R on Blood Bowl Scholarly blogs

The Super League: Tournament Blood Bowl online

Thu, 21 Sep 2023 00:00:00 +0000

Yet another Blood Bowl post! If you don’t know about Blood Bowl and/or FUMBBL, See my previous blog posts on Blood Bowl for more background and stats.

This one is to introduce the Super League, where top (and not so top) coaches compete with each other online on FUMBBL. The Super League is a system of interconnected leagues, in a pyramid-like system. This is not uncommon in sports, a well known example is the English Football league system, also called the “football pyramid”. The creator and tireless promotor of this league, the infamous Tripleskull from Denmark, asked me to do some data analysis on the SL, with this blog as a result.

How does it work? Each season coaches play a match against all the other coaches in their division / tier (“single round-robin”). At the end of the season coaches are promoted to a higher division, remain, or are relegated to a lower division, based on how well they did. New coaches have to start in the lowest division and work their way to the top. The highest division is called the “Premier League”, where some of the world’s best Blood Bowl coaches compete for the crown (well ring in fact, designed and 3D printed by Flashman1234).

In a post on Discord, current head admin Storr explains the design of the Super League:

The Super League is designed as a tiered league with a pyramid structure to allow coaches to play against opponents of their own level. This is done intentionally in contrast to the TSD*, where we have completely random divisions each season with a playoff at the end. We do have a rather high number of relegations and promotions to allow everyone to pass rather quickly to the tier they “should” be in, but we need to try to keep the balance between the two. I know it is a bit unfortunate to join now and having to take the long way to the top, but that’s what it is in a tiered league. There will not be a reset in the Super League, as that would be against the very nature and idea of the league. There is, however, the option to make the lower tiers a bit wider, as we don’t follow through with the pyramid structure through all tiers at the moment. The reason for that is that the league grew quite large rather quick, and a decision was made to implement the divisional structure as it is now, possibly because it fit the participant numbers well (i.e. “historical reasons). We will think about expanding the pyramid structure, especially if the league grows even further, as we also don’t want to handle a giant swiss division at the bottom (ask LemonheadWallenstein why), but I won’t make any promises.

(*TackleZone Star Division, another competitive league/tournament on FUMBBL)

At this point we should mention TheFear, who together with TripleSkull started it all, and was sole admin for Season 0 and 1. His historic blog post on FUMBBL introducing the Super League can be found here on FUMBBL.

To illustrate the (somewhat) pyramid-like structure of the SL I made a schematic:

Each rectangular block is a division with 8 coaches, except for the lowest division, which uses a Swiss pairing system (i.e. coaches are initially randomly matched and after that are matched based on their performance so-far). At the time of writing the league is going into its third season, with 188 coaches competing.

A “season 0” Swiss tournament was played to fill the “empty” pyramid with coaches according to their performance in S0. This filled the top half of the pyramid, after that new coaches were divided randomly across the remaining divisions (D3 and D4), hence this season was nicknamed “the season of shortcuts” by Lorebass. In November 2022 Season 1 kicked off. Going into S2 (march 2023) and now S3 (sept 2023), the system is working as intended with coaches moving through the pyramid via promotion and relegation.

Ranking the stars!

So what can be learned from two seasons of Super League competition play? As this is about competition, let’s try and create a ranking over who did best over those two seasons.

{width=100%} Here we use the ranking primarily to summarise / show part of the data. In creating the ranking, I made several choices that can be considered arbitrary, I mention these in parentheses.

The ranking is constructed as follows: Main idea is that competing in the Premier League is valued the most (Arbitrary Decision #1, division level as main score). This puts Strider84 and RolexWL firmly at the top as the only coaches that ended up, and managed to stayed in the Premier League for all three seasons. Here, Strider84 is clearly number one, since he also managed to win the PL (Arbitrary Decision #2, trophies as first tie breaker). Next we have a block of six coaches that played twice in the PL, and once in the league directly below (Championship A/B). Using trophies as a tie breaker puts Purplegoo in third place with two trophies, and Storr in fourth place with one trophy.

How to rank the remaining four coaches? For this I looked at placement within the division, averaged over S1 and S2. This puts Stimme in fifth place, as his average placement was 2. (Arbitrary decision #3, average placement over season disregarding division level as second tie breaker). Finally I cut the list at a threshold of score 12 or less (I award 6 points for playing one season in the PL, 5 points for a single season in the CH etc). This resulted in one Dutch Coach making the list, and that honour goes to Schwifty, who managed to accomplish this playing Nurgle! Respect.

Rising to the top

As all new coaches have to start at the bottom (with Season 1 being an exception where coaches could be randomly placed in D2, halfway in the pyramid), we can change perspective. Let’s have a look at ALL coaches who in the upcoming season (S3) play in the top two divisions and how they got there. The top two divisions are Premier League and Championship A/B, containing 24 coaches:

{ width=100%} In the Premier League, we see two new names: Junior84 and siggi. They rose with lightning speed to the top division, and show what is possible for the rest of us. Good luck up there competing with the best! In the Championship A/B division, we have a group of five coaches that steadfastly marched their way up from Division 2. Here we have a Dutch coach as well, go Edwin!

What teams do successful coaches pick?

Note that the league placing belongs to the coach. Coaches can swap teams (between seasons) and retain their ranking. So what teams did successful coaches pick in Season 2?

In season 2 the World Cup ruleset was used, with one exception: no star players. Now, we already know the World Cup win rates by team, thanks to Mike Davies and his awesome Tableau stats site. Here I show the top 10 teams with the highest win rate at the World Cup:

Top five teams in the WC ruleset are Underworld, Dark Elf, Amazon, Lizardmen and Shambling Undead. Indeed, the same teams did well in the Super League:

{ width=100%}

Here I looked at teams from Coaches that played in the top three divisions in Season 2, and managed to either remain in their league, or that got promoted to a higher league.

Interestingly, two coaches managed to do well with Imperial Nobility, and also low tier teams such as Nurgle, Chaos Chosen and Black Orc has some degree of success. Note that the absence of Old World Alliance in the Super League is likely because its success in the World Cup depended heavily on Star players.

Final remarks

If you’re interested in following along head over to Discord or FUMBBL. If you would like to join you have to wait a few months, Season 4 will likely start in Feb/March 2024. If you’re interested in the data analysis itself, both data and R code are available on my GitHub, the repository fumbbl_super_league is a good place to start.

Visualizing Blood Bowl Tournament rosters

Mon, 27 Feb 2023 00:00:00 +0000

(Photo by Erik Cats)

Yet another Blood Bowl post! This one is to warm up to the upcoming World Cup, analyzing variation in Roster choices by top coaches. For the uninitiated: In my spare time I like to play Blood Bowl, a chess-like board game where two teams play a match of fantasy football. It is set in the Warhammer universe, with teams populated by Orcs, Dwarves, Elves etc. See my previous blog posts on Blood Bowl for more background and stats. To get an estimate of the player base: a popular Blood Bowl channel on Youtube called “Bonehead podcast” currently has some 16K subscribers.

So the cool thing of Blood Bowl is that it has a highly competitive Tournament scene. With the ultimate tournament of course being the World Cup! The previous World Cup was held in 2019 in Dornbirn, Austria, and attracted 1400 coaches. This year the World Cup is hosted by Spain, and at least as many coaches are expected in Alicante to battle it out.

Blood Bowl tournaments typically come with a “Rule pack”. This describes which teams are allowed, how much gold coaches have to buy players, and how much skills can be allotted to players to improve their ball handling, or ability to knock other players down.

It turns out that the World Cup is pretty similar to the Eurobowl ruleset, so we can learn from the Eurobowl rosters to prepare for the World Cup. Bad roster choices will put you at a disadvantage even before the actual game has started. So let’s check out what the pro’s are using!

Online Blood Bowl Tournaments on FUMBBL

Blood Bowl can also be played online. Here we focus on FUMBBL.com, consisting of a Game client (a Java app), linked to a website that performs match making, calculates rankings, provides a forum for discussions etc.

There are many things great with respect to FUMBBL. Here I like to mention three key elements that allow for this post:

Online Blood Bowl tournaments are organized on FUMBBL that use exactly the same rules as large tabletop tournaments such as Eurobowl,
FUMBBL has an API where we can fetch the data we need for roster analysis,
FUMBBL stores full replays of all FUMBBL matches, so we can observe how successful rosters can be played

The 2022 Road to Malta Online NAF tournament

Last November, Eurobowl took place in Malta. To get an impression check this vid on Youtube. A few months before Eurobowl 2022, the Nuffle Amorical Football (NAF) organized an online Tournament “Road to Malta” on FUMBBL to warm up to this event. The Online Tournament used the exact same ruleset as the Eurobowl itself, and the Rulepack for the 2023 World cup is highly similar.

The rulepack for the Road to Malta can be found here.

This blog post’s main goal is to share the analysis results. Check my Github for the technical stuff (R/Python web scraping and visualization of FUMBBL data).

I’ll discuss the most popular races, and leave the rest for the interested reader. I created a PDF rosterbook containing all 29 different races that participated in the tournament. For each race I made a plot (a clustered heatmap) displaying the roster as a colorcoded matrix, where teams that are similar (based on a cost weighted similarity score) are put side by side. To give some information about the performance of the various builds, I added the tournament points scored behind the coach name.

Of course, roster build is only one factor influencing tournament results, with skill being much more important, so we have to keep that in mind. Occasionaly, I’ll compare with the Eurobowl rosters and their performance, for this I use the great Tableau resource compiled by Mike Davies.

Top 10 races in Road To Malta

The Road to Malta tournament featured 46 teams of 4 coaches each, supplying a total of 184 rosters for our analysis. Here we focus on the top 10 most popular teams:

The two most popular teams with 16 rosters each were Underworld Denizens and High Elf.

Underworld Denizens (Tier 2)

We start with Underworld Denizens. Even after the nerf to UW by GW in November 2021 they kept their popularity. And 10 out of 16 rosters include a star player! All rosters pick the full allowance of Blitzer, Thrower, Runner and three clan rats. From the plot below, we see that the Morg plus Troll build was most popular, but that there were various other “Big guy + star player” builds, with the big guy being either a Troll or the Rat Ogre, and the star player being either Varag, Glart, or Kreek.

Surprisingly, there are no Hakflem builds ! This might be due to the additional skill cost, although Morg too had an additional skill cost, leaving only a single skill left, that was unanimously used to gain an extra re-roll by putting the leader skill on the thrower.

{width=150%} The roster with the best performance (by Sandune) was the Varag plus Troll build.

High elf (Tier 4)

UW shared the top spot for most popular team with High Elves. High Elf’s recent popularity is likely related to their low tier placement, for this tournament it was Tier 4. This allowed for 8 primary skills and 1 secondary skill (or 6 primary skills and 2 secondary skills), on a 12 player line up. From the rosters, it appears that there are two roster variants popular. Both variants have the same lineup wrt players (2 blitzers, 3 catcher, 1 thrower). Coaches either choose three rerolls without an apo, or two rerolls with apo. The two-reroll builds all use leader to compensate for the “missing” reroll.

The best performing roster here was by Hartl78, who fielded a Sneaky Git/ Dirty player lineman, a most unelvish thing to do :) For the World cup, High elves moved from Tier 4 to Tier 2, so performance of the team will likely suffer.

{width=100%}

Necromantic Horror (Tier 2)

Necro was in tier 2, allowing for seven primary skills. All rosters went for this skill pack. Skill wise, well, a lot of guard and block! For Necro, there where three variants popular:

One werewolf, 6 zombies, 3 rerolls
Two werewolves, 4 zombies, 2 rerolls
Two werewolves, 3 zombies, 3 rerolls

Performance-wise, no clear differences to be seen.

{width=200%}

Shambling Undead (Tier 1)

The Shambling Undead rosters are all surprisingly similar:

2 mummies with guard,
2 wights (one guard, one tackle or mighty blow),
4 ghoul runners (two with block, 1 wrestle), and
three rerolls.

The only noticable variation is whether to complement that lineup with 5 zombies, 4 zombies plus 1 skeleton, or 3 zombies plus 2 skeletons. Performance wise there is little to guide that last judgement, as all three variants performed similar.

{width=200%}

Dark Elf (Tier 1)

All coaches selected the skill pack with 6 primary skills. Skill wise, there appears to be strong consensus: dodge on the blitzers, block and wrestle on the Witch Elves, and if you pick a runner put leader on it for that extra reroll.

There is some variation in the positional choices, we can distinguish three builds:

No assassins, 1 runner, 4-5 linemen
1 assassin, 1 runner, 3 linemen
2 assassins, no runners, 3 linemen

No clear performance differences between the three builds.

{width=200%}

Chaos Dwarf (Tier 1)

For chaos dwarves, the big question is whether to take a minotaur or not. The teams without minotaur did slightly worse. Without a minotaur, the full set of five hobgoblins can be fielded, as well as an additional reroll and apothecary. Skill wise, we can see that the three rosters with sure hands on a hobgoblin underperformed, suggesting that this skill choice is subtoptimal here. We can see that the most common skill allocation is to put block on the bull centaurs and guard on the dwarf blockers.

Interestingly, the highest scoring build by Liam has an uncommon skill choice, with guard on the bull centaurs and two blockers with Mighty Blow.

{width=200%}

Skaven (Tier 2)

Skaven did very well during the Eurobowl, with three Skaven teams in the top 10 best performing coaches. Two of those coaches also participated in Road to Malta, Olivierdulac and Sokratez (Tank).

First the commonalities: All rosters have the Rat ogre (Block or Juggernaut) and at least three gutter runners, of which one has strip ball. For Skaven I highlighted three variants:

Star player builds without blitzer (Either Kreek or Glart, this leaves 4 skills)
3 reroll builds without Thrower
2 reroll build with thrower (+ leader)

{width=200%}

Lizardmen (Tier 1)

For lizardmen coaches life is simple. 6 saurusses, 5 skinks, krox and an apo. Put block on all saurusses, done. That leaves time to bicker on about bigger questions in life, such a whether or not to swap one regular skink for a Chameleon skink :) Just as with all the previous roster variants, performance wise there is no clear winner. Some coaches swap one to three blocks for other skills such as guard, tackle or wrestle.

{width=200%}

Orc (Tier 1)

Orcs also did very well in Malta, with three teams in the top 10. The orcs show some serious roster variation! Agreed, all rosters take four blitzers with mighty blow and tackle, and four big uns with block and guard, but apart from that we see rosters with and without the Troll, with and without an apo, with and without a Thrower, and with varying amounts of goblins. Interestingly, the most exotic roster, with three goblins, and no thrower, performed best here.

{width=200%}

Wood Elf (Tier 1)

Wood elf coaches always take the treeman, two wardancers, and at least two catchers. Typical skills choices are one wardancer with tackle, and one wardancer with strip ball.

As the wood elf players are expensive, to get at least 2 rerolls requires a lineup with only 11 players. Variation is in the number of catchers (2-4) and taking a thrower, versus taking ordinary linemen. On one end, we see rosters that take all four catchers and a thrower, with only three linemen. At the other end, we see rosters that take only two catchers and a thrower, or three catchers, and take five linemen. This allows for three rerolls or two rerolls and an apothecary.

{width=200%}

Final remarks

For now my curiosity on roster variation has been satisfied, and I really looking forward to playing more and learning the finesses of this great game. The World cup in Alicante will be my first international tournament, hope to see you all there!

Classifying Blood Bowl teams using clustered heatmaps

Tue, 01 Nov 2022 00:00:00 +0000

(Photo by Erik Cats)

“If you graph the numbers of any system, patterns emerge” is one of my favorite movie quotes (from Darren Aronofsky’s cult movie about mathematics $\pi$ (“pi”)). In this post we’ll graph the numbers from the Blood Bowl Fantasy football game, and see what patterns emerge. Blood Bowl is a board game that can be summarized as “fantasy-chess-with-dice”, but this would hardly do the game justice. For example, in chess both players play with the same pieces, but in Blood Bowl, almost 30 different teams (e.g. orcs, elves, etc) are available to choose from, each team with different skills that require different playing styles. In addition, Blood Bowl coaches must assemble and paint their playing pieces themselves, making it a creative hobby as well.

For this blog post, we have a look at similarities and differences between the different Blood Bowl 2020 teams, and see where the newly introduced Black Orc and Khorne teams fit in. Using data analysis, we can cluster teams that have similar (average) match performance statistics and graph the data using heatmaps.

Heatmaps are a graphical representation of the data, with for example darker colors representing higher numbers. This allows patterns to emerge visually, and deviations on the patterns are also easy to spot. (Fun fact: a hundred years ago, people already “shaded matrices” but did not call it a heatmap yet (Wilkinson and Friendly 2009).)

We use match performance data from FUMBBL.com where Blood Bowl 2020 can be played online. For a previous blog post that describes the process of scraping the data see here. I made the match data (currently from August 2020 up to June 2022) publicly available in a Github repository.

Team play style categories

The most common way to classify the 25-30 different Blood Bowl teams is to distinguish four categories:

Bash (e.g. Orcs)
Agile (or Dash) (e.g. Wood Elf)
Hybrid (e.g. Humans)
Stunty (e.g. Halflings)

To formally classify a team we can use the follow decision tree, taking as input the most common roster choices for a team:

Does the team roster have a lot of Stunty players and a few Big Guys with negatrait?
- yes classify as Stunty
- no continue
Does the team roster have 4+ players with Strength skill access but < 4 players with Agility access?
- yes classify as Bash
- no continue
Does the team roster have 4+ players with Agility skill access but < 4 players with Strength access?
- yes classify as Agile / Dash
- no classify as Hybrid

As an example, Shambling Undead are typically played with 2 wights and 2 mummies (4 players with Strength access), but also with four Ghoul runners (4 players with Agility access), so this team is classified as Hybrid.

Main source for this scheme is (Dode 2017), but the bash/dash/hybrid/stunty categorization is widespread, for example at (Breidr 2015), (Amiral 2017) and (Schlice 2018).

It will be interesting to compare the patterns in the data with this categorization. In the next section we’ll discuss the various match statistics available from FUMBBL, but first we need to prep the data.

Read and prep the data

We start with reading in the scraped FUMBBL match data, see my previous blog posts mentioned above for details.

# Load packages
library(tidyverse)
library(ggfortify)
library(ggrepel)

df_mbt <- read.csv(file = "fumbbl_datasets/df_mbt.csv")

race_types <- unique(df_mbt %>% 
                       select(race_name, race_type) %>%
                       filter(race_type != "") %>%
                       arrange(race_type))

This blog post focusses on the Blood Bowl 2020 ruleset, for this we need the “Competitive” division from FUMBBL. (I performed the analysis for the older divisions using the 2016 ruleset as well, the plots can be found at the end of this blog post. )

A blog post from Schlice (Schlice 2018) got me interested in BB team classification using data. In his post, he makes heavy use of functional programming using R’s purrr package. This allows us to write a function and have this function work in parallel on a list of objects, and have it return the results also in list form.

As this was new to me, I decided to adapt his code to process the four divisions simultaneously. To do so, I wrote a function filter_division() that takes the source data and selects only matches from a particular division:

divisions <- c("Competitive", "Blackbox", "Ranked", "Regular_league")

filter_division <- function(div_name){
  df_mbt %>% 
  filter(division_name == div_name) %>%
  filter(race_name != "Treeman") %>%
  filter(race_name != "Simyin")
}

data_tables <- purrr::map(divisions, filter_division)
names(data_tables) <- divisions

data_tables now contains a list of four datasets, one for each division.

Next step is to select the variables we are interested in. Again we use the purrr::map() function to apply our selection function at once to all four datasets:

select_stats <- function(df) {
  df %>% 
  select(race_name, race_type, team_score, away_team_score:away_cas_rip) %>%
  select(-(home_cas_bh:home_cas_rip), -(away_cas_bh:away_cas_rip))
}

data_tables <- map(data_tables, select_stats)

We end up with the following set of variables:

`1`	`data_tables$Competitive %>% colnames()`

##  [1] "race_name"       "race_type"       "team_score"      "away_team_score"
##  [5] "home_comp"       "home_pass"       "home_rush"       "home_block"     
##  [9] "home_foul"       "home_cas"        "away_comp"       "away_pass"      
## [13] "away_rush"       "away_block"      "away_foul"       "away_cas"

A quick summary of the FUMMBL match statistics

The match statistics that are made available by FUMBBL are all related to important events during Blood Bowl gameplay. Of course, teams with similar stats can still be different on some aspect not captured in the data, so we keep that in mind.

The most obvious one is scoring Touchdowns (team_score and away_team_score). As this involves getting the ball in the end zone of the opposing player, stats for actions that contribute to scoring such as passing the ball are also present. There is the number of completed passes (home_comp and away_comp) as well as the net passing distance (with distance towards the opposing endzone counted positive and counted negative for passes in the opposite direction) (home_pass and away_pass). As passing is risky, running the ball into the endzone is the most common way to score. For running with the ball FUMBBL has the rushing statistic. This name was chosen some 20 years ago, and now may lead to confusion as under the BB2020 rules moving extra squares is called “rushing” (previously called “Going-for-it” or GFI).

The rushing statistic is described in a 2004 (!) FUMBBL forum post by SkiJunkie (the author of the predecessor of FUMBBL):

running/passing in the wrong direction counts negative. 
So if you run one step forward then one step back, your net rushing is 0.

Running/passing up and down neither adds nor subtracts. Getting pushed/thrown does not count. 
Only movement made during the players move who has the ball counts.

So, you can end up with negative rushing/passing.

This is great! The rushing statistic captures both ball possession as well as the maximum movement of typical ball handlers in each team. For example, in a Skaven team, the ball is typically handled by Gutter Runners, that have a maximum movement of 9. Compare this to the Dwarf team, where the Dwarf runner only has movement 6.

Finally, we have the statistics related towards taking out opposing players. This involves blocking, leading to blocking casualties, as well as fouling (illegally hitting players that are already down). For blocking, the total number of blocks thrown is tracked (home_block and away_block), for casualties we have the number of players who suffer a casualty (home_cas , away_cas) and for fouling we have the number of fouls performed by each team (home_foul, away_foul). For casualties, we use both the number of casualties inflicted as well as casualties suffered (home_cas and away_cas). The average amount of casualties suffered quantifies the vulnerability of a team, and will likely be related to the average armour value (AV) of a team.

As we are interested in statistics at the team level (not for individual matches), we take the average of all the statistics by team:

divisions <- c("Competitive", "Blackbox", "Ranked", "Regular_league")

df_agg <- data_tables$Competitive %>% 
  group_by(race_name, race_type) %>% 
  summarise(Blocks = mean(home_block), 
            Completions = mean(home_comp),
            Fouls = mean(home_foul),
            Distance_passed = mean(home_pass), 
            Running_w_ball = mean(home_rush), 
            Cas_inflicted = mean(home_cas),
            Cas_suffered = mean(away_cas), 
            Touchdowns = mean(team_score),
            size = n())

## `summarise()` has regrouped the output.
## ℹ Summaries were computed grouped by race_name and race_type.
## ℹ Output is grouped by race_name.
## ℹ Use `summarise(.groups = "drop_last")` to silence this message.
## ℹ Use `summarise(.by = c(race_name, race_type))` for per-operation grouping
##   (`?dplyr::dplyr_by`) instead.

`1`	`dim(df_agg)`

## [1] 28 11

This gives us 11 numbers for 28 different BB2020 teams.

So which teams have similar stats?

Next, we want to compare the different Blood Bowl teams on their match statistics with each other, and see which races have comparable stats.

If we have only one statistic, it is easy: we can just plot the different teams on one axis, the value of the statistic on the other axis, sort them and see which races are closest by.

Lets do that for the number of blocks made:

options(dplyr.summarise.inform = FALSE)

ggplot(df_agg, aes(x = reorder(race_name, Blocks), y = Blocks, size = size, col = race_type)) +
         geom_point() + scale_size_area() + coord_flip() + labs(x = "team name")

The different colors visualize the Team classification described above. We can see that the average number of blocks is already pretty predictive of team type.

But races can be similar if we look at blocks, but dissimilar if we look at e.g. passing. So lets make a few plots with each plot comparing two different metrics (code hidden for readability):

1
2
3

library(patchwork)

(p1 | p2)/(p3 | p4) / (p5 | plot_spacer()) +   plot_layout(guides = 'collect')

## `geom_smooth()` using formula = 'y ~ x'

Looking at the plots, we can see that there are clear patterns here. Some metrics are inherently related: Teams that make more passes during a match also cover more passing distance, that is only logical. But there are also patterns that are related to what makes teams different: teams that make more blocks tend to pass less (because they are less good at it).

Given these patterns, we also see teams that do not conform to the pattern: For example, in the top left plot, the Goblin and Snotling teams tend to make less blocks, but also less passes compared to other teams. And for the top middle plot we see that Halflings are an exception: Given how much they pass, we would expect a higher passing distance. (A quick look at their roster explains this: Halflings have dedicated Catchers (making passing attractive), but also the Stunty trait, making long distance passes more risky, decreasing the average passing distance).

Finally, which teams are similar depends on which plot we look at! Take for example the Black Orc team: Based on fouling and distance passed (Bottom left plot), it is very similar to the four Stunty teams. However, In the plot that combines Fouls and Blocks (lower right), it suddenly becomes unique!

PCA versus clustered heatmaps

So how can we compare the teams on all stats simultaneously? We can try to reduce the number of variables by looking at correlations between variables. For example, we observed a strong (negative) correlation between the number of blocks and the number of passes. We could take these two variables together and map (“project”) each team on a single new “axis” (the straight line in the top left plot). For this a commonly used technique is PCA (“principal component analysis”). The original variables are combined into “principle components” along which the most variation occurs. A nufflytics blog post by Schlice (Schlice 2018) applies this technique to classify the Blood Bowl teams of the online Blood Bowl 2 variant of the game. Another example of this approach is the master thesis project of Tadas Ivanauskas (Ivanauskas 2020).

With PCA, a common approach is to focus on the first two principal components, as these can be visualized on a 2D coordinate system (i.e. a scatter plot). However, a big disadvantage of PCA is that the new dimensions are difficult to interpret, because the original data is transformed and combined, and therefore no longer recognizable. In the above mentioned blog post, this becomes apparent when the author puts in serious effort to understand the first four principal components, see his “diving deeper” follow up post. Because each component is a (different) linear combination of ALL 17 variables (!), interpreting the principal components becomes very difficult. To do so requires a high cognitive load, and it is unclear where the signal ends and noise starts.

An attractive alternative is performing a cluster analysis on the original data, and using the cluster order to directly visualize the data as a heatmap. This technique is called clustered heatmaps, and is widely used in bioinformatics to display patterns in data.

Two of the most popular clustering algorithms are Hierarchical clustering and k-means clustering. For example we can ask the computer to cluster the 28 teams into e.g. 5 clusters using the K-means algorithm. This assigns each team to one of $k = 5$ clusters, with the cluster centers chosen such that teams within a cluster are close by the cluster center and far away from the other cluster centers. However, this forces us to pick the number of clusters, AND it does not provide information on team similarity WITHIN a cluster.

Clustered heatmaps for the win

Hierarchical clustering circumvents both drawbacks as it clusters ALL the observations in such a way that the most similar observations end up next to each other, and thus provide a complete ordering of all observations. All that remains is to plot the data in this order on a colored grid, and BAM the patterns hit us right between the eyes!

The basic functionality to make a heatmap is provided in the R package ggplot2 by the geom_tile() geometric object. Many wrapper functions around geom_tile() exist to make clustered heatmaps, however they all combine the clustering step with the plotting step.

I am currently experimenting with my own wrapper function ggorder_heatmap(), which I turned into an R package called ggoheatmap. See for details the GitHub repository for the ggoheatmap R package. Note that it is unstable, under development, and possible not even a good idea, so do not use it yourself. Instead use pheatmap by Raivo Kolde, available from CRAN, with the official repo here.

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# pak::pak("gsverhoeven/ggoheatmap")
library(ggoheatmap)

varnames <- c("Blocks", "Fouls", "Cas_inflicted",  "Distance_passed", "Completions", "Running_w_ball", "Cas_suffered")
yorder <- data.frame(yorder = 1:length(varnames), variable = varnames)

df_long <- df_agg %>%
  pivot_longer(cols = !c(race_name, race_type, size), names_to = "variable") %>%
  group_by(variable) %>%
  mutate(sd_value = scale(value)) %>%
  left_join(yorder, by = "variable")

df_long <- ggoheatmap::hclust_order(df_long, 
                        xvar = "race_name", 
                        yvar = "variable", 
                        value_var = "value",
                   clust_method = "complete",
                   dist_method = "euclidean")

library(patchwork)

p1 <- ggplot(df_long, aes(x = reorder(race_name, cluster_order), y = 1, fill = race_type)) +
  geom_tile() + 
  coord_flip() + 
  labs(x = "", y = NULL) + 
  scale_y_discrete(labels = NULL, breaks = NULL) 

p2 <- ggorder_heatmap(df_long, 
                xvar = "race_name", 
                yvar = "variable", 
                col_var = "sd_value", 
                order_var = "cluster_order",
                yorder_var ="yorder",
                legend = FALSE,
                label_var = "value", round.digits = 1) + coord_flip() +
  scale_y_discrete(position = "right")

p1 + p2 + 
  plot_layout(widths = c(1, 6), guides = "collect") + 
  plot_annotation(title = 'BB2020 team typology')

And there we have it, our main result.

First, we see that the existing team categorization does a good job, with the different categories showing different patterns in the match statistics. Note that I did not pre-specify the sort order here, it emerged naturally from the match statistics by the hierarchical clustering algorithm. So the quote from the Pi movie was correct (at least for Blood Bowl): we graphed the numbers, and patterns emerged!

What else can we learn from this plot? The difference between Bash and Hybrid appears not so clear cut: for example: Are Norse (the old BB2020 roster, not the new one) and Chaos Chosen bash or hybrid? Agile and Stunty teams on the other hand are very distinct. Underworld Denizens are an interesting edge case: In BB2020, they cluster right in between Goblin and Skaven teams, I ended up classifying them as Agile. Finally, it is apparent that Vampires indeed are a category of their own.

Conclusions

And now for the question we set out to answer: Where do the new Khorne and Black Orc teams fit in?

Khorne are clearly a hybrid team (because of the lack of skills and frenzy reduces base contact and thus blocks). Black Orcs appear to be quite unique, being a combination of Bashy Hybrid and Fouling.

Finally we produce the clustered heatmap for all four divisions:

source("make_heatmap.R")

for(i in 1:length(divisions)){
  print(make_heatmap(data_tables[[divisions[i]]], division_name = divisions[i]))
}

References

Amiral, Taureau. 2017. “Blood Bowl Teams Strength.” Blood Bowl Strategies, November 24. https://bloodbowlstrategies.com/en/relative-strength-of-teams/.

Breidr. 2015. “Cyanide - ‘Bashy’ Race Choice for League Play.” Blood Bowl Tactics Forum. https://bbtactics.com/forum/threads/bashy-race-choice-for-league-play.4961/.

Dode. 2017. “What Is the Bashiest Agility Team After Dark Elves? :: Blood Bowl 2 General Discussions.” https://steamcommunity.com/app/236690/discussions/0/135512931355543270/.

Ivanauskas, Tadas. 2020. BloodBowl 2 race clustering by different playstyles. http://urn.kb.se/resolve?urn=urn:nbn:se:mau:diva-41540.

Schlice. 2018. “Bash/Dash/Hybrid by the Numbers.” February 27. http://www.nufflytics.com/post/bash-dash-hybrid-by-the-numbers/.

Wilkinson, Leland, and Michael Friendly. 2009. “The History of the Cluster Heat Map.” The American Statistician 63 (2): 179–84. https://doi.org/10.1198/tas.2009.0033.