Nufflenomics: The Fouling Game

Who doesn’t love a good foul? Elf coaches, probably. But the rest of us love seeing a wee little player take a run up and jump on top of that big mean wardancer!

There are inducements you can take that will help you with this and I want to chat about them all, along with a couple of other things around kicking them while they’re down.

Bribes

Firstly, and probably most commonly, there is a bribe. The cost of this is 100k, reduced to 50k for teams with the Bribery & Corruption rule. I wanted to consider what the actual value of a bribe is.

In simple terms, you’ve got a 5 in 6 chance of keeping your player on the pitch. Therefore, mathematically, the actual value of the bribe can be calculated as the value of the player multiplied by 83% and then compared against the cost of the bribe.

For a goblin team, the mathematical value of a bribe if used on a Fanatic (their most expensive secret weapon) is therefore 58k. It is therefore slightly advantageous. The only other player where you win mathematically is the ‘Ooligan if they get sent off on a foul.

Let’s look at the value of keeping a select range of players on the pitch:

Unsurprisingly, the larger the value of the player, the better value you get from a bribe. The value of a bribe combined with Lord Borak (if you can somehow get them both into your team) then you will be effectively doubling your investment in the bribe. The likelihood of a bribe being needed is slightly reduced due to him having the Sneaky Git skill.

There are obviously other factors to consider other than the mathematical side of things. Using a bribe on a foul means the player kept on the pitch. You therefore don’t go down on numbers (which is even better if you get a removal) and you also prevent a turnover.

A bribe can also allow you to use a strategy that you might not normally follow. One of the results on the kick off table (Get the Ref on a 2 – a 3% chance of occurring) means that both teams will get a bribe. This might allow a team to foul when they previously wouldn’t as their players are too expensive or important to risk. A similar thing applies to inducing a bribe – a team who might not want to risk their players getting sent off might be able to play differently if they take a bribe at the start of the game.

The final thing to consider is the effect on your opponent. They may be (justifiably) scared of you fouling their better players off the pitch. They might then keep them away from the action when they might otherwise have got them stuck in. Alternatively, as you are more likely to keep your dangerous players on the pitch, your opponent may focus on trying to injure your secret weapons and fouling players which could leave them out of position and put you at an advantage.

Should I argue the call before using a bribe?

This can depend on exactly what you are trying to achieve. The order of actions is that if you want to argue the call, you must do it prior to using a bribe. Therefore, if your goal is to simply keep the specific player on the pitch, then it makes sense to take the best option statistically.

The odds of keeping a player on the pitch by arguing the call is 1 in 6 as it only succeeds on a 6. This is a 17% success rate.

If you argue and then bribe, the success rate increases to 72%. This is worked out because a 1 on the first roll will fail, preventing you from using a bribe, so there is always 1/6 chance of failure. There is then also a 1/6 chance of failing the second roll if you rolled between 2 and 5 for the argue. This gives us a total of 10 chances in 36 of failing (or 26/36 of succeeding).

The Bribery and Corruption rule which allows you to re-roll the first 1 you roll on arguing the call will increase this percentage to 77% assuming you haven’t already taken advantage of this special rule.  

A straight up bribe will give you a 5 in 6 chance (or 83%) of keeping the player on the pitch. This is a higher chance than if you argue the call first because you don’t have two chances of failing. A 1 on either roll will send off the player, so reducing the number of chances to roll a 1 will reduce the likelihood of getting sent off.

If your goal is either to keep the player on the pitch, or to prevent a turnover (as you haven’t left the foul until the final action of your turn), then a using a bribe straight away is definitely the best option. This changes if you have a biased referee (see the next section).

However, if your goal is to keep as large a number of players on pitch as possible, then a bribe becomes more useful if it is available later. For example, if you chose to argue the call first, you might be 11% less likely to keep this player on the pitch but if you expect the referee to send off multiple players (or the same player multiple times) then it makes sense to argue the call every time.

Let’s assume that during the game you will have two sending off incidents. If you argue the first one, you’ve got a 2 in 6 chance of the bribe being available for the second sending off if the argue either succeeds, or the coach is sent off. This increases the chance of keeping a second player on the pitch to be 37% compared to 17%. This is calculated by looking at whether both the coach and bribe are available for the second sending off.  And if the bribe is available for future instances when it is needed then it obviously improves the odds of future players not being sent off.

The total chance of both players being available if a bribe is used straight off on the first sending off is 14%. The chance of both players being available if you argue the first sending off is 26%.

What is the value of a biased referee?

The Biased Referee inducement costs 120k (or 80k for Bribery & Corruption teams) and has two functions – sending off opponent’s players when they foul (a roll of 5+ will send off a fouling player who is otherwise not sent off) and keeping your players on the pitch when they are sent off (with a +1 bonus to arguing the call). Let’s look at each of these components separately.

Mathematically, you will be sending off your opponent’s fouling players 1 in 3 times when they are not spotted. Let’s assume they are only fouling when they are expecting to break armour.  A foul where the armour is broken should result in a sending off 31% of the time (1 in 6 on the armour followed by 1 in 6 on the injury, less the chance of getting a double on both rolls).  Having a biased referee means that the likelihood of a sending off is increased to 54%. This means that players will be sent off roughly every 1 in 2 fouls compared to 1 in 3 previously.

Will this stop your opponent fouling? There are probably several factors that will affect this, with the two key ones being your opponent’s mentality and the team they are playing. It might dissuade an elf coach from doing that key foul on one of your important players, but it is unlikely to really affect a goblin coach’s game plan.

The affect on your arguing the call is twofold. The +1 modifier means that the coach can never be sent off, meaning you can argue the call repeatedly. This is obviously of most benefit when you are expecting significant numbers of players to be sent off.

It means you are twice as likely to successfully argue the call – increasing from 17% to 34%. If you combine this with a bribe, there is a total chance of your player staying on the pitch 89% of the time. In this instance it is the sensible option to argue the call first unless you specifically need to avoid a turnover.

The payback on the biased referee is difficult to calculate, however I think the best instance to attempt to do so is to look at the goblin team. If we assume we have all the secret weapons (without your opponent removing them) and the ‘Oogligan does one foul which is spotted, then the payback will be 17% of each of their values due to the additional 1 result on the die which will prevent a sending off.

Their total values are 220k, meaning that the straight payback is 37k on your 80k investment. However, this can’t be seen in isolation as for any successful argue, you will potentially have another argue further down the line, and it doesn’t consider the fact that the coach will never be sent off. Even so, it would take a lot for it to balance out to the amount spent. For some teams it might be worth the expenditure if it is tactically advantageous even if it only gets one of these players back on the field.

The payback of stopping your opponent fouling is even more complex. If they decide not to foul when they otherwise would have done, that may actually not be advantageous to you as a play on their team who might have previously be sent off is now not sent off. To calculate the benefit of catching someone fouling, then we would be looking at the increased chance of being sent off on a foul (which is 23%) multiplied by the value of the player removed. As teams generally will be fouling with their cheapest player, it won’t be a large value. Assuming they are fouling with a standard strength 3 lineman who costs 50k, then the value you are getting from your Biased Referee is just 11k.

There are a number of Named Biased Referees. They are Jorm The Ogre, The Trundlefoot Triples, Thoron Korensson and Ranulf ‘Red’ Hokuli. I want to consider each of these briefly to see how they compare to the regular Biased Referee Inducement.

Jorm The Ogre

Jorm will spot an opponent on a 4+ which is better than the regular biased referee. The effect is different, which is that he will attack the player doing the fouling rather than sending them off – which even with Mighty Blow is worse as it doesn’t guarantee they will be removed from the field of play. He also doesn’t allow you to argue the call on a 5+, so he is likely not as good

The Trundlefoot Triplets

The Triplets work in a similar way to the regular biased referee when an opposing player commits a foul, however they do not add the bonus when you argue the call. They do also have the risk of the victim of the foul being sent off one in every six times. There is not much value here in taking them over the regular biased referee. The only instance when it would be useful is for any Halfling Thimble Cup Teams (who can get them for 40k when the regular biased referee would be 120k) when they expect their opponents to do a lot of fouling.

Thoron Korensson

Thoron offers a bonus to the biased referee if you want your opponents to not foul against you. He works in the same way when an opponent commits a foul, but this time they are sent off on a 4+. He also prevents bribes being used when a player is sent off in this manner, and if the coach rolls a 1 on an argue roll then another player gets sent off as well! Thoron would be a good inducement if you’re coming up against a fouling team as he will really help to thin out their numbers.

Ranulf ‘Red’ Hokuli

Like Thoron, Ranluf will spot fouls where no double is rolled on a 4+. Three options are given to the coach of a player spotted this way – they can use a bribe, accept their player being sent off, or have an armour roll made against them with a 2+ modifier (if the armour fails then they’re sent off). This puts your opponent in an interesting quandry. He could potentially be worth inducing against fouling teams, although anyone who is happy to accept the risk of injury to their fouling player might not be deterred.

Fink da Fixer

Another inducement you can take that affects fouling is Fink Da Fixer. Fink allows the re-roll of a failed bribe and also allows you to argue the call on a 5 or 6. This can be combined with the biased referee inducement to allow you to argue the call on a 4+. This could be potentially useful for any teams that can afford both and are really keen to keep their players on the pitch.

Sneaky Git

Sneaky Git could be seen as effectively half a bribe. It will guarantee that you don’t get sent off if you roll a double on the armour roll. However, unlike a bribe, you get to use it on multiple occasions and, mathematically, there isn’t the 1 in 6 chance of failure like there is with a bribe.

The value of Sneaky Git on any one occasion can be worked out a roughly: Player value x 1/2.

It will keep your player on the pitch once in every 2 sendings off on the assumption that double rolls are evenly split between the armour and the injury roll. You would expect to survive on average 1 sending off per game.

The cost of having Sneaky Git on your team in terms of team value is somewhere between 10k (random primary) and 40k (chosen secondary).  The table below shows the break even point in terms of team value.

	Cost	Value of Player to Break Even
Random Primary	10	20
Chosen Primary/Random Secondary	20	40
Chosen Secondary	40	80

 

Another bonus to sneaky git is that you might chance a foul with no assists, knowing that there’s only a chance of a sending off if you’re causing at least a stun to your opponent. It’s difficult to measure that and obviously it’s not always the smart choice if you’re stood next to someone big and mean who will just get up and wallop you if you don’t hurt them.

Dirty Player

What is the value of Dirty Player? In terms of breaking armour, it is very much dependent on how many assists you have for the foul. In pure numerical terms it makes it more likely that armour will break by a minimum of 6% and a maximum of 17%.

Roll needed	Extra likelihood with DP
5	11%
6	14%
7	17%
8	14%
9	11%
10	8%
11	6%

I think the real value comes in when you’re able to surround your opponent with enough assists to guarantee (or nearly guarantee) an armour break. With 3 assists, a player with armour 8 will be injured 72% of the time. The likelihood then is that you will be able to use Dirty Player on the injury roll. This then increases the chance of at least a knockout by 17% (to a total of 58%) and a casualty by 8% (to a total of 28%).

Reasons not to foul

Are there any reasons to not foul? I guess the most obvious is not risking losing a player yourself (see below section). There are a couple of others reasons I can think of – perhaps you need to be able to jump over your prone opponent to get one or more of your players where you want to be, and obviously you may not want to suffer a turnover which is why it is often the last action of any turn.

The EV of fouling

The expected value (EV) of an action is what, on average, you would expect to gain or lose based on the mathematics. There has been talk in the BloodBowl community of the expected value of a foul. This can be calculated by looking at the likelihood of a removal compared to the likelihood of a sending off. Let’s look at this without any thoughts of bribes, skills or arguing the call.

The chance of you losing your player is 31% as discussed earlier.  The chance of your opponent suffering a casualty is 20%. To calculate the EV of any one foul, you would need to compare two values: The cost of your player x 31% and  the cost of your opponent’s player x 20%.

If 31% of your player’s cost is higher than 20% of their player’s cost then the foul will have a positive expected value. A formula to calculate this is:

Break Even Value = Your Players Value x 31% x 5

To give a guide on what value you should be looking for, here are some comparison numbers.

Cost of your player	Break Even Value
15	23
30	47
40	62
50	78
70	109

One thing to note is that if you’re fouling a player with regen then you will want to double these values as they have a 50% chance of being placed in the reserves box instead.

Obviously adding in any factors such as bribes, sneaky git, dirty player, biased referees will skew this more in your favour. If you as a coach have been sent off, and whether they have an apothecary to use will skew it slightly in your opponent’s favour. However these numbers should provide a good basis for calculating the EV of a foul.