Crusader King III Man-at-Arms analysis¶

About Crusader Kings¶

I am a big fan of optimization. This extends into many aspects of my life, including recreation. Because of this, I am big fan of min-maxing in video games, and one of my favorites is a game called Crusader Kings III.

In CK3 you control a dynasty throughout the Middle Ages, playing as your ruler until they die, and then continuing as their heir. There is no real goal, besides making sure you always have an heir. Despite not having an explicit goal, generally you want a strong military to support both the defense of your realm and the conquest of others.

Military Overview¶

In CK3 there are two main types of military, levies and Man-at-Arms (MaA). Levies are essentially peasants drafted to fight, whereas MaA are trained soldiers that cost gold to both recruit and maintain, even when unraised. Basically, levies are cannon fodder and MaAs are what win you battles. There are also seige MaA, but they arent used in battle and therefore out of the scope of this analysis.

There are 6 main types of damaging MaA,which are:

Light Infantry
Heavy Infantry
Pikemen
Light Horsemen
Heavy Horsemen
Archers

each with their own strengths and weaknesses. There is a countering system ( Light Infantry > Heavy Infantry > Pikemen > Horsemen > Archer > Light Infantry) but countering becomes less effective as the number of one type of unit increases (i.e. 100 archers effectively counter 100 light infantry, but 100 archers fail to counter 1,000 light infantry.) This drop-off in countering effectiveness means when facing the game's AI you can effectively build only one type of unit and be safe from countering.

Each MaA also potentially has terrain bonuses or weaknesses (ex. horsemen generally perform worse in mountainous terrain). This generally is only important if your domain is located in an area that is predominantly one type of terrain. Otherwise, general performance is more important.

Each MaA has a unit size, which is 100 soldiers per unit for all but a few types of MaA (Heavy Calavry, Elephant Cavalry, and a few other outliers).

Finally, and most importantly, each MaA has 4 battle stats and a cost and upkeep:

Damage: How much damage a soldier deals during the battle phase
Toughness: How much damage a soldier can take before dying/retreating
Pursuit: How much damage a soldier deals against retreating enemies (retreat phase)
Screen: How much damage a soldier can block while retreating (retreat phase)
Cost: How much gold it takes to train one unit of the MaA
Upkeep: How much gold it takes to maintain the unit (raised and unraised)

Each MaA has different values for each of these stats, and these are what make the real difference in a battle.

The Problem¶

There are 38 MaAs that are tied to different cultural traditions in the game. The goal of this analysis is to figure out which is the best, to guide the player in wisely choosing these traditions to grant access to the best MaA.

Getting started¶

First we need to get the data for all the MaAs we are interested in comparing. This data can be conveniently found on the CK3 wiki. Let's grab the data using pandas. We use selenium since the webpage will throw errors with javascript disabled.

import time
from io import StringIO
from selenium import webdriver
from selenium.webdriver.chrome.options import Options
import pandas as pd
from IPython.display import display, HTML
import rich

# Display helper function to hide index
def show_df(df):
    display(HTML(df.to_html(index=False)))

# Get dataframe from the webpage
options = Options()
driver = webdriver.Chrome(options=options)
url = "https://ck3.paradoxwikis.com/Army"
driver.get(url)
html = driver.page_source
driver.quit()

df_list = pd.read_html(StringIO(html))
df = df_list[2] # We want the third table on the page
df.style.hide(axis=0)
show_df(df[:10])

Regiment	Type	Cost	Raised maintenance	Provisions used	Unnamed: 5	Unnamed: 6	Unnamed: 7	Unnamed: 8	Counters	Favorable terrain	Unfavorable terrain	Tradition	Era
Abudrar	NaN	81.0	0.81	3	20	16	20	10	NaN	NaN	NaN	Mountain Herding	Tribal
Ayyar	NaN	75.0	1.08	12	35	22	12	0	NaN	NaN	NaN	Futuwaa	Tribal
Bush Hunters	NaN	55.0	0.66	3	30	12	5	0	NaN	NaN	NaN	Bush Hunting	Tribal
Druzhina	NaN	117.0	1.56	7	40	30	0	30	NaN	NaN	NaN	Druzhina	Tribal
Gendarmes	NaN	240.0	2.52	21	125	40	20	10	NaN	NaN	NaN	Chanson de Geste	Late Medieval
Goedendag Militia	NaN	45.0	0.45	3	13	18	0	16	NaN	NaN	NaN	Polders	Late Medieval
Guinean Uplanders	NaN	54.0	0.54	3	18	14	0	20	NaN	NaN	NaN	Upland Skirmishing	Tribal
Horn Warriors	NaN	45.0	0.45	7	12	16	0	20	NaN	NaN	NaN	Mountain Skirmishing	Tribal
Horse Archers	NaN	135.0	1.35	7	45	20	40	30	NaN	NaN	NaN	Horse Lords	Tribal
Huscarls	NaN	115.0	1.53	3	40	26	0	24	NaN	NaN	NaN	Hirds / Coastal Warriors	Tribal

Cleaning¶

Now that we have the data, we have to clean up a couple of things. First let's add the proper column names.

df.rename(columns={"Unnamed: 5": "Damage", "Unnamed: 6" : "Toughness", "Unnamed: 7" : "Pursuit", "Unnamed: 8" : "Screen"}, inplace=True)
df.columns

Index(['Regiment', 'Type', 'Cost', 'Raised maintenance', 'Provisions used',
       'Damage', 'Toughness', 'Pursuit', 'Screen', 'Counters',
       'Favorable terrain', 'Unfavorable terrain', 'Tradition', 'Era'],
      dtype='object')

Now let's fix the NaN values. For the purposes of this analysis we do not care about counters, favorable terrain, or unfavorable terrain, so we can drop those columns. However, we need the unit type for the unit size. To do this I will just manually input the unit sizes in the Type column.

# drop NaN columns we don't need
df.drop(columns=["Counters", "Favorable terrain", "Unfavorable terrain"], inplace=True)

unit_types = {
    "Abudrar": 'light infantry',
    "Akritai": 'heavy infantry',
    "Ayrudzi": 'light cavalry',
    "Ayyar": 'heavy infantry',
    "Ballistrai": 'archers',
    "Bondi": 'pikemen',
    "Bush Hunters": 'archers',
    "Conrois": 'heavy cavalry',
    "Druzhina": 'heavy infantry',
    "Gendarmes": 'heavy cavalry',
    "Goedendag Militia": 'light infantry',
    "Guinean Uplanders": 'light infantry',
    "Horn Warriors": 'light infantry',
    "Horse Archers": 'horse archers',
    "Huscarls": 'heavy infantry',
    "Kataphraktoi": 'heavy cavalry',
    "Khandayat": 'heavy infantry',
    "Konni": 'light cavalry',
    "Lenkas": 'heavy infantry',
    "Longbowmen": 'archers',
    "Metsänvartija": 'archers',
    "Monaspa": 'heavy cavalry',
    "Mountaineers": 'heavy infantry',
    "Mubarizun": 'heavy infantry',
    "Mulathanūm": 'light cavalry',
    "Nile Archers": 'archers',
    "Palace Guards": 'heavy infantry',
    "Picchieri": 'pikemen',
    "Sarawit": 'heavy infantry',
    "Schiltron": 'pikemen',
    "Shomer": 'light infantry',
    "Skoutatoi": 'pikemen',
    "Tarkhans": 'heavy cavalry',
    "Tawashi": 'light cavalry',
    "Varangian Veterans": 'heavy infantry',
    "Vigmen": 'archers',
    "Zbrojnosh": 'heavy infantry',
    "Zupin Spearmen": 'pikemen'
}

unit_sizes = {
    "Abudrar": 100,
    "Akritai": 100,
    "Ayrudzi": 50,
    "Ayyar": 50,
    "Ballistrai": 50,
    "Bondi": 100,
    "Bush Hunters": 100,
    "Conrois": 50,
    "Druzhina": 100,
    "Gendarmes": 50,
    "Goedendag Militia": 100,
    "Guinean Uplanders": 100,
    "Horn Warriors": 100,
    "Horse Archers": 100,
    "Huscarls": 100,
    "Kataphraktoi": 50,
    "Khandayat": 100,
    "Konni": 100,
    "Lenkas": 100,
    "Longbowmen": 100,
    "Metsänvartija": 100,
    "Monaspa": 50,
    "Mountaineers": 100,
    "Mubarizun": 100,
    "Mulathanūm": 100,
    "Nile Archers": 100,
    "Palace Guards": 100,
    "Picchieri": 100,
    "Sarawit": 100,
    "Schiltron": 100,
    "Shomer": 100,
    "Skoutatoi": 100,
    "Tarkhans": 50,
    "Tawashi": 100,
    "Varangian Veterans": 100,
    "Vigmen": 100,
    "Zbrojnosh": 100,
    "Zupin Spearmen": 100
}

df['Type'] = df['Regiment'].map(unit_types)
df['Size'] = df['Regiment'].map(unit_sizes)
show_df(df[:10])

Regiment	Type	Cost	Raised maintenance	Provisions used	Damage	Toughness	Pursuit	Screen	Tradition	Era	Size
Abudrar	light infantry	81.0	0.81	3	20	16	20	10	Mountain Herding	Tribal	100
Ayyar	heavy infantry	75.0	1.08	12	35	22	12	0	Futuwaa	Tribal	50
Bush Hunters	archers	55.0	0.66	3	30	12	5	0	Bush Hunting	Tribal	100
Druzhina	heavy infantry	117.0	1.56	7	40	30	0	30	Druzhina	Tribal	100
Gendarmes	heavy cavalry	240.0	2.52	21	125	40	20	10	Chanson de Geste	Late Medieval	50
Goedendag Militia	light infantry	45.0	0.45	3	13	18	0	16	Polders	Late Medieval	100
Guinean Uplanders	light infantry	54.0	0.54	3	18	14	0	20	Upland Skirmishing	Tribal	100
Horn Warriors	light infantry	45.0	0.45	7	12	16	0	20	Mountain Skirmishing	Tribal	100
Horse Archers	horse archers	135.0	1.35	7	45	20	40	30	Horse Lords	Tribal	100
Huscarls	heavy infantry	115.0	1.53	3	40	26	0	24	Hirds / Coastal Warriors	Tribal	100

Analysis¶

Now that the data clean we can get to the analysis. Firstly, let's look at what the most effective MaA is for winning battles with no consideration for maintenance or what happens in the retreat phase. There are a few things we must first note:

Damage: During a battle, there are several rounds until one side is entirely routed. During each of these round, 1 soldier does 0.03 damage per point of Damage stat.
Toughness: Each point allows for a soldier to take 1 total damage before routing.
Combat Width: An extra consideration is something called combat width, computed with the equation (Enemy units + Ally units) * (0.5 * Terrain Width). This gives an advantage to smaller unit sizes in a head to head. For the purposes of this analysis we will assume the terrain width is 1.

These will come in handy later, but initially let's just look at damage * toughness * unit size

Era considerations¶

Since some MaAs only become available much later into the game, they have zero utility for most of a play-through. Because of this I will limit the comparisons to only tribal era MaAs, which are always available to be recruited.

df = df[df.Era == 'Tribal'].copy()
df['Era'].value_counts()

Era
Tribal    33
Name: count, dtype: int64

Damage x Toughness x Size¶

df['d*t*s'] = df['Damage'] * df['Toughness'] * df['Size']
df.sort_values(by='d*t*s', ascending=False, inplace=True)

show_df(df.loc[:,['Regiment', 'Type', 'd*t*s']][:10])

Regiment	Type	dts
Kataphraktoi	heavy cavalry	210000
Tarkhans	heavy cavalry	165000
Monaspa	heavy cavalry	165000
Varangian Veterans	heavy infantry	135000
Conrois	heavy cavalry	125000
Druzhina	heavy infantry	120000
Mubarizun	heavy infantry	112500
Mountaineers	heavy infantry	104000
Huscarls	heavy infantry	104000
Lenkas	heavy infantry	92400

Well, there's the top 10, but let's see if we can get a better sense of the distribution.

import plotly.express as px
from plotly.colors import named_colorscales

fig = px.strip(
    df, 
    y="d*t*s", 
    hover_name='Regiment', 
    hover_data=['Type', 'Size'], 
    labels={'d*t*s': 'Damage x Toughness x Size'},
    title="Most Effective MaA in CK3", 
    height=600, 
    width=800, 
    template='plotly_dark', 
    color='Type'
)
# fig.update_traces(marker_color="yellow")
fig.show()

We can see that the top contenders are all heavy cavalry, even before taking into account the combat width advantage they have for having a smaller unit size.

This shows Kataphraktoi as our top performer by Damage x Toughness x Size.

Damage x Size vs. Toughness x Size¶

There are some instances, however, where Toughness or Damage may be more or less preferable than the other. For example, if you are running an army with a lot of levies (cannon fodder), toughness is less important since more of the enemy's damage is soaked up by your levies. Conversely, if you are running a Knight heavy army (Knights are special characters, not nameless troops), toughness may be more important to protect your relatively fragile but high damage knights. Let's look at a scatter plot comparing damage and toughness.

import plotly.graph_objects as go

df['t*s'] = df['Toughness'] * df['Size']
df['d*s'] = df['Damage'] * df['Size']
fig = px.scatter(
    df, 
    x='d*s', 
    y='t*s', 
    labels={'d*s':"Damage x Size", 't*s': 'Toughness x Size'},
    title='Damage vs. Toughness Trade-off',
    hover_name='Regiment',
    color='Type', 
    height=600, 
    width=800,
    template='plotly_dark')

fig.add_shape(
    type='line',
    xref='x',
    yref='y',
    x0=0,
    y0=0,
    x1=df['d*s'].max()*1.1,
    y1=df['t*s'].max()*1.1,
    opacity=0.5
)

fig.show()

This paints a slightly more nuanced picture. Now it seems like there are two main contenders for "best" depending on what the player is prioritizing. This now shows Varangian Veterans as the best if the player prefers toughness, and Kataphraktoi if the player prefers damage, based on the rest of their army composition.

On Pursuit and Screen¶

Pursuit is how much extra damage can be done after winning a battle. This can be useful for what is called "stack-wiping," which is when you kill or capture the entire enemy army, with no one able to escape. While this can be very useful it requires winning the battle first. Therefore, I will not be considering it in these analyses. Similarly, screen is only useful when you have lost a battle to protect from further loses. Our goal is to not lose any battles, so we will not be considering it in these analyses either.

Gold Considerations¶

MaA cost gold to recruit, and to maintain. The cost to recruit is a one time cost, and only really matter in the very early game, since once recruited you will never have to pay that cost again. For this reason, we will not consider it in these analyses. Maintenance, however, is important for much longer. There are severe penalties for fighting battles while in debt, and a massive army of the most expensive troops can lead a player to financial ruin very quickly. In the late game maintenance is less of a concern since your economy should be strong enough to support any army, but for much of the game this is still an important consideration. Lets look at the same two graphs as before, but now weighted by maintenance costs.

df['d*t*s/gold'] = df['d*t*s'] / df['Raised maintenance']
df.sort_values(by='d*t*s/gold', ascending=False, inplace=True)
show_df(df.loc[:,['Regiment', 'Type', 'Raised maintenance','d*t*s/gold', 'Era']][:10])

Regiment	Type	Raised maintenance	dts/gold	Era
Mubarizun	heavy infantry	1.44	78125.000000	Tribal
Kataphraktoi	heavy cavalry	2.73	76923.076923	Tribal
Druzhina	heavy infantry	1.56	76923.076923	Tribal
Mountaineers	heavy infantry	1.44	72222.222222	Tribal
Ayrudzi	light cavalry	1.05	71428.571429	Tribal
Conrois	heavy cavalry	1.78	70224.719101	Tribal
Huscarls	heavy infantry	1.53	67973.856209	Tribal
Varangian Veterans	heavy infantry	2.00	67500.000000	Tribal
Horse Archers	horse archers	1.35	66666.666667	Tribal
Khandayat	heavy infantry	1.44	64166.666667	Tribal

This changes up the top 10 quite a bit. With cost considerations heavy infantry MaAs are now preferable, with Mubarizun coming out on top. Let's look at the strip chart.

fig = px.strip(
    df, 
    y="d*t*s/gold", 
    hover_name='Regiment', 
    hover_data=['Type', 'Size', 'Raised maintenance'], 
    labels={'d*t*s/gold': '(Damage x Toughness x Size) / Maintenance Cost'},
    title="Most Effective MaA in CK3 Per Gold", 
    height=600, 
    width=800, 
    template='plotly_dark', 
    color='Type'
)
# fig.update_traces(marker_color="yellow")
fig.show()

We can see that the ranking are much more linear. There are some interesting outliers. For one, almost all the light cavalry is terrible, except the Ayrudzi. This is likely a balancing effort by the developers of the game to make light cavalry more attractive, since Ayrudzi are a recent addition. Another outlier is longbowmen, which are the lowest of the bunch. The reason for this is, unlike other MaAs, they improve with each era, becoming very formidable by the end of the game. Unfortunately as mentioned before, late game has limited utility since the player will likely already be very strong, and can field almost any force.

Damage vs Toughness per Maintenance¶

Now finally lets look at Damage and Toughness individually, while considering Maintenance.

df['t*s/gold'] = df['t*s'] / df['Raised maintenance']
df['d*s/gold'] = df['d*s'] / df['Raised maintenance']
fig = px.scatter(
    df, 
    x='d*s/gold', 
    y='t*s/gold', 
    labels={'d*s/gold':"Damage x Size / Maintenance Cost", 't*s/gold': 'Toughness x Size / Maintenance Cost'},
    title='Damage vs. Toughness Trade-off per Gold',
    hover_name='Regiment',
    color='Type', 
    height=600, 
    width=800,
    template='plotly_dark')

fig.add_shape(
    type='line',
    xref='x',
    yref='y',
    x0=0,
    y0=0,
    x1=df['d*s/gold'].max()*1.1,
    y1=df['t*s/gold'].max()*1.1,
    opacity=0.5
)

fig.show()

This paints an interesting picture. Our usual suspects (Heavy Cavalry and Heavy Infantry) aren't found on either extreme. Instead, we have two archers on the far right (Damage heavy per maintenance) and a single light infantry on the top (Toughness heavy per maintenance). None of the specialist appear in the top 10 damage x toughness charts, but may prove valuable if the player:

Has very little income
Has a ton of levies (High damage preference) or Very strong Knights (High toughness preference)

However, situation 1 and situation 2 very rarely happen together, since if you have a lot of levies or a lot of strong knights, you likely will have a lot of income too.

Conclusion¶

So, what does all of this mean? Essentially, it can be boiled down to these questions.

Do you have a lot of extra income? If so max out on Kataphraktoi.
Do you have a good amount of extra income, but can't support a lot of Kataphraktoi? If so max out on Mubarizun.
Are you struggling with gold and can't support max stacks of Mubarizun or other comparable Heavy Infantry? If so Ayrzudi have the lowest maintenance in the Damage x Toughness / Gold top 10.
Are you struggling with gold but want a damage or toughness focused composition? Max out Nile archers for damage, or Shomer for toughness.

Further work¶

I plan to come back to this and compute the optimal MaA for adventurers (who use provisions for maintenance, not gold), as well as to add a head-to-head simulation to better capture the effects of terrain width. Taking terrain width into account will boost up heavy cavalry more, and perhaps make them the best choice given a budget as well. Additionally, it would be interesting to do a pairwise 1v1 tournament to see who has the most wins and loses when considering unit type counters.