More Ranged Math! Elves vs Kingdoms of Men

I figured it was time I subject you all to a little more math. Unfortunately, I’m no mathematician. Fortunately, I don’t need to be, because someone has done all the hard stuff for me and set this wonderful thing up. Scroll down to the end for the TL;DR read summary.

Swordmaster’s most recent battle got a few comments from folks on the forum wondering, “Hey, could Kingdoms of Men run a MSU list like Swordmasters?” Well, if you’re reading this, you likely know the answer. “Technically yes, but you need to be more talented than this TastyBagel guy. He hasn’t won a game yet.”

The truth hurts, but thank you for your honesty.

For those wanting to one-up me and run a successful MSU-style Kingdoms of Men army, and in particular wanting to emulate Swordmaster's typical list, I figured I'd compare some similar-looking units between the Elves, Twilight Kin and Kingdoms of Men. 

I'm not a fan of strict emulation of a list. There are lots of factors as to why an army would take a unit, and it's important to understand the reasoning behind the choices, rather than just copy blindly. 

Take, for example, this thread, relating to Twilight Kin MSU. The third post provides a link to the list (sorry for the runaround). For those who haven't clicked the link... the list contains an oddity right from the start with two Troops of Spearmen. If you copied that list over into another army, and just added in two troops of your spear units, you'd do ok, but likely miss the point of those units. In looking at the rest of the army list, their inclusion makes more sense. There are only two options in the army that are cheaper: Lower Abyssals and Gargoyles. Both are odd models, and Kpzelenski might not a) own them, or b) want to run them in this army (for fluff reasons). So what are those Spearmen doing in the list? They are chaff. Look at your army; if you have better options for chaff, use those instead of Spearmen troops.

Alright, enough words! On to the numbers! 

First, we’ll be looking at some supporting ranged cavalry units! The first table is the expected damage to a unit with a given defense. (Yes, this includes Elite and Vicious. And yes, they apparently have the same mathematical impact, as explained in the thread. Math!)

Unit (Troop)	De3	De4	De5	De6
Silverbreeze Cavalry	2.7	2	1.4	0.7
Heralds of Woe	2.7	2	1.4	0.7
Scouts & Bows	1.6	1.2	0.8	0.4
Scouts & Carbines	1.9	1.6	1.2	0.8

Pretty easy table. The elvish rerolls and extra skill in hitting generate way more wounds per unit than the Kingdom of Men's Scouts with Bows. Our Scouts with Carbines do better (yay Piercing!), but still barely catch up against targets with defense 5 or 6. Alright, we get it. Trope confirmed. But the point ears cost more, right? Like, a lot more. How does that factor in? Let's take the points spent on the unit, and divide it by the expected damage the unit would inflict:

Unit (Troop)	Points	Value De3	Value De4	Value De5	Value De6
Silverbreeze Cavalry	145	53.70	72.50	103.57	207.14
Heralds of Woe	145	53.70	72.50	103.57	207.14
Scouts & Bows	100	62.50	83.33	125.00	250.00
Scouts & Carbines	115	60.53	71.88	95.83	143.75

So, the elvish units are again identical to each other, and again, trounce the Kingdom of Men's Scouts and human-made bows (and that's not even exploring the other value of the elves, with higher defense themselves, and higher Nerve than the humans). Point for point, the elves are just great. But something interesting happens with the Kingdom of Men Scouts when they stop emulating the elves and reach for a gun. The Piercing value of the Carbines start compensating for the poor skills of the puny bumbling humans. Now, this isn't the best comparison, as the bows have a longer range than the Carbines... but it's something to note and keep in mind. Humans appear to do better when we do our own thing.

Now, onward, to the tables of the infantry shooters!

Unit (Troop)	De3	De4	De5	De6
Elf	3.1	2.3	1.6	0.8
Moving Elf	2.1	1.6	1	0.5
KoM Bowmen	1.8	1.3	0.9	0.4
KoM Bowmen (Moving)	0.9	0.7	0.4	0.2
KoM Crossbowmen	2.2	1.8	1.3	0.9
KoM Arquebusiers	2.2	2.2	1.8	1.3

Oddly enough, the three Elvish shooters that I was investigating (Kindred Archers, TK Crossbowmen and TK Shadows) are all armed with bows, and have the same attack stats... so all their numbers match. Rather than inflate the size of the table, I compressed it for this one. Probably as expected, the skills of the average elf go a long way. 

Again though, Piercing starts to catch the human up, and it is important to note that an elf on the move is statistically outperformed by both the KoM Crossbowmen and Arquebusiers against all Defense values. Cool. So, what about the points? Let's take a look. This time, the various pointy-eared units are differentiated.

Unit (Troop)	Points	Value De3	Valeu De4	Value De5	Value De6
Kindred Archers	115	37.10	50.00	71.88	143.75
Kindred Archers (Moving)	115	54.76	71.88	115.00	230.00
TK Crossbowmen	115	37.10	50.00	71.88	143.75
TK Crossbowmen (Moving)	115	54.76	71.88	115.00	230.00
TK Shadows	130	41.94	56.52	81.25	162.50
TK Shadows (Moving)	130	61.90	81.25	130.00	260.00
KoM Bowmen	75	41.67	57.69	83.33	187.50
KoM Bowmen (Moving)	75	83.33	107.14	187.50	375.00
KoM Crossbowmen	85	38.64	47.22	65.38	94.44
KoM Arquebusiers	100	45.45	45.45	55.56	76.92

Yikes! So many numbers...

Let's do a few quick comparisons. TK Shadows had the same stats as the other elves, but cost more points (due to gaining Pathfinder and Vanguard... for 15 points). So, they will have the worst value for the elves. And Kindred Archers and TK Crossbowmen cost the same points... had the same stats, so they are identical. They are the best "Team Pointy" can do.

Most of "Team Pointy" wipes the floor with KoM Bowmen. Stationary TK Shadows are about about good as stationary KoM Bowmen, though the Shadows quickly surpass them if both are on the move.

What about KoM Crossbowmen and KoM Arquebusiers? Well, this is where stuff gets fun. From the start at De3, Crossbowmen points efficiency surpasses moving elves, and are pretty comparable to stationary elves. From De4 and up, the Crossbowmen are more points efficient across the board. The Arquebusiers aren't a great buy against De3, since you're paying for two Piercing and only need one... but from De4 and up they are more points efficient than the elves too, and even edge out their Crossbow-toting brothers. More Piercing is apparently great. 

That said, there are some caveats to this all this math and how it relates to these armies. 

The first is that these numbers are only dealing with efficiency with outbound damage. we're ignoring the units own Defense and Nerve values. Like it or not, even if KoM can deal out more efficient damage per point in some cases, the elves are better on defense.

The second is that internal balance in each army is actually pretty good. A more expensive unit in a given army probably has a cheaper alternative... but also lacks some stats boosts or rules. That's how it goes in this game. Some units are arguably better than others, but just about everything can find a niche to be worth taking. It usually comes down points available or to player preference. 

So with those limitations in mind, what can we say about elf and human shooting?


-----------------------------------------------------------------------------TL;DR-----------------------------------------------------------------------------

Elves have better stats across the board than their human counterparts. However, they are limited to using only their bows. Point for point or unit for unit, the elves statistically outperform their human counterparts when it comes to bows. Even the low points costs of humans doesn't even the playing field. Boo.

So, what should a KoM player do?

In short, learn the lessons that each army plays differently, and to utilize the strengths of your particular army. In this particular case, if a KoM player wants to go toe-to-toe with an Elf player of any kind from afar... bringing some units that have Piercing might be good investments.

War Machines of the Kingdoms of Men

I was thinking about the Battle for the Brew, and decided I was actually pretty happy with how my Ballistae performed. They solved a troop shortage problem for me, and did their supporting fire job well. I’m going to continue running and experimenting with them, but I decided I was also curious enough to do some additional (basic) number crunching on my other War Machine options.

So, first up we have just the stats, and then the average hits that would be generated from the Blast column.

Pts	Machine	Blast	Pierceing	Av. Hits
60	Ballista	D3+2	Pierce 3	4
85	Cannon	D6+1	Pierce 4	4.5
90	Artillery	D6+2	Pierce 3	5.5

All the war machines hit on a 5+ and have the Blast rule for generating hits, so it should be a little more honest to compare them to each other using this simple math than it was comparing them to the infantry shooters.

Like we’ve done before, we’ll take the average number of hits, factor in Pierce, and see what the expected damage caused would be against the various defensive stats.

Machine	Def3	Def4	Def5	Def6
Ballista	3.33	3.33	3.33	2.67
Cannon	3.75	3.75	3.75	3.75
Artillery	4.58	4.58	4.58	3.67

The high Pierce value on all the war machine options kinda makes for a boring table. Piercing 3 means that everything up to any targets with Defense 6 will be hurt on the same rolls. In the case of the cannon, with Piercing 4… even targets with Defense 6 will be hurt on 2’s. Our war machines can really reliably dish out the damage… if we hit with the shot.

Now, we take the points cost of the machine, and divide it by the expected damage. It looks my spreadsheet may be rounding a bit for the Ballista (should be 18.01 or .02), but hey, close enough. You get the general idea. As expected, when the expected damage vs defense isn’t really changing… factoring in the points also doesn’t really change anything.

Machine	Def3	Def4	Def5	Def6
Ballista	18.00	18.00	18.00	22.50
Cannon	22.67	22.67	22.67	22.67
Artillery	19.64	19.64	19.64	24.55

So… the big question, what does this mean?

Well, the biggest takeaway for me is that the war machines for Kingdoms of Men are very comparable mathematically. All hit similarly, and all have a good pierce value. There is no choice mathematically superior enough to simply dismiss the other choices. So run what you have or want and let your inner fluff bunny guide your picks if you want. With that said, here's how I'm looking at each option.

The Ballista is a great “all around” option. Mathematically-speaking, it deals the best overall damage for the points invested, but it’ll be a little less effective against Defense 6 targets, and deal a little less damage on average when it hits. It’s an effective, basic war machine option, and will be my go-to choice I think.

Siege Artillery is also a mathematically good choice overall, but with Indirect Fire, can do a few things the Ballista can’t. It can fire over a crowded battlefield or into forests to harm enemy targets without being hampered by cover penalties. If you play with lots of forests or often find yourself needing 6's to hit with your war machines... taking some Siege Artillery can really help you out.

The Cannon, with Piercing 4 will be a little overkill against many targets (you don't need that much Piercing, usually), but if you find yourself facing down a lot of Defense 6 units on a regular basis, running a Cannon seems like a good choice for dealing damage to everything.

Initially, I was bummed that I went to the trouble of examining these units, and didn’t find anything that mathematically stood out. While thinking more, and writing this post though, I'm actually really happy about that.

They are comparable enough that as a player, fluff can rule, and I'm not given a slew of false options if I were to try to run a "good" or "competitive" list. That's huge, but it goes a little further. Each choice also has a bit of a niche, that would let it shine a little brighter depending on your play style or your meta. That's.... really cool, and really well done game design.

The more I dig into the mechanics of this game the more I am impressed with it. Bravo Mantic!

Kingdoms of Men Shooting 2: Ballista!

So, all of this math stuff started with me trying to build some 1k lists, and being frustrated that I kept running out of Troop choices. I noticed I wasn’t taking much by way of heroes or monsters.. so I thought to myself, well… what if I take a few Ballistae instead of the Troop-sized infantry shooters I was trying to squeeze in? That’s save me some points, and free up some Troop slots.

I really liked my solution, and drew up a new MSU list I am very eager to test out and report back on… (hopefully in the next week). While my solution fixed some list concerns, I found myself wondering how how mathematically viable my solution was.

Like I said before, I am not a statistician. So I tried to break my question up into more manageable components, part one was figuring out what shooters I should hypothetically be running, with was <this post>.

I was surprised to find that an Arquebusier Horde is not only the deadliest ranged infantry unit, but also the most points efficient. In general Hordes are the best “bang-for-your-buck,” but most troop choices aren’t too far behind, and you’re paying for troops to increase your list’s versatility (more drops) or in-game options (splitting fire among enemy units/avoiding overkill). So, after all that math, I had my go-to unit to compare against the Ballista: the Arquebusier Troop.

So…. Arquebusier Troop vs Ballista!

I had no idea how to compare the two, since the Ballista fires one shot, and if it hits, turns into multiple hits. Perhaps my approach is totally wrong, but I decided to break it all down to a single attack to start.

Both have a 5+ to hit, or a .33 chance to hit with each volley. The Arquebisuer has Pierce 2; the Ballisa Pierce 3, so assuming a hit, their chance to wound for a few Defense stats is below.

	De3	De4	De5	De6
Arq. Troop	0.83	0.83	0.67	0.50
Ballista	0.83	0.83	0.83	0.67

Again, this may be wrong approach, but I decided to look at the chance each attack would have to wound, taking the (.33 chance to hit with a shot) multiplied by (a single hit’s chance to wound, shown above). That gives me the chance of each shot to hit and wound (I think):

	De3	De4	De5	De6
Arq. Troop	0.28	0.28	0.22	0.17
Ballista	0.28	0.28	0.28	0.22

That looks right to me. The impediment is hitting, but provided that the shot hits, the solid Pierce values help insure that it wounds. The pseudo plateau also makes since, since Pierce eventually gets the roll to one or below, but rolls of 1 are always failing.

Alright, so that looks like the math for each shot. But the hard stuff is still to come… Units have multiple shots per volley… and the bonkerz D3+2 Ballista is still wrecking my noggin.

The unit stuff should be easier? I think? We take their communal shots, and multiply that number versus our values, to get the average damage per volley. That comes out to what I came up with previously, so uh… I think that’s good? Probably wasted some time here.

	De3	De4	De5	De6
Arq. Troop	2.22	2.22	1.78	1.33

The Ballista is harder… it has one shot, but Blast means a connecting shot will result in D3+2 hits against the unit. If we take the chance to hit for a shot, and then assume 4 hits (the average roll for D3+2), we get the following average wounds per volley.

	De3	De4	De5	De6
Ballistae	1.1	1.1	1.1	0.9

Fortunately, at this point, my google-fu turned up this thread after a few hours spent looking at various probability tutorials trying to figure out how to proofread my approaches. I was happy to see Blast was messing with people far more math savvy than I!

http://vb.manticforum.com/forum/kings-of-war/general-discussion-aa/251752-kings-of-math

As discussed in the thread, the math is a little off, as this is looking at an average, not probability. Two thirds of the time, the Ballista will miss and do nothing. Averaging out volleys shot and damage dealt would get the values above.

So, it’s imprecise. But, it is largely comparable to the values I was using before. Since we were looking to compare the two units, that’s a win.

So, Like before, let’s take the points of each unit, and divide it by the number of average damage dealt, and see what unit will, on average, give us the best efficiency. Again, a lower number is more desirable, as you are spending fewer points to deal the damage.

	De3	De4	De5	De6
Ballista	54.55	54.55	54.55	66.67
Arq. Troop	45.00	45.00	56.25	75.00

Man, these Arquebusiers! The Arquebusier Troops coming out on top against Def3 and Def4 shouldn’t be a surprise, given our previous testing and some mental math (the Arquebusiers being on average about twice as deadly as the Ballista for less than twice the points). Also to be expected is a little “overkill” factor, as any additional Pierce you have after getting the damage roll down to a 2+ is just wasted points.

So, the Ballista really only comes into its own mathematically once that extra Pierce can kick in, which happens at Def5. Still, that is a tiny difference.

I still want to experiment with a few Ballistae in my MSU lists to free up Troop slots… but man, it sounds like for someone just starting their army, Arquebusiers are a good buy.

Kingdoms of Men Shooting

I was getting frustrated the other day, as I was trying to sketch out my next few lists of MSU stuff to test, but felt like they were all just terrible.

In my experience with MSU, if you don’t have reliable, efficient, top-notch melee units (which Kingdoms of Men does not), you probably need to invest more in ranged support elements. I was looking at our ranged infantry options, as they are generally our cheapest shooting options, and something I was keen to test them out more... but I was just not thrilled.

Conventional wisdom stated that the bigger the unit, the more damage it would potentially deal (more attacks!), and the more cost-effective magic items (Jar of the 4 winds, or Brew of Keen-Eyeness) would be (since more attacks would be influenced by the item). That made sense. However, taking bigger units is not what MSU is all about, and in some ways, I felt that spending points on these items was throwing good points after bad for KoM.

Conventional wisdom also says that Pierce is a powerful special rule, which we as KoM, do have ready access to. Our Arquebusiers have Pierce -2, which makes them the deadliest of our infantry shooters.

However, the Ranged attack stat for our shooters is 5+ across the board… not great. We’re going to be missing (statistically) 2/3 of our shots, more if the unit is in cover.

It was swiftly becoming a mental quagmire. Our shooters are not accurate, but we can be deadly with our ready access to Pierce, so we want to compensate with more shots, but then we’re spending lots and lots of points….

I decided I was wandering in circles. So I sat down and decided to try my hand at some basic math, to see if I could use numbers to help me identify a “least bad” option for us (not knocking KoM, but we are just human), or highlight anything I may have overlooked or just wasn’t seeing. Uh, strap in, friends. This is a lengthy post.

I was quickly bewildered. I’m no statistician (so apologies, if I misuse any terminology. If you know better, tell me and I can edit the post for correctness.) Eventually, I resorted to my high school physics approach to solving problems, and tried to start oversimplifying as much as humanly possible. 

Thanks, physics!

The first oversimplification I would make, is that I wanted the deadliest units possible. We won’t try to factor in the benefits of having a higher Nerve into these calculations at all.

Part 1: Points per Shot

I started out slow. I knew all shots were not created equal. But... just for curiosity, how many points I would be spending to take each shot?

Bows	Shots	Points	Points per shot
Troop	8	75	9.375
Regiment	10	100	10
Horde	20	165	8.25
Crossbows	Shots	Points	Points per shot
Troop	8	85	10.625
Regiment	10	115	11.5
Horde	20	190	9.5
Arquebusiers	Shots	Points	Points per shot
Troop	8	100	12.5
Regiment	10	135	13.5
Horde	20	225	11.25

Ok, so I remembered basic math. We're off to a good start! The number of shots are equal, so the more expensive weapon option at each level (Arquebusiers) should be paying more per shot at each unit size... which they totally are. 

Additionally, if you are looking for the best “bang for your buck” so-to-speak, the Regiment is the least points efficient option you can pick. If you’re taking a Regiment, you are absolutely taking it for its boosted Nerve value, not the increased damage potential, because at equal points, regiments have fewer attacks than their troop counterparts. Hordes come out on top of everything, though looking at the points per shot, not by as far a margin as I guess I was intuiting. These non-linear unit costs, man…

Part 2: How Many Hits?

I assumed that while the bowmen could hypothetically move, ideally they would not, as they have a decent range, and moving would further reduce their accuracy by 50%. Crossbowmen and Arquebusiers have Reload! so would need to not move in order to shoot… so hey, we'll have everything stand still. We’re doing ok on the oversimplification front so far! More good news is that each unit has the same range, number of attacks, and target roll (5+) needed to hit. Nice. So, across the board, assuming no cover (and no moving), on average:

·         Troops would land 2.66 hits per volley

·         Regiments would land 3.33 hits per volley

·         Hordes would land 6.66 hits per volley

Neat. While the math is easy to follow, I was guess I was surprised at how few shots from a Horde would (statistically) hit. I don’t think I had previously thought the math through before, so it was a kind of stark realization for me. Again, the non-linear unit sizes are messing with my intuition.

Part 3: Average “Deadliness”

Since the number of shots/hits would be the same, I figured that the Arquebusiers would be the most killy unit type (dealing the most wounds), and the Horde of Arquebusiers, the most killy ranged infantry unit overall. That was actually correct. Phew. My intuition is correct there! Piercing is no joke folks. 

	Defense 3+	Defense 4+	Defense 5+	Defense 6+
Bowmen Troop	1.7777777778	1.3333333333	0.8888888889	0.4444444444
Bowmen Reg.	2.2222222222	1.6666666667	1.1111111111	0.5555555556
Bowmen Horde	4.4444444444	3.3333333333	2.2222222222	1.1111111111
Crossbow Troop	2.2222222222	1.7777777778	1.3333333333	0.8888888889
Crossbow Reg.	2.7777777778	2.2222222222	1.6666666667	1.1111111111
Crossbow Horde	5.5555555556	4.4444444444	3.3333333333	2.2222222222
Arquebus. Troop	2.2222222222	2.2222222222	1.7777777778	1.3333333333
Arquebusier Reg.	2.7777777778	2.7777777778	2.2222222222	1.6666666667
Arquebus. Horde	5.5555555556	5.5555555556	4.4444444444	3.3333333333

Part 4: Efficiency and mitigating excessive pierce

you may have noticed Crossbows Hordes and Arquebusier Hordes tie in their deadliness against units with Defense 3+. So in that one instance, the Arquebusiers aren't the most efficient tools use of points.

That "overkill" intrigued me, so I tried to take one more step in this exercise, and circle back to looking at how points factor in compared to deadliness, to see if there were any sweet spots for unit deadliness. This table takes the unit points and divides it by the corresponding number above. A lower number is therefore better, as you're spending fewer points to deal that anticipated damage.

  

	Defense 3+	Defense 4+	Defense 5+	Defense 6+
Bowmen Troop	42.1875	56.25	84.375	168.75
Bowmen Regiment	45	60	90	180
Bowmen Horde	37.125	49.5	74.25	148.5
Crossbow Troop	38.25	47.8125	63.75	95.625
Crossbow Regiment	41.4	51.75	69	103.5
Crossbow Horde	34.2	42.75	57	85.5
Arquebusier Troop	45	45	56.25	75
Arquebusier Regiment	48.6	48.6	60.75	81
Arquebusier Horde	40.5	40.5	50.625	67.5

Part 5: Findings

Well... generally speaking:

• Arquebusiers are the deadliest option for ranged infantry... which should be a no brainer, as they have Pierce -2, the best of these options. What surprised me is that they came out as being the most points efficient way to deal damage of these options too. I would have assumed that we would pay a little premium for the extra oompf (meaning they might not be be best choice in every situation), but point for point they appear to be our best investment for infantry shooters.
• Taking Hordes of our infantry shooters is the most points-efficient way to take our shooters (most shots per points spend and most expected damage for points spent, compared to the same unit type of different  sizes).
• Troops are the second best choice if you are looking for points efficiency. Going big seems to be mathematically encouraged, and you pay a little extra for the extra drop/deployment/unit, as it gives you more options in game (splitting fire among two targets, for instance).
• Unless you really really really want that extra nerve, avoid regiments.
• As for dealing with your opponent... these are the most points efficient ranged infantry options to take against units with these defenses.

	Best	2nd Best	3rd Best
Defense 3+	Crossbow Horde	Bowmen Horde	Crossbow Troop
Defense 4+	Arquebusier Horde	Crossbow Horde	Arquebusier Troop
Defense 5+	Arquebusier Horde	Arquebusier Troop	Crossbow Horde
Defense 6+	Arquebusier Horde	Arquebusier Troop	Arquebusier Regiment

I gotta say, I was surprised by what I found.

I should also say, that I'm fighting off a cold. If someone notices any math errors, please speak up and I can amend the post. Thanks!