Method for Estimating the Outcomes of Large Battles

@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.

After reading a few times CalvinandHobbesliker PDF and others, I wonder if the Stack formula, instead of being N units^2 would be more accurate like these:

MetaPower= (number of Hits) (number of Units) (Power of stack: sum all individuals to hit number)/(number of Units)**
this can be simply put like this:
MetaPower= (number of Hits) (number of Units) (average Power per unit)**

This is the basic formula to compare Metapower between two stacks:

MetaPower= (number of Hits) (Punch: sum all individuals to hit number)*

If I’m correct, the only difference between Punch and stack formula is whether an addition or a multiplier.

Punch:
1 Tank A3 + 4 Infantry A4 provide A7 + 5 hits = 12 Punch points

Stack: A7 * 5 hits = 35 Metapower

Compared to 4 Infantry D2
Punch: D8 + 4 hits = 12 Punch points

Stack: D8 * 4 hits = 32 Metapower

Overall %*: A. survives: 63.2% D. survives: 32.8% No one survives: 4.1%

It appears that multiplier is more accurate to determine the outcome.

Trying to apply in a different way the 6 pips division of Mock battle formula (or Very Low Luck mechanism)
If you take 35/6 = 5 5/6 and 32/6 = 5 2/6,
5 5/6 - 5 2/6= 3/6
Would that means that attacker will win with his A3 Tank most of the time?

Another previous example:

First version of stack formula:
For instance, comparing these 2 fleets on defense:
2 CV + 4 ftr, + 2 DDs + 10 subs
(16 hits + 2*1.618034)^2 * 34 pips/18 units = 698.9

2 CV + 4 ftr, + 8 DDs + 2 subs
(14 hits + 2*1.618034)^2 * 38 pips/16 units = 705.6

New formula, (easier to calculate than above):

2 CVs D4, 4 hits + 4 ftr D16 + 2 DDs D4 + 10 subs D10
20 hits * 34 pips = 680 Metapower

2 CVs D4, 4 hits + 4 ftr D16 + 8 DDs D16 + 2 subs D2
18 hits * 38 pips  = 684 Metapower

If divided by 6,
680/6= 113 2/6
684/6= 114
The only difference is 4/6

Does this would imply that on average the Destroyers fleet is stronger by a single Fighter D4?

Third test:
1 Inf A1 + 9 Tanks A27 attacking 11 Infantry D22.

A28* 10 hits = 280 metapower

D22*11 hits = 242 metapower

280/6= 46 4/6 compared to 242/6= 40 2/6

Difference: 6 2/6 What happen?

Average AACalc:
Overall %*: A. survives: 68.1% D. survives: 30.5% No one survives: 1.4%
Avg 3 Tanks survived.

Does it imply that √(6 2/6)= 2.6 revealed the average number of units surviving?

Third test:
9 Tanks A27 attacking 11 Infantry D22.

A27* 9 hits = 243 metapower

D22*11 hits = 242 metapower

243/6= 40 3/6 compared to 242/6= 40 2/6
Difference:
Even combat.

√1/6 = 0.41

Any idea?

My first post about axis in more than 15 years lol so ill tell my A&A story a little bit.

Ive started playing A&A IRL in 1991 with players that already knew the game for a couple of years. I failed at least 2 trimesters (here we have 2 years college between high school and university) because of my A&A addiction  :wink:

I played a few hundreds game on Hasbro the few first years it came out and had a relapse a couple years later when i heard there was a patch and now im having another relapse because of triplea . I remember using a site that were using omaha,utah and kremlin database (never heard of sword, or anzio until this year) No tech I had a 90% win rate and with 75% tech while mostly playing vs players with at least a 50% win rate. In those time i knew only 2 other players with similar records one was IIRC Robert Brink or something close to it I remember him because this guy wrote good strategy articles and played more games than me (wich was quite rare), I remember many were laughing about Don “essays” and we had no respect for the PBEM crowd. I did play some pbem but i tought the players were much weaker, maybe there was some really good PBEM players but usually those who play more get better faster and it was just easier to play more game with Hasblow than via pbem so AFAIK at that time IMO the strongest players were playing hasblow not pbem.  IIRC almost all the games played on our database used the 3 min timers so you need to know what you were doing.

Anyway he is the classic expert wisdom I used in the early 90’s and computer era.

lets say you need to kill 69 basic units (or close to 60+ 4 fighter +1/2 bomber) a 20% lead in $$ could lead you to a 45% and a 25% lead in cash$ was a 60% battle.

The key is that you really want to avoid losing tanks in the first 2 rounds and have no too much remaining inf alive after 2 rounds all thing being equal its better to have too much inf than losing tanks in r2. (with econ power tech its 3 rounds, you dont want to lose tanks in the first 3 rounds but want as little inf as possible alive after 3 rounds)

So you expect opp to make 23 hits for round one and usually you do slightly less hit than your opponents. Lets say you expect to make 21 hits so that we get another round number. opp expected power for the 2nd round is 48 “infs” for 16 hit . So right from the start you need at least 21+16= 37 infs, its quite bad to have less than 37 infs since it mean that you are favorite to lose tanks in the 2nd round wich isnt cost effective. Its costlier to lose tanks on the 2nd round that to have inf remaining on the 3rd so for safety the optimum is 38-40 infs.

Now the $$ part is that you need 20% more in $$ for a 45% battle and 25% more $$  for a 60% battle than your ennemy stack (worth in inf$).

69 infs = 207 X 1.20 = 248 So lets test it.

41 infs,+25 tanks vs 69i = 40%  (we have exactly 248$ but we have too much inf)

38i+27t vs 69i = 42% (we have 249$ bu we dont have enough inf and losing tank in the 2nd round is painful)

39+26 =38%  at 247 we are under 1.20 and its painful.

So it look like 20% is not quite enough for 45% but its close. If instead of a pure 69 infs there is some fighters we might do better.

Now 207 X 1.25 = 258.75

39inf+28 tanks (257$)= 59%
41+27 = 61%  (258$) = 62%
38i+29t (259$) = 63%

So it look like the old adage of +25% is doing well for the 60%.

With art its tricky, if im looking at offensive power i felt the same principles of wanting your art to be “effective” in the first 2 rounds but not have any inf left for the 3rd round woudl be good but i dont see that in the results. I initially tought art would render tank obsolete but many in revised use tanks rather than art wich i tought was weird. (i only play a couple of revised game and i think the map is just super lame, but art with classic maps will be interesting)

we want to kill 99 inf and we have 351 budget.

65+39+0 = 60% (too many inf left alive after r2 ?)

61+42+0 = 61% (better but some art arent efficient)

59+41+2 = 61% (less inf mean less art are efficient on r2)

59+36+6 = 351= 61%

59+30+10 = 351=61%

59+26+14=351 = 60% we make 29.833 hits but after suffering 33 hit(r1) and 23(r2) we will have 3 inf alive.

55+29+14=351 = 60% we make 30.66 hits but after suffering 33 hits, 7 of our art will not work full time in the 2nd round.

57+30+12 =351 = 61%

58+28+13 = 61% (i tought this would be the sweet spot, not too many inv alive after r2, all the art seems to be working rest is tank)

58+23+17 = 58% (i tought this would be the sweet spot, not too many inv alive after r2, all the art seems to be working rest is tank)

57+25+16 = 351 = 58%

Not easy to draw conclusion from those.

A few observations:

In response to benlessard:

1. In your simulations, you are valuing tanks at 5. Most of the players on this thread work with a cost of 6.

2. In your list of battles vs. 99 infantry, wouldn’t the 60% battle with the most infantry be optimum considering that you want as strong a defense as possible for your stack after you take the territory in question? If the battle will win the game, then obviously this isn’t a consideration. However, most battles don’t win the game and so for general strategy, shouldn’t weigh the stack towards infantry?

You’ve obviously read Don’s essay on the infantry push mechanic. That’s essentially the idea I’m repeating here. In modern A&A, mobility is a lot more important and that is certainly a factor as well, but it’s a complicating factor and our discussion about these calculations hasn’t reached that scope yet. So my question is do you agree with the infantry push mechanic and weighing the stack towards infantry?

In response to Baron Munchhausen:

Your point is taken with regard to having the thread in a place where everyone can see it. The problem is that all of these threads are just going to get moved to player help! I don’t think that’s really the best place for them, do you?

I was pragmatic, assuming creating a specific forum or sub-forum is like selling popsicle to an Inuit (Eskimo? ).
As moderator seems willing to move similar threads strictly about calculations and values of units in Player Help, I get into the bus while it is moving.

I hope it may develop a new habit for some player to look in this Player Help forum.

Compartmentalizing properly is hard for many threads.
A lot start about a specific game roster then compare to others or House ruled units.

I knew tank went to 3-3 in revised (wich made some sense since they were a little weak in classic) but i still tought they were somewhat inferior to art wich is somewhat confirmed with the calculators, but at 6 cost they just make no sense. Add to that that in revised you can get 2 art per transport but only one tank and they are lol purchase for UK/jap/USA.

I read that in one version if you attack with a tank if they are matched to an inf they cancel an ennemy hit is that in this version that they cost 6 ?

@larrymarx:

A few observations:

You’ve obviously read Don’s essay on the infantry push mechanic. That’s essentially the idea I’m repeating here. In modern A&A, mobility is a lot more important and that is certainly a factor as well, but it’s a complicating factor and our discussion about these calculations hasn’t reached that scope yet. So my question is do you agree with the infantry push mechanic and weighing the stack towards infantry?

I’m just catching up the impact of skew (power distribution) when adding a few fodders into an existing stack with higher hitters.

In itself, cheaper units are usually stronger.
But if you don’t look at cost, only formula, skew increase odds of winning for a similar Punch.
However, starting from scratch, higher hitter are always less cost effective.

Subs D1 C6  compared to DDs D2 C8 on defense creates a few aberrations due to low cost of Sub.
DD was meant to be fleet fodder but the game mechanics still favor Subs for fleet stack.
The Subs as fodder and the air, DD, Sub triangle was one of my favorite issues to try to solve.

@benlessard:

I knew tank went to 3-3 in revised (wich made some sense since they were a little weak in classic) but i still tought they were somewhat inferior to art wich is somewhat confirmed with the calculators, but at 6 cost they just make no sense. Add to that that in revised you can get 2 art per transport but only one tank and they are lol purchase for UK/���/USA.

I read that in one version if you attack with a tank if they are matched to an inf they cancel an ennemy hit is that in this version that they cost 6 ?

2 hits Tank is 1914 OOB.
C6 Tank A3 D3 M2, are still useful because of mobility and blitz.
It also gives +1A to Tactical Bomber A3-4 D3 M4 C11.

To be of same strength, Artillery A2 D2 M1 should cost 4.5 IPCs
while Tank A3 D3 M2 should be 5.5 IPCs.
(See above Lanchester table cross-referenced 2 and 3 : 9:11 or .82, 11:9 or 1.22)
If A2 worth 4$, then A3 worth 41.22 = 4.88$, If A3 worth 5$, A2 should cost 5.82= 4.10$)

So, from developer POV, 5 is too strong and 6 a bit too much.
Hence, giving a few special niche abilities.

What happened with Tank A3 D3 C5 M2 vs Art A2 D2 C4 M1?
Same combat strength, except for Infantry boost.
So, you buy plenty of Tank because you also get mobility.

With C6 Tank, you are in a dilemma: attack power of cheaper Art vs mobility of costlier Tank.

On Global map, mobility is important.

Mechanized Infantry A1-2 D2 M2 C4 is great with Tank because of mobility and defense factor.
Same as Artillery but M2.
It can blitz if paired 1:1 with Tank.
And it can catch up early moved Artillery for a better impact than Tank per cost ratio.

Germany cannot win G40 without MIs.

I understand that tank can create dead zone on both side, and that for the final blow you sometimes will buy tanks only but in general in don’t think they are bread and butter units.

For example lets say that i need a stack of units capable of attacking and winning against a stack costing 99$ but my stack need to be able to resist an attack vs a stack costing 140$. I dont think there will be a lot of tanks (any ?) in my stack.

Be more explicit, please.
Where come from this 140$?

We are in Player Help, and when I returned to older posts of my own and not focused (with same state of mind), the more details, the better understanding and catching up.

Against 110 Inf, 330 IPCs, you need 90 Tanks, 540 IPCs.
For 33 Infs, 99 IPCs, you need 27 Tanks, 162 IPCs.

This is 1,64 times costlier in armor.

But if you need 34 units attacking at 2, this cost Inf C3+ Art C4, 7 IPCs for 2 units.
17 pairs * 7 IPCs = 119 IPCs.
This is 1,20 times costlier with Artillery Infantry combo.
More exactly, 16*7+ 4 = 116 IPCs
1,17 times costlier to be even with 33 Infantry.

@benlessard:

I understand that tank can create dead zone on both side, and that for the final blow you sometimes will buy tanks only but in general in don’t think they are bread and butter units.

For example lets say that i need a stack of units capable of attacking and winning against a stack costing 99$ but my stack need to be able to resist an attack vs a stack costing 140$. I dont think there will be a lot of tanks (any ?) in my stack.

Another interesting thing to notice about Tank at 6 IPCs and Mechanized Infantry at 4 IPCs is the Skew effect:
Suppose 24 IPCs of units.
8 Infantry   A8 D16 M1,  Metapower: 88= 64  / 168 = 128
6 Artillery   A12 D12 M1, Metapower: 126= 72 / 126=72

6 Mech Inf  A6 D12 M2,   Metapower: 66= 36 /  126= 72
3 MI+ 2 Tk A9 D12 M2,   Metapower: 95= 45 / 125 = 60
4 Tanks      A12 D12 M2, Metapower: 124= 48 / 124= 48

If looking at offense for M2 units, 4 Tanks (48) seems better than 6 MIs (36) or 3MIs+2Tks (45)

But AACalc reveals it is true compared to 6 MIs vs 4 Tanks,
A. survives: 29.6% D. survives: 68.4% No one survives: 2%

But 3MIs+2Tks vs 4 Tanks are better:
A. survives: 52.6% D. survives: 41.3% No one survives: 6.1%

So, mixing M2 units is optimal for mobility and firepower.

Can 3MIs+2Tks (45) vs 6 Artillerys (72) on Offense be better? Nope.
A. survives: 26.7% D. survives: 70.7% No one survives: 2.6%

But 3MIs+2Tks (45) compared to 8 Infantry (64) on Offense is not better either.
A. survives: 38.6% D. survives: 59.6% No one survives: 1.8%

So, still in that case, high Skew seems to be add around 20% to the basic Metapower,
45*1.20= 54 is better than 48 metapower.

In fact, this 3MIs+2Tks (54?) slightly beats 5 Infantry D2 (D10*5= 50 Metapower).
A. survives: 52.3% D. survives: 43.7% No one survives: 4%

And is superior to a 54 Metapower (3Inf D6+3 Bmb D3) in which the D2 is taken as first casualty.
A. survives: 59.1% D. survives: 39.1% No one survives: 1.8%

But inferior to a 54 Metapower with moderate skew (3 Infs D6+3 Bmb D3) in which the D1 is taken as first casualty.
A. survives: 39.5% D. survives: 57.4% No one survives: 3.2%

Now about defense, does 3 MI+ 2 Tk D12 (60) can beat 6 Art or MIs D12, (72)?
Assuming a 10% bonus for moderate Skew, this would place Metapower 60 similar to 66.

AACalc reveals that:
A. survives: 40% D. survives: 56.4% No one survives: 3.7%

But what about 3 MI+ 2 Tk D12 (60) against 8 bombers in Defense (D8*8= 64)?
A. survives: 51% D. survives: 47.4% No one survives: 1.6%

So, based on this case, 10% seems right bonus for a moderate Skew.

Let’s suppose 36 IPCs of units.
12 Infantry    A12 D24 M1,  Metapower:  1212= 144  / 2412 = 288
6 Arts+4 Infs A20 D20 M1,  Metapower:  2010= 200 / 2010= 200
9 Artillery      A18 D18 M1,   Metapower:   189= 162 /  189= 162

9 Mech Infs  A9 D18 M2,   Metapower:   99= 81 /  189= 162
6 MI+ 2 Tk A12 D24 M2,   Metapower: 128= 96 / 248 = 192
3 MI+ 4 Tk A15 D21 M2,   Metapower: 157= 105 / 217 = 147
6 Tanks      A18 D18 M2,   Metapower: 186= 108 /186= 108

Does 6 MI+ 2 Tk A12 D24 M2 (96) / (192)
can beat 6 Tanks A18 D18 M2  (108) / (108) ?

A. survives: 50.9% D. survives: 45.4% No one survives: 3.7%

It is the case, then 96 1.20=  115.2 ?
961.15= 110.4 ?
Probably 115 because against 11 bombers D1 (11*11)= 121
AACalc: A. survives: 46.5% D. survives: 52.5% No one survives: 1.1%

3 MI+ 4 Tk A15 D21 M2,   Metapower: 157= 105 / 217 = 147
beats both, here it is against 6 Tanks.
A. survives: 56.2% D. survives: 39.6% No one survives: 4.2%
So, what can be the modifier for moderate skew?
1051.20= 126 ?
1051.15= 120.75 ?
105*1.10=115.5 ?

Probably 126 because against 11 bombers D1 (11*11)= 121
A. survives: 52.2% D. survives: 46.7% No one survives: 1.1%

IMO, a moderate skew like 3 vs 1 fodder allows for 20% bonus on Metapower from Stack formula.

No Distribution (all attacking/defending at same number): no bonus
Slight Distribution (equal 1s and 2s, or equal 2s and 3s): 5% bonus
Equal Distribution (equal 1s and 3s, equal 1s, 2s, 3s, and 4s): 10% bonus
Fodder Distribution (equal 1s and 4s): 15% bonus

So, I’m not sure about OP Skew bonus, the fodder distribution seems too low.
At least all numbers need +5 or +10%:
No Distribution (all attacking/defending at same number): no bonus
Slight Skew Distribution (equal 1s and 2s, or equal 2s and 3s): 10% bonus
Equal Skew Distribution (equal 1s and 3s, equal 1s, 2s, 3s, and 4s): 20% bonus
Fodder Skew Distribution (equal 1s and 4s): 25% bonus

By the way, the best optimized M2 combos for offense and defense is:
6 MI+ 2 Tk A12 D24 M2,   Metapower: 128= 96 / 248 = 192
Giving near (+20%) 115 offense and (+10%) 211 defense Metapower

Baron for fun I challenge you to find a 360$ stack that will do well vs in attack vs 100inf (300$) and do reasonnably well in defense vs a 420$ stack. Use tanks at cost 5$ if we can show that tanks are not super useful at cost 5 we can deduce that they are poor at cost 6 (wich imo is obvious for this specific task).

So you need to create a purely offensive 420$ stack and a two way stack (middle stack) and I will do the same.

We add our win ratio when our middle stack kill the 100i (ex 57%) and when our middle stack defeat the
the opponent 420$ stack. (ex 420 vs middle win 56% so 44%+57% = 101)

@benlessard:

Baron for fun I challenge you to find a 360$ stack that will do well vs in attack vs 100inf (300$) and do reasonnably well in defense vs a 420$ stack. Use tanks at cost 5$ if we can show that tanks are not super useful at cost 5 we can deduce that they are poor at cost 6 (wich imo is obvious for this specific task).

So you need to create a purely offensive 420$ stack and a two way stack (middle stack) and I will do the same.

We add our win ratio when our middle stack kill the 100i (ex 57%) and when our middle stack defeat the
the opponent 420$ stack. (ex 420 vs middle win 56% so 44%+57% = 101)

I’m not sure to understand…
Does 60 Infs 180 IPCs
30 Artillery    120 IPCs
12 Tanks C5   60 IPCs

Fit the bills?
A. 70% vs D. 30%

Does 60 Infs 180 IPCs
30 Artillery    120 IPCs
10 Tanks C6   60 IPCs

Fit the bills?
A. 53% vs D. 46%
But lose 99.8 vs 0.1% against 60 Infs +60 Arts.

This last example is also a way to show how skew get an important impact on outcomes.

Does 60 Infs 180 IPCs
45 Artillery    180 IPCs

Fit the bills too?
A. 70% vs D. 30%
But lose 98.7 vs 1.3% against 60 Infs +60 Arts.

The quote below also showed that nothing can beat an Artillery+Infantry combos.
You suggested a 360 IPCs, just multiply by 10 numbers below.

10 Infantry    A10 D20 M1,  Metapower:  1010= 100  / 2010 = 200

@Baron:

Let’s suppose 36 IPCs of units.
12 Infantry    A12 D24 M1,  Metapower:  1212= 144  / 2412 = 288
6 Arts+4 Infs A20 D20 M1,  Metapower:  2010= 200 / 2010= 200
9 Artillery      A18 D18 M1,  Metapower:  189= 162 /  189= 162

9 Mech Infs  A9 D18 M2,  Metapower:  99= 81 /  189= 162
6 MI+ 2 Tk A12 D24 M2,  Metapower: 128= 96 / 248 = 192
3 MI+ 4 Tk A15 D21 M2,  Metapower: 157= 105 / 217 = 147
6 Tanks      A18 D18 M2,  Metapower: 186= 108 /186= 108

Does 6 MI+ 2 Tk A12 D24 M2 (96) / (192)
can beat 6 Tanks A18 D18 M2  (108) / (108) ?

A. survives: 50.9% D. survives: 45.4% No one survives: 3.7%

It is the case, then 96 1.20=  115.2 ?
961.15= 110.4 ?
Probably 115 because against 11 bombers D1 (11*11)= 121
AACalc: A. survives: 46.5% D. survives: 52.5% No one survives: 1.1%

3 MI+ 4 Tk A15 D21 M2,  Metapower: 157= 105 / 217 = 147
beats both, here it is against 6 Tanks.
A. survives: 56.2% D. survives: 39.6% No one survives: 4.2%
So, what can be the modifier for moderate skew?
1051.20= 126 ?
1051.15= 120.75 ?
105*1.10=115.5 ?

Probably 126 because against 11 bombers D1 (11*11)= 121
A. survives: 52.2% D. survives: 46.7% No one survives: 1.1%

IMO, a moderate skew like 3 vs 1 fodder allows for 20% bonus on Metapower from Stack formula.

No Distribution (all attacking/defending at same number): no bonus
Slight Distribution (equal 1s and 2s, or equal 2s and 3s): 5% bonus
Equal Distribution (equal 1s and 3s, equal 1s, 2s, 3s, and 4s): 10% bonus
Fodder Distribution (equal 1s and 4s): 15% bonus

So, I’m not sure about OP Skew bonus, the fodder distribution seems too low.
At least all numbers need +5 or +10%:
No Distribution (all attacking/defending at same number): no bonus
Slight Skew Distribution (equal 1s and 2s, or equal 2s and 3s): 10% bonus
Equal Skew Distribution (equal 1s and 3s, equal 1s, 2s, 3s, and 4s): 20% bonus
Fodder Skew Distribution (equal 1s and 4s): 25% bonus

By the way, the best optimized M2 combos for offense and defense is:
6 MI+ 2 Tk A12 D24 M2,  Metapower: 128= 96 / 248 = 192
Giving near (+20%) 115 offense and (+10%) 211 defense Metapower

No, lets just say that you can get slightly more than 70% when attacking, more than 1% when your defending and more than 71% when you add the two together.

Do you want to use one attacking unit must live on or off ?

@benlessard:

No, lets just say that you can get slightly more than 70% when attacking, more than 1% when your defending and more than 71% when you add the two together.

Do you want to use one attacking unit must live on or off ?

Does 50 Infs 150 IPCs
15 Artillery     60 IPCs
30 Tanks C5   150 IPCs

Fit the bills?
A. 58% vs D. 42%
But lose A. survives: 98.3% D. survives: 1.7% No one survives: 0%
against 60 Infs +60 Arts.

Does 60 Infs 180 IPCs
25 Artillery     100 IPCs
16 Tanks C5   80 IPCs

Fit better the bills?
A. survives: 68.5% D. survives: 31.4% No one survives: 0.1%
But lose A. survives: 96.8% D. survives: 3.2% No one survives: 0%
against 60 Infs +60 Arts.

Does 65 Infs 195 IPCs
30 Artillery     120 IPCs
9 Tanks C5   45 IPCs

Fit better the bills?
A. survives: 71.3% D. survives: 28.6% No one survives: 0.1%
But lose A. survives: 96.6% D. survives: 3.4% No one survives: 0%
against 60 Infs +60 Arts.

But the best is 68 Infs and 39 Arts?
@MrMalachiCrunch:

Is this a sign I have too much time on my hands?

I had as my target army 100 Inf.  I started out using a ratio of 8:3 Infantry to Artillery and did runs with 5000 trials.  I then varied the ratios while maintaing the same exact IPC value ratio of 300:360 IPCs defense:attack, my results

Offensive Force Win %
80 Inf+30 Art 64.1, 63.9, 64.8, 64.9, 63.4
76 Inf+33 Art 67.9, 66.6, 66.7, 66.0, 66.5
72 Inf+36 Art 69.8, 68.6, 69.5, 69.4, 69.9

68 Inf+39 Art 70.5, 70.3, 69.9, 70.4, 70.3

64 Inf+42 Art 69.5, 69.5, 69.4, 69.1, 69.5
60 Inf+45 Art 68.2, 68.1, 68.3, 68.8, 68.2
?

So it seems the ratio 68:39 or round it off to 7:4 which is closer to 2:1 than 3:1.

A side note, take the case of:

68 Inf+39 Art 70.5, 70.3, 69.9, 70.4, 70.3
Trade 10 IPC in the form of 2 Inf and 1 Art for 2 tanks.  Run the 5 trials and you get:

66 Inf+38 Art+2 Tanks 70.6, 69.8, 70.2, 70.6, 71.6
64 Inf+37 Art+4 Tanks 70.4, 71.9, 71.4, 69.8, 71.3
62 Inf+36 Art+6 Tanks 70.6, 70.4, 70.7, 71.3, 71.1
60 Inf+35 Art+8 Tanks 70.5, 70.3, 70.5, 70.8, 69.7
58 Inf+34 Art+10 Tanks 70.1, 71.3, 70.5, 70.8, 70.6
56 Inf+33 Art+12 Tanks 69.9, 70.0, 69.5, 69.6, 69.3
54 Inf+34 Art+14 Tanks 69.4, 69.5, 67.8, 69.7, 67.8

It would seem against lots of infantry at least, that attacking with mostly Inf and Art in a ratio of about 7:4 is best.  Having a few tanks doesn’t seem to hurt but as you add more tanks at the cost of Inf+Art your odds of success go down.  At least in this isolated scope!

Malachi

From what I can understand about ground units ratio, it seems you need almost 16 Inf for 8 Art for 1 Tank.
This would optimized the metapower, according to AACalc.

So, by building 8 Inf for 4 Art, it gives you a 1 Inf slack to remain 7:4 for the main battle.
By keeping this 2:1 ratio it allows you to split your offensive stack and to adapt.
It includes a low skew on offense (1A1, 2A2), probably a 10% bonus
For 10 IPCs, you get a metapower of A53=15  / D63=18
Ratio: 1.5/IPC*1.1= 1.65/IPC  /  1.8/IPC.

Also, the stack formula showed that a defensive picket is more cost effective when you put 3 Infs for 9 IPCs.
You get D6*3 = 18 Metapower for 9  IPCs, 2 metapower for each IPC.
For 3 IPCs, you get 2 metapower, 0.67/IPC.
For 6 IPCs, you get 8 metapower, 1.33/IPC.
For 9 IPCs, you get 18 metapower, 2.00/IPC.

Of course a 12 IPCs stack gets higher metapower ratio: D8*4=32, 2.67/IPC, but it is no more like a picket fodder.
At least, we can see that metapower simply increase by increment of 2 pips/1 Inf unit per 3 IPC = 0.67.

For picket tactic, maybe we can agree that 2 Infs per TT might be more cost effective because you get .33 above the 1 metapower/ IPC invested. This increase the probability that trading TT will take an enemy casualty.
Also, the increase in metapower, from 2 for 1 unit, to 8 for 2 units is *4 multipler (400%), while 18 for 3 units is *2.25 multiplier (a 225% increase). So the highest boost in defense is gained with 2 Infantry as picket.
Since your metapower/IPC increase is the same 0.67 per each additional unit.
That’s my 2 cents on picket line.

Did you ever use 2 Infs instead of only 1 as blocker?