Nufflenomics: The Master Chef

I played a practice tournament game this week and my Master Chef stole zero re-rolls from my opponent for the entire game. Obviously disappointed to have wasted 100k on this, I thought I’d take a look into the economics of hiring a Master Chef.

The Economics

Taking a chef is an almost automatic part of any halfling team but it is also a gamble. There are a few things that I want to talk about when considering the economical effect of a chef. Let’s start however by dissecting the basics.

At the start of each half, after the kick has deviated, you get to roll 3 D6 and for each one with a value of 4 or more, you receive one of your opponent’s re-rolls.

This means that each dice has a 50:50 chance of success. If we look at each roll of 3 dice, it then works out to be:

Re-rolls stolen	Probability
0	12.5%
1	37.5%
2	37.5%
3	12.5%

This applies each half, so in total for the game, the total number of re-rolls you gained on a percentage basis is:

Re-rolls stolen	Probability
0	1.6%
1	9.4%
2	23.4%
3	31.3%
4	24.4%
5	9.4%
6	1.6%

Now this is just probability of how many times you will roll four or more on six dice. What we want to do is put monetary values on it. For halflings, team re-rolls are 60k each. You need to remember that you get to use these once per half, so each time you steal a re-roll you are effectively gaining 30k of value.

We can multiply each probability by the reward and then add up all of these results to get the expected value of our gain.  For example, for two re-rolls across the game, the total reward is 60k (the cost of one re-roll), multiply this by the probability of 23.4% we get a result of 14.06. The table below shows all of these.

Re-rolls stolen	Probability	Reward	EV
0	1.6%	0k	0
1	9.4%	30k	2.8
2	23.4%	60k	14.0
3	31.3%	90k	23.1
4	24.4%	120k	28.1
5	9.4%	150k	14.1
6	1.6%	180k	2.8
TOTAL			90

By adding up all of the EVs, we can expect to gain 90k worth of value from the master chef which, as halflings costs us 100k. This could be calculated more easily by looking at the dice rolls, knowing that on average you will pass one in every two and then multiplying that by half a re-roll cost. That’s the way for people who like doing things simply. We will make use of the above data later again anyway.

It might therefore look like we are losing 10k with each purchase of a MasterChef, however what you need to remember is that for each re-roll you take, your opponent will lose a re-roll and you are therefore going to gain even more by changing the Team Value (TV) differential.

Different teams have different re-roll costs, and these are split out as below.

50k: (8teams) Dark Elf, Dwarf, Elf Union, High Elf, Human, Skaven, Slaan, Wood Elf

60k: (10 teams) Amazon, Black Orc, Chaos Chosen, Goblin, Halfling, Khorne, Norse, Orc, Snotling, Vampire

70k: (11 teams) Chaos Dwarf, Chaos Renegade, Imperial Nobility, Lizardmen, Necromantic Horror, Nurgle, Ogre, Old World Alliance, Shambling Undead, Tomb Kings, Underworld Denizens

Please note that the case of two halfling teams with master chefs coming up against each other complicates the calculations immensely and would need a much longer article to dissect than is below.  

For each re-roll you steal, you will therefore effectively be reducing your opponent’s TV by 25k, 30k or 35k depending on the cost of their re-rolls, up to the point that they have stealable (ie non-leader) re-rolls. If your opponent doesn’t have three re-rolls on their roster, then it changes how much the differential will move by. The tables below are therefore a little complex. This table shows the value lost by your opponent with 50k re-rolls depending on how many re-rolls they have rostered (either none, one, two and three or more).

Re-rolls stolen	0 re-rolls	1 re-roll	2 re-rolls	3+ rerolls
0	0	0	0	0
1	0	25	25	25
2	0	50	50	50
3	0	50	75	75
4	0	50	100	100
5	0	50	100	125
6	0	50	100	150

You can see that the total value stolen is capped at the total amount that your opponent spent on re-rolls. If we layer the probabilities on top of this, we get the EV for the value lost for your opponent depending on how many re-rolls they have.

Re-rolls stolen	0 re-rolls	1 re-roll	2 re-rolls	3+ rerolls
0	0	0	0	0
1	0	2.3	2.3	2.3
2	0	11.7	11.7	11.7
3	0	15.6	23.4	23.4
4	0	11.7	23.4	23.4
5	0	4.7	9.4	11.7
6	0	0.8	1.6	2.3
EV Lost	0	46.9	71.9	75.0

As you can see, there is no loss if they have no re-rolls, almost the entire cost of a re-roll if they have 1 and not too much difference between two and three or more re-rolls.

Now we can combine this loss to our opponent with our gain to get the total movement in TV.

Re-rolls stolen	0 re-rolls	1 re-roll	2 re-rolls	3+ rerolls
0	0	0	0	0
1	2.8	5.2	5.2	5.2
2	14.1	25.8	25.8	25.8
3	28.1	43.8	51.6	51.6
4	28.1	39.8	51.6	51.6
5	14.1	18.8	23.4	25.78
6	2.8	3.6	4.4	5.16
TOTAL EV	90	136.9	161.9	165

What we can see here is that as long as your opponent had at least 1 roster, investing 100k into a Master Chef should, on average, yield a larger reward in moving the TV differential than your investment.

I will leave out the interim tables and present you with the equivalent of the above table for teams with 60k and 70k re-rolls.

60k re-rolls

Re-rolls stolen	0 re-rolls	1 re-roll	2 re-rolls	3+ rerolls
0	0	0	0	0
1	2.8	5.6	5.6	5.6
2	14.1	28.1	28.1	28.1
3	28.1	46.9	56.3	56.3
4	28.1	42.2	56.3	56.3
5	14.1	19.7	25.3	28.1
6	2.8	3.8	4.7	5.6
TOTAL EV	90	146.3	176.3	180

70k re-rolls

Re-rolls stolen	0 re-rolls	1 re-roll	2 re-rolls	3+ rerolls
0	0	0	0	0
1	2.8	6.1	6.1	6.1
2	14.1	30.5	30.5	30.5
3	28.1	50.0	60.9	60.9
4	28.1	44.5	60.9	60.9
5	14.1	20.6	27.2	27.2
6	2.8	3.9	5.0	6.1
TOTAL EV	90	155.6	190.6	195.0

To create a nice little summary table of all these expected values:

Rostered Re-rolls	50k Re-rolls	60k Re-rolls	70k Re-rolls
0	90	90	90
1	136.9	146.3	155.6
2	161.9	176.3	190.6
3 or more	165	180	195

Now it is very important to remember that these are expected values. You won’t get this every time – sometimes you may get a lot more, sometimes you might get a lot less (or nothing as I got the other day). Obviously taking the chef is a gamble, but in most circumstances, one that will pay off.

Tournament Play

In tournament play you have no additional information before you make the decision to hire the chef or not. It will usually be beneficial and if your opponents don’t have no rostered re-rolls, will have a positive expected value.

Using data from the 2023 World Cup provided by Mike Davies (sann0638) the average number of re-rolls taken was 2.494. Obviously this is only one ruleset however it was a fairly standard ruleset and also had over 2,000 coaches so we can safely assume that this number is a good average of what you're likely to come up against in tournaments.

We can assume therefore that the majority of coaches will have either 2 or 3 re-rolls on their roster. There will be outliers - the average Slaan coach takes 4 re-rolls, for example - but this is good enough to base our decision on when thinking about taking a chef to a tournament.

Working out the average spend per re-roll, we get 60.8, which is close enough for us to base our workings on the 60k re-roll tables above. In a game where your opponent has 2 re-rolls your TV gain is 176.3 and when they have 3, your TV gain is 180. Therefore in a tournament, by taking a master chef you will expect to have a return on investment of around 75-80 TV each game. 

League play

In league play, you will know in advance how many re-rolls your opponent has as well as the cost of those re-rolls so you can see the expected value before making the decision to purchase the chef. There may well be other alternatives that might benefit you more in terms of the specific game you are due to play, and you can take a call on how much risk you are willing to take. The tables above should help you work out the expected return on the investment.

In leagues, a halfling coach may strategically keep their team value low and use the masterchef as a go to in order to be able to create the TV disparity. 

One thing to note here is that if your opponent has no re-rolls (probably unlikely in league play but not impossible) then the Extra Team Training Inducement which is also 100k might be worth considering instead if you wanted to guarantee one extra re-roll each half. The chef will on average get you more, but it is more of a risk.

Intangibles

There are also then intangible factors – how important are re-rolls to your opponent. If they are playing a team that relies heavily on re-rolls, such as Vampires or Slaan then the chance to deny them some is probably more important than if you are playing an elf team with lots of built in re-rolls. However these teams are likely to start with a large number already, so your chef would really need to be on form to be able to have a significant impact on the game. 

Another factor to consider is whether having fewer re-rolls will make your opponent play differently or not. It is likely that an experienced player might opt to play more cautiously if they don’t have their full complement of re-rolls. Do you want that to happen? An inexperienced coach might burn through their reduced number of re-rolls quite quickly and leave themselves exposed later on in each half. 

The psychological impact is something to think about. Wiping out a team's entire stock or re-rolls could potentially deflate your opposition and if you manage to do this in the first half, it could give you a significant psychological advantage. It will also mean that they are much more likely to suffer turnovers at inconvinient points, which could give you a significant strategic advantage. 

It will also limit the strategic options that your opponent has. They may no longer be able to play as aggressivley or boldly knowing that they don't have the safety mat of having re-rolls to save them. 

It could also open up your own strategic options - a halfling team who has stolen 3 re-rolls will be able to take many more chances than one who haven't stolen any. 

It can also give you a chance in matchups that you might not have had a chance in previously. Dwarves are a bad match up for any stunty team, but if they are playing with no re-rolls then when something goes wrong they have no chance to recover. This could give you an opening, for example a way into their cage or to pick up a loose ball, that might not have been a possibility if they still had re-rolls available. 

The calculations above look at re-rolls stolen over the course of the game but don't look at when they are stolen. Taking 3 re-rolls across the course of the game would be your average result, however if you take them all in one half, it will likely have a more significant impact than if you took 2 in one half and 1 in the other. 

Non-Halfling teams

The one thing we’ve not yet discussed is hiring a Master Chef on a team which isn’t Halflings. The cost of doing so is 300k, so you would need the EV to be higher than this to make it worth doing. The only possibility of this coming close to a good return on investment is if both yours and your opponent’s re-rolls cost 70k and your opponent has three or more rostered. In that circumstance the EV of the TV shift is 210k, so you would be buying this at a 90k loss.

I would therefore only recommend doing this in specific circumstances. For example, you may really want to deny your opponent re-rolls, or you have far more inducement cash than you could ever spend. It is not a good return on your investment, so you would need to have a really good reason for taking this choice. 

Conclusion

A Master Chef is a mainstay for halflings in tournaments and the expected value will be better than the amount spent on the chef unless your opponent has no re-rolls on their roster. This is always a possibility and should be factored in before finalising your decision. It is very unlikely that the meta of a tournament will favour no re-roll builds, but it is something to consider. Will I take a master chef if playing halflings in a tournament? Yes, always. The only time I might reconsider now after running the numbers is if a significant number of other coaches are likely to take no re-roll builds. 

In league play you are able to make more of a considered decision before making the choice, and hopefully the information above may guide you in making that decision.

Please feel free to let me know if there is anything you think I’ve missed here. I think the maths side of it is fine, and I have added a bit more commentary after discussions with other coaches but it can always be expanded on.