wishmaster3 wrote:
Stronger than 2x axes + 1x GB axes I presume you meanArmour_US wrote:
So yes, 2x axes + 1x GB maces is stronger than 2x axes + 1x GB maces.
Good work though!
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This isn't accurate in a real hammer, but the strength boost for each is an identical percentage, so it doesn't really matter.
Smithy boosts are not a flat rate, they vary from troop to troop. It’s partially a percentage of the base attack, and partially a flat boost based on wheat consumed. So troops that are less efficient in attack/upkeep get a bigger boost.
Level 20 clubs have 52.36 attack so a 30.9% boost
Level 20 axes have 75.36 attack so only a 25.6% boost
This leads to all sorts of interesting facts:
EI/EC are more powerful than steppes/marauders
Level 20 scouts have some attack even though level 0 scouts have none
Level 20 Teuton scouts are about 10% worse at scouting and 20% worse at detecting a scout attempt than any other tribe. 
Tineren wrote:
This isn't accurate in a real hammer, but the strength boost for each is an identical percentage, so it doesn't really matter.
Level 20 clubs have 52.36 attack so a 30.9% boost
Level 20 axes have 75.36 attack so only a 25.6% boost
This leads to all sorts of interesting facts:
EI/EC are more powerful than steppes/marauders
Level 20 scouts have some attack even though level 0 scouts have none
Level 20 Teuton scouts are about 10% worse at scouting and 20% worse at detecting a scout attempt than any other tribe.
So I was originally just planning to say, "Bias the numbers a couple toward maces, it'll be fine!" But I plugged in a test case just to be certain. To my surprise, the difference is pretty extreme (about 50% longer for axe payoff). The two are already so close in damage output / time that little differences make a big change for simulations like this. So I redid the whole thing. If a kind mod wouldn't mind copypasting this over the old post, I'd be ever grateful
Armour_US wrote:
This is going to a be a long post. Jacopo got me thinking about an ageold argument. I've always had my beliefs, but I've never done the math. Today, I decided to do the math.
Maces or Axes?
Axes vs maces is an interesting argument that I've seen thrown around the forums since the beginning of the game, but I've yet to see someone do out the math. What makes a hammer "better" than another hammer? When it comes down to it, it's fighting strength that matters. In a perfect world, everyone trains maces. They gain a good bit more strength per hour, so it makes sense to only make them. They are also considerably cheaper to keep queued than axes. They're faster. They raid better. Some like to use axes for hammerstands, but that's a hybrid strategy so it's hard to quantify so I'll ignore that for this analysis. The only reason that people use axes is for the improved attack / wheat consumption. This (besides the defense) is quite literally the ONLY advantage axes hold over maces. Players who can keep maces/TK queued 24/7 with a small trainer in the GB/GS all game are few and few between. It typically requires an alliance dedicated to hiding the EGH in little pieces around the map. Even when using them as a working hammer, just the sheer buildcost is rather steep. So I'd like to suggest that there are several "tiers" of Teut hammer based off what the player can afford to keep queued and what they can afford to feed. For this exercise, I'll be making a few assumptions. First, I'll assume that the player's res production is constant. Any troops that are fed are fed with res that otherwise would go towards the hammer. Second, I'll use some typical latemid game numbers for queuing times. That means this: 8% recruitment bonus and tier 2 infantry and cavalry helms. This gives us 76 seconds for maces and 127 seconds for axes. All strength values are also based off of level 20 smithy and a tier2 weapon. This gives us 56.4 strength for maces and 79.4 strength for axes. I've accounted lower stats when mixing troops. An odd assumption is that you queue up all of the troops from "axe savings" at the end of our measured period. This is the absolute optimal build for maxing axes' utility in our equations, and is a bit odd, but still demonstrates the absolute minimum time needed for axe payoff.
Now, moving on, what are these "tiers"? They are several hammer styles, ordered by feed cost and strength /time (the two go hand in hand). These styles are:
Axes, barracks only
Maces, barracks only
Axes, barracks (2x arti) only
Maces, barracks (2x arti) only
Axes, barracks (2x arti) + Axes, GB (1x)
Axes, barracks (2x arti) + Maces, GB (1x)
Maces, barracks (2x arti) + Axes, GB (1x)
Maces, barracks (2x arti) + Maces, GB (1x)
Axes, barracks (2x arti) + Axes, GB (2x arti)
Axes, barracks (2x arti) + Maces, GB (2x arti)
Maces, barracks (2x arti) + Maces, GB (2x arti)
A couple notes. Yes, you lose off on some of the weapon bonus if you mix. But the weapon bonus is smaller than most think. Maces get a 9.5% boost from a Tier3 weapon. Axes get a 6.6% bonus. However, switching one queue from axes to maces grants an 11.4% boost in strength. So yes, 2x axes + 1x GB maces is stronger than 2x axes + 1x GB axes. The bonus for sticking to one troop type is much less than people usually assume.
Another important note: if you raid, maces are significantly better than they appear in the following list. Maces are the third best raiding unit in the game, behind TT and EI. A mace hammer lets you easily be a top raider. So if you like to raid, give a little bias to maces over what is here.
One more note: I don't include TKs here, but TKs should always be added to the 2x arti or GS after the same tier of infantry is filled out. Infantry are always more resource efficient, so there isn't much analysis to be done there.
More notes: I've included tier2 weapons for infantry in this model. However, if you have the same amount of axes and TKs queued, you would use a TK weapon over the axe weapon. Do some math to find out if you build would use an axe weapon or not. If not, bias set the breakeven times up a few days.
Yet another note: what does 1x GB mean? Clearly, all my tiers with a GB already have a 2x arti, so the troops are actually in 2x time. The jump from no GB to 24/7 GB is a huge leap, so I use 1x GB as an intermediate. It represents a player who runs a 2x GB 50% of the time.
So what good is this tier list? What does it measure? It measures absolute strength /time. However, there are other factors that play into building a hammer. As a hammer grows old, the cost of maintaining it cuts into the resources that could be used to build it further. A hammer may eventually reach the point that production must be dropped in order to continue to feed it. It's possible that choosing a low tier could have resulted in a larger hammer by nature of the hammer being more feedable. There are also other factors that go into making a hammer. How do you intend to use it? Is it an EGH that will sit around doing nothing all round? What if you want to build a working hammer? What if you only think it will last 30 days, max? The goal is to have the max strength for what your resources can build at the time it splats, whatever the use is. Any savings over the course of a hammer build can be reinvested into more troops. But how long do each of the choices you make take to pay off? To figure this out, we need a mathematical model of the hammerbuilding process.
The Details
The Model
One way to represent a hammer build is as a total cost. If we use total cost, we can first find the breakeven point for axes vs maces. Then, we can find how long it takes for the savings to outmatch the reduced strength / resource of the axe hammer. Our total cost can be thought of like this: total cost = buildcost + feedcost. Build cost is a linear function of either troops or time, so we represent it as either buildcost = x * basecost or buildcost = t * basecost / trainingtime, where x is the number of troops or t is the time. Feed cost can be represented by series(x(y)) where y goes from 0 to h, h is the total number of hours, and x(y) is the number of troops at time y. This can be simplified to series(y/trainingtime), where y goes from 0 to h and h is the total time in hours. This, in turn, is the mathematical equivalent of .5 * (h^2 + h) / trainingtime, where h is the total time in hours. Combining these, we get that total cost = h * basecost / trainingtime + .5 * (h^2 + h) / trainingtime.
Now that we have a model, let's take a look at the barracks. If resources is you limiting factor, how long do you need your hammer to survive before the axe hammer gets you extra strength? Let's say that you have enough to queue up your barracks 24/7 at 1x speed with either troop. Your limited extra resources are being invested into either a 2x trainer arti your alliance has access to, or, a GB. This gives us three scenarios to test. 1) you invest extra res into axes in the GB, 2) you invest extra res into maces in the GB, and 3) you invest extra res into the 2x trainer, and make the same type of troops.
First, we need to find the barracks resource breakeven point. Before this point, there are no "axe savings," which is the variable we will test. Experience tells me that the answer is 9 days, which we can verify mathematically (I've used hours for all timeunits):
Maces = (9*24)*250/(76/3600) + 0.5*(216^2+216)/(76/3600) = 2557895 + 1110126 = 3668021
Axes = (9*24)*490/(127/3600) + 0.5*(216^2+216)/(127/3600) = 3000189 + 664327 = 3664517
However, maces are naturally 11.4% stronger than axes in the same training time. So we've spent the same amount of resources on a worse hammer. For this reason, the breakeven point we should care about is the strength breakeven point. The result will be different for each of the three cases I gave above. Let's do the math real quick.
First, we need a new model. An inverted model, to be exact. Before, we knew how long we had, the question was how much did it cost. Now, we need to know how long we can train for given a certain level of "axe savings." The easiest was it to sub in the constants for simpler ones. Let's say basecost / trainingtime = a and .5 / trainingtime = b. In that case, we have totalcost = a * h + b * (h^2 + h). This can be rearranged to a quadratic (hey, I knew I'd use high school algebra at some point!). 0 = b * h^2 + (a + b) * h  totalcost. A quick plug into the quadratic formula gives that h = (sqrt((a + b)^2 + 4 * b * totalcost)  (a + b)) / (2 * b). At this point, the math is little too complex, so I'll keep most of the work to my spreadsheet (don't worry, I'm not doing all of this by hand! I'll put a download link at the end of the post if you're curious.) From h, we can get that the bonus strength from axes savings is h * basestrength / trainingtime.
Tier 1
Case 1) 1x barracks production. The "axe savings" go into 1x GB axes. Result: 45.4 days
After 45.4 days of hammerbuilding, your axe savings will let you queue up GB axes for the last 203 hours of your 45.4 days. This means you now have 30886 barracks axes + 5769 GB axes, netting you 36655 axes for 2910458 fighting strength, equal to the amount you could get from the 51612 maces you could train in this same time with the same resources.
Case 2) 1x barracks production. The "axe savings" go into 1x GB maces. Result: 38.1 days
After 38.1 days of hammerbuilding, your axe savings will let you queue up GB maces for the last 155 hours of your 38.1 days. This means you now have 25920 barracks axes + 7352 GB maces, netting you a total of 2443321 fighting strength, equal to the amount you could get from the 43313 maces you could train in this same time with the same resources.
Case 3) 1x barracks production. The "axe savings" go into 2x arti barracks axes. Result: 21.4 days
After 21.4 days of hammerbuilding, your axe savings will let you queue up 2x barracks axes for the last 95 hours of your 21.4 days. This means you now have 14558 barracks axes + 2715 2x arti axes, netting you 17273 axes for a total of 1371580 fighting strength, equal to the amount you could get from the 24328 maces you could train in this same time with the same resources.
So that's if you only have the resources for 1x training in the barracks. These numbers are a good guideline for beginners, for longterm hammers, axes win out strongly. Even without a trainer arti, even GB axes pay off over maces in 45 days. That said, a working hammer should figure out how long they expect to survive. If they expect to only last 30 days, a mace hammer is probably a better call without a 2x arti.
Tier 2
What about if you are a few tiers higher? What do you do then? Let's say that you have access to a 2x trainer arti and can keep your barracks queued. Spare res go to either 4) GB axes or 5) GB maces. How long does it take for this to pay off?
Case 4) 2x barracks production. The "axe savings" go into 1x GB axes. Result: 49 days
After 49 days of hammerbuilding, your axe savings will let you queue up GB axes for the last 458 hours of your 49 days. This means you now have 66150 barracks axes + 12988 GB axes, netting you 79138 axes for a total of 6283583 fighting strength, equal to the amount you could get from the 111410 maces you could train in this same time with the same resources.
Case 5) 2x barracks production. The "axe savings" go into 1x GB maces. Result: 42 days
After 42 days of hammerbuilding, your axe savings will let you queue up GB maces for the last 356 hours of your 42 days. This means you now have 56700 barracks axes + 16867 GB maces, netting you a total of 5385855 fighting strength, equal to the amount you could get from the 95494 maces you could train in this same time with the same resources.
For working hammers, axes are not going to be worth it in this tier unless you believe you will survive more than 45 days. GBing axes is an especially bold move as the payoff is nearly 50 days.
Tier 3
Let's go up one more tier. You have already decided based on the above what you will be producing in your 2x barracks. Therefore, I won't make that a factor here. You want to know what you should be producing 24/7 in your 1x GB. Any axe savings will go directly into 6) 2x GB axes. Is axes worth it?
Case 6) 1x GB production. The "axe savings" go into 2x GB axes. Result: 67.9 days
After 67.9 days days of hammerbuilding, looking only at your GB, your axe savings will let you queue up 2x GB axes for the last 634 hours of your 67.9 days. This means you now have 91665 1x GB axes + 17995 2x GB axes, netting you 109660 GB axes for a total of 2209663 fighting strength, equal to the amount you could get from the 154383 GB maces you could train in this same time with the same resources.
This seems like a very strange choice for a working hammer. The only reasonable use of 1x GB axes is in an EGH. A working hammer should not survive 50 days if used right. Maces are the way to go here.
Tier 4
If you're at a higher tier, it's really up to you. If you can keep 2x barracks + 2x GB queued up all round long, you are truly a master of resource management and raiding. If you think you can survive maces or plan to be a working hammer, do maces. If you're unsure or want to do an EGH, axes may be the better choice. A mace hammer at this tier is about 20% stronger than an axe one.
Making You Own Decision
So what should YOU build? There a couple things to consider. First, what do you think you can handle? 1x barracks? 2x barracks? 2x barracks + 1x GB? 2x barracks + 2x GB? As a beginner, you probably won't be queuing fulltime 2x barracks unless you have some great mentors. If you've played before, you know what you're capable of. Are you a working hammer? If so, get a number that is how long you think your hammer will survive for. Go to what tier you think you can support. Pick based on the breakeven points there. If resources is you main constraint, the "Days to Strength BreakEven" is how many days until the resources saved by an axe hammer will get you enough extra troops to reach the same strength as a mace hammer. If the hammer survives longer, axes are better than maces for this use. There are other things to consider as well. Are you a raider? Bias toward maces. Maces are amazing raiders and the extra res will net you extra strength. This depends on how good of a raider you are of course. Are you an EGH? First, figure out if you alliance will feed parts your EGH for you. Figure out how much extra wheat you can expect to gain from storage. Factor this amount into whether you can afford to go up a tier. If you are confident you can maintain 2x maces in the barracks and GB, you already know that's the best. If not, axe 2x barracks + mace 2x GB is a good compromise. If not that, axe 2x barracks + 2x axe GB. If still not, fill in as much as possible with axes.
BreakEven Points Summary
TIER 1: 1x barracks 24/7
Axe Savings go to: Days to Strength BreakEven GB Axes 45.4 GB Maces 38.1 2x Barracks Axes 21.4
TIER 2: 2x barracks 24/7
Axe Savings go to: Days to Strength BreakEven GB Axes 49 GB Maces 42
TIER 3: 2x barracks + 1x GB 24/7
(This assumes that you have already chosen the troop you will produce in the barracks. Therefore, this table only factors in the troops you produce in the GB)
Axe Savings go to: Days to Strength BreakEven 2x GB Axes 67.9
TIER 4: 2x barracks + 2x GB 24/7
If you can push a tier 4 hammer you are an amazing player. The best hammer in the game is Tier 4 mace hammer. The cost will be absurd. There is no breakeven if you can maintain it. A tier 4 axe hammer is 10% weaker but is much more manageable after resource breakeven at 22 days. Maces in the GB and Axes in the barracks is a compromise hammer that sits between the two.
My Spreadsheet
Armour

Thanks for taking the time to redo it


I'll post in this in earnest once the current S1 round is over.
Jonothan Crane wrote:
Patients suffering delusional episodes often focus their paranoia on an external tormentor. Usually one conforming to Jungian archetypes. In this case, a scarecrow.

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