Method for Estimating the Outcomes of Large Battles

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?

For reference, here where I get the Punch formula:
@taamvan:

Nice presentation sir!

I think that odds calculators (novel or traditional) are a bit of a crutch, also, they are ruthlessly accurate but accuracy does not reflect all the special rules or anomalous outcomes that occur in a wargame vs a flat probabilities system.   I personally do not use them, permit them to be used in hosted games, and they are not allowed in tournaments.

Just as you propose, I therefore need a way to determine odds for large battles.   I much prefer to sketch this out rather than even reducing the odds to a series of estimates the reason for this is pretty basic;

A formal system of analysis may give you a rough estimate, but since luck has an overwhelming influence when you are rolling a few dice, it will not really advise optimal commits.   1 man and 1 tank should beat 1 man most of the time but you really should send at least 3 ground units against 1 to ensure a take no matter how crap your odds…there are also situations where you can “win” a battle when you really “lost” it, such as when the attacker kills a big stack, and then retreats leaving you with like 1 tank or 1 fighter.   He did not “win” or take the territory, but it was a huge attritional win for him within the context of the whole game.  An example of a failed “win” is one where the enemy kills all your warships but is in turn killed by you and therefore your transports survive.   These are examples of non-binary outcomes, which the game permits (often) but the calculator cannot necessarily account for or counsel for.

One important thing to remember is that “regression to the mean” has a bigger effect the more dice you roll.  that means that the smaller editions of this game (1941) where the average dice rolled per battle and total are smaller than global (for example) are actually more subject to luck than the bigger ones.   This is also true of larger battles vs smaller ones;  a really big battle you are rolling 30+ dice per round and so crazy anomalies (like rolling 3 6s with just 3 dice) are less likely to happen in a way that affects you catastrophically.   The odds of rolling triple boxcars on any 3 dice remain the same…but if you only have 3 units rolling trip boxcars is devastating whereas if you have 30 units, you just missed with some and hopefully hit with others since you’re rolling all those dice.

**Anyways, the best system for me is usually just adding up all

(Attacker HP + total attack pips) = total attack power
(Defender HP + total defense pips) = total defense power**
compare the two gross numbers, if they are roughly equal, so are your odds.   If one exceeds the other by 10 or more, then the odds are probably closer to 60% or worse.

You still have to analyze whether and which units you can afford to lose in every situation (ground vs air) based on whether you must take that zone (and or block it) or not in order to know what the optimal commit is.   For this reason, I consider reliance on the calculator to be misguided (except where you are jamming the odds for big battles, and then its not misguided, its just a crutch lol)

And moreover, sometimes you just have to say #$.   Even when the odds are against you, you may still need to do a risky attack in order to win.   If you are discouraged by the odds and wait (never tell me the odds!), you are guaranteed to lose the game because you relied on a calculator rather than your moxy.

Finally, I have found that while luck matters very much, the rules for the attacker in this game (attack as much as you please, retreat whenever, maneuver wherever) mean that most offensive battles are intentional blowouts.  It is only those few battles where risks must be taken that determine the outcome of the game (not the minibattles picking off 1 infantry).   You can trade risk for time by waiting to attack, the odds on some subsequent turn may be better as more of your units coalesce into that zone.

The quintessence of the game is therefore knowing when to attack now at great risk (but to speed things up) vs wait a turn or two for the odds to become more blowout-like.   The Axis especially cannot just wait for premium odds and 95% victory chances as the Allies power grows over time…

no calculator can tell you when to seize the day ;)

OK, I found the correct Stack formula when there is A0 or D0 unit from AAA or Carrier or Battleship:

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

MetaPower= (number of 1 Hit units with combat value) (Punch: sum all individuals to hit number) +
1.62 (number of units with no combat value)* (Punch: sum all individuals to hit number)**

For instance, what would be the outcomes of 1 Inf+2 Art + 1 Tank vs 5 AAAs + 2 Infs ?
Punch says: A9 and 4 hits vs D4 and 7 hits,
9+4 = 13 points vs 4+7 = 11 points
So, the attacker might win according to Punch formula.

According to stack formula:
4 hitsPunch A9 = 36 points vs (2 hitsD4 = 8 points) + (1.625 hitsD4= 32.4 ) = 40.4
Considering skew, we can add 5% to attacker stack: 36*1.05 = 37.8

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

But still, defender will win on average according to Stack formula (based on Lanchester’s laws).

AACalc: Overall %*: A. survives: 41% D. survives: 55.1% No one survives: 3.9%
http://calc.axisandallies.org/?mustland=0&abortratio=0&saveunits=0&strafeunits=0&aInf=&aArt=&aArm=&aFig=&aBom=&aTra=&aSub=&aDes=3&aCru=1&aCar=&aBat=&adBat=&dInf=&dArt=&dArm=&dFig=&dBom=&dTra=5&dSub=&dDes=2&dCru=&dCar=&dBat=&ddBat=&ool_att=Bat-Inf-Art-AArt-Arm-Sub-SSub-Des-Fig-JFig-Cru-Bom-HBom-Car-dBat-Tra&ool_def=Bat-Inf-Art-AArt-Arm-Bom-HBom-Tra-Sub-SSub-Des-Car-Cru-Fig-JFig-dBat&battle=Run&rounds=&reps=10000&luck=pure&ruleset=AA1942&territory=&round=1&pbem=

Another example, what would be the outcomes of 1 Inf+1 Art + 2 Tank vs 5 AAAs + 2 Infs ?
Punch says: A10 and 4 hits vs D4 and 7 hits,
10+4 = 14 points vs 4+7 = 11 points
So, again here, the attacker might win according to Punch formula.

According to Stack formula:
4 hitsPunch A10 = 40 points vs (2 hitsD4 = 8 points) + (1.625 hitsD4= 32.4 ) = 40.4
Again the skew might be considered for attacker’s stack (+5%) : 40 * 1.05 = 42

So, again, even on average according to Stack formula (based on Lanchester’s laws), it remains very similar.
And the AACalc results are according to the formula:
AACalc: Overall %*: A. survives: 49% D. survives: 46.9% No one survives: 4.1%
http://calc.axisandallies.org/?mustland=0&abortratio=0&saveunits=0&strafeunits=0&aInf=&aArt=&aArm=&aFig=&aBom=&aTra=&aSub=&aDes=2&aCru=2&aCar=&aBat=&adBat=&dInf=&dArt=&dArm=&dFig=&dBom=&dTra=5&dSub=&dDes=2&dCru=&dCar=&dBat=&ddBat=&ool_att=Bat-Inf-Art-AArt-Arm-Sub-SSub-Des-Fig-JFig-Cru-Bom-HBom-Car-dBat-Tra&ool_def=Bat-Inf-Art-AArt-Arm-Bom-HBom-Tra-Sub-SSub-Des-Car-Cru-Fig-JFig-dBat&battle=Run&rounds=&reps=10000&luck=pure&ruleset=AA1942&territory=&round=1&pbem=

Played a fair amount of games during confinement and quite often my opponents had poor ratios of land units for the big battles.

Ive played more with artillery and the system is not that different from what ive post 3 years ago.
(assuming tanks cost 5$)

1- do not lose art or tanks in the first 2 rounds
2- have as less as possible infantry alive after the first 2 rounds.
3- between 1 & 2 give rule 1 the priority***.
4- number of artillery should be as close as possible of the number of infantry that survive round 1, aim for more rather than less. (having not enough art hurt more than having too much of them wich implies less tanks***)
5- rest are tanks if they cost 5$.

ex
60inf to kill. you estimate they are going to make 20 hits on R1 (you will make at least 15) so hopefully they will make 35 hits or less in the first 2 rounds. So you need around 35i and 15a (35-20)
cost of total units 211$

35 inf+14 art+10 tanks(211$) = above 52% & close to 80% for LL.
34i + 11a + 13t = above 48.5% and slightly under 46% for LL (not enough art)
33i + 13a + 12t = slighty under 51% and 74% for LL.
36i + 12a + 11t = 50.5% and under 56% for LL
36i + 17a + 7t = 54.5% & 90.7% for LL
36i + 22a +3t = 55.5% and 85% for LL

210$
34i+12a+12t = 48% and under 39% for LL
35i+15a+9t = 51% and under 71% for LL

• ***There is a small “bonus” for having more units especially when the stacks make less than 25 hits on the first round.