High FF

[Mestari]

[quote="High & Mighty]


To me it seems that after reaching a certain TR, controlling the oscillation of the TR value becomes the most important team management aspect.

Yup. That's where I'm trying to get to, too. I still think the system of equations above is correct with one caveat that your comment made me look through again.

Sacking and dying players still plays no part whatsoever in reducing your team rating per se. The act of firing and hiring the same player achieves the same goal as freebooting a wizard or a star player. It's money thrown away. Since a full roster is a fixed variable, it only goes to 15/16 and every coach has 15/16 at their max. But what does need to be constrained are the SPPs. In the equation above, I was taking average SPPs per game and multiplying by the number of games.

So over time, the equation doesn't allow for losing SPPs which those dead/retired players had. So instead of average spps times number of games, the correct variable is average spps times players on the roster (since if you multiply the average spps times the total players you get total spps). This will effectively cap the spps and model the coach trying to manage the roster by maintaining a constant level of average spps per player to suit their taste (so say a coach trying to keep all his players at an average of two skills).
[/quote]

But firing a player and not hiring him instantly again does reduce your TR. Momentarily. But over long term, I agree that the team is going to have a fixed number of players.
Your reasoning behind using the average SPP's is a correct one.

Coaching ability is admittedly hard to factor in, but you could consider how you could use the Coach Rating (CR) concept used by the NAF for that purpose.

Anyhow, looking at your theorem from the team management point of view, that aspect would become:

Coach aims for the following equation to be true at all times:

X + Y <= Z,

WHERE

X = the base of the team and is composed of the following factors:
X = P + C + TRR + FF
P = TR in players (P=p*av.price(p))
C = TR in coaching staff and money
TRR = TR in team rerolls
FF = TR in fan factor

Y = nr of players *the average spp per player = p*av.spp(p)

Z = our "plateau phase"-limit, which is a function of coaching skill, fan factor, team type and the league the team is in.
Z = f(FF,CR,race,League)

Resulting in a final team management formula of

p*(av.price(p) + av.spp(p)) + C + TRR + FF <= f(FF,CR,race,League)

This is what the coach would be looking at. This formula is, however, useless without knowing the function f as it gives the value that the coach is attempting to avoid (for the most of the time) - as going over it would result in the oscillation being out of his control and could result in a steeper downfall. He also tries to balance the aspects on the left side of the equation in a fashion that gives the most effective team possible.

The function f in the ideal case gives a TR value that can be described as follows:
-Should the teams TR exceed that value, a controlled oscillation of TR becomes in practice impossible to maintain. Instead the team will unavoidably go through a more radical "Downfall"-phase that restores the balance.
Also, as f is a function of FF, coaching ability and the league, the value of f will change over time.

If I can express some wishes concerning your theorem, I would like it to approximate the ideal function f above. To give an approximation of where to keep a teams TR.
Coaching ability, the team race and the League are at least to some extent factored in the "winning percentage" of the team, so I think you're really getting close.
What would also be needed is a decision on what is a sufficient level of income for a team... which naturally must vary with the teams race. It would also have to depend on the leagues attrition level so that the income level is sufficient to replace the casualties in addition to allowing you to replace players with too high SPP counts.

I'm too tired to think clearly anymore... I'll try to read the above text tomorrow and check if it makes any sense... good night everyone.

[Zombie]

It's a great article. I read it when you posted in on TBB and again when you mentioned here a couple days ago to compare my own numbers.

Thanks, but i actually posted it at BBC. Still remember that place, right?

About the FF, remember that when you win, you don't always get 2 TDs and 2 CAS. When you lose, you may get both, one or neither. You have to take that into account.

For example, when you lose, you'll go down in FF more than every 6th game. Even if we assume that you don't get the -1 for 10+ FF (which we shouldn't even assume), then you go down
2 in 6 if you don't get 2 TDs and 2 CAS
1 in 6 if you get one but not the other
1 in 6 still even if you get both (remember the natural 1)

The same principle applies when you win.

It doesn't make the calculation all that much harder to make though. You just gotta be careful.

[High & Mighty]

Thanks, but i actually posted it at BBC. Still remember that place, right?

Geez, even when I want to type its name, I can't. BBC is what I meant. Entertaining place...back when people went there.

About the FF, remember that when you win, you don't always get 2 TDs and 2 CAS. When you lose, you may get both, one or neither. You have to take that into account.

For example, when you lose, you'll go down in FF more than every 6th game. Even if we assume that you don't get the -1 for 10+ FF (which we shouldn't even assume), then you go down
2 in 6 if you don't get 2 TDs and 2 CAS
1 in 6 if you get one but not the other
1 in 6 still even if you get both (remember the natural 1)

The same principle applies when you win.

It doesn't make the calculation all that much harder to make though. You just gotta be careful.

Yup, it's in there, but imagine for a minute that instead of living with probabilities, you simply gain or lose a fraction of ff each game you play so that we can analyze the dynamics over time without getting bogged down in each single change. In that case

FF mod=[Win%+2td%+2cas%-Losing%-(>10mod)]/6

and since your Losing%=(1-Win%), we can sub that in and get

FF=[2*Win%+2td%+2cas%-(>10mod)-1]/6

So if you win the game, score 2 tds and 2 cas and have ff=15, we get:

FF=[2*1+1+1-1-1]/6=2/6

So your FF has increased by 1/3 for playing the game. To think of it in terms of the probabilities when rolling the dice, you have a +3 modifier to your roll (win, 2 TDs, 2 Cas) and a -1 modifier for being over 10 FF for a net mod of +2. This means the probabilities of your roll are:

1/6=decrease
3/6=increase
2/6=stay the same

So if you were to play the same way for six games, you would expect your FF to go up 3 times, down once, and stay the same twice, for a net of 2. So if you now multiply the bold equation above by 6 to represent you having played six games, you get a total of 2. Plug in any scenario you want and the equation works.

So essentially, you can predict your FF at any point in the future if you know:

What percent of your games you will win
What percent of your games you will score 2+ TDs
What percent of your games you will score 2+ Casualties
What percent of your games you will be above a multiple of 10 FF

As two side notes to now add to the discussion on creating an overall 'Economy of Bloodbowl' model:

1) If you could also assume that you score 2+ TDs in X percent of your wins and Y percent of your losses and the same for casualties, then you could actually predict with 100 percent accuracy what your FF will be in the future by knowing only your win percentage and the percent of games you are over a multiple of 10FF

2) It's that last part that was a real bugger until I figured it out a little bit ago for the reason that the negative ff modifier is not continuous. If you have a ff=15, you don't have a modifier of -1.5, just -1. A real problem if you want an equation.


Mestari wrote:

Resulting in a final team management formula of:
p*(av.price(p) + av.spp(p)) + C + TRR + FF <= f(FF,CR,race,League)

I agree with everything on the left hand side of your equation and this is what I have used. It's the right-hand where I have a problem. f(x) above is a function describing your win percentage perhaps, not necessarily where you would optimally want to be. But I think if you were to change your focus regarding the below, you would be dead on:

What would also be needed is a decision on what is a sufficient level of income for a team... which naturally must vary with the teams race. It would also have to depend on the leagues attrition level so that the income level is sufficient to replace the casualties in addition to allowing you to replace players with too high SPP counts.

Yes, you want to have "sufficient" income as a coach to buy everything you want. I don't know whether your background is straight math or economics, but since this is an econ question, the answer is very easy.

Q: How much money do I want?
A: All of it.

Or short of that, as much as I can possibly have. The question of whether the money is sufficient is entirely subjective based upon what I want to spend it on (just like you may not be able to live on less than $200,000 a year while I'm happy to live on $20,000). One may not be sufficient for you but fine for me, but the factors in determining what our salaries will be are 100% identical. Salary=years of schooling+being born into a rich family, or something like that.

Which is a big insight into how the economy of Blood Bowl works. I promise to post it all this weekend once I correct it with the suggestions and corrections you've given me, but in a nutshell:

1. We know how to calculate team rating
2. We know how to calculate fan factor
3. From 2, we know how to calculate the gate
4. From 1 and 3, we know how to calculate the winnings

Therefore, winnings are determined solely on the basis of the equations governing team rating and fan factor

This means when all is said and done, we know how to calculate what income we are making with our team at any point in the future, knowing only:

what we purchase to start our team (players, rerolls, fan factor),
how much more we will spend on players and rerolls,
how skilled we will keep those players,
how many games we play,
fan factor of other coaches in the league (the only iffy link)
and what our win percentage is.

Not only that, but we can say exactly what effect starting with 9 FF or 1 FF has on your winnings in the Nth game, and more generally, whether it is possible to, when properly managed, grow forever as a blood bowl team, and if so, what exactly do you need to do and what is more important.

OK, now it's my bed time.

[Mestari]

Looks good, H&M - keep up the good work.

-Teemu

BTW. Pure mathematics - I'm not that far in my studies yet, so I don't know what I'm going to choose as my specialisation field - probably statistics or theoretical mathematics. My math studies are my second major - I'm trying to graduate from computer science first...

[Dangerous Dave]

One important point to add to this. Winnings can be hugely affected by tournament winnings. Sure only 2 teams (winner and runner up) are going to win this but nonetheless that will clearly make the best teams better. If winnings are circa 200k with coming second at 100k this has a very big effect - obviously the length of the season does impact this but if it is say 8 to 10 games that is 20k minimum extra per game.

Further, in the play offs (semis / finals), your winnings is likely to go up since you get extra on the dice.


Dave

[High & Mighty]

One important point to add to this. Winnings can be hugely affected by tournament winnings. Sure only 2 teams (winner and runner up) are going to win this but nonetheless that will clearly make the best teams better. If winnings are circa 200k with coming second at 100k this has a very big effect - obviously the length of the season does impact this but if it is say 8 to 10 games that is 20k minimum extra per game.

Further, in the play offs (semis / finals), your winnings is likely to go up since you get extra on the dice.


Dave

Well, yes and no. Tournament winnings will make the best teams better without a doubt, AND it can be a sizeable chunk of change AND it has a bigger impact when the season is shorter because it averages out to more money per game as you point out.

BUT, you could award all the money in the world to a team, but if they blow all their money on freebooting Morg, that money has zero impact on the earnings potential of the team. At the end of the day, you can only use money to buy rerolls, players, coaches and cheerleaders so only ACCUMULATING those things will decrease your generated income.

("accumulating" because if you fire a player and hire him right back as a way to launder your money, you could have had the same ultimate effect if you just freebooted a wizard or star player. The only difference is that firing players is a way to keep your total SPPs down, as Mestari noted I was doing wrong before.)

The only impact will come if you get so much money that you cannot or do not want to blow it all freebooting for one game. Then it sits in your treasury and does potentially lower your winnings if it pushes you into the next threshold. (But only temporarily since again, if you spend the money to freeboot one player in each of the next 10 games, at the end of the 10 games, you will be right back to where you were in terms of earning potential before you won the tournament. So again, it only makes a difference ultimately if you spend that money on something that stays on your roster.)

So that means that if it is possible to level your team rating off so that you no longer move to the right on the winnings table, winning a tournament could potentially shift you to the right (100,000=10TR; 200,000=20TR) and decrease your winnings by at most 10k per game (since you need to go up by 25TR to move through each level) and so the cost of winning the tournament is 10k per game for every game until you have freebooted enough to get back to the level where you started. (So if you won 100k but got a -1 mod on winnings for the next five games until you have freebooted enough to get back to where you were, you actually netted 50k from winning the tournament.)

Well thanks for making me think about that since I was pretty much ignoring it until now.

Further, in the play offs (semis / finals), your winnings is likely to go up since you get extra on the dice.

On that point though, you don't get a playoff mod on the winnings roll (but if I remember correctly, you used to before LRB, correct? It's been so long.) but you do get the mods on the fan factor roll which may prove to be important since a short 6-8 game season will have a number of those mods and could help ensure healthy ff growth.

HEY! Thanks! You just made me realize how to make Mestari happy and include league format in all this. The shorter the league season, the higher the potential for FF growth through the tourney mods.