SBR comparative analysis, a real breakthrough

[baron Münchhausen]

Hi Private Panic,
what is bolded is the results for each type of SBR.
The rest is the last calculation which allowed me to get the final result.
I do not want to provides all other undigestible pre-existent tables (but needed to reach the results) because it brings too small benefits for analysis to go to such details when talking on a wider scope.

About offense/defense cost ratio, it provides a number to compare with what happen in OOB regular combat. All A&A game is based on giving an advantage to defender by making the same attack number a bit costlier. For instance, all OOB regular ratio goes from 1 to 1.667 costlier for the same attack value than defender.

So, when comparing OOB G40 SBR, you get an incredible 0.632 off/def cost ratio.
A completly reverse situation than any other aspect of regular combat between units!
The attacker needs a very little investment to reach an even odds to score damage against defender.
It shows why StBs are clearly OP compared to other SBR values.

[Ozymandiac]

You both seem to arrive at the same outcome for StB A1 doing SBR against 2 Fgs D1:
G40 D6+2: +4.85 - 5.056 = -0.206 IPCs damage/SBR*9= -1.854

But Baron Munchausen multiplies this 0.206 with a factor of 9. Could you explain this BM, I’m not sure what this factor is doing there.

EDIT: I strongly discourage using math like it’s used in that line. Equations like +4 - 5 = -1 *9 are false, and hurt my feelings. [/math-nazi]

[Private-Panic]

Ozymandiac is right - it is Baron’s multiplication by 9 which throws me off. Thanks for spotting that Ozy.

Thanks for your reply Baron. Still not got it but will look again tomorrow. My brain hurts!

[baron Münchhausen]

@Ozymandiac:

You both seem to arrive at the same outcome for StB A1 doing SBR against 2 Fgs D1:
G40 D6+2: +4.85 - 5.056 = -0.206 IPCs damage/SBR*9= -1.854

But Baron Munchausen multiplies this 0.206 with a factor of 9. Could you explain this BM, I’m not sure what this factor is doing there.

EDIT: I strongly discourage using math like it’s used in that line. Equations like +4 - 5 = -1 *9 are false, and hurt my feelings. [/math-nazi]

To find the break even point, you need to reach a near zero either positive or negative (+ or - 0.0).
To get it you must confront a positive occurance such as 2 StBs vs 1 Fg against a negative occurance, sometimes a 1 StB vs 1 Fg or 1 StB vs 2 Fgs.
Then you must find the multplication factor for each which gives a sum near zero odds of comparative damage offender vs defender.
For example, G40 OOB needs 9 times 1 StB against 2 Fgs to get a total negative value of -1.854 IPCs for a single run of 9 StBs against 18 Fgs.
Adding 1 StB vs 1 Fg gives positive value of +1.819.
So when 10 StBs raids against 19 Fgs, you reach a break even point.
Small variation is negligeable but due to difference between both odds: -0.035 and not a perfect zero.

[baron Münchhausen]

@ShadowHAwk:

You cannot scale this type of attack effectively without taking into account the maximum damage you can do.

Max damage is 20 so 4 bombers will damage maximum the rest will add 0 damage.

It is still true my calculation doesn’t consider this ceiling factor.
It is not perfect but it still provides a good basis for comparing between SBRs values.
Once you know what is the break even point in a given set of values, you know when you plan an SBR, if the odds still on your side even it is a borderline riskier attack.

In 1942.2, it can be less than 20, 16 even 6 or 4.
It will be the next challenge to consider this 20 maximum damage factor per IC at most.

However, the scale of break even poit is still true.
If you have 5 StBs attacking 9 Fgs, for a 0.556 StB/Fg value, it is above .526 StB/Fg.
So it stay in favor of the attacker.
Now, the attacker must approximate is odds of maxing out the IC.
A rough calculation gives 5 StBs * 5.5 = 27.5 IPCs.
But, it will probably loose one StB. 4 StBs *5.5 = 22 IPCs. A max out result.
It can be possible to use my OOB tables to get a more precise result, to find a more on the line limit.
1 vs 1: + 5.486 - 3.667 = +1.819 IPC damage/SBR
StB A1 doing SBR against 2 Fgs D
1 vs 2 : + 4.85 - 5.056 = - 0.206 IPCs damage/SBR
*4 = +19.4 - 20.224 = - 0.824 IPC

5 vs 9 Fgs: +24.886 - 23.891= +0.995
So, 2 bombers will be lost (near -24 IPCs).
3 StBs will roll dmg, for 5.5 = 17.5 IPCs. No maxout.
Or 1 Fg will be destroyed +10 IPCs, it gives around 15 IPCs damage on IC.

So, my point is that no matter the high number of units implied in the calculation, break even point it still correct. The adjustment on 20 IPCs CAP must be made in game to evaluate if a given situation  gives good odds for attacker and if it still worth even if it gets max out on IC.

Also, defender should ask if it worth the case to intercept.
For instance,
1 StB doing SBR without interceptor
Sum: +4.583 - 2 = +2.583 IPCs damage/SBR
*5 = +22.915 -10 = +12.915 IPCs damage/raid
The difference is near 12 IPCs (12.915-0.995), so intercept reduces the benefits of the attacker.

But no intercept brings a 20 Max out on avg, +22.915. So, instead of 13  IPC, it will be a +10 IPCs on the defender due to cap at 20. A calc maniac can then compare pros and cons of intercept.

[simon33]

So you are saying for G40 OOB, there is a positive expected value for a bombing raid with 2SBs vs 3 ftr intercepting? Interesting that’s not what I expected.

I get for the IC:

• 40% chance of doing 11 damage average
• 15% chance of doing no damage
• 45% chance of doing 5.5 damage average
  For the fighters:
• 1/36 chance of doing 20 damage
• 10/36 chance of doing 10 damage
  Average damage = 10.22 total.

For the bombers:

• 15% chance of 24 damage
• 45% chance of 12 damage
  Average damage for attacker: 8.95

Interesting. Seems you’re correct.

[baron Münchhausen]

@simon33:

So you are saying for G40 OOB, there is a positive expected value for a bombing raid with 2SBs vs 3 ftr intercepting? Interesting that’s not what I expected.

I get for the IC:

• 40% chance of doing 11 damage average
• 13% chance of doing no damage
• 47% chance of doing 5.5 damage average
  For the fighters:
• 1/36 chance of doing 20 damage
• 10/36 chance of doing 10 damage
  Average damage = 10.31 total.

For the bombers:

• 13% chance of 24 damage
• 47% chance of 12 damage
  Average damage for attacker: 8.76

Interesting. Seems you’re correct.

You will find on this link the whole post on OOB G40 SBR calculations.
http://www.axisandallies.org/forums/index.php?topic=35614.msg1391984#msg1391984

The mistake in your calculation is that Fgs interceptors can roll 3 dices to kill up to 2 StBs.
16/216 to kill 2 bombers. 7.4%
75/216 to kill 1 bomber. 34.7%
125/216 no kill. 57.9%

StBs roll /Fgs roll / AAA roll
1/3616/21636/36  = 96/46656  2 StBs killed by Fg vs 2 Fgs
1/3675/2166/36 =  75/46656  1 StB killed by Fg and 1 StB killed by AAA vs 2 Fgs
1/3675/21630/36 =   375/46656  1 StB by Fg vs 2 Fgs
1/36125/2161/36 = 125/46656  2 StBs killed by AAA vs 2 Fgs
1/36125/21610/36 = 1250/46656  1 StB killed by AAA vs 2 Fgs
1/36125/21625/36 = /46656  no casualty vs 2 Fgs

10/3616/2166/6 = 960/46656    2 StBs killed by Fg vs 1 Fg
10/3675/2161/6= 750/46656  1 StB  by Fg and 1 StB killed by AAA vs 1 Fg
10/3675/2165/6 = 3750/46656 1 StB by Fg vs 1 Fg
10/36125/2161/6 = 1250/46656  1 StB killed by AAA vs 1 Fg
10/36125/2165/6 = 6250/46656  no casualty vs 1 Fg

25/3616/2166/6 = 2400/46656   2 StBs killed by Fg vs no casualty
25/3675/2161/6 = 1875/46656    1 StB by Fg and 1 StB killed by AAA vs no casualty
25/3675/2165/6 = 9375/46656  1 StB by Fg vs no casualty
25/36125/2161/6 = 3125/46656   1 StB killed by AAA vs no casualty
25/36125/2165/6 = 15625/46656  no casualty at all

Results:
Bombard on IC: 22500/46656 * ((1+2) +(6+2) IPCs)/2= +11 IPCs) = + 5.305 IPCs
Bombard on IC: 18000/46656 * ((1+2) +(6+2) IPCs)/2= +5.5 IPCs) = + 2.122 IPCs
Killing 2 Fgs: 1296/46656 +20 IPCs =  + 0.556 IPC  2.778%
Killing 1 Fg: 12960/46656 +10 IPCs = + 2.778 IPCs  27.78%
1 StB killed: 18000/46656-12 IPCs = - 4.630 IPCs  38.58%
2 StBs killed: 6156/46656-24 IPCs = - 3.167 IPCs  13.19%

Sum: +10.761 - 7.797 = + 2.964 IPC damage/StB

THIS TABLE NEED EDITION SORRY…

[simon33]

No mistake. I’m including the AAA.

If one is shot down by the interceptors, it has a 1/6 chance of not living to tell the tale.
If none are shot down by the interceptors, they have a 1/36 chance of neither living to tell the tale.

So that makes the chance of two not proceeding to bomb the IC:

7.4% + 37.7% / 6 + 57.9% / 36 = 15.3%

Without rounding I get 14.8%.

(5/6)**5 = 40% though. That is the chance of them both getting through.

I’ll correct my odds above.

[baron Münchhausen]

@simon33:

No mistake. I’m including the AAA.

If one is shot down by the interceptors, it has a 1/6 chance of not living to tell the tale.
If none are shot down by the interceptors, they have a 1/36 chance of neither living to tell the tale.
So that makes the chance of two not proceeding to bomb the IC:

7.4% + 37.7% / 6 + 57.9% / 36 = 15.3%

Without rounding I get 14.8%.

(5/6)**5 = 40% though. That is the chance of them both getting through.

I’ll correct my odds above.

I forgot to include this little AAA vs 2 StBs in my last post, I will correct this soon.

The result must be +1.613 IPC/raid
+1.819 - 0.206 = +1.613

1 StB A1 doing SBR against 1 Fg D1
D6+2: + 5.486 - 3.667 = +1.819 IPC damage/SBR
StB A1 doing SBR against 2 Fgs D1
D6+2: +4.85 - 5.056 = -0.206 IPCs damage/SBR

[simon33]

I see where you’ve gone wrong. If two planes get through the interceptors, there are two AAA shots, not one.

[JDOW]

Hi guys,

my opinion on that. If you guys just enjoy doing such analysis - which I would really understand - then this stuff is nice.

However the practical use of that analysis in the game close to zero.

The following things are not btw calculated:
1. 1 SBR damage =/= 1 IPC cost. In many cases UK or Russia does not need to repair each damage immediately and can perfectly build even with 1-5 damage
2. Next to the cost of a bomber the threat projection must be considered in case a bomber gets lost. A bomber not being on the map after the German turn can significantly influence the Allied options. Thus, the attacker risks more than 2 IPC from each shot at1 against the bomber.

In a nutshell: A&A is a complex game and such analysis will not provide significant support to master the game.

However, if this is just fun for you guys, of course something I appreciate. I didn’t want to sound rude, just express that I find this aspect isnt’t crucial to master the game.

Cheers,
Tobias

[simon33]

I don’t completely agree, JDOW. It does have some effect on the game to know that you can be outnumbered and still have a positive expected value from an SBR. I would not have expected that.

One person I play with reckons that SBR would be more reasonable if the air raid part all rolled at a 2 rather than a 1. That way you could do a more effective counter to it.

[baron Münchhausen]

@JDOW:

Hi guys,

my opinion on that. If you guys just enjoy doing such analysis - which I would really understand - then this stuff is nice.

However the practical use of that analysis in the game close to zero.

The following things are not btw calculated:
1. 1 SBR damage =/= 1 IPC cost. In many cases UK or Russia does not need to repair each damage immediately and can perfectly build even with 1-5 damage
2. Next to the cost of a bomber the threat projection must be considered in case a bomber gets lost. A bomber not being on the map after the German turn can significantly influence the Allied options. Thus, the attacker risks more than 2 IPC from each shot at1 against the bomber.

In a nutshell: A&A is a complex game and such analysis will not provide significant support to master the game.

However, if this is just fun for you guys, of course something I appreciate. I didn’t want to sound rude, just express that I find this aspect isnt’t crucial to master the game.

Cheers,
Tobias

@simon33:

I don’t completely agree, JDOW. It does have some effect on the game to know that you can be outnumbered and still have a positive expected value from an SBR. I would not have expected that.

One person I play with reckons that SBR would be more reasonable if the air raid part all rolled at a 2 rather than a 1. That way you could do a more effective counter to it.

I would add that some people play also 1942.2 on both TripleA and boardgame.
If played according to OOB, it is not the same as Global SBR, neither TripleA SBR nor SBR board version.

And all the factors you mentionned still apply in this three different SBR rules.

The breakthrough is about summarizing in a single ratio to remember.
No one clearly explained it to me that way.
All my tables made for each SBR rules were giving many numbers for a lot of different situations.
Absolutely not easy to manage.

[simon33]

Follow on question. What is the ratio of interceptors to unescorted bombers needed to justify interception?

SB continuing to target does:
average damage = 5/6 * 5.5 - 12*1/6 = 2.5833

So where ‘a’ strat bombers are intercepted by ‘b’ fighters
damage saved = 2.5833 * b - 1/6 * 10 * a

Netting to zero damage saved:
10/6 * a = 2.5833 * b

a/b = 2.5833 * 6 / 10 = 1.55

So two interceptors are very slightly ahead against 3 unescorted strat bombers, assuming that all damage will be repaired and will not max out the IC’s damage.

[baron Münchhausen]

@simon33:

Follow on question. What is the ratio of interceptors to unescorted bombers needed to justify interception?

It depends on which SBR system used.
G40 OOB SBR is quite hard on interceptors because it usually increase the average odds of damage if a Fighter try to intercept when there is the same number of escorting Fg.

The chart is below.
From a direct SBR on IC to 1 intercepting Fg, it reduce the odds from 2.583 to 1.819.
But 1 StB and 1 escort is more destructive if you intercept with 1 Fg:
3.972 IPCs/raid.
As defender, you should not intercept because you help your opponent:
giving 3.972 (intercept) - 2.583 (no intercept) = +1.389 additional damage on average!

If you have 1 Fg and 1 StB against 2 interceptors, the odds will be quite similar than no intercept:
2.445 compared to 2.583.
So, only when you have 3 interceptors against 1 Fg and 1 StB that it really worth to risk Fgs on intercept mission.

G1940 OOB SBR:
1 StB doing SBR without interceptor
Sum: +4.583 - 2 = +2.583 IPCs damage/SBR run

1 StB A1 against 1 Fg D1 damage: 1D6+2
Sum: + 5.486 - 3.667 = + 1.819 IPC damage/SBR run

1 StB A1 doing SBR against 2 intercepting Fgs D1
Sum: + 4.85 - 5.056 = - 0.206 IPCs damage/SBR run

1 StB & 1 Fg A1 doing SBR against 2 intercepting Fgs D1 damage: 1D6+2
Sum: +7.775 - 5.33 = + 2.445 IPCs damage/SBR run

2 StBs A1 doing SBR against 2 intercepting Fgs D1 damage: 1D6+2
Sum: +10.973 - 7.334 = + 3.639 IPCs damage/SBR run

1 StB & 1 Fg A1 doing SBR against 1 intercepting Fg D1
Sum: + 7.639 - 3.667 = + 3.972 IPCs damage/SBR run

2 StBs A1 doing SBR against 1 intercepting Fg D1
Sum: +11.459 - 5.666 = + 5.793 IPCs damage/SBR run

---

[baron Münchhausen]

@simon33:

Follow on question. What is the ratio of interceptors to unescorted bombers needed to justify interception?

SB continuing to target does:
average damage = 5/6 * 5.5 - 12*1/6 = 2.5833

So where ‘a’ strat bombers are intercepted by ‘b’ fighters
damage saved = 2.5833 * b - 1/6 * 10 * a

Netting to zero damage saved:
10/6 * a = 2.5833 * b

a/b = 2.5833 * 6 / 10 = 1.55

So two interceptors are very slightly ahead against 3 unescorted strat bombers, assuming that all damage will be repaired and will not max out the IC’s damage.

On your numbers, I’m not sure about your methodology.
Here is what the previous chart gives:
1 StB vs 2 Fgs = -0.206
1 StB vs 1 Fg = +1.819
Sum 2 StBs vs 3 Fgs = +1.613 IPCs average damage per raid.

[simon33]

1 SB + 1ftr vs 2ftr, it is about line ball, I agree. Going to be based on whether the defending fighters are worth more or less than the attacking ones.

I’m looking at the scenario of 3SB vs 2 ftr. Is it worth intercepting? It seems that it is line ball.

Actually, I got it slightly wrong. It is very slightly not worth intercepting 3SB vs 2ftr. But 4 vs 3 will be worthwhile so long as IPCs are worth the same to each side and the major IC is undamaged.

[baron Münchhausen]

@simon33:

1 SB + 1ftr vs 2ftr, it is about line ball, I agree. Going to be based on whether the defending fighters are worth more or less than the attacking ones.

I’m looking at the scenario of 3SB vs 2 ftr. Is it worth intercepting? It seems that it is line ball.

Actually, I got it slightly wrong. It is very slightly not worth intercepting 3SB vs 2ftr. But 4 vs 3 will be worthwhile so long as IPCs are worth the same to each side and the major IC is undamaged.

StB A1 against 1 Fg D1 damage: 1D6+2
Sum: + 5.486 - 3.667 = + 1.819 IPC damage/SBR run
AND
2 StBs A1 doing SBR against 1 intercepting Fg D1
Sum: +11.459 - 5.666 = + 5.793 IPCs damage/SBR run
Total: 3 StBs vs 2 Fgs
+16.945 - 9.333 =+7.612
3 StBs vs no interception 3*2.583=
+13.749 - 6.000 = + 7.749
So it is a very small difference. (-0.137 against no interception)
You can decide to destroy some more StBs with your interceptors at that point indeed.
If the ratio is 1.5 StB/Fg or below you have a go for interception indeed.

[baron Münchhausen]

Sorry,
I bumped this thread to get more easily some datas.
Baron

[baron Münchhausen]

Now it has been put arbitrarily in HR forum (for no apparent reason), I bump it again.

I need to reintroduce another important concept in first post: offence/defense cost ratio.

It will explain a possible issue about A0 C5 StBomber D6 damage.