Method for Estimating the Outcomes of Large Battles

@taamvan:

Anything that uses “formulas” other than

HP vs HP
ATT POWER vs DEFENSE POWER
6 / 36 / 108 style statistics for d6

is too complex to be used during any game, or too abstract to counsel any action.  No battle calcs are allowed at the tourney, which is fine, because they are totally unnecessary.    Counting and addition are all that is necessary.

? 108 ?
6+36= 42
636= 216
3636= 1296

Any link which explains this Punch formula?

Cell phone are not allowed?

Before judging a formula is too difficult for mental calculations, you have to ask how accurate is a formula.
Then, you can tell which is too cumbersome but accurate compared to inaccurate but come in handy.

People used clock and compass, or sun and moss but now GPS.
There is many ways to Rome.
Some faster but costlier, other longer but funnier.

While looking for Punch formula, I found the first occurrence of the OP stack formula based on Lanchester’s Law posted by akreider2 in 2007:
@akreider2:

I’m not sure if I ever mentioned this before, so I apologize if I did.

The formula comes from my experience playing the old version, where I used buy almost entirely infantry (and no tanks).  I ran several simulations and came up with this formula.  Frankly, it’s really amazing that it works and that I just figured it out inductively.

My sister actually proved this formula - but I don’t recall the proof.

The Formula
Power= (number of units) (number of units) (number you need to hit)**
IF attacker power = defender power THEN you have a 50% chance of winning.  
This only works if you are using units of the same level of strength (both attacker and defender - eg the defender can use a mixture of artillery and infantry as they both defend on ‘2’).

So this formula is useful for calculating your odds of winning.

For instance
A:  3 inf  - power = 331 = 9
D: 2inf  - power= 222 =8
So the attacker has an advantage.

Frood Says
A. survives: 50.7%   D. survives: 45.5%   No one survives: 3.7%

A: 40 inf - power = 1600 (40401)
D: 20 fig - power = 1600 (20204)
A. survives: 48.2%   D. survives: 51.4%   No one survives: 0.4%

Now using larger numbers helps:
A: 80 inf
D: 40 fig
A. survives: 49.3%   D. survives: 50.5%   No one survives: 0.1%

As you move to infinity units, it approaches 50% to 50%.  There is some kind of limit/calculus going on.  It seems to be approaching 50% from below (for the attacker, eg the attacker odds start at 48% and increase to 50% as the number of units increases) which is weird.  If I didn’t get timeouts on Frood when using 10,000 simulations, and if it allowed me to use more than 100 units, then I might have better luck proving whether the approach is random (and purely due to standard deviation) or from below.

I think it’d be possible to create a formula for calculating odds in general, but I’m not a complete math genius.

When are people going to learn that no single calculation is going to win them the game.  There is no silver bullet.

Further - many battles I fight are losing propositions(in terms of calculations,  but strategic and necessary evils, because guess what folks - winning the war requires sacrifice

Lanchester’s Table for Axis and Allies 2nd Edition

I made it on AACalc then I revised numbers by applying this formula derived from above Stack formula:
√(P2 / P1) = N1 / N2

This might be another way to estimate outcomes…

@Baron:

It is my first shot working with Table feature of the Forum.
If someone can do better, I will appreciate.
With this small 6 x 6 table, you get in a glimpse what is the 50%-50% break even according to the ratio of units and the average power of a given stack.
This table may also be written in decimal instead of a ratio, below. But ratio are better to understand how each ratio is paired to another one which is simply reversed.

A more complete table would include average power: 1.5, 2.5 and 3.5
But, in F-2-F, you can always round up the average enemy’s power, so you do a safer battle to be sure you are above break even ratio.

X means the ratio is 1:1. I did not want to overcharge this table with obvious infos.

| Power
1
2
3
4
4, 2hits | 1
X
12:17
11:19
1:2
7:23

| 2
17:12
X
9:11
12:17
4:9
| 3
19:11
11:9
X
13:15
10:19
| 4
2:1
17:12
15:13
X
5:8
| 4, 2hits
23:7
9:4
19:10
8:5
X

| Power
1
2
3
4
4, 2hits | 1
X
0.70
0.58
0.50
0.30
| 2
1.42
X
0.82
0.70
0.43
| 3
1.73
1.22
X
0.87
0.53
| 4
2.00
1.42
1.15
X
0.63 | 4, 2hits
3.33
2.30
1.87
1.60
X

This table can also be memorized with the main 5 basic ratios and you can reverse at will, or 6 if we include 1:1 ratio, the @4 with 2 hits for BBs might not be very relevant in F-2-F:

1:2 (4 vs 1), 11:19 (3 vs 1), 12:17 (4 vs 2 or 2 vs 1), 9:11 (3 vs 2), 13:15 (4 vs 3)

OR same order but from reverse ratio:
2:1 (1 vs 4), 19:11 (1 vs 3), 17:12 (1 vs 2 or 2 vs 4), 11:9 (2 vs 3), 15:13 (3 vs 4)

For instance, reading from left row to the right, if you have an average power of 1.3 for 17 units and the defender has 2.2 average power for 10 units.
You may cross-referenced the 1 row with the 2 column, saying you need  17: 12 ratio, or 1.42 more units than defender.
So, your 17:12 ratio or 1.42 more units  give an above 50-50% odds of winning.

Of course, during battle, ratio of units and average power may changes, especially when fodders are done.
So, you may decide at critical moment to recheck your odds of success.

For example, the defender may have only 3 Tanks left, while attacker only 4 Infs and 1 Tank.
This give a 5: 3 units ratio. And the 1 (A7 / 5= 1.4) row compared to 3 column, says: 19 to 11 or 1.73.
If looking the 2 row, it says 11:9 or 1.22.
So, assuming this rounding up or down, it reveals you are above or below the break even point.
Thus, it is still near 50%-50%.
In fact, AACalc says:
Overall %*:   A. survives: 70.6%    D. survives: 24%    No one survives: 5.5%

But, Punch formula might be better to anticipate results?
A7 for 5 hits vs A9 for 3 hits.

7+5 =12  or 9+3 =12, so still even…?

Stack formula is more revealing IMO:
251.4=  35  vs 93= 27

Clearly the 5 attackers units are on winning side. |

|

Why all these charts, calculations,and complications?

What’s wrong with:
Attack Power divided by 6 = X Hits
vs
Defense power divided by 6 = X hits,

and then seeing where that will take each side hit points after X rounds.

This is really simple stuff guys.

@Gargantua:

Why all these charts, calculations,and complications?

What’s wrong with:
Attack Power divided by 6 = X Hits
vs
Defense power divided by 6 = Y hits,

and then seeing where that will take each side hit points after Z rounds.

This is really simple stuff guys.

What is the name of this method? Pips substraction?
Is it different from Punch formula?

Once all methods identified, it can be easier to compare relevance and accuracy.

One thing, I saw with multiple divisions and substractions is that it overloads short term memory and you have to repeat for defender the process, at each step.

Stack formula use almost only multiplications and may avg power more intuitively.

You can call it whatever you want .

How about common sense or axis and Allies  101.

No one’s brain is getting overloaded with that method, as opposed to the calculus phone books some of the guys posted earlier.

I thought that since it is an old ways of predicting outcomes, it would have been named or nicknamed by someone.

Common Sense Formulas.

@SS:

Common Sense Formulas.

We should add a .com  and make this a website

@zooooma:

I actually run through a mock battle in my head where both sides get average hit.

• Total the attack strengths of the units (on one side)

• Divide by 6

This will give you the average or expected number of hits for each side.  Apply the hits, and repeat this process with the remaining units until the battle is decided.  This is a little rough, but it will tell you which side is favoured and how much you can expect that side to survive with.  This is particularly important if you are trying to calculate the result of successive allied attacks against a single defender.

Sometimes you have to round.  6 tanks + 7 Infantry have a total combat strength of 25.  This averages 4.167 hits.  For simplicity, just take the attack strength of 24 (4 hits), and add the remainder to that side’s combat strength on the next round.

You can take some short cuts.  
If the total combat strengths are close, but one side has considerably more units (hot points), the advantage goes to the larger army.
Likewise if combat strength and army size are similar the force with big and small pieces will beat the force with all average pieces.
But if you want a sense of what survives, it’s better to play the whole battle through.

It might seem like a lot of work - especially if you like drinking the beers or smoking the pretzels.  But it goes pretty fast once you get used to it.

Zoooma:

I actually run through a mock battle in my head where both sides get average hit.

@SS:

Common Sense Formulas.

What about Mock battle formula?
Pretty similar to low luck procedure.

First example, the defender may have only 3 Tanks left 9 pips,
while attacker only 4 Infs and 1 Tank for 7 pips.
First round, attacker does 1 hit, defender 1.5 hits
Second round, attacker does 1 hit, defender 1 hit.
Third round, attacker does 1 hit (4 less, rnd up), defender  .5 hit

Attacker is winning with 1 Inf and 1 Tank remaining.

But, Punch formula (brought by taamvan and ShadowHAwk) might it be better to anticipate results?
(Attacker Hit points + total attack Pips) = total Attack Strength
(Defender Hit points + total defense Pips) = total Defense Strength
Also,
Determine high Skew or even distribution: if defender will lose heavy hitters before attacker does or the other way around.
(Attacking with Infs + Arms  against Infs, defender will lose power quicker, if attacker use Inf/Art combos, both will lose power at the same pace.)

Attacker should win if he got more units and more power.
Attacker get good chance if he got more power and less units but high skew: distributed high/low where the defender is all middle or even distribution.
Attacker still get good chance if he got less power and more units and high skew: distribution is high/low against even distribution.
COW provided a few combinations which considered the Skew effect to determines if GO or No Go:
3 Infantry + 1 Artillery vs 3 Infantry allows around 70% success.
A6, 4 hits vs D6, 3 hits
2 Infantry + 1 Armor vs 3 Infantry is just below (45%) break even 50-50%,
A5, 3 hits vs D6, 3 hits
4 Infantry + 2 Armor vs 6 Infantry is just on break even 50-50% odds.
A10, 6 hits vs D12, 6 hits

And this last case allows for highest Skew distribution to add +20% on Metapower (see Stack formula below).
10/26^2= 60  compared to 26^2= 72

A more extreme case of Skew effect:
15 Infantry A15 + 4 Bombers A16, A31 19 hits is 50-50% with
(31/19)19^2=3119= 589
19 Infantry D38.
19^22= 722
23% increase in metapower.
Overall %: A. survives: 49.5% D. survives: 49.2% No one survives: 1.2%

Example:
A7 for 5 hits vs D9 for 3 hits.

7+5 =12  or 9+3 =12, so still even…?

Stack formula is faster to reveal IMO:
75 =  35  vs 93= 27

Clearly the 5 attackers units are on winning side.

Overall %*:   A. survives: 70.6%    D. survives: 24%    No one survives: 5.5%
It also says, 60% survival with at least 1 unit and 64% to conquer.

Second example, the defender may have only 3 Tanks left, while attacker only 3 Infs and 1 Tank
First round, attacker does 1 hit, defender 1.5 hits
Second round, attacker does 1 hit, defender 1 hit.
Third round, attacker does .5 hit, defender  .5 hit
Fourth round, .5 vs .5 hit.
Attacker is winning with .5 Tank remaining, by a small margin.

You can still use the 50-50 or break even in Lachenster table.
Assuming @2 needs to 11 vs 9 for @3,
Being 4 units vs 3 units, or 12:9 ratio, you are just above the ratio, but you don’t have @2 avg, a bit lower.
So, you  can say you are on this break even points.

Punch formula might be better to anticipate results.
3 Infantry+ 1 Tank vs 3 Tanks
A6 for 4 hits vs A9 for 3 hits. Telling 10 vs 9, just a bit in front of defender.

4^21.5= 24 vs 3^33= 27, Stack formula says not a totally even match.
Only skew, loosing fodder @1 first, might put balance toward attacker.
You get a better skew if Power distribution is not even.
In that case, as Cow cases showed above, you can add +20% metapower for max Skew distribution.
Hence, 24*1.2= 28.8
And this battle is another near 50%-50% odds which is slightly in favor of the attacker.
In number of units, max skew means like adding 15% more units on average or 1.15 multiplier on Lanchester table.

The mock battle seems to take longer time and more mental operations to reach the goal.

@Baron:

Lanchester’s Table for Axis and Allies 2nd Edition

I made it on AACalc then I revised numbers by applying this formula derived from above Stack formula:
√(P2 / P1) = N1 / N2

This might be another way to estimate outcomes…

@Baron:

It is my first shot working with Table feature of the Forum.
If someone can do better, I will appreciate.
With this small 6 x 6 table, you get in a glimpse what is the 50%-50% break even according to the ratio of units and the average power of a given stack.
This table may also be written in decimal instead of a ratio, below. But ratio are better to understand how each ratio is paired to another one which is simply reversed.

A more complete table would include average power: 1.5, 2.5 and 3.5
But, in F-2-F, you can always round up the average enemy’s power, so you do a safer battle to be sure you are above break even ratio.

X means the ratio is 1:1. I did not want to overcharge this table with obvious infos.

| Power
1
2
3
4
4, 2hits | 1
X
12:17
11:19
1:2
5:16

| 2
17:12
X
9:11
12:17
4:9
| 3
19:11
11:9
X
13:15
10:19
| 4
2:1
17:12
15:13
X
5:8
| 4, 2hits
16:5
9:4
19:10
8:5
X

| Power
1
2
3
4
4, 2hits | 1
X
0.70
0.58
0.50
0.30
| 2
1.42
X
0.82
0.70
0.43
| 3
1.73
1.22
X
0.87
0.53
| 4
2.00
1.42
1.15
X
0.63 | 4, 2hits
3.33
2.30
1.87
1.60
X

This table can also be memorized with the main 5 basic ratios and you can reverse at will, or 6 if we include 1:1 ratio, the @4 with 2 hits for BBs might not be very relevant in F-2-F:

1:2 (4 vs 1), 11:19 (3 vs 1), 12:17 (4 vs 2 or 2 vs 1), 9:11 (3 vs 2), 13:15 (4 vs 3)

OR same order but from reverse ratio:
2:1 (1 vs 4), 19:11 (1 vs 3), 17:12 (1 vs 2 or 2 vs 4), 11:9 (2 vs 3), 15:13 (3 vs 4)

For gameplay, you can easily replaced a 11:19 or 19:11 with 4:7 or 7:4.
And 17:12 or 12:17 with 3:2 or 2:3, in fact the real number is √2 and 1/√2. 1.5 vs 1.42 and 0.7 vs 0.75.
The ratio is not too different, just less accurate.
But in game, it is easier to calculate it with 1 digit number.

| Power
1
2
3
4
4, 2hits | 1
X
2:3
4:7
1:2
3:10

| 2
3:2
X
9:11
2:3
4:9
| 3
7:4
11:9
X
13:15
10:19
| 4
2:1
3:2
15:13
X
5:8
| 4, 2hits
10:3
9:4
19:10
8:5
X

| Avg Power
1
1.5
2
2.5
3
3.5
4
4, 2hits | 1
1.00
0.82
0.70
0.63
0.58
0.53
0.50
0.30
| 1.5
1.22
1.00
0.87
0.77
0.70
0.65
0.63
0.38
| 2
1.41
1.15
1.00
0.89
0.82
0.76
0.70
0.43
| 2.5
1.58
1.29
1.12
1.00
0.91
0.85
0.79
0.50
| 3
1.73
1.41
1.22
1.10
1.00
0.93
0.87
0.53
| 3.5
1.87
1.53
1.32
1.18
1.08
1.00
0.94
0.58
| 4
2.00
1.63
1.41
1.26
1.15
1.07
1.00
0.63
| 4, 2hits
3.33
2.64
2.30
2.00
1.87
1.73
1.60
1.00

| Avg Power
1
1.5
2
2.5
3
3.5
4
4, 2hits | 1
1:1
9:11
12:17
5:8
4:7
10:19
1:2
3:10

|
1.5
11:9
1:1
13:15
7:9
12:17
9:14
5:8
3:8
| 2
17:12
15:13
1:1
9:10
9:11
3:4
12:17
4:9
| 2.5
8:5
9:7
10:9
1:1
10:11
5:6
4:5
1:2
| 3
7:4
17:12
11:9
11:10
1:1
19:20
13:15
10:19
| 3.5
19:10
14:9
4:3
6:5
20:19
1:1
20:21
4:7
| 4
2:1
8:5
17:12
5:4
15:13
21:20
1:1
5:8
| 4, 2hits
10:3
8:3
9:4
2:1
19:10
7:4
8:5
1:1
|

|

|

|

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Guys…  you’re killing me here.

Why would you stick with an arbitrary formula determined prior to the battle start?  and not quickly re-asses probabilities after each round?

The multiple rounds and option to retreat give the attacker a number of calculatory advantages and opportunities - use them!

@Gargantua:

Guys…  you’re killing me here.

Why would you stick with an arbitrary formula determined prior to the battle start?  and not quickly re-asses probabilities after each round?

The multiple rounds and option to retreat give the attacker a number of calculatory advantages and opportunities - use them!

This is exactly the idea here, how being able to re-assess quickly probabilities to decide on whether or not retreat or push his luck or not?

In all cases, you have to sum up Pips and Hits on both side.

Then you may have more than 1 path to find if the odds or on your side or not.
And to choose, if you push your luck or if you go conservative.

IMO, it is because you use the Mock combat formula for a long time that you are trained to it and probably very fast with it.

But a competitive beginner may decide which method he want to learn to get the best out of the habit he will get from one method or the other.

You seems to already have made up your mind about what is better. But it can be a very subjective POV or that matter may revealed to be only a matter of taste, IDK.

It is the first time that all 4 methods (Stack, Punch, Mock Battle or Lanchester Table) are in the same thread.
I’m genuinely inquisitive about pros and cons of each methods.
There can be many criterias to judge them:

• accuracy, (Stack formula being more accurate than Punch formula)

• learning curve, (Punch being easier to learn than Stack)

• numbers and difficulty of mental operations required (Punch additions being easier to process than Stack multiplications),

• time to get an answer, (Punch and Stack being straight forward while Mock Battle formula seems to take longer)

• infos you get out of it (odds, number of units remaining, etc. : Mock Battle provides an idea of units remaining on winning side).

It is also the first time, I get a synthesis of all even odds in a single 5x5 or 6x6 table (named Lanchester Table).
No one has never done it before.  
During the years, I used a few ratios 9:11 or 11:9, mostly about 3 vs 2 or 2 vs 3. That I discovered tinkering about Mech Artillery HR unit, and 5 IPCs or 6 IPCs Tank. Because I was trying to find a balanced cost for combat values, hence playing with 50-50% odds for various cost and power.
IDK what can be done with this Lanchester Table. Is there something to do F-2F in game knowing these relationships?

And for the 5th one, one side of Vann formula and table, I came to the conclusion that it does not have enough relevance on this matter of assessing odds. So, it is discarded and kept for HouseRuled units and customization. In which it gets its own specific purpose.

It’s all about sharpening tactical combat skills on this matter. Sometimes, a retreat is better, sometimes going on 1 more round, even with below avg odds, is the thing to do. Reducing the enemy’s ground fodders before retreating sometimes is enough to force him to wait for ground reinforcement before launching an attack. And the stack formula clearly showed that number of units are a more important factor than power factor.

Experience also provides a lot of intuitive assessment without requiring any mental calculations too. You literally “feel” which stack is ahead in a given battle.

Here is my method for estimations. The minimum to attack is 3 inf 1 arty for every 3 inf defending (for 70%+ greater odds). 2 inf 1 armor vs 3 inf is roughly 47% odds 4 inf 2 armor vs 6 inf is 50/50 roughly. So the more cannon fodder and big hitters the better the odds. So you can eyeball math this way really fast.

I mean let us face it, there comes a time when you have to attack Russia regardless of the odds, because time is not on your side. If time is on your side and Japan is massive, Germany is massive, push south and make the money, strangle russia out of resources and keep increasing those odds. Very simple. So estimating outcomes like this rarely comes into play in global.

This is more of a AA50 and 42 variants type of thing to do.

You can pretend it is low luck, add up your attack power (6 inf = 1 hit 2 tank = 1 hit etc) and add up his defense power, then from there simply do the battle low luck style to see who wins rounding up on hits (1remainder is a hit for this assumption). Then you can see just how close or far ahead / behind you are. Because if you win with low luck odds you are looking at 55% or better odds of winning… which means GO GO GO.

There have been a few dismissive remarks on this thread, and I recall a few more on previous threads that I participated in. Regardless of how well you are able to use them in your games, I say that the ideas on this thread are worthwhile for the simple reasons that they shed new light on mathematical principles that underlie this game and that those mathematical principles are the exact same principles that underlie actual modern warfare.

It may well be that none of this math works as well as just dividing total punch by 6 and running through the rounds in your head. On the other hand, it is possible that going down this path will lead to insights that allow players to assess the situation even more quickly than that. Maybe the insights will not lead to greater speed, but will help players craft their overall strategies better.

I understand that many players (perhaps a majority) will not be swayed by this argument, and don’t want to be troubled with formulas and calculations. Can we be respectful of both sides and create a new section on this forum dedicated to exploring these principles further? Does anyone else think that’s a good idea?

Added benefit: if we had a board devoted to the mathematics of Axis and Allies, then the four threads about the VANN FORMULAS would not currently be clogging up the player help board.

I don’t think this hypothetical Sub-forum will be that popular to justify the need.
(A dedicated sub-forum for HRs on 1914 might be much more useful and visited.)
IDK  if heuristic thread has been intended for Player’s help forum.
However, once sound results are achieved, it can be worthwhile to make a specific thread to explain things in the best way possible.

As long as we stay outside HR discussion in this thread, G40 forum is a very popular places for hard core players with lot of experience.
Many can read and bring their 2 cents.
For instance, Cow post was an interesting rule of thumb which point straight at skew impact and how non-homogeneous are not included in Stack formula.  And may create misleading results as Cow showed with his 2 examples.

@larrymarx:

There have been a few dismissive remarks on this thread, and I recall a few more on previous threads that I participated in. Regardless of how well you are able to use them in your games, I say that the ideas on this thread are worthwhile for the simple reasons that they shed new light on mathematical principles that underlie this game and that those mathematical principles are the exact same principles that underlie actual modern warfare.

It may well be that none of this math works as well as just dividing total punch by 6 and running through the rounds in your head. On the other hand, it is possible that going down this path will lead to insights that allow players to assess the situation even more quickly than that. Maybe the insights will not lead to greater speed, but will help players craft their overall strategies better.

I understand that many players (perhaps a majority) will not be swayed by this argument, and don’t want to be troubled with formulas and calculations. Can we be respectful of both sides and create a new section on this forum dedicated to exploring these principles further? Does anyone else think that’s a good idea?

Added benefit: if we had a board devoted to the mathematics of Axis and Allies, then the four threads about the VANN FORMULAS would not currently be clogging up the player help board.

Thanks for posting Larry; great peacemaking comments.

Now lets dispense with the pleasantries and get down to business! Having respect for lunacy is never a good idea; Just look at the president (who wildly entertains us all daily).

Then look at some of these proposed calculation systems…  equally ridiculous.

The Cow Method (aka the common sense or low luck method) is the best, fastest, and most versatile method.  It is exactly what I discussed above; and will not fail you or anyone.

And if you use the Cow method, you will ultimately get Cow Dice - which is a legitimate and real phenomenon on these boards (ask around).

As for shedding “new light” on the game from this “new religion” of “new formulas” which seem to appear every “formula wednesday”;  I was INSPIRED by all the nut job calculations to create a new house rule.

Assumption:  Seeing how much some people are enjoying all the math, square roots and derivatives; I have created a fun new house rule for those people to use; one that they can continue to lose their minds over.  These insane calculations reminded me of an insurance conference I once attended… and voila…

BATTLE INSURANCE has been born!

For 4 IPC’s each turn you can buy reroll tokens.  One token can be placed in each battle you fight; and can be used for a straight reroll of the entire battle.  If the reroll is not used the token is LOST.  Tokens can also be placed for the “defence” of a territory and are eliminated at the next purchase units phase.  Because whats more mathematically fun than INSURANCE?!?!

Now that we’ve arrived at this conclusion- let’s move this preposterous thread to the house rules forum where it will die a slow and painful death. PLEASE.

Funny Gargantua,
you’re line of thinking is analogous to people saying :
why are you looking for a different painkiller pill, my Aspirin does the job?

Such way of thinking would not have produce: Tylenol, Advil and Motrin.
All 4 painkillers provides different ways to interact with the same issue and with various benefits and side effects. Physicians are now really glad to have these options to help patients.

IMO, at the theoretical research level of A&A combat mechanism, we cannot tell what can be all the practical  outcomes of a given mathematical relationship found out.

@Baron:

Funny Gargantua,
you’re line of thinking is analogous to people saying :
why are you looking for a different painkiller pill, my Aspirin does the job?

Such way of thinking would not have produce: Tylenol, Advil and Motrin.
All 4 painkillers provides different ways to interact with the same issue and with various benefits and side effects. Physicians are now really glad to have these options to help patients.

IMO, at the theoretical research level of A&A combat mechanism, we cannot tell what can be all the practical  outcomes of a given mathematical relationship found out.

Lanchester’s Tables for Axis and Allies 2nd Edition

I made it on AACalc then I revised numbers by applying this formula derived from above Stack formula:
√(P2 / P1) = N1 / N2

| Avg Power
1
1.5
2
2.5
3
3.5
4
4, 2hits | 1
1.00
0.82
0.70
0.63
0.58
0.53
0.50
0.31
| 1.5
1.22
1.00
0.87
0.77
0.70
0.65
0.62
0.38
| 2
1.41
1.15
1.00
0.89
0.82
0.76
0.70
0.43
| 2.5
1.58
1.29
1.12
1.00
0.91
0.85
0.79
0.50
| 3
1.73
1.41
1.22
1.10
1.00
0.93
0.87
0.53
| 3.5
1.87
1.53
1.32
1.18
1.08
1.00
0.94
0.58
| 4
2.00
1.62
1.41
1.26
1.15
1.07
1.00
0.62
| 4, 2hits
3.33
2.64
2.30
2.00
1.87
1.73
1.62
1.00

| Avg Power
1
1.5
2
2.5
3
3.5
4
4, 2hits | 1
1:1
9:11
12:17
5:8
4:7
9:17
1:2
3:10

|
1.5
11:9
1:1
13:15
7:9
12:17
9:14
5:8
3:8
| 2
17:12
15:13
1:1
9:10
9:11
3:4
12:17
3:7
| 2.5
8:5
9:7
10:9
1:1
10:11
5:6
4:5
1:2
| 3
7:4
17:12
11:9
11:10
1:1
19:20
13:15
9:17
| 3.5
17:9
14:9
4:3
6:5
20:19
1:1
20:21
4:7
| 4
2:1
8:5
17:12
5:4
15:13
21:20
1:1
5:8
| 4, 2hits
10:3
8:3
7:3
2:1
17:9
7:4
8:5
1:1

@Arthur:

You would need 8.5 subs to defend against 6 destroyers and have a 50% chance of surviving (6*1.41 ~ 8.5).  Hence the sub build would be slightly inferior if you only looked at defense, but obviously 8 subs is better than 6 destroyers on offense.

In real situations when you have a mixed fleet with a range of ships, the extra subs will provide equal benefit as the smaller number of destroyers.  On offense they are obviously way better because they can absorb extra hits and do extra damage.  Japan needs more destroyers since they have to hunt down convoy-raiding Allied subs.  The Allies can primarily focus on subs with just enough destroyers for ship blocking purposes.

For instance, the 1.41, which is a 17 to 12 ratio, produced from out of nowhere by Arthur Bomber Harris, is just a special case of @1 vs @2 combat. And this ratio for a 50% chance of survival is the same when @2 fight @4 units or @1.5 avg stack power vs @3 units. The Lanchester tables above give all the basic cases you can encounter in A&A.

Even more, with 1.5 line you can consider this as the Sub @1 first strike value.
The exact number would be 1.33, but for practical purpose, 1.5 is a good approximate.

On the other hand, this cannot provide the number of surviving units. Hence, mental Mock battle give both.
zooooma provides a nuanced judgement about it:

@zooooma:

I actually run through a mock battle in my head where both sides get average hit.

• Total the attack strengths of the units (on one side)

• Divide by 6

This will give you the average or expected number of hits for each side.  Apply the hits, and repeat this process with the remaining units until the battle is decided.  This is a little rough, but it will tell you which side is favoured and how much you can expect that side to survive with.  This is particularly important if you are trying to calculate the result of successive allied attacks against a single defender.

Sometimes you have to round.  6 tanks + 7 Infantry have a total combat strength of 25.  This averages 4.167 hits.  For simplicity, just take the attack strength of 24 (4 hits), and add the remainder to that side’s combat strength on the next round.

It might seem like a lot of work - especially if you like drinking the beers or smoking the pretzels.  But it goes pretty fast once you get used to it.

One possibility when you have a fractional number of expected hits is to round down for your units and to round up for the opposing units, to give a more conservative estimate of whether you can win the battle with a certain number of units left.

|

|

@taamvan:

Anything that uses “formulas” other than

HP vs HP
ATT POWER vs DEFENSE POWER
6 / 36 / 108 style statistics for d6

is too complex to be used during any game, or too abstract to counsel any action.No battle calcs are allowed at the tourney, which is fine, because they are totally unnecessary.    Counting and addition are all that is necessary.

Do you think printed Lanchester’s tables on paper would be prohibited?
It may appear like any other player’s aid provided in OOB.