Nufflenomics: Bloodweiser Kegs

The Bloodweiser Keg (or its less politically correctly named predecessor) has been a mainstay of BloodBowl inducements for as long as I’ve been playing the game. If you’ve got some inducement cash and you don’t know what to do with it, one or – even better – two kegs is the way to go. It is a great fallback, and never really looked upon as a bad decision. I was wondering if the maths backs this up.

What is it?

A Bloodweiser Keg allows you to add +1 to any knockout recovery rolls made during the game, meaning a player returns to the game on a 3+. If you buy two, then you are making all recovery rolls on a 2+. Each keg costs 50,000, and you cannot buy more than 2. 

The mathematics

Each keg you buy means you will return a player to the game on average 1 in 6 times more than you would have previously. Looking at this on a percentage basis, we have the following return rates:

Number of kegs	Returns on	Return %
0	4+	50%
1	3+	67%
2	2+	83%

Each keg gives you an increased change of recovery, and the percentages look good especially if you have 2 kegs.

To calculate the value of the first keg we need to remember that 50% of the time that player would have returned anyway due to having rolled a 4 or more, so we are simply looking at the number of times a 3 is rolled for recovery rolls, therefore each purchase only affects 1 in every 6 dice rolls.

The first will mean that a 3 brings back a player and the second that a 2 brings back a player. As such, each keg is effectively only useful 16% of the time as any higher roll would have provided the recovery anyway.

For the keg to prove its value it will need to return, on average, over 50k of players to the pitch each game.

To calculate the value of players returned to the pitch, we need to consider that a player’s purchase price is for the whole game. If a keg returns a player at half time, then you get their value for the second half, and therefore we should probably only take into account half of their value as being the benefit of the keg.  

You also need to consider how many players you are expecting to get knocked out each game. Some teams are more likely to have players knocked out than average – stunty players, for example. Some teams are less likely to have players knocked out, such as those with thick skull.

A stunty player is knocked out on a 7 or 8 (31% of injury rolls) compared to an 8 or 9 for most players (25% of injury rolls). In actuality, the likelihood is even higher for some stunty teams as their players have lower armour than the average Blood Bowl player.

A player with thick skull is only getting knocked out on a 9 (11% of injury rolls), meaning a keg is significantly less useful for teams with a large amount of the Thick Skull skill.

It is always going to be more valuable for you to have a key positional recover from a knockout roll than a lineman because of their importance to your team. There will be some teams where the number of returns is more important than returning the key players. Snotlings are probably more worried about getting as many players back as possible than about any one specific snotling, however a wood elf team is probably more likely to want their wardancer back over just any returning line elf.

It turns out that the maths behind calculating this payback is pretty difficult as it becomes very situational.  There will be a lot of factors to consider when making the decision to hire a keg.

1.       Can you live without a key player (if no, then an apothecary which allows you to guarantee a KO recovery is a better choice)

2.       Does your opponent have a lot of mighty blow? This means both you’re more likely to get injured, and more likely to get knocked out (depending on when the skill is used).

3.       Does your team have anything that modifies how often you’ll get knocked out (stunty, thick skull)?

4.       How many kick offs do you expect there to be?

5.       Are you playing against a team likely to repeatedly foul your key players and therefore you are likely to have more players removed?

The number of expected rolls is more important than the number of players.

The best way to try to calculate a keg’s expected payback is to assume that both teams will play to score on turn 8 of their drive, and therefore there will be 3 chances at KO recovery. These will be Turns 8/7, 8/8 and 15/16.

It is important not to discount the final turn of the game if you are the team with a one turn touchdown attempt at the end of the game, although if you are a team with little to no possibility of this then you may wish to discount it.

We need to look at each team’s turn individually when doing the maths on this, so rather than a player being back for x/16 turns, we need to look at them being back for x/32 turns.

A player returning on turn 8/7 will therefore be back for 17/32 turns, for example. The table below shows the number of turns at each of the expected return stages.

Turn	Turns available	% of game
8/7	17	53%
8/8	16	50%
15/16	1	3%

Note: we are assuming no further removal of this player partly because it would make the maths super complex and also because otherwise we need to factor in the likelihood of removal of every player throughout the whole game. It’s just not worth it for this purpose. It seems fairly safe to assume that chance of removal has been factored into a player’s initial costing.

Each keg has, as we’ve seen earlier, a 16% effectiveness rate for each KO roll. We can therefore multiply the percentages above by 16% to get the % of a player’s value that we can recover at each time with each keg. That gets us the below table.

Turn	Turns available	% of game	Keg effectiveness
8/7	17	53%	9%
8/8	16	50%	8%
15/16	1	3%	1%

To simplify things, let’s assume that Turn 16 is going to be irrelevant - the percentage is very low and the impact on the game will probably be minimal.

We are therefore only looking at players knocked out in the first half. They will then have 2 chances of recovery – turn 8/7 and 8/8. If they recover on the first of these, then we can ignore the second.

This gives us the table below. Whilst we know that on each individual roll, a keg will improve our recovery chances by 16%, when you look at multiple rolls – as you would expect at the end of the first half – then each keg has diminishing returns.

Kegs	Turn 8/7 recovery	Turn 8/8 recovery	Total Recovery Chance	Increase
0	50.0%	25.0%	75.0%	n/a
1	66.7%	22.2%	88.9%	13.9%
2	83.3%	13.9%	97.2%	8.3%

Let’s start to translate these into value. The first keg makes it 14% more likely that you will have a player on the pitch for the second half.  This is roughly 1 in ever 7 rolls.

With an average value player, say a 50k lineman, you will then be getting 25k for the second half. This will happen on 14% of occasions from the first keg, meaning the keg’s effective value is 25k x 14% - 3.5k. You would therefore need 14 knocked out players for the keg to repay its value. This is obviously not going to happen, and so the conclusion would be that a Keg is inefficient value.

Average player value (k)	Keg 1	KO Rolls for Payback	Keg 2	KO Rolls for Payback
15	1.0	48.0	0.6	80.0
30	2.1	24.0	1.3	40.0
40	2.8	18.0	1.7	30.0
50	3.5	14.4	2.1	24.0
60	4.2	12.0	2.5	20.0
70	4.9	10.3	2.9	17.1
80	5.6	9.0	3.3	15.0
90	6.3	8.0	3.8	13.3
100	6.9	7.2	4.2	12.0

It obviously isn’t quite as clear cut as this, and I have simplified it significantly to be able to get a vague approximation of the payback. 

You may be expecting to play with or against a high scoring team – this will mean that there are more kick-offs meaning more KO rolls can be made, thus improving a keg’s effectiveness. You may be playing a lower armoured or stunty team – or against a hard-hitting team - and expecting more players to be knocked out, and the slightly improved chance of recovery is worth the risk to try to keep numbers on the pitch.  

If you have thick skull, then the keg will be less effective as more than half the time that a player would be knocked out, they will be staying on the pitch anyway.

And the number of KO rolls for payback reduces as your player cost increases. Any team with high-cost linemen such as dark elves or high elves where the base cost is 70k will need significantly fewer KO rolls for the keg to be financially viable. They also may not be able to afford additional players with their remaining cash so a keg might be the smart play in this circumstance.

The long and short of it, however, is that kegs are not strictly value for money and in a tournament/exhibition game almost all of the time you would be better served spending the money on an additional player (or players if they’re very cheap) if you are able to do so.

They are probably a better investment in league play when you can’t spend your inducement cash on additional players. Their value then could be seen relative to the other available inducements – and will likely come off quite favourably. I will compare inducements for league side by side at some point I’m sure.