) strength x HP x FP

For example, a howitzer attacks as 12 x 2 x 3 = 72. Mechanized infantry defend as 6 x 3 x 1 = 18.

You can figure in the multipliers for veteran status, terrain, fortifications, etc., at any point because all the factors are multipliers.

Note that this will give you the relative strength of the units, not the chance of winning the battle. The stronger unit has a higher probability of winning the battle than you'd think just by comparing the unit strengths. For example, a horseman (2 x 1 x 1 = 2) attacking a warrior (1 x 1 x 1 = 1) has somewhat better than a 2-to-1 chance of winning.

On average, the winning unit will lose strength proportional to the losing unit's share of this equation. For example, horsemen attacking warriors will, on average, lose 1/2 of their strength. Chariots attacking warriors will lose, on average, 1/3 of their strength. Howitzers attacking fortified mech infantry will lose 27/72 of their strength. This varies widely through random variation, though.

No, this is not handy! This is wrong.

See Marquis' ../../Forum1/HTML/001944.html?25. But he didn't add an example yet, see Civ2 General: What is the difference between firepower and attack points? (may be on second page of the forum now), posted April 10, 2001 14:50.

You need two numbers to describe combat abilities of a unit:

quote: Originally posted by deebest But FOR ALL PRACTICAL PURPOSES, my formula is accurate.

a) combat (see info: Combat Modifiers and Calculation for the definition of "combat")
warrior vs. vet warrior:
wins of combat are divided with a ratio 1 : 2, not 1 : 1.5
A big difference!

Second example: if the defenders are veteran pikemen, they get another 1.5 bonus for an unmodified defense strength of 10.125, and a modified strength of 14.25(!). So I would expect to lose two elephants and have a third damaged to about 50% for each defending pikeman.

quote: Originally posted by DaveV on 04-20-2001 01:41 PM ...with a unmodified defense strength of 6.75 ... Since my elephants attack at 6, the modified defense strength is 7.5. ...for an unmodified defense strength of 10.125, and a modified strength of 14.25(!).

DaveV,

I think I understand your unmodified attack and defense strengths. They seem to be just the debeest method and seem to include every factor I can think of. How are you getting your "modified" defense strengths?

Marquis de Sodaq's accurate but complex SUMn(COMB) method favors combat value over hitpoints. I assume Sodaq's method favors the vet pikemen over newbie musketeers when defending against elephants.

debeest's method ranks them as equivalent:
pikemen = 2 def * 2 vsMount * 1.5 vet * 1.5 river * 1.5 fortified = 13.5 debeest strength
musketeers = 3 def * 2 doubleHitPoints * 1.5 river * 1.5 fortified = 13.5 debeest strength

Your method ranks the newbie musketeers slightly better:
pikemen = 14.25 DaveV strength
musketeers = 15 DaveV strength

DaveV's formula resembles the true odds calculation for each combat round, but not for the total battle. As such, it also reflects the likely winner, and tends more strongly in the direction of the odds.

The maddeningly complex looking formula is actually workable with a calculator, it just takes a large amount of key punching to finish. What is great about it is that it stands up to playtesting, time and again. A good bonus is that it also accounts for wounded units.

I will soon include in the Combat thread MarkoPolo's example comparing artillery and armor versus riflemen. First I'll try to put together either an Excel spreadsheet or a VB macro to do the number crunching (I've contacted Eggman about getting his, but have not yet heard back). If that is successful, I'll take the time to make an actual combat odds matrix. Like a multiplication table, but civ units in place of numbers. Then nobody will need to think at all, they can just peek at the odds posted next to their Oedo year list...

Yes, that's exactly what I'm saying. If you just want to know which unit is stronger, my formula is quick, easy, and functionally accurate; if you want to know the odds precisely, you have to do complicated statistics. Have I not stated emphatically enough that the relative strength of the units is not the same as their relative frequency of winning a battle? The stronger unit has a disproportionately greater likelihood of winning the battle, because multiple rounds reduce the effects of random variation. Therefore, for practical purposes, I will choose to fight battles where I have any substantial edge (say, 3:2, where I might win 90%), and avoid battles where I have any disadvantage or even a very small advantage (say, 8:7, where I would lose nearly half). How often would I really need to know the precise odds?

My formula does'nt take into account the tie-goes-to-the-defense advantage, and that does seem to matter more than I had thought. I wonder if multiplying the defender's strength by 9/7 would balance that?

"A good bonus is that it also accounts for wounded units."

My formula can also account for wounded units. Just plug in an estimate of the unit's remaining % strength as a multiplier for its HP. So simple.

quote: Originally posted by debeest I wonder if multiplying the defender's strength by 9/7 would balance that?

Sorry , I must repeat my disagreement. I will aim attention to your former post here:

quote: Originally posted by debeest, posted April 19, 2001 01:49 For example, horsemen attacking warriors will, on average, lose 1/2 of their strength.

DaveV, I can't see how you derived your formula. As a representation of the likelihood of winning the BATTLE, it seems fairly good, but I don't see the specific logic behind it. Is it calculated based on theoretical calculation, or is it estimated based on observation?

quote: Originally posted by debeest on 04-23-2001 02:35 AM DaveV, I can't see how you derived your formula. As a representation of the likelihood of winning the BATTLE, it seems fairly good, but I don't see the specific logic behind it. Is it calculated based on theoretical calculation, or is it estimated based on observation?

I didn't derive it; I'm indebted to Xin Yu, who pointed it out and convinced me of its validity. The stronger unit bonus is mathematically based on the battle odds formula, not just an empirical estimate.

quote: I note that several people in this thread have identified the pikeman-versus-mounted bonus as 1.5, whereas someone else gave it as 2, which is what the Civilopedia says and what I've always assumed to be true. (I almost never build them, so I haven't had much opportunity to evaluate their strength.) (I tried cheat-testing by giving all the units 10-fold HP, and reached the conclusion that the bonus didn't actually exist at all; then I realized that it only applied against 1-HP units. The perils of modifying any condition at all....) Is the Civilopedia wrong on the 2-fold pikemen bonus, as it is on so many other points?

It was I who repeated the misinformation that pikemen get a x2 bonus against mounted units. The Civilopedia is obviously AI propaganda intended to confuse humans.

Marquis de Sodaq,

Your goal of making a cheat sheet for combat odds is laudable. However, when I considered making one for just full strength ancient units, I found that once you include terrain, vet ,etc. modifiers there are 250 or so possible attack and defend strength combinations! And this was after dropping duplicates.

I don't think Xin Yu's formula

strongUnit+(strongUnit-weakUnit) vs. weakUnit

is accurate; it's just more accurate than a raw

combatValue*hitpoints*firepower

formula.

Xin Yu's formula is going in the right direction in that the difference in the two units' strengths is more important than straight numbers would lead one to believe. However, according to Marquis de Sodaq's combinatorial equation as I understand it, the combat values are more important than hitpoint & firepower values. Any realistic simplification of the combinatorial formula has to treat combat value and firepower/hitpoints differently.

My guess is that an approximation of strength would be more like

(combatValue^2)*firepower*hitpoints

Strength Ratio=[Strong+(Strong-Weak)]*HP_Strong*FP_Strong/[Weak*HP_Weak*FP_Weak]

Roughly speaking, if Strength Ratio>1, then the strong unit wins, if <1, then the weak unit wins.

Example: 'a=8 hp=1 fp=3' unit attack 'd=3 hp=2 fp=1' unit behind city walls on forest:

Strong=3*3*1.5=13.5, HP_strong=2, FP_strong=1 (defending unit);
Weak=8, HP_weak=1, FP_weak=3 (attacking unit).

Strength Ratio=[(13.5+5.5)*2*1]/(8*3*1)=38/24=19/12. Defense unit wins.

Edward, DeBeest, et al, instead of squaring the attack or defense value, the real calculation multiplies by 8. This treats it differently than the HP and FP. For simplicity (who's going to sit a calculate the probability during game time, anyway?), the calculation of small p is a good indicator - it gives the odds of the attacker winning each combat round. Higher odds mean higher chance of winning. See the combat thread for other considerations - namely that more HP and FP and higher attack/defense values tend to favor the stronger unit.

Stronger attacker: p = ((Attack * 8) - 1)/(2 * (Defend * 8)
Stronger defender (or equal): p = 1 - ((Defend * 8) + 1)/(2 * (Attack * 8)

quote: Originally posted by Xin Yu April 25 Strength Ratio=[Strong+(Strong-Weak)]*HP_Strong*FP_Strong/[Weak*HP_Weak*FP_Weak]

Ah, I see. Compare the combat value difference before multiplying in the hitpoints and firepower. This would give better answers. I misunderstood that when I read DaveV's April 20 post. So a catapult attacking a musketeer would be:
catapult combat value = 6 att
musketeer combat value = 3 def
[catapultCV + (catapultCV - musketeerCV)] * 1 singleHP vs. musketeerCV * 2 doubleHP
[6 + (3)] vs. 3 * 2
9 vs. 6
Not bad odds for the catapult (which I incorrectly chided East Street Trader for using a while back. I stand shamefully corrected.)

Versus my favorite method - the easy to calculate debeest ratings:
catapult = 6 att * 1 singleHP = 6
musketeer = 3 def * 2 doubleHP = 6
Equal ('though incorrect) odds.

However, if I understand it correctly, Xin Yu's formula ignores the tie-to-the-defender advantage. For example, suppose a 3-strength unit attacks a 2-strength unit. Xin Yu's formula says that the exact probability per round is (3 + (3-2))/2, or 4/2. The two-dice method says (3*8) * (2*8) = 384 possible combinations. Of those, the weaker unit gets a higher score 15*(15+1)/2 = 120 times. The scores are equal 16 times. The stronger unit gets a higher score the other 248 times. If the tie scores were split evenly, the totals would be 256 and 128, Xin Yu's ratio. So, Xin Yu's formula (somehow) gives the exact probability of each unit's winning a single round, EXCEPT for a modest-to-negligible defender advantage. Is that right, Xin?

And then, of course, the combinatorials formula kicks in and the stronger unit whups ass very disproportionately, because the randomness reduces toward zero the more hit points (divided by firepower) are involved in the battle.

Stronger attacker: p = ((Attack * 8) - 1)/(2 * (Defend * 8), or,
p = 23/32, or almost 3/4. Xin Yu's 2/3 (4:2) is similar, but far from exactly the same.

"So, Xin Yu's formula (somehow) gives the exact probability of each unit's winning a single round" - It looks like what Xin's formula does is not find the single round odds, but approximate the big P. In this case, at least, Xin Yu's formula gives the ratio of favorable possible outcomes of the battle.

If this applies generally to any combat, it could be a good simplification to use. Obviously, testing Xin's number against P with a slew of different combat ratios would show if this is true. If so, small p and Xin's number would be opposite sides of the same coin.

quote: Originally posted by Marquis de Sodaq on 05-02-2001 10:31 AM Stronger attacker: p = ((Attack * 8) - 1)/(2 * (Defend * 8), or, p = 23/32, or almost 3/4. Xin Yu's 2/3 (4:2) is similar, but far from exactly the same.

Actually, isn't the stronger attacker formula p = 1 - (Defend*8+1)/(Attack*8*2)?

Xin Yu's formula is an inexact ('though quite good) guess as the victor of a whole battle. It does NOT factor in the defender tie advantage.

Marquis de Sodaq's recently posted formula is exactly correct for the victor of a single mini-combat round. It does (as it must) factor in the defender tie advantage.

quote: Originally posted by debeest on 05-02-2001 03:35 AM So, Xin Yu's formula (somehow) gives the exact probability of each unit's winning a single round, EXCEPT for a modest-to-negligible defender advantage. Is that right, Xin?

Yes! The defense bonus diminishes as attack and defense (adjusted for terrain, etc.) factor increases. Roughly speaking, the chance for a tie is

minimum (attack, defense)/(8* attack * defense).

Which means, if the attack factor is weaker than the defense factor, then the chance for a tie is

1/(8 * defense);

if the attack factor is stronger than the defense factor, then the chance for a tie is

1/(8 * attack).

If you want an exact formula, the attack side should substract, and the defense side should add, half of this tie value, or minimum (attack, defense)/(16* attack * defense), from one-round odds (not the odds ratio given by my formula, but the odds. Odds ratio= win/lose, odds=win/(win+lose), they are different).

How big is the chance of a tie? A warrior against a warrior, both non-vet, the defender on grass unfortified will give 1/16, or 0.0625; if either of the two is a vet then the chance gets down to 1/24, or 0.0417. I think we can ignore it if the chance of tie is below 0.01 (happens when either the attack or defense (adjusted for bonus) is 7 or up), unless the attack factor and defense factor are equal.

quote: Originally posted by Edward on 05-02-2001 12:27 PM Yes. As Marquis de Sodaq pointed out... Xin Yu's formula is an inexact ('though quite good) guess as the victor of a whole battle.

Arrrgh. No, it's the odds for a single round. A slight advantage in the Xin Yu formula will result in a much greater advantage in a battle.

quote: It does NOT factor in the defender tie advantage.

If you're going for accuracy, subtract 1/8 from the attacker and add 1/8 to the defender. So in the example above, 3 strength unit attacking 2 strength unit, the odds are 3.875:2.125, or 31/48, exactly the same result as the other formula. I think it's easier to calculate without the fractions, and close enough if you bear in mind that the defender has the advantage in close contests (see your point below).

quote: Yes, they give similar results, but then again, all of the proposed formulas give somewhat similar results. (A good reason for one to use a formula that's inaccurate but simple for one to calculate on-the-fly.)