Number of BG in the Britcon armies

[lawrenceg]

I haven't ploughed through all the posts on the thread, but if no one has
done it, I suggest doing some statistical significance testing.

Null hypothesis (i.e. thing to be tested)

All armies have the same chance of winning or losing regardless of number of
BGs. In this context, win means inflict an army rout without being routed
and lose means suffering an army rout without inflicting one (i.e. someone
got the 5 point bonus).

First step is to find the chance of winning or losing or drawing simply as
the proportion of all games that are 25 points. Call these W, L and D.
W+L+D = 1.

Then I suggest two things you can do.

a) divide the games up into classes by army size difference in BG. For each
game, the larger army is army 1. For equal games assign army 1 randomly or
by alphabetical order whatever.

Then you have

BG difference: number of games : Army 1 win, draw, army 1 lose with
observed and expected number of results for these three.

Then I think you can do a chi squared test with degrees of freedom of 2 x N
where N is the number of classes.

IIRC chi squared = sum of ((observed-expected)^2 )/ expected. There might be
a 1/n in there somewhere where n is the number of games. Anyway it is bound
to be on wikipedia. Given the value of chi squared and the degrees of
freedom, excel will tell you the probability.

This is the probability that, if the hypothesis were correct, the observed
results would have fitted that badly or worse by chance.

b) divide up games by absolute size in BG of the army. Again look at
observed and expected number of each kind of result for each size of army.

You might need to think about the degrees of freedom in this case as it
looks as though each game would end up being counted twice so the results
are not indepedent.

[lawrenceg]

General comments on numbers of BG.

a. Two BGs of 4 can stick together and do everything that one BG of 8 can
do. Or they can split up, which gives them more tactical options than the
single BG of 8. Therefore the two 4's have an advantage.

b. DBM analogy. In FOG you get 1 move per BG (might be a double move, but
we'll count that as one). In DBM you get 1 move per PIP (OK, some cost more,
but we'll ignore that). So if you play FOG with 18 BGs against a 12 BG, you
are in effect playing DBM with 18 PIPS per bound vs an opponent with 12 PIPs
per bound. Who would you bet on?

These are simplifications that ignore a lot of the issues, principally that
BG combat effectiveness is a function of size. However, there is a prima facie case that large numbers of BGs give an advantage.

Whether this is counteracted by the combat disadvantage of small BG size
is a good question. I'm not convinced that there is much disadvantage in
practice, as the various rounding effects tend to cancel each other out.
e.g. 4-bases shot at test at -1 if they take 1 hit per 2, but don't test at
all with 1 hit per 3. In general small BGs have less bases fighting against
them, so take less hits, so benefit more from the +2 on death rolls. Etc.

[rbodleyscott]

So at 800 points the maximum break point is 16. You can have 22 BGs (for example) but your army will still break when it hits 16 attrition points. A 14 BG army would still break on 14 attrition points.

I think this is an exteremly elegant solution - thanks Richard.

Encouragingly there is nothing to stop competition organisers introducing this now if they wished to.

This could be introduced now, but it would take decades on how to agree to alter the scoring system....

Why? It would be calculated on proportion of army break point reached, as at present.

The form just needs altering to specify what the maximum break point is for the (army size of the) tournament.

[philqw78]

People don't need to agree, it just needs to be enforced. Nobody asked my permission to write FoG, but now I have to abide by the rules.

[guthroth]

Why? It would be calculated on proportion of army break point reached, as at present.

The form just needs altering to specify what the maximum break point is for the (army size of the) tournament.

Well, it gets my vote !

Maybe an email from the authors to the BHGS asking for this to be adpoted would bring results.

[hammy]

General comments on numbers of BG.
Whether this is counteracted by the combat disadvantage of small BG size
is a good question. I'm not convinced that there is much disadvantage in
practice, as the various rounding effects tend to cancel each other out.
e.g. 4-bases shot at test at -1 if they take 1 hit per 2, but don't test at
all with 1 hit per 3. In general small BGs have less bases fighting against
them, so take less hits, so benefit more from the +2 on death rolls. Etc.

This has been discussed before. There is an argument that two BGs of 4 bases are just as effective in a straight up fight as one BG of 8. I have not got the tools to do a full simulation but there are a number of problems for the 2 4 base BGs which seem to hurt quite a lot in my actual game experience.

Probably the biggest issue is the speed at which a 4 base BG can evaporate. Lose one roud of combat and you are in with a fair chance of losing a base and disrupting which halves your combat power and puts you at a big minus for future tests. An 8 base BG is less impacted by a base loss and disruption.

I know that for example in the Santa Hermandad army where you can have spear in BGs of 4, 6 or 8 that the armies that do best are those with them in 8s.

[expendablecinc]

General comments on numbers of BG.
Whether this is counteracted by the combat disadvantage of small BG size
is a good question. I'm not convinced that there is much disadvantage in
practice, as the various rounding effects tend to cancel each other out.
e.g. 4-bases shot at test at -1 if they take 1 hit per 2, but don't test at
all with 1 hit per 3. In general small BGs have less bases fighting against
them, so take less hits, so benefit more from the +2 on death rolls. Etc.

This has been discussed before. There is an argument that two BGs of 4 bases are just as effective in a straight up fight as one BG of 8. I have not got the tools to do a full simulation but there are a number of problems for the 2 4 base BGs which seem to hurt quite a lot in my actual game experience.

Probably the biggest issue is the speed at which a 4 base BG can evaporate. Lose one roud of combat and you are in with a fair chance of losing a base and disrupting which halves your combat power and puts you at a big minus for future tests. An 8 base BG is less impacted by a base loss and disruption.

I know that for example in the Santa Hermandad army where you can have spear in BGs of 4, 6 or 8 that the armies that do best are those with them in 8s.

I agree.


In a straight up fight it is always better to have a large BG o f8 fighting 2 of 4.

ignoring Bg size they have identical chances of losing bu tit either Bg of 4 have to test they are far more likely to get a -1. Also they have twixe as many AP at stake so double to risk for the same reward. In a noone loses combat situation the 2 BGs of 4 have a slight advantage as they are unlikely to have to tet as the +2 for deatch rolls may mean neith has to test. on the other hand there is also more likelihood of two bases being lost reath than one.

the value of the two smaller Bgs is bulking the army out - only of beneift if they darent riboth risked. and the increased oppornunitites for flank attachs and/or flank protection.

anthony

[hazelbark]

But its not about straight up fight only.

The big BG that starts getting one charge on the flank and the other in the nose has headaces.
- POA for 2 directions
drop a level
++ POA for the impact in flank.

Or 2 big BGs. 1 is screened off by 1 small BG and then 3 BGs go mug the other one.

[ethan]

the value of the two smaller Bgs is bulking the army out - only of beneift if they darent riboth risked. and the increased oppornunitites for flank attachs and/or flank protection.

It is also in the death roll. An 8 element BG can take a significant death roll even if it wins - especially when fighting other solid troops. Rolling 8 dice and winning/drawing and taking 5 hits means you have a 50-50 chance of taking a casaulty.

If you are two four base BGs and take 5 hits, say two on one BG and three on the other you have a 1 in 6 chance of taking a casaulty. Now that casaulty might be more problematic but that is an awful big difference in casaulty rates.

[hammy]

the value of the two smaller Bgs is bulking the army out - only of beneift if they darent riboth risked. and the increased oppornunitites for flank attachs and/or flank protection.

It is also in the death roll. An 8 element BG can take a significant death roll even if it wins - especially when fighting other solid troops. Rolling 8 dice and winning/drawing and taking 5 hits means you have a 50-50 chance of taking a casaulty.

If you are two four base BGs and take 5 hits, say two on one BG and three on the other you have a 1 in 6 chance of taking a casaulty. Now that casaulty might be more problematic but that is an awful big difference in casaulty rates.

Only if both small BGs actually draw the combat......

It is more likely that in a 'drawn' overall combat one of the small BGs will lose and the other will win than there will be an exact even combat along the line. In this case the losing small BG will not have a +2 so the actuall chance of either side losing a base will be pretty much the same if not in the larger BGs favour.
If two BGs of 4 draw 5 hits to 5 with a BG of 8 there is indeed a 50% chance the 8 base BG will lose a base but there is a far higher than 16% chance that one of the 4 base BGs will lose one.

[hammy]

Hmm, this is actually really a rather hard probability problem.

Some numbers of note (assuming even POA and average troops):
8 dice will generate 4 hits 27.34% of the time
these hits will be split 2 and 2 between sets of 4 14.06% of the time
in the situation where exactly 4 hits are inflicted then there is a 48.57% chance that one of the smaller BGs will have lost the combat

Overall I think this means that the smaller BGs are less likely to lose a base in a drawn fight but that it is nowhere near as bad as Ethan sugested and there is a fairly high chance that one of the smaller BGs will be taking a cohesion test as well as having a chance of losing a base.

[babyshark]

But its not about straight up fight only.

The big BG that starts getting one charge on the flank and the other in the nose has headaces.
- POA for 2 directions
drop a level
++ POA for the impact in flank.

Or 2 big BGs. 1 is screened off by 1 small BG and then 3 BGs go mug the other one.

Of course, the other side of that is that if the 2 small BGs don't quite manage to coordinate their flank-and-front attack. If the big BG gets to pick on them one at a time it will likely be ugly for the small BGs.

In short, I don't think there is an easy comparison to make between one 8 base BG and two 4 base BGs. There are too many other variables.

Marc

[ethan]

Overall I think this means that the smaller BGs are less likely to lose a base in a drawn fight but that it is nowhere near as bad as Ethan sugested and there is a fairly high chance that one of the smaller BGs will be taking a cohesion test as well as having a chance of losing a base.

It is pretty easy to see that the 2 small BGs will on average lose less bases in a drawn fight.

Assuming 4 hits from each side the options for the 2 BGs are as follows:

4 hits on on BG: Small and Large side have exactly the same chances of losing a base
Split 3/1: Small has a 1 in 6 chance of losing a base, large has 2 in 6
Split 2/2: Small has 0 chance of losing a base, large has 2 in 6

There are no other options for splitting the hits on the small BG side.

The exact distribution isn't that hard either actually. Basically if each hit is equally likely to hit either BG then it is just like flipping a coin. Heads it hits BG A and tails it hits BG B.

4 hits on on BG is 12.5%
3 and 1 is 62.5%
2 and 2 is 37.5%

So the small BG side loses

(.125)*(2/6)+.625*(1/6)=0.15 bases

versus the large side losing 2/6 of a base or .33 bases.

So the large BG takes twice as many casaulties.

[ethan]

Doint a quick Excel simulation with 5000 trials I get an 8 Base BG on average losing 0.3 bases per combat and each 4 base BG losing 0.1 bases per combat BUT this ignores the possibility that 8 base BG could lose two bases which ups there casaulties a bit.

The 8 base BG loses 40% of the time, the 4 base BGs lose about 35% of the each.

On the Morale side

Only 8 base BG loses the combat about 30%
Only one 4 base loses about 40%
Both 4 bases lose about 14%
One 4 base and the 8 bases loses about 8%
Draw about 7%

One thing I take away is that morale on the small BG side matters. Each individual BG is basically just as likely as the large BG to takea check but combined the side with the small BGs is more likely to take at least one check than the side wiht the 8. On the big BG side numbers seems to matter more as they are likely to take more casaulties.

So for high quality BGs small is perhaps better, for lower quality BGs go with size.

[Cerberias]

Bigger BG's get more effect from generals, and cant be 'picked off' one at a time like smaller BG's can.

[lawrenceg]

General comments on numbers of BG.
Whether this is counteracted by the combat disadvantage of small BG size
is a good question. I'm not convinced that there is much disadvantage in
practice, as the various rounding effects tend to cancel each other out.
e.g. 4-bases shot at test at -1 if they take 1 hit per 2, but don't test at
all with 1 hit per 3. In general small BGs have less bases fighting against
them, so take less hits, so benefit more from the +2 on death rolls. Etc.

This has been discussed before. There is an argument that two BGs of 4 bases are just as effective in a straight up fight as one BG of 8. I have not got the tools to do a full simulation but there are a number of problems for the 2 4 base BGs which seem to hurt quite a lot in my actual game experience.

Probably the biggest issue is the speed at which a 4 base BG can evaporate. Lose one roud of combat and you are in with a fair chance of losing a base and disrupting which halves your combat power and puts you at a big minus for future tests. An 8 base BG is less impacted by a base loss and disruption.

I know that for example in the Santa Hermandad army where you can have spear in BGs of 4, 6 or 8 that the armies that do best are those with them in 8s.

Quantifying the combat disadvantage of 2x4 vs 8 is a difficult probability problem and comparing that to the tactical options disadvantage of 1 x 8 is intractable. In practice we are comparing, e.g 18 BG of various sizes to 12 BG of various sizes anyway.

That is why I suggested applying some statistical rigour to the game results analysis.

By the way, one ought to separate out the round 1 BRITCON results as they may have a longer or shorter time limit depending on when the game started.


Has anyone looked at how the game results would change with the suggested upper limit on AP needed for an army rout.

[hammy]

Ethan,

You are I think missing my point.

If the 8 base BG gets 4 hits all from the right hand half and the 4 base BGs get 4 hits in total spread across the line then one of the 4 base BGs has lost so does not get the +2 to its death roll.

If you assume that all the combats are drawn then I agree 100% that the 4 base BGs take fewer losses BUT with the overall combat drawn there is a higher chance that one of the 4 base BGs will have lost and the other one won than that both 4 base BGs actually drew.

If A and B are the 4 base BGs and X is the 8 base one you have to consider:

In all the following cases BGX has a 33% chance of losing a base.

With all the hits delivered by one small BG we get:

A4+B0 vs Xa4+Xb0 -> straight draw, BGA has a 33% chance of losing a base
A4+B0 vs Xa3+Xb1 -> BGA and BGB have a 16% chance of losing a base
A4+B0 vs Xa2+Xb2 -> BGX and BGB have a 33% chance of losing a base
A4+B0 vs Xa1+Xb3 -> BGB has a 50% chance of losing a base
A4+B0 vs Xa0+Xb4 -> BGB has a 66% chance of losing a base

With the hits split 3-1 between the small BGs:

A3+B1 vs Xa4+Xb0 -> BGA has a 66% chance of losing a base
A3+B1 vs Xa3+Xb1 -> straight draw, BGA and BGC have a 16% chance of losing a base,
A3+B1 vs Xa2+Xb2 -> BGB has a 33% chance of losing a base
A3+B1 vs Xa1+Xb3 -> BGB has a 50% chance of base loss
A3+B1 vs Xa0+XB4 -) BGB has a 66% chance of base loss

You then have to consider the hits on BG X split 2-2, 1-3 and 0-4

I suspect that you are right with your montecarlo that the small BGs take fewer base losses but not by the margin you originally proposed.

[hammy]

That is why I suggested applying some statistical rigour to the game results analysis.

By the way, one ought to separate out the round 1 BRITCON results as they may have a longer or shorter time limit depending on when the game started.

Has anyone looked at how the game results would change with the suggested upper limit on AP needed for an army rout.

While the scoresheets are somewhere in the BHGS archive there is no record of the AP lost in each game in the results file. It would be a lot of work to go through and look at what effect upper limits on AP lost would have on the scores. It would be doubly difficult when you consider that if a player's army will break at 16AP rather than 19 then once they reach 12 or 14 AP lost they may well chance to a defensive mode of play.

I agree that the round 1 games may be an anomaly.

I would be happy to send anyone who wants them the who played who and number of BG information. At present I don't have the time to do any further analysis.

[shadowdragon]

I suspect that you are right with your montecarlo that the small BGs take fewer base losses but not by the margin you originally proposed.

Assuming 1 X 8-base BG versus 2 X 4-base BG, in toe-to-toe close combat with no POA to either side and all BG being average - I went through the tedium of calculating the probabilities for possible outcomes for one round of combat.

The calculations take into account a -1 CT modifiers for (a) >= 1 hit per 3 bases and (b) at least 2 or more hits received than inflicted in close combat.

Winning, drawing or losing makes a big difference on both base cohesion loss, but the calculations are simplified since each BG must wins, draws or loses based on hits inflicted / received against all opponents. So....with both sides having 8 bases involved in melee for 8 dice (or equivalently 2 dice for 4 bases in contact on each side for impact)...

8-Base BG:
Potential hits received / inflicted vary from 0-8 hits.
Probability of losing / drawing / winning = 40.2% / 19.6% / 40.2%
Probability of not losing a base / losing 1 base / losing 2 bases = 82.3% / 17.4% / 0.3%
Unconditional probability of losing the cohesion test = 26.5% (i.e., 66% if the combat is lost)

Single 4-base BG (facing 4 bases of the 8-base BG):
Potential hits receive / inflicted vary from 0-4 hits.
Probability of losing / drawing / winning = 36.3% / 27.3% / 36.3%
Probability of not losing a base / losing 1 base = 93.5% / 6.5%
Unconditional probability of losing the cohesion test = 22.9% (i.e., 63.1% if the combat is lost)

Note the increase in the probability of drawing which is due to the coarser resolution of outcomes with 0-4 hits versus 0-8 (i.e., the greater the number of potential hits the less likely a draw). For an extreme case of this, consider 1 base versus 1 base in melee. The chance of a draw would be 50% and the chance of losing drops to 25%. This is important since it affects whether or not a unit takes a CT or significantly reduces the chance of losing a base.

Now for both 4-base BG together – since the 8-base BG’s dice are allocated to each 4-base BG prior to rolling for results, the results for each 4-base BG are independent of the other and the joint outcomes for the pair of BG are:

Lose-Lose: 13.2% (i.e, both 4-base BG lose)
Lose-Draw: 19.9%
Lose-Win: 26.4%
Draw-Draw: 7.5%
Draw-Win: 19.9%
Win-Win: 13.2%
Probability of not losing a base / losing 1 base / losing 2 bases = 87.5% / 12.1% / 0.4%
Probability of both BG maintaining cohesion / one BG dropping in cohesion / both BG dropping in cohesion = 59.4% / 35.4% / 5.3%

Comparing these results to those for the 8-base BG we see that, as written above there is much less chance of losing a base with the 2 X 4-base BG, but the chance of both 4-base BG maintaining their cohesion is far less than for the 8-base BG.

For idle interest, the probabilities for all outcomes for all BG are:

W-LL: 13.2% (i.e., the 8-base BG wins and both 4-base BG lose)
W-LD: 19.9%
W-LW: 7.1%
D-LW: 12.2%
L-LW: 7.1%
D-DD: 7.5%
L-DW: 19.9%
L-WW: 13.2%

Now that should numb a few minds.

[hazelbark]

Now that should numb a few minds.

yep. I think i rather play FoG. Its more fun that calculating this cr**