Combat Explained....

Dragonlord said:

To the contrary: they're both saying the same thing with different words: First Strike makes a difference to winning when the units are at equal or almost equal strength

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The math of the post I quoted reveals first strike to be most valuable to units that have a high strength ratio to the units they fight. Because they can steamroll (or brickwall, which I guess you could term the defensive equivalent).

DeepO does say that first strikes are a benefit in a close fight, but it's a smaller perk than in a not-close fight. He emphasizes the greater benefit in a mismatch.

Arathorn says they make the most dramatic difference in close fights.

Those are differing (not to be construed "opposite") opinions.

Good thread!

I think the article you posted is more concerned with HP left over after the battle than the odds of winning. The battle odds involving a ~2 strength ratio can't really be improved but I'm sure the odds of winning without HP loss increase greatly with first strikes. For equal strength ratio, with first strikes or not you're pretty much going to loss HP unless you can survive ~5 coin flips in a row.

No offense to DeepO, but I think he's wrong. Let's look at some numbers.

Close Fight

I'm going to take "close fight" to mean 1.1 vs. 1. We can disagree on what close fight means, but a 10% difference in strength is reasonably minor and not too infrequent. We can look at the difference first strike makes in this case and compare it with a "blow-out" fight. For the sake of reading (and without loss of generality), we'll assume the attacker is the stronger unit.

No First Strike

Attacker wins: 68.0%
Attacker wins flawlessly (0 damage): 3.9%
Average hp remaining when winning: 44.5
Average hp of defender remaining when attacker loses: 44.1
Average damage taken: 69.7 hps
Average damage done: 85.9 hps

Attacker 1 first strike

Attacker wins: 73.8%
Attacker wins flawlessly (0 damage): 5.8%
Average hp remaining when winning: 48.6
Average hp of defender remaining when attacker loses: 41.0
Average damage taken: 64.3 hps
Average damage done: 89.3 hps

Defender 1 first strike

Attacker wins: 62.2%
Attacker wins flawlessly (0 damage): 2.0%
Average hp remaining when winning: 40.1
Average hp of defender remaining when attacker loses: 46.2
Average damage taken: 75.1 hps
Average damage done: 82.5 hps

Attacker +10% strength (1.21 vs. 1)

Attacker wins: 73.3%
Attacker wins flawlessly (0 damage): 4.9%
Average hp remaining when winning: 49.5
Average hp of defender remaining when attacker loses: 40.1
Average damage taken: 63.7 hps
Average damage done: 89.3 hps

Conclusions for close

A single first strike makes about a pure 2% difference in surviving unscathed (taking no damage). But that's relatively about a 50% difference -- an attacker is 1.5 times as likely to come out without damage with a single first strike advantage as without. And the difference in winning is notable, too -- a pure 5.8%, which is a relative 8.5%. These are pretty significant differences. First strike also makes about a 5 hp difference in the "average" case -- 5 more hps are done and 5 fewer hps are taken. That has healing time repercussions.

I added the +10% strength for comparison's sake. Note that the single first strike is better across-the-board here.

Blowout Fight

Again, you can argue with the definition, but for simplicity of example's sake, I took a blowout fight to mean 2 vs 1. So the attacker (again, for simplicity) is twice as strong as the defender.

No first strike

Attacker wins: 99.1%
Attacker wins flawlessly (0 damage): 19.8%
Average hp remaining when winning: 72.8
Average hp of defender remaining when attacker loses: 30.3
Average damage taken: 27.9 hps
Average damage done: 99.7 hps

Attacker 1 first strike

Attacker wins: 99.4%
Attacker wins flawlessly (0 damage): 26.3%
Average hp remaining when winning: 77.1
Average hp of defender remaining when attacker loses: 28.1
Average damage taken: 23.3 hps
Average damage done: 99.8 hps

Defender 1 first strike

Attacker wins: 98.8%
Attacker wins flawlessly (0 damage): 13.2%
Average hp remaining when winning: 68.5
Average hp of defender remaining when attacker loses: 31.3
Average damage taken: 32.3 hps
Average damage done: 99.6 hps

Attacker +10% strength (2.2 vs 1)

Attacker wins: 99.4%
Attacker wins flawlessly (0 damage): 22.3%
Average hp remaining when winning: 76.9
Average hp of defender remaining when attacker loses: 27.4
Average damage taken: 23.6 hps
Average damage done: 99.8 hps

Conclusions for blowout

A single first strike makes about a pure 6.5% difference in surviving unscathed (taking no damage). But that's relatively only about a 33% difference -- an attacker is 1.33 times as likely to come out without damage with a single first strike advantage as without. The difference in winning is pretty microscopic -- a pure 0.3%, which is a relative 0.3%. These are differences that will be very hard to detect. First strike still makes about a 5 hp difference in the "average" case -- 0.1 more hps are done but 4.6 fewer hps are taken. That has healing time repercussions, but not as pronounced as in the close fight case.

Again, though, we note that first strike is better than +10% strength universally. It's better in every category we noted.

Overall Conclusions

First strike does make a bigger pure percentage difference of coming out of a battle without injury in the blowout case. If your units are WAY stronger and you want to be blitzing a lot and care almost exclusively about not being injured at all, first strike helps. I didn't show the numbers, but it helps more than an extra 10% of strength, too (even more than an extra 25% strength, but, again, ONLY for coming out completely uninjured).

When looking at overall damage taken, first strike is quite minor in the blowout case, even moreso than in the "close" case. Total expected healing time (what I find to be the most compelling statistic) is affected more by first strike in the close case than in the blowout case.

Also, first strike is quite compelling. It's better than +10% strength in these cases. Of course, if the +10% strength can put you over a "jump point", it's going to clearly be better. If not, though, first strike is pretty strong. It just makes more difference in the "close" case than the blowout case.

Arathorn

First post updated -- including a graph of various combat odds with first strikes.

I'm working on something about damage done/taken and # of units of x strength that will be expected to kill a unit of y strength, but those will be a while. It's hard to do too much over lunch here at work.

Arathorn

The conclusion were not different. Lord Chambers was just confusing to differnt points both were making. First strike can affect 2 things:

Chance of Winning:
DeepO: Only good if close strength makes little difference in lopsided fights
Arathorn: Only good if close strength makes little difference in lopsided fights

Maintaining health after fight:
DeepO: In lopsided fights the heavily favored unit will first strike will sustain great reduced health loss (possible none).
Arathorn: In lopsided fights the heavily favored unit will first strike will sustain great reduced health loss (possible none).


Both uses are of great importance at various times of the game, and the posts on both sites were great.

Takatok

good aricle but seems not clear about combat promotion
the bonus is just influenced by the strength when the unit is born such like
one knight with combat 4 promotion now has only 1.1strength but at the battle it will be 1.1+40%*10=5.1 so it can be more powerful even when it is
not healthy, Yet the combat level si very improtant when your unit grow up
PS: I'm a new comer to here from china

I don't think you're right about combat promotions. They only seem to affect the unit's CURRENT strength, not its overall total strength. So, an injured knight with combat 4 promotion with 1.1 strength will only fight at 1.1*1.4 = 1.51, not at 5.1. But I should verify to be certain.

Arathorn

Yeah combat promotions work off the current strength. I had a 0.1str swordsman with Combat 3 and the odds when I dragged him over a barbarian sword was 0.1 vs 6.0

Do you have any information on how XP is assigned? I haven't been able to find any details in the manual. (Not that I've looked very hard. )

Fantastic article, by the way. Thanks.

Do you have any information on how XP is assigned?

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Sorry, not yet. It's on the "To do" list, but it's quite a ways down at this point. All that's known is the hard cap of 10 vs. barbs (you can't get more than 10 xps by fighting barbs), and vague discussions of "more" for attacking and "more" for winning upset battles. And something vague about less for dominating victories.

Arathorn

Hi All -

I've enjoyed reading the info in the forum - thanks. Based on Arathorn's equations for damage and the odds of winning an individual round as I understand them, I put together a Monte Carlo simulation in Excel. Basically it runs 100,000 simulated encounters for each value of R (from 1.00 to 2.00 in 0.01 increments). The odds of winning are approximated by the number of times (out of 100,000) that the attacker was successful, by being the first to "score" 100 points of damage. The curve plots the odds of winning as a function of R ratio, and shows some breakpoint features similar to what has been described... I only spent a couple hours on it, so please don't take it as gospel. If anybody's interested, please PM me, and I'll email the spreadsheet.

- Pete

Short Timer said:

The curve plots the odds of winning as a function of R ratio, and shows some breakpoint features similar to what has been described.

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Nice work. What does the graph look like below 50%? Is it a 180-degree rotation around [1.00,50%] ?

Great analysis, thanks, Arathorn!

The first conclusion I can draw is that we should try all the means to get R>1 in the 1st place. This should become much harder at higher levels.

The 2nd idea is to get every new unit at least 6 XP: 4 from barracks, 2 from vassalage or theocracy. 6 XP means 2 promotions before battle, giving much larger chance to make R>1 than 4 XP.

The 3rd: could I say that combat promotions are in general the best ones? Not only because they are generically useful, but also because their effect is not necessarily worse than a specific +20% or +25% promotion. A classical example is sword vs. archer in a hill city. The defensive bonus gotten by archer is 50% from city, 25%+25% from hill, 25% from fortifying, altogether 125%, then reduced by sword's born city attack 10%, therefore 115%. Now, if the sword has combat 1, A=6*(1+0.1)=6.6, D=3*(1+1.15)=6.45, R=A/D=1.023. OTOH, if it's city raider 1, A=6, D=3*(1+1.15-0.2)=5.85, R=1.026. The net difference in combat result is rather small, but combat 1 will be also useful when not attacking city.

Mathematically, it's comparing (1+a)/(1+d) and 1/(1+d-b), where a is the combat promotion bonus, d is total defensive bonus, b is specific promotion bonus. This is assuming d-b>=0, which is in most cases true. To make the 2 things equal, we get (1+a)(1+d-b)=1+d, therefore a(1+d)=(1+a)b, b=a(1+d)/(1+a), or b/a=(1+d)/(1+a). So, the bigger d is, the bigger we need b/a to make specific promotion better. In another word, the smaller d is, the more profitable is specific promotion.

When to begin assign drill promotion (first strike)? I guess it should be after making sure you get a large enough R, then you mainly care about remaining health. As you show, a first strike helps more than combat 1 when R=1.1, but that's because 1.21 from 1.1 doesn't make a jump. And if the defender gets combat 1 rather than drill 1, he will make R=1, thus turn the tide. Before getting large R, I guess it's still wise to add more strength than first strike.

For defense, you talk briefly about forests/jungle. But what happens when your defending unit is on a forest/jungle hill? (Or a coastal tile.) Do you get the sum of 25%+50% (or 10%+50%) or do you just get the max for that particular tile?

Same question with regards to applying a fort to a tile. Does a fort on a hill grant 25%+25%, or just 25%?

Terrain bonuses stack, so a forested hill is worth 75% defense (+25% for hill, +50% for forest). I should really write those up in more detail. This is all work in progress, though, and there are so many things on my "to do" list just for combat, it's not even funny.

I don't know how forts work for certain yet. I'd guess they stack as well, but that's just a guess until I can do some actual testing on it.

Arathorn

One question, are you certain that the bonuses are applied to the defender, because that makes for an interesting situation

assume that unit 1 has a net +25 v. unit 2 (unit 1s bonus - unit 2's bonus)

If unit 1 attacks, then the R in unit 1s favor goes up by 33% (the defender is reduced to 75% and there fore the attackers bonus is the inverse 3/4 -> 4/3)

If unit 2 attacks the R in unit 1s favor only goes up by 25% (unit 1 is the defender so the bonus applies dirsctly to it.)

This means that if
1) combat is inevitable
2) either unit has a net bonus against the other unit
3) there are no 'defender only' bonuses (fortification, culture, terrain, garrison)

You always want to be the attacker

especially leading to interesting situations, for example Quechas with 100% v archers reduce an archer to 0 strength if they attack it while it has no defense bonuses (so a Quecha/Pikeman/Explorer v. Archer/Knight/Animal unfortified in the open field will kill its target and suffer no damage)

If unit 1 attacks, then the R in unit 1s favor goes up by 33% (the defender is reduced to 75% and there fore the attackers bonus is the inverse 3/4 -> 4/3)

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Nope. For "negative" bonuses for the defender, the absolute value of it is added to 100% and the defender's strength is divided by the total. So, in this case, the defender has a -25% bonus, so its strength is divided by 1.25. Coincidentally (no, not really), that's the same as multiplying the defender's strength .8 (4/5). That leads to R = 1/.8 = 1.25. Exactly the same!!! It's only when there ARE other defender bonuses that the system doesn't do exactly what it claims.

So, that quencha attacking an archer in the open will show a strength of 2 vs. 1.5. The archer's defensive bonus is -100%, but to prevent the situation you talked about, negative percentages don't really subtract. The archer's strength is 3/(1+1) = 3/2 = 1.5. That's the same as if the quencha were given its 100% bonus to be 4 on 3, like you'd expect.

Put that archer in a forest, though, and you get a -50% defender bonus, so the archer's strength is 3/(1.5) = 2 and you get 2 vs. 2, which is worse for the attacker than the expected case of 4 vs 3.75.

Fortify the archer on a hill in a city, and the defender bonus is 25% (fortification) + 50% (hill for archer) + 50% (archer city bonus) - 100% (quencha bonus) = 25%. The attack is 2 vs. 3.75 or R = .53. The "expected value" is 4 vs. 6.75 or R = .59, so the attacker is again worse off than the way the desciption normally would lead one to believe.

HTH,
Arathorn

Arathorn said:

Nope. For "negative" bonuses for the defender, the absolute value of it is added to 100% and the defender's strength is divided by the total. So, in this case, the defender has a -25% bonus, so its strength is divided by 1.25. Coincidentally (no, not really), that's the same as multiplying the defender's strength .8 (4/5). That leads to R = 1/.8 = 1.25. Exactly the same!!! It's only when there ARE other defender bonuses that the system doesn't do exactly what it claims.

So, that quencha attacking an archer in the open will show a strength of 2 vs. 1.5. The archer's defensive bonus is -100%, but to prevent the situation you talked about, negative percentages don't really subtract. The archer's strength is 3/(1+1) = 3/2 = 1.5. That's the same as if the quencha were given its 100% bonus to be 4 on 3, like you'd expect.

Put that archer in a forest, though, and you get a -50% defender bonus, so the archer's strength is 3/(1.5) = 2 and you get 2 vs. 2, which is worse for the attacker than the expected case of 4 vs 3.75.

Fortify the archer on a hill in a city, and the defender bonus is 25% (fortification) + 50% (hill for archer) + 50% (archer city bonus) - 100% (quencha bonus) = 25%. The attack is 2 vs. 3.75 or R = .53. The "expected value" is 4 vs. 6.75 or R = .59, so the attacker is again worse off than the way the desciption normally would lead one to believe.

HTH,
Arathorn

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I see, makes sense. (in some sense making it easier because its only about adding/subtracting all the necessary bonuses.)

Krikkitone said:

I see, makes sense. (in some sense making it easier because its only about adding/subtracting all the necessary bonuses.)

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Yes, once you get past the situations when a "negative bonus" is applied to a defender (and the subsequent formula change), it does make it easier.