What's an AC/CL worth?

[mattgslater]

Here's the question: under what circumstances is an AC/CL a good bargain?

This is a little long and rambly; I may edit it later, but right now, it's just a musing.

Part one: The Value of Winning.
5/36 of all Kickoff Table rolls yield a "Brilliant Coaching" (BC) result, and another 5/36 yield "Cheering Fans" (CF) result. When this comes up, one or both sides will get a Re-Roll counter (RRC). A Re-Roll counter is worth half a Team Re-Roll (TRR), if it is likely to be used. Some Re-Roll counters are not likely to be used, or are impossible to use, and are worth less than this, because all RRCs provided by TRRs are likely to be used, for our purposes.

TRRs have a "hard value" in GC, which is convenient for comparison to the 10k value of the AC/CL. But the "hard value" of the TRR isn't so hard, really. It's got one value in starting cash and TV, another value in inducement cash, and a third value in winnings. If a team can't take enough TRRs, or if it induces Extra Training, a TRR can be considered to be worth 100k, so a meaningful RRC is worth 50k. If a team has enough TRRs, but one more would be meaningful, the value of a TRR is its listed cost of 50k-70k, so the RRC is worth 25k-35k. If one more TRR would not be meaningful, the value of an RRC is 0k. In a league, your opponent's RRC is worth 50k if you play a lot of novice teams and 25k if you play a lot of veteran teams (they will average 30k, but some won't be able to use the RRC). If you regularly induce a Master Chef, set your opponent's RRC to 50k value.

Let's call a drive where a "free" RRC would be valuable to both teams a meaningful drive (MD). If only one team could use the RRC, the drive is "meaningful for" (M+) or "meaningful against" (M-) you, and only one side of the equation is considered. If you gain a TRR in a drive that's meaningful for you, it's a Re-Roll gain (RR+); if you deny your opponent a meaningful gain, you get a Re-Roll stop or denial (RR-). Successful grind teams tend to play an average of two or three meaningful drives, possibly also with one more drive that's meaningful against, but not in favor. Struggling agility teams tend to play three or four meaningful drives, with maybe one of them that's not meaningful against. Successful agility teams, and teams with major offensive problems (especially the ones that have a hard time securing the ball) may play more meaningful drives.

Putting this together, we have the value of getting an automatic RR+ on all BC or CF rolls equal to the value of a usable RRC for your team (25k-35k or 50k), times M+ (the number of drives where you'd like a free re-roll), times 5/36 (for each type). The value of an automatic RR- on BC/CF is equal to the value of an enemy RRC (25k or 50k) times M-, times 5/36. So that's one part of the equation.

Example I: An Amazon team is playing in a perpetual league. That team has all the TRRs and players and cash it wants, and its TV is still nice and low. The team plays a lot of three-drive games, with higher-scoring games usually blowout wins, so the coach settles on MD=3.5 drives as an average even though the actual average is higher. The team has slightly fewer meaningful drives than its opposition, so the coach settles on M+=3, M-=4. The value of an automatic RR+ for this team on all BC rolls is 25k x 3 x 5/36 = 25k x 5/12 = 125k/12, or just over 10k GC.

Now, since this team is playing perpetual, you have to assume that your opponent will have what he wants most of the time, so you probably want to consider 25k or 30k for the median TRR value. Let's use 30k. 30k x 4 x 5/36 = 50k/3, or 16.667k. So the total value of automatically winning all BC rolls is RR+ plus RR-, or 26.667k, by comparison to automatically losing.

Example II: A runaway High Elf team is smashing its way through the same league, embarrassing all and sundry, and racking up an average of 3.5 TDs per game. [/i]Every drive is meaningful except post-turn-8, so MD, M+ and M- are all about 5.5, which the coach brings down to 5, because there are fewer drives in most of the important games. The team tries to stay lean by running 3x TRR, but its runaway value means that it tends to give up a Master Chef sometimes, and its opponents tend to have all their stuff together. All the team's money goes to replacements, and cash is at a premium. So all RR counters for are worth 50k, and against are worth 25k. Since M+ and M- are the same, the condensed formula is (50k or 25k) x 5 x 5/36, or 75k x 25/36, or (5^4)k/(36 or 18), or 625k/18 (34.7k) for RR+ and 625k/36 (17.4k) for RR-.

Part 2: The Odds

The odds of "winning" the CF or BC roll are based an opposed D3 + FAME + (CL or AC). There are three variables in there: the opposed d3 roll, relative FAME, and the opponent's coaching squad.

FAME is a little more complex than the others: let's start by introducing a concept called FAME weight, or your estimation of the odds of getting any given FAME outcome. I'll leave to you the odds calculation; it's rather back-of-the-envelope. But put a percentage or fraction on the general odds of each possible FAME result (+2, +1, 0, -1, -2).

For instance, if nobody's got any FF, you've got 159/1296 each for +2 and -2 and 146/1296 for 0. That's about 12% each with an equal chance of being over or under, so we can approach FAME as one-third +1, one-third -1, and one-ninth each +2, 0 and -2. But if you're FF7 in a perpetual format, and you're usually a 3-point underdog, you'll almost always be FAME -1, so that's the only one you have to consider.

Next, the opponent's coaching squad. I think it's smartest to use the league median here, or 0 if you don't suspect otherwise. Whatever you use, subtract that from your FAME for each FAME being weighted for.

Finally, the die roll.

Net FAME: RR+ odds; RR- odds — Advantage
+4: 1; 1 — 0
+3: 1; 1 — +1/9 (all stop)
+2: 1; 8/9 — +1/3 (1/9 gain, 2/9 stop)
+1: 8/9; 2/3 — +5/9 (2/9 gain, 1/3 stop)
+0: 2/3; 1/3 — +5/9 (1/3 gain, 2/9 stop)
-1: 1/3; 1/9 — +1/3 (2/9 gain, 1/9 stop)
-2: 1/9; 0 — +1/9 (all gain)
-3; 0; 0 — 0

As you can see, the "point of diminishing returns" comes at +1 NAC/NCL if you're trying to earn TRRs, and at +2 if you're primarily trying to stop them. There are a few ways to calculate the value of the AC or CL based on this. Let's start with an example, and work our way back.

That High Elf team, with its 34.7k RR+ value and 17.4k RR- value, has a FAME that tends to be higher than its opponents. Its opponents aren't slouches, though, and it's hard to double up on them, so most games the High Elf team has FAME +1, say 60% of the time, with 20% FAME +2 and 20% FAME -1.

One AC would have a 60% chance of adding 1/9 gain and 2/9 stop, plus 20% of 1/3 gain and 2/9 stop, and 20% of 1/9 stop. So that's 2/15 gain and 1/5 stop on average. 2/15 x 34.7k is 4k x 34.7 /30, or about 4.65k for gain. 1/5 x 17.4k is 3.48k for denial. 3.48k plus 4.65k makes 8.13k, or about 5/6 of the cost. That's too close to make a definitive statement. If you find you or your opponent always running out of RRs early, take one. If not, don't.

Let's say the Amazons have a pretty average FF, and their FAME may be +1 or -1, with equal odds. If automatically winning CF rolls is worth about 26.667k, or 80k/3, then buying one CL pays off if it generates a positive result 3/8 of the time. If you have a 50% chance of being -1 and a 50% chance of being +1, you have the average of 1/3 and 5/9 of a TRR, or 4/9. 4/9 > 3/8, so one AC/CL is clearly worth it.

I'll leave the rest to y'all. I haven't proofed this. Feel free to attack my assumptions.

[Carnis]

Think it sums pretty nicely up to, less than 10k . So if you're worried about inducements, never buy one. We did these calcs differently, but with the same ideas some years ago.

[mattgslater]

I really don't think it's always less than 10k.

Going into this, I anticipated that the most important variables would be the lesser of the expected FF+staff scores and the expected number of drives.

If you really want to be scientific about it, build an Excel document listing each opponent with its FF, AC and CL. For each opponent, Compare your FF to your opponent's, and compare each to twice the other, to determine what net FAME results to expect. You're only looking for results that will crop up a lot. Let's use 25% or more as our benchmark.
* Net FAME +2 is likely if your FF > their FFx2.
* Net FAME +1 is likely if your FF > their FF-3.
* Net FAME 0 is never likely.
* Net FAME -1 is likely if your FF < their FF-2.
* Net FAME -2 is likely if your FF < their FFx2.

Then, for each likely FAME for each coming opponent (maybe the next three games, or whatever your tolerance for 10k winnings is), subtract from that the number of ACs the team has to get the net ACs, used to calculate the "BC bonus" (and ditto for CLs, net CLs and "CF bonus") for that outcome. This bonus is the value of +1 on a BC/CF roll, in ninths of a success per BC/CF result, and is indicated in two numbers separated by a comma: the chance of gain, and the chance of denial. You don't need to track them separately if you think they're both equally valuable; in that case, just add them up. So if +1 were likely, you could note 1,2 (gain,denial) or 3 (gain+denial).

Net AC: BC bonus
+2: 0,1
+1: 1,2
+0: 2,3
-1: 3,2
-2: 2,1
-3: 1,0
-4: 0,0

If there is more than one likely FAME outcome, average them. For instance, if your opponent's FF is 5 and your FF is 4, then +1 and -1 are likely. If he has one AC, then 0 and -2 are the likely net AC results, so the net AC bonus is (2+2)/2,(3+1)/2, or 2,2. That means that one AC will win about 2/9 of BC results for you. Average your coming opponents. So you'll have four numbers between 0 and 3: your BC bonus for gain, your BC bonus for denial, your CF bonus for gain, and your CF bonus for denial.

Next, the expected number of BC/CF results. First you need the number of drives per game: for that, add up the scores you expect, for and against (if you're the grinder type, you'll see about 2-3 scores per game; if you're the speedster, you'll see 3-4 on average), and add +1 for the chance of a meaningful drive to end on turns. Multiply this by 5/324, and multiply each of your BC and CF bonuses. This is your expected RR per match for or against, for each match, for one AC, and for one CL.

Then, to figure out if it's worth the TV/cash, compare it to the value of a TRRC in 10k increments.

So let's say you're going into a 1M rookie format and are expecting everybody to have FF0, and you have 20k to dump. Should you just bank it? Perhaps, but what's an AC worth to you?
* +1 and -1 are the likely results on BC That's 1,2 and 3,2, for an average of 2,2. There are no ACs in your calc, because you don't think anybody else (or at least not many) will have them.
* You're all rookies. Everybody wants a RR, nobody can get enough. Use the high value on TRRs (100k, so 50k for an RRC). A denial is as good as a gain, and vice versa, so your BC bonus of 2,2 can just be thought of as 4.
* You expect two or three scores per game, for an average of, say, 3.5 drives. Multiply that by your BC bonus, and you get 14. 14 x 5/324 is 70/324 of an RRC. An RRC is worth 50k, so an AC is worth 350/324 its weight in RRs.

[Joemanji]

... one shiny penny.

[mattgslater]

Really? Want to buy ACs in bulk? PM me for my address.

Models not included.

[Carnis]

Your problem is meaningful drives definition. Normally it's 3 meaningful drives on a 2-1 grind, or 2 meaningful drives in an outright steal. The drives where you go from 2-0 to 3-0 are NOT meaningful at all. So 3.5 drives is very optimistic..

[mattgslater]

Hmm... Let's see if we can condense that a bit into a formula you can use on the fly, for whether your AC or CL will help you for your next opponent. The more I think about it, the more I think we can consider the value of an RR for as equal to an RR against, because usually there is more than one likely result, so there are frequently the same odds. Let's just aggregate all that stuff. Let's also assume we're competing with 0 AC/CL, and note that we're describing the net value of winning the AC/CL comparison by one. That simplifies it significantly.

Now we're down to this:

Winning the AC comparison by +1 is valuable if the BC bonus times the drive bonus times the RRC value is at least (>10k•5÷324) 650k gold.

BC Bonus: Compare your FF to your opponent's FF (OFF).
If your FF > (2xOFF)+10, BCB = 1
If your FF = (2xOFF)+(4-10), BCB = 2
If your FF > OFF+2, BCB = 3
If your FF = OFF±(1-2), BCB = 4
If your FF > OFF/2, but your FF < OFF-2, BCB = 5
If your FF = OFF/2-(0-10), BCB = 4
If your FF < OFF/2-10, BCB = 3

Drive bonus: I'd imagine a formula for "how many meaningful drives per game" would be great. Obviously, 2 is a minimum. Let's generally use 3, but note that some teams should use 4.*

So most of the time, you'll be multiplying some number that's about 3-5 by some other number that's about 3 or 4, for some number that's somewhere between 9 and 20. Then multiply that by the value of a net RRC, compare to 650k and you're good to go!

For a net RRC, use (TRRV+OTRRV)÷4. TRRV is either your listed TRR cost or 100k, depending on how badly you want it. OTTRV is the other guy's listed TRR cost or 100k, depending on how badly he wants it. If you don't know, use 80k for your opponent, so the net RRC value is 20k plus your TRRV/4. So you'll be multiplying a number between 25k and 50k by a number between 9 and 20. 25k•9 = 225k, well under the 650k threshold. 50k•20 = 1,000k, well over the 650k threshold. So it's not a question with a "yes" or "no" answer. With an assumed drive number of 3 for a grind team, it's somewhere between 225k and 750k, depending on how much of an overdog you are. Assuming you're not a FF outlier and you don't have all the TRRs you could want (or you tend to induce them), then you're looking at between 600k and 750k, which is pretty much a wash. If you're the kind of coach with either a high-flying elfy defense or real offensive problems that lead to backfield cage failures, you might multiply by one and a third. if you don't really need the RRC, you might cut it by 30% (70kTRR) to 50% (50kTRR).

*I disagree with Carnis about the value of the final drive in a 3-0 game. There are objectives that transcend winning, like SPP and possibly tiebreaker status (SPP is the big one). I'd also point to the agility strategy of turning over on defense, with just enough time to force an offensive hurry. if there's a proper pressure attempt in the second half, this can lead to as many as 5 meaningful drives. But that's not actually what we're measuring. What we're measuring is drives in which the BC/CF winner is likely to get a benefit from a RRC. For instance, say your opponent stalls on you, scoring in T8. Let's say you have a viable 1TTD mechanism and a TRR left. The drive is very meaningful, in win/loss terms, but it doesn't count for BC/CF terms, because nobody could use the RRC.

[Darkson]

If I wanted a game to be all about number crunching, I'd play 40k...

[Smeborg]

Matt - teams and rosters are not equal in this regard.

Some teams (Vampires for example) are re-roll hungry, others are keen to deny re-rolls to the opponent (e.g. 'Flings).

Some teams (bash, mainly) plan to play no more than 3 meaningful drives per game.

Successful teams with a combination of high roster cost and high FF may fear inducements more than they fear the opponent getting re-rolls during the match.

I have not bothered with AC/CL on any of my teams this season, and have not regretted it. The case seems particularly clear for my successful DE and Nurgle teams, both of which have relatively high cost, high FF and tend to be short of money.

I have long felt that you have to have good reason to take more than 1AC/1CL. Your number crunching seems to support this.

All the best.

[Joemanji]

Really? Want to buy ACs in bulk? PM me for my address. Models not included.

I'm happy to send you one shiny penny for every AC I buy for a league team. Just don't hold your breath ...

[mattgslater]

Matt - teams and rosters are not equal in this regard.

I think I accounted for each of these.

Some teams (Vampires for example) are re-roll hungry, others are keen to deny re-rolls to the opponent (e.g. 'Flings).

That's factored into RRC value. If you've got such a team, set your (or your opponent's) RRV to 50k, or half an Extra Training. For example, Halflings always use 40k (60k for their own TRRs, plus 100k for the other guy's, divided by 4). I could see increasing that by 10k on the grounds that opposing TRRs come at a premium, especially if the Halfling team doesn't run a lot of RRs or knows it will get the key inducements either way. On 3 meaningful drives with a Halfling-style record (BCB 5), that's 3x5x50k=15x50k=750k > 650k, worth it by a good margin.

Successful teams with a combination of high roster cost and high FF may fear inducements more than they fear the opponent getting re-rolls during the match.

If your opponent doesn't get disproportionate gain from the TRR and you have all you need (but would still like another if it were cheap), your TRRVs will be based on the actual RR cost. If you're a FF overdog, you'll have a low BCB/CFB.

Some teams (bash, mainly) plan to play no more than 3 meaningful drives per game.

Yeah. Bash teams should think in terms of 3, or even 2.75, drives per game. Does this mean that faster teams get more for AC/CL than slower ones do?

I have long felt that you have to have good reason to take more than 1AC/1CL. Your number crunching seems to support this.

I suspect you're right, but I haven't really looked at the value of +2. I'd imagine if you were always a FF underdog you'd get better mileage from the second point than from the first. Certainly, there's little benefit from having +3 as opposed to +2 after FAME.

If I wanted a game to be all about number crunching, I'd play 40k...

The beauty of BB is that it's not "all about" any one thing. If you really want to master 40k, you become a master number-cruncher, marry yourself to one tactical idea, and voila, there you are. That's not enough in BB, though it sure helps. So here's a lot of work on condensing a small number to a handy formula that has a few variables you can plug in as needed. This math suggests that ACs and CLs are a good bargain if you meet three of the following criteria:
* You don't have enough re-rolls
* Your opponents don't have enough re-rolls
* You score or give up a lot of TDs
* You have a lower-than-average FF, preferably above 1 or 2.
* You have either a very similar TV to most of your opponents, or a very different one from most of your opponents (so the 10k inducements don't matter as much).

[mattgslater]

Okay, more on how to break down RRC value.
RRC base value
* If you're desperate on TRRs, use 60k or 70k as a base RRCV. This is very subjective, yes.
* If you would take an extra TRR for a 100k TV add given the circumstances, use 50k as a base RRCV.
* If you'd take an extra TRR for the money, but you wouldn't pay the full inducement value, use 40k.
* If you don't need the extra RR so badly you'd give up its weight in TV for it, use half your list TRR cost, 25k-35k.
* Add to this your opponent's RRV based on the same calculus. If you don't know your opponent in advance, or if cash is at a premium and you can't hire or fire staffers all the time, assume 30k, 40k or 50k, depending on format.

If you're a bash team and you're playing a lot of 1-0, 2-0 and 2-1 games, assume 3 drives per game. Sometimes there will be fewer, but never fewer than 3-1, while occasionally there will be more than 3+1, and there tend to be 3+ drives in the closest matches. If you're giving up a lot of TDs, use score +1 instead.

If you're an agility team, you want to assume you'll get at least 3 drives in each game. If you can't, that fact probably ups the value of a TRR significantly, so hey. If you're playing really good defense, you might get stretches with 5 or 6 drives; assume 4. If you're getting mobbed and beaten, you may end up with a smaller average than this, and may want to assume 3.5.

Both of those factors are strongly influenced by team race. Assuming a default 40k opposing RRC value and tack on +10k to the half-TRR-cost formula on your own, we could probably work out both RRV factors for each race, give or take.

[mattgslater]

So, then, if we have average CFB/BCB times expected drives times 25k+TRRcost/4: 37.5k, 40k, or 42.5k. If you don't know your CFB/BCB, assume 4. So if you're talking about an average team with an average 60k TRR that can expect 3 drives per game, you've got 4 x 3 x 40k = 640k = wash.

So yeah, one AC/CL is absolutely a good deal for teams that are off somewhere on that equation. If you really need the TRR or you really need to deny the TRR, if you have a poor record and a corresponding FF in the 3-6 range, if you have a tendency to run up the score. All of those will add to the value of an AC/CL.

[Ullis]

I like this. In a small league you could have an arms race between AC's and CL's. Obviously getting to stay out of it (while guaranteeing that you'd lose all RR's from kick offs) would give one team a comparative advantage to the others.

[Smeborg]

I like this. In a small league you could have an arms race between AC's and CL's. Obviously getting to stay out of it (while guaranteeing that you'd lose all RR's from kick offs) would give one team a comparative advantage to the others.

I have observed leagues in several different countries and cities. Each league has habits of one kind or another. The coach who knows this and goes against the local trend gains an advantage, and exposes the habit as a bad one.