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.
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.
-----------------------------------------------------------------------------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.
Hello!
ReplyDeleteThanks a lot for a very interesting post! This time I need to warn you, however, that there are things I don't necessarily agree with and I will try to demonstrate my point of view. I hope it will add to the discussion though!
There are some players who just love using some simple calculations to back up their opinions on what is "optimal" choice. Personally, I find the results of such calculations helpful (if correct!) but at the same time I remember saying: "Lies, big lies, statistics!". :)
As I hope to find time to write a more detailed article about my own army and why I have in it what I have, I will simply focus on the shooting elements you discussed here. The general assumption I would like to make is that KoM army you are interested in is going to follow MSU concept and will have elements that participate in all phases of the game, i.e. movement, shooting and melee.
This time we discuss shooting and compare it to some of the options in other armies.
1. Probability of success vs averages.
Personally, I think that operating with averages is often misleading. First, averages tend to introduce error to the calculations (be it by rounding fractions or not including the error bars of the end result) and second, people tend to treat them as what they supposed to get every time they roll the dice.
That is obviously not true.
Simple probability of a success has the advantage of being a more accurate number and it is just an information you need to make a decision. It is not a guarantee of a success at all but tells you about the odds.
Please, note, that the same chance of a success will be considered as worth taking a risk or not depending on the situation. E.g. at the beginning of the game you may not want to spread your fire as fresh units are not going to worry about a few shots they receive. But in the end game, even a single wound would result in nerve test and that may mean the difference between the victory or defeat. In that situation 5% chance of success may still be worthy taking.
One day I hope to calculate the probabilities of different number of successes in particular circumstances or even the whole spectrum of results. E.g. what is the probability that out of 7 attacks I will obtain 1,2,3 or more points of damage (up to 7) for Silver Breeze cavalry. But for now I prefer simply to operate with the probability that a single shot results in a single point of damage.
From that point of view the results for Silver Breeze cavalry are:
Def 3 ~ 39%
Def 4 ~ 29%
Def 5 - 19%
Def 6 ~ 10%
Now Scouts with Bows (with carbines in brackets) have:
Def 3 ~ 22% (27%)
Def 4 ~ 16% (22%)
Def 5 ~ 11% (16%)
Def 6 ~ 6% (11%)
It is obvious that scouts with the bows do not have the same chances to inflict the same damage but these with carbines are getting close (but at the price of being more expensive and having a shorter range). We knew it before the calculations, the results simply show the difference.
Mr. TastyBagel suggested then another method to help in assessing each unit. And this is another approach I am not that happy with :)
Greetings Swordmaster!
DeleteI hope you find the time to write that army article sometime. It would be a great read!
This is actually my 3rd mathy post, and I will absolutely concede that math has plenty of shortcomings when discussing a wargame. Wargames are complicated, so these posts really look at just two things: outgoing damage (the first table) and dividing that by points to establish a "yardstick for killiness" if you will (second table). Far more facets of a given unit are ignored than covered in a post like this.
To your first point: I actually like averages, as it gives me a ballpark idea for the unit, both before it hits the table and in-game. Expecting the average result every time is indeed folly, but knowing the number isn't a bad thing. It can help people understand the game better, and realize when they are rolling particularly hot or cold. I had a 9th Age game the other week, flank charged some depleted dwarves with some chaos warriors... rolled up 1 wound, lost combat, broke, and was run down by stunties. Hyperbolic annectodal evidence says I should probably just never play dwarves again; stats and averages say the charge was a good call and I just rolled incredibly terribly! You are correct in noting that dealing with averages can warp people's expectations, but that more on them than the math. :)
I am really intrigued by your alternative approach here; just looking to find the chances of getting in that one point of damage... I've finally been hobbying on my Scouts/Bows a bit, and will keep it in mind when they hit the table. I think they are the only thing that I think I can reliably use to chase down and force Nerve checks late game. I'm looking forward to trying them out!
2. Point value vs number of units for the some price.
ReplyDeleteSome players go as far as to assuming that the unit needs to prove its value by routing the enemy which is worth at least their own point of value. I think that this approach is behind the Value factor used in the above calculations.
I don't like it because it does not take into account units have different roles, that they might perform duties that are not directly transferred into pure damage but still contribute to the victory etc.
Obviously different units cannot cost the same. That is the foundation of point based systems I think. It is unfair then to compare a unit that costs less with the one that costs more and assess it based on the same characteristics. The result will always show the more expensive unit performing better.
I suggest different approach, that takes into account not only pure damage the unit can do. I want you to compare the units based on how many of them you can have for the same cost.
Since my army was one of the initial reference points let's take my own investment in fast cavalry. I have 2 units of Silver Breeze that cost 290 points. For just 10 points more you can get 3 units of mounted scouts.
I think that the comparison looks much more interesting now. On one hand you have 2 units that hit on 4+ with Elite and have total of 14 attacks. On the other, 3 units that hit on 5+ but have 21 attacks total. The number of hits Elven units can get on average is only slightly better than that of Mounted Scouts. However, Scouts potentially can inflict 15+ points of damage, something Elven fast cavalry cannot do at all.
What is more, the Scouts can divide their shooting better and having one more unit means they have advantages elsewhere. More deployment drops, more units to interfere with enemy's movement etc. Basically, more flexibility.
3. Conclusions.
It is easy to assume that units are not as good as what enemy has by simple comparison of their characteristics. I hope I demonstrated it is not fair but also may limit the options for the player who may choose these seemingly worse troops.
I agree that every army needs to play to its own strengths (as much as obvious it sounds!). The example of 2 Silver Breeze vs 3 Mounted Scouts is a great one and shows exactly what I mean. After all we don't play with fixed amount of units but fixed points.
So, what should KoM player do? Use his flexibility of choices. It may mean taking 3 units of Mounted Scouts. Or 2 with carbines but with the addition of a cheap but useful character. Or maybe magic artifact to boost some capabilities.
Cheers!
Your second point here starts to highlight the limitations of applying math to a wargame, as tactics can change everything, but I feel like you've misunderstood the math my friend. I am already (mathematically at least) doing the approach you advocate when I go to the second table in these posts.
DeleteThe get the numbers in my second "value" table, I take the points of a unit and divide by the number of average damage dealt. If we scaled up, looking at the damage multiple units would do, and then divide the total points of those units by the total average damage dealt... the second table wouldn't change at all.
Plug the numbers into the Kings of Math spreadsheet (or just appropriately multiply my first table above by the number of units for each choice) and we'll see the average damage output with your multiple unit suggestion. Against De3, two troops of Silverbreeze still deal more damage (5.4) on average than three Scout/bow troops (4.8). If we take points spent on those units divided by average damage dealt, we will see the same values as displayed in the second table, which again, is unchanged.
The second table I use creates a yardstick so I can compare apples to apples as best I can (again, math has a lot of shortcomings when applied to a wargame). Here, the second table shows that the elves, on average, are more killy for the points invested that their KoM equivelant.
...All that said, there are a ton of variables at work while a game is in progress. Just because the elves are arguably better buys in an instance... doesn't mean they'd be top choice all the time. Continuing with the Silverbreeze/Scout comparison, maybe it is more desirable to disperse my ranged attacks over more units, have that extra deployment, etc. What is the "better" choice is (suprisingly, coming from WHFB) up for debate.
One thing that I have noted when writing up these mathy posts, is that there are no real "bad buys" for units. The mathematical differences are usually small, and costs tend to scale well. You spend more points for a unit, and it becomes more efficient but less flexible (fewer deployments; fewer choices in-game). Mantic's points system is very well designed, and gives players a lot of freedom.
Hello!
ReplyDeleteFirst of all, my apologies for 2 comments in a row. I run out of space for one and had to continue with the second.
I will reply to both of yours in a single one (hopefully!) as otherwise we may find we are talking about too many things at the same time!
I would like to add that I don't think that mathematics has any shortcomings if applied to wargaming. In my opinion, it is always about the user and what formulas he uses, how he uses them and what conclusions he draws from the results.
There is absolutely nothing wrong with using averages. It is just one of the methods people may employ to help them with decision making. I simply stated I prefer percentages and I will come back to why a bit later in this reply.
My second point highlighted the situation that often takes place when people try to draw the conclusions which are not necessarily supported by the results of their calculations. :)
Again, there is nothing wrong in using your own methods and values to analyze the units and their potential. What I am disagreeing with is the notion that the unit must "return" the point value the player invested in it. I got the impression that this is where you are pointing at.
And this is the moment where I don't like averages. You mentioned the results from King of Math where you obtained damage from 2 units of Silver Breeze vs 3 units of Mounted Scouts. For me the interpretation where 5.4 points of damage is even discussed is not acceptable. First of all, you don't have fractions on dice rolls. So what does it really mean when you get 5.4 as a result? Enter personal interpretation, not mathematical shortcomings.
Second, when you round the fractions 4.8 and 5.4 result in the same value - 5. So from that point of view 2 units of Elven fast cavalry is the same as 3 units of human horse archers. That example illustrates why I prefer to operate with probabilities of certain results rather than averages.
Last but not least, while comparisons between units are valuable, I would not do it to learn which unit is "better". Because I always want to follow up with "better for what and better for whom?". For one player it may be more desirable to have fewer units that deal more damage per shot (scouts with carbines), while for another it is more advantageous to have more units that can either focus or disperse their fire accordingly to the circumstances.
Mathematics can provide wonderful tools, also for wargamers. The conclusion I would prefer to choose, based on these calculations and based on the question "can KoM do what Elves do?" is that "yes they can but with their own twist" :)
I am always impressed with your posts. Lots of good thoughts above, and that's the takeaway I was fumbling toward. Each army has to play the game their own way.
DeleteThanks for posting! It is very much appreciated!