Drill promos analysis

[Um the Muse]

I've always felt that Drill promotions are the most interesting. A recent thread has brought them back up, so I decided to do some of the math around them. If you're mathphobic, you might want to just give this a miss. EDIT: It looks like this has all been done before, and by people much better at math than I am. Check out some of those other links below
First of all, I guess I should explain how first strikes work. A first strike is a "round of immunity". In other words, you're not affecting the roll of the dice but you get a re-roll for each first strike you have. That, in turn, means that a first strike only does something if you lose the round.

If the attacker is the one with the first strike advantage, the odds that the first strike will change the outcome of the round is D/(A+D) where D and A are the modified strength of the defender and the attacker, respectively. If the defender is the one with the first strike advantage, the odds are reverse: A/(A+D).

First strike chances work counter-intuitively, so I'm just going to spell out their relative values:
D1 is worth FS/2 (where FS=D/(A+D); see above formula)
D2 is worth FS; D1+D2 is: D1 + FS=1.5*FS
D3 is where this gets a little weird. More first strike chances hurt the odds of previous first strike chances because the odds of getting 1 first strike is the same as the odds of getting 2 first strikes. Thus, while D3 adds 2/4*FS in value, it lowers the value of D1, from FS/2 to FS/4. So, the net gain is 1/4*FS. It actually adds half the value that D1 does! The total value is FS/4+2/4*FS+FS=7/4*FS.
D4 is worth 2*FS, for a total of 15/4*FS

Most (all?) units that can get D1 start out with a FS. Just add one FS to the above results. Also, notice that the equation changes for a unit that can go up the drill promotion line and starts out with a first strike chance. For these units, D1 is actually equal to 1/3*FS (for a total of 2/3*FS) and each first strike chance when you reach D3 is 1/5*FS (so that D1+D3+the original first strike chance becomes worth 4/5*FS). Yes, that really does mean that your extra first strike chance is basically meaningless once you hit Drill III. You might want to consider a different path

So, how does this compare to Combat promotions? First, we'd need to calculate how big of a difference Combat makes. Yay, more math! Actually, this part has all ready been done and is in the War Academy. Basically, there are certain key values where your relative strengths (between your unit and theirs) pass certain thresholds. Passing these means you have to do more (fewer) rounds of combat than you would if you were a tiny bit stronger (weaker). Passing just one of these thresholds is strictly better than one FS. If the two units are more or less the same strength, you can just about guarantee that you can pass a threshold. FYI, catapults automatically get one of these free passes whenever they withdraw from combat since they never actually completely kill an enemy . One more reason why collateral damage rocks.

Conclusions and generalizations:
The most important one is that FS matters more the weaker the unit with FS is compared to its foe.
The next is that D3 is always worth exactly one FS, no matter how many first strike chances the unit started with, meaning those that start out with extra should not waste their time on Drill promotions. By the time that you get D3, D1 is absolutely worthless D1 and D3 are both weak promotions (not counting their affect on collateral).
D4 really is worth more than all three drill promotions put together.

 

[OneLeggedRhino]

One of the main benefits of Drill is exp and GG points.

Exp is determined based on the relative strength of attacker and defender. It doesn't take first strikes into account. So, for whatever range of combat odds you normally fight with, your Drill archers will get more exp and GG points than your combat archers.

 

[VoiceOfUnreason]

D3 is worth FS/3+2*FS/3+FS=2*FS (This is where is gets a little weird. More first strike chances hurt the odds of previous first strike chances)

Your math is either wrong or your notation is bad, I'm not sure which.

With Drill I, Drill II, and Drill III, you have 1 first strike (from Drill II) and 3 first strike chances. First strike chances are converted to first strikes uniformly, so on average you are looking at 3 chances -> 1.5 strikes.

So the naive math is

DI : 0.5
DII: 1.5
DIII: 2.5
DIV: 4.5

Put another way - each first strike chance adds an average of .5 first strikes.

That's the flat rate; as UtM notes, there are some diminishing returns effects to consider.

Consider Davy and Goliath, where Goliath can kill Davy in one blow, but Davy has a 1% chance of hitting Goliath in any given round.

First thing to notice is that with no first strikes on either side, the numbers are pretty small for Davy. Depending on how many decimal places you want to carry around, you can reasonably say either

(a) there's a 1% chance that Davy hits Goliath once or...
(b) there's a .99% chance that Davy hits Goliath once and a .01% chance that Davy hits Goliath twice or...
(c) there's a .99% chance that Davy hits Goliath once and a .0099% chance that Davy hits Goliath twice, and a .0001% chance that he hits Goliath three times...

For illustration, I'll use (a) here.

So with no first strikes, Davy has a 1% chance of scratching Goliath.

With 1 first strike, Davy has a 1.99% chance of scratching Goliath, almost doubling his chances.

With 2 first strikes, Davy has a 2.97% chance of scratching Goliath.

In other words, one first strike doubles Davy's chances of accomplishing something, and two first strikes triples it, but in each case there's a small diminishing returns penalty. We started at 1%, 1 first strike added .99%, a second first strike added .98%

Thus: at extremely low odds, each additional first strike adds roughly the same benefit.

Now let's turn this around, and give Goliath the first strikes

With no first strikes, he has a 99% chance of killing Davy without being scratched

With 1 first strikes, he has a 99.99% chance of killing Davy without being scratched; he's improved his chances .99%

With 2 first strikes, he has a 99.9999% chance of killing Davy without being scratched; that extra first strike is only improving his chances by .0099%

Thus: at extremely favorable odds, the returns of first strikes diminish almost immediately.

 

[barbertje]

I've always felt that Drill promotions are the most interesting. A recent thread has brought them back up, so I decided to do some of the math around them. If you're mathphobic, you might want to just give this a miss.
First of all, I guess I should explain how first strikes work. A first strike is a "round of immunity". In other words, you're not affecting the roll of the dice but you get a re-roll for each first strike you have. That, in turn, means that a first strike only does something if you lose the round.

If the attacker is the one with the first strike advantage, the odds that the first strike will change the outcome of the round is D/(A+D) where D and A are the modified strength of the defender and the attacker, respectively. If the defender is the one with the first strike advantage, the odds are reverse: A/(A+D).

First strike chances work counter-intuitively, so I'm just going to spell out their relative values:
D1 is worth FS/2 (where FS=D/(A+D); see above formula)
D2 is worth D1 + FS=1.5*FS
D3 is worth FS/3+2*FS/3+FS=2*FS (This is where is gets a little weird. More first strike chances hurt the odds of previous first strike chances)
D4 is worth D3+2*FS=4*FS

If you have drill 1+2+3, you get 1, 2, 3 or 4 First strikes with equal chances for each possibility. this is 2.5 first strikes on average. If you look at my last post in the recent thread about galleys, you can see that drill 3 has about the same effect as drill 2, and about double the effect of drill I.
drill 1+2+3+4 gives 3-6 first strikes, so 4.5 on average.

 

[Mantic0re]

There was an analysis a few years ago (I think VoU contributed) that looked at the Drill line from a slightly different perspective. Instead of talking solely about combat chances (which the human brain does not handle particularly well) the thread considered the expected health of a unit after combat, and a series of combats.

iirc, the Drill line performed very well at preserving unit HP for multiple combat rounds (defense) and quick mobile attacks (less healing).


A very simplified (and rough) conclusion posted late in the thread was that Drill is best used for clean-up duties after siege combat or on units facing earlier era material.


This isn't the thread I remember but close enough, contains graphs / quick conclusions / discussion:
http://forums.civfanatics.com/showthread.php?t=301160&highlight=drill

 

[coanda]

http://forums.civfanatics.com/showthread.php?p=7720659#post7720659
Seems like a useful post to include in this discussion, so I guess I'll just stick it here.

With 2 first strikes, he has a 99.9999% chance of killing Davy without being scratched; that extra first strike is only improving his chances by .0099%

Thus: at extremely favorable odds, the returns of first strikes diminish almost immediately.

Yees... but not quite. A couple points. First, it takes truly ridiculously favorable odds to have a 99% chance of winning without even taking damage against undamaged units (you can express it precisely: you will need to win 2 rounds of combat, at (attacker strength / total strength) odds in each round.
So...
.99 <= (A/(A+D))^2
.995 <= A/(A+D)
.995D <= .005A
Which is to say, you'd need an attacker with 200 times the strength of the defender if both units were starting undamaged. It's significantly nicer if the other unit can start damaged, say from a cannon attacking first. A modern armor attacking a 5% health war chariot has about a 99% chance of winning undamaged, for reference. A more plausible unbalanced matchup (e.g., axeman defending against warrior on flat terrain) has about a 50-50 chance of winning undamaged.

Second point: Always compare to other options. You may not get much of a boost from taking Drill II (which, if the other unit is starting below 61HP and isn't immune to FS, pushes you from 99% chances of being undamaged to 99.995%), but it's a better boost than it'll get from taking Combat II (going from 99% to 99.2%). If for some reason your units are pulling this far ahead in strength from the AIs, (1) congratulations on winning your game, and (2) Drill is the way to go. Note that this works the other way too - giving the weaker units Drill II will do more to increase his odds of at least doing a little damage before the battle ends than giving the poor guy Combat II.
Conceptually, look at it this way: highly unbalanced battles will generally be over in just a couple rounds of combat, while close matches will tend to drag on for more rounds (as neither side can win quickly). If there are many rounds of combat, a promotion that effects every round of combat is more significant; if there are few rounds of combat, a promotion that adds extra chances to do damage adds more.

 

[Um the Muse]

One of the main benefits of Drill is exp and GG points.

Exp is determined based on the relative strength of attacker and defender. It doesn't take first strikes into account. So, for whatever range of combat odds you normally fight with, your Drill archers will get more exp and GG points than your combat archers.

Thank you. That's a very good point that I didn't realize.

Your math is either wrong or your notation is bad, I'm not sure which.

Bad math, I'm afraid. I only sometimes remembered about the chance that none of the chances pan out. Thanks for pointing that out. I hope I got everything right this time

 

[VoiceOfUnreason]

I hope I got everything right this time

Doesn't look like it.

1 First strike chance -> 0/2 + 1/2 = 1/2

3 First strike chances -> 0/4 + 1/4 + 2/4 + 3/4 = 6/4 = 3/2

D1 = 1/2
D2 = 1/2 + 1
D3 = 3/2 + 1
D4 = 3/2 + 3

 

[D666]

I always thought that Drill was made weaker than Combat so as to 'pay for' the reduced collateral damage effect. Which makes it kind of lame for galleys.

OTOH, it definitely has the coolness edge, "combat, baby, combat" just doesn't have the same ring.

 

[Keilah]

one nice thing about combat is that combat 3 unlocks march which is can be nice if you've got 1 or a few uber-units that you'd like to see combat as often as possible.

Drill is awesome, you get sooooooo much xp and you are way more likely to end a fight at high-full HP with drill than strength. Also the protection from collateral and +10% vs mounted is IMO better than healing extra in neutral/enemy lands.

 

[Iranon]

Also: Most test scenarios assume healthy units. Having either side wounded increases the usefulness of first strikes relative to strength boosts.

My experience was that compared to Combat, Drill is bad for winning fair fights, but good at almost everything else (chance to wear down much superior opponents, preserve health when you are advantaged, cleanup after collateral, resisting collateral, xp farming...)

 

[VoiceOfUnreason]

Having either side wounded increases the usefulness of first strikes relative to strength boosts.

That doesn't sound right to me at all.

Yes, in some (most?) situations where a combat promotion pushes you across a jump point when healthy units fight, lack of health on one side will slide the jump point outside the range of the promotion.

However, the opposite case holds as well -- when the jump point is outside the range of the combat promotion for healthy units, wounding one of the units can slide the jump point into the promotion range.


I'll buy that hit points changes the relative value of first strikes and strength boosts, but if you are going to claim that the effect always favors the first strikes, I think you're going to have to show some work.

 

[coanda]

That doesn't sound right to me at all.

Yes, in some (most?) situations where a combat promotion pushes you across a jump point when healthy units fight, lack of health on one side will slide the jump point outside the range of the promotion.

However, the opposite case holds as well -- when the jump point is outside the range of the combat promotion for healthy units, wounding one of the units can slide the jump point into the promotion range.


I'll buy that hit points changes the relative value of first strikes and strength boosts, but if you are going to claim that the effect always favors the first strikes, I think you're going to have to show some work.

I doubt Drill is always improved vis-a-vis Combat promos at lower HP (and I'm not sure Iranon claimed that in the first place). But when units have lower HP there will be fewer rounds until the combat is over. Fewer rounds until the combat is over = fewer rounds in which the per-round bonus of Combat promos can aggregate to outperform the flat bonus of Drill promos. Which means generally lowering HP should be more likely to favor a Drill promo rather than a Combat promo.

 

[Keilah]

yah that makes a lot of sense, imagine if both units had 1hp - drill would be >>> strength.