You are correct Carolina 🙂
Method for Estimating the Outcomes of Large Battles

For several nations, particularly Germany, Russia, Japan, and the US, a large chunk of their strategy revolves around positioning their large stacks of units. A critical part of this maneuvering is determining if the enemy stack can crush your stack, or vice versa. Calculators make this really easy, but take a lot of time when playing in person. I aim to demonstrate a way to mentally calculate which stack would win, and by how much.
To start with, I want to emphasize that I know this game is a dice game. Dice are random. However, because dice tend to average out over larger numbers of throws, estimation is still possible and fairly accurate for large stacks. This method won’t work well for determining if 3 infantry will beat a tank  that battle has a lot of luck involved; but then again, you probably know the results of small battles without any calculations.
Finally, I know that some people will consider this cheating or against the spirit of the game. For me, little feels worse than for the US to have 6 hours of naval builds wiped out because I misjudged a stack!
Ok now that that’s out of the way, let’s get to the fun part. The foundation of this is in Lanchester’s Laws, which describe a set of equations for how 2 forces battle each other. The output is rather intuitive: In modern combat, with ranged units firing at each other from a distance, the guns can attack multiple targets and can receive fire from multiple directions. Lanchester determined that the power of such an army is proportional not to the number of units it has, but to the square of the number of units.
If you think about it, you probably already knew this. When 10 artillery attack 5 artillery, more than 5 attacking units survive, like we would expect if power was linear. It’s around 8.
“Suppose the blue army is three times the size of the red army. This means that it is concentrating three times as much firepower on red as red is firing. Just as importantly, red’s firepower is diluted over three times as many blue units. The combined effect of these two conditions is that blue is nine times as strong as red although it only has three times as many units. Similarly, the number of units remaining at the end of the battle is the square root of the difference between the squares of the numbers of units on each side. “ (quoted from gamasutra, which explains this far better than I would).
Going back to our 10 vs. 5 artillery example… if you plug that in to the formula, you see that sqrt(10^2  5^2) ~ 8 units!
Now, there is a key difference between AAA and Lanchester’s problem  Lanchester’s equations assume continuous combat, whereas AAA is discrete both in terms of time (each round of battle) and of units (you can’t have half a unit). However, we’re only looking to model large stacks with large numbers of units, so above a certain stack size (> 10) this doesn’t have a big impact.
The time bit has more of an impact. The way to think about it is that with discrete salvos (rounds of combat), the defender keeps more of its forces around longer than if it was a continuous fight. For a simple example, think about 10 fighters vs 1 fighter; in a continuous battle, that single fighter would be dead so quickly it wouldn’t get a shot off. But in AAA, it gets a full round of combat nonetheless, and will likely take out another fighter on its way out. The consequence is that the defender has a slight advantage over the model, and that advantage grows somewhat for higher defense values. Looking at the model vs actual AAA battles, it’s around 5%10%. It’s not huge, and because it’s predictable we can somewhat take that into account though.
So far, we’ve been dealing with units of equal power attacking each other. Of course, there are multiple different units in AAA. Lanchester’s laws dictate that if A attacking units of strength α hit B defending units of strength β, the number of attacking units remaining is described by this formula: A_final = sqrt(A^2  B^2 * β / α) . Here, α and β are just the unit attack/defense values, so an artillery is 2 and a tank is 3 etc, and an infantry is 1 on attack and 2 on defense.
So if 30 attacking infantry hit 15 defending artillery, we get:
sqrt(30^2  15^2 * 2 / 1) = 21 units surviving
In a AAA simulator, we see that the median occurrence is 20, which is pretty close. And remember, we were expecting to see a defender advantage of around 5% than the model.An interesting thing you see with the square root is that a small change in MetaPower can make a big difference in the average outcome. If you’ve ever simulated a close Moscow with adding 1 or 2 infantry on either side, you know what a big impact they can make!
Before people start saying that they can’t do square roots in their head… if you just want to know if you’ll win, you can just calculate if α * A^2 > B^2 * β. Since A is number of units, and α is the unit’s power, then you can also calculate this as A * (α * A), where α * A is the Total Power of your units. This value is probably familiar, since it is displayed in the tripleA simulator  2 tanks have 6 total power, 5 inf and 5 art have 20 total power, etc. With lack of a better term, I’ll define the quantity “MetaPower” = Number of Units * Total Power of Units. If your MetaPower is greater than their MetaPower, you’ll probably win. The nice part about this number is that it’s fairly easy to calculate, and has good predictive qualities.
You can do other things with MetaPower besides just seeing if you’d win. For example, your MetaPower will be different for attack and defense, so you can see if your stack would be better positioned to attack or defend. Also, you can see if adding 3 subs and 3 destroyers would help out to a giant Pacific battle more or less than a loaded carrier.
One note for ships  for 2 hit units like carriers and battleships, you would count the unit twice in the unit count, and only once in the power.
Now, one factor we’ve been ignoring is that compositions aren’t uniform. Intuitively, we know that this makes a difference  a tank and an inf is better than an inf and an art even though both have the same power, and we’ve all been burned when German bombers + mech decimated a larger Russian stack of infantry.
Unfortunately, this is the end of where math takes us. There’s no closedform solution past this point. However, I modeled keeping power and number of units the same and changing composition, and I came up with the following bonuses to MetaPower based off composition:
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% bonusIn summary,
 MetaPower = Number Units * Total Power of Units * (distribution bonus + defense bonus)
 If your MetaPower is higher, you have a good chance of winning
 Remaining Units after a battle ~ sqrt(MetaPower1  MetaPower2)
Here are some examples:
Example 1: Can Russia safely move 35 inf and 20 art next to a german stack that can hit with 10 inf, 10 mech, 5 art, and 15 tanks, and 10 bombers?
German attack MetaPower = 50 units * 120 power + 10% dist bonus = 6600
Russian defense MetaPower = 55 units * 110 power + 5% def bonus = 6350Remaining German units = sqrt(66006350) = 16
Sim results: mean case is 11 remaining German units, median case is 17 remaining German units. Probably a bad move for Russia.
Example 2: How many destroyers can I beat with 8 subs and 12 bombers?
MetaPower = (8+12) * (8*2 + 12 * 4) * (1 + 15%) = 1472
26 destroyers have MetaPower = 1420, and 27 have 1530, so I’d bet between 26 and 27.
Sim results: 26: 55% attacker win, 27 and it’s a 43% battle. Not too shabby a prediction.
Now… is this easier than plugging numbers into a simulator? It might depend how fast you are at math. I’m not sure how useful any of this is, but I find it interesting that with a little arithmetic and an algorithm used to model combat in WW1, you can get a decent handle on AAA combat.
For more info on Lanchester’s Laws:
https://en.wikipedia.org/wiki/Lanchester’s_laws
http://calhoun.nps.edu/bitstream/handle/10945/38442/inc_hughes_MORS_1995_v1n3_F_ADA3213372.pdf?sequence=1
http://www.gamasutra.com/view/feature/130536/the_designers_notebook_kicking_.php 
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 nonbinary 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 powercompare 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 blowoutlike. 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

Interesting read, I was surprised how you’ve connected that model to the game and how well it equates simulation results.
I/we’ve been estimating battles based on comparing 3 factors listed in order of importance.
 number of units
 power
 ‘skew’, the concept that you can throw away lowpowerunits (infantry) while the defender has to sacrifice highpowerunits (infantry).
This was heavily influenced by an old article on these forums by HolKann, and still the way I determine whether to attack/defend/retreat (http://www.axisandallies.org/forums/index.php?topic=19854.0). Interesting difference between your ideas is that HolKann claims the number of units is more important than the power; while your calculations show metapower is a multiplication of power and the number of units so they are symmetrically important.

Interesting read. I/we’ve been estimating battles based on comparing 3 factors listed in order of importance.
 number of units
 power
 ‘skew’, the concept that you can throw away lowpowerunits (infantry) while the defender has to sacrifice highpowerunits (infantry).
This was heavily influenced by an old article on these forums by HolKann, and still the way I determine whether to attack/defend/retreat (http://www.axisandallies.org/forums/index.php?topic=19854.0). Interesting difference between your ideas is that HolKann claims the number of units is more important than the power; while your calculations show metapower is a multiplication of power and the number of units so they are symmetrically important.
Hmm, well in the case of 3x attacking inf vs 2x defending tanks, OP’s calculation would give the attacker a metapower of 9x^2 against 12x^2 for the defender, and the calculator gives 99% chance to the defender. This supports OP’s model over HolKann’s.
To further prove the model, at least in the case where every unit on each side has the same power, assume the attacker has x units of power i, and the defender has y units of power j. On the first round of battle, the attacker will kill xi/6 defending units, and the defender will kill yj/6 attacking units. If one side loses a greater proportion of its units than the other side, they will clearly lose the battle (on average), so the equilibrium, when the battle has a 5050 chance, is when (xi/6)/y=(yj/6)/x, or when ix^2=jy^2. Note that ix^2 and jy^2 are the metapowers of each side, respectively.
I’m working on calculating the number of rounds of battle and remaining attacking units assuming the attacker has more metapower. I’ll try to post that tomorrow.

Interesting read. I/we’ve been estimating battles based on comparing 3 factors listed in order of importance.
 number of units
 power
 ‘skew’, the concept that you can throw away lowpowerunits (infantry) while the defender has to sacrifice highpowerunits (infantry).
This was heavily influenced by an old article on these forums by HolKann, and still the way I determine whether to attack/defend/retreat (http://www.axisandallies.org/forums/index.php?topic=19854.0). Interesting difference between your ideas is that HolKann claims the number of units is more important than the power; while your calculations show metapower is a multiplication of power and the number of units so they are symmetrically important.
These factors are more important then the points made in the OP posts.
OP asumes forces of equal strength where this is not the case in A&A, it only works if you are attacking with (for example )art vs inf because both have attack/defence of 2 ( there are other combinations but they are all unrealistic )You should take into account that both the attacker and defender have some fodder units that they will sacrefice first as to preserve their attacking power and hence their amount of hits.
Skew does this, not idealy but to a certain degree. And you cannot really do this at the table in your head.For F2F games the best thing you could do is.
Count number of hitpoints
Count nummer of attacking/defending point.
Determine if you will lose heavy hitters before they do or the other way around ( you attacking with inf + arm vs inf he will lose power quicker, if you attack with inf/art you will lose power at the same pace )If you got more units and more power you should win.
If you got more power but less units but they are distribute high/low where the other is all middle ( inf ) you have a good chance.
IF you got less power and more units and distribution is high/low against his middle you have a good chance. 
i like this article. have not gone through the math yet, but it seems similar to what I calculated back in the day. My calculation was “What is the minimum ‘power’ needed to win an attack”.
So, If There is X tanks defending, how many infs do I need to win the attack.
http://www.axisandallies.org/forums/index.php?topic=38497.msg1578268#msg1578268
The math in mine is Correct, but It does not contain an estimate on how many units you will have left.

I think the most important thing isnt 100% accuracy here but easy to do quicky.
If your at the table you dont want to spend 30 minutes with a pen/paper and a calculator to determine your odds you want to do a quick check.

Thanks so much for all of your interest and comments on this!
Note that ix^2 and jy^2 are the metapowers of each side, respectively.
Yup! It’s pretty intuitive though was fun to work through it. Let me know if you find anything about the number of rounds  this source said the following: itech.fgcu.edu/faculty/pfeng/teaching/Lanchester Equation for MAT 5932.ppt
rounds = 1 / (2 sqrt( α * β )) * ln( (1 + B/A sqrt( α / β) ) / (1  B/A sqrt( α / β) ))
 Derivation left as an exercise to the reader.
Now, that formula doesn’t count nonhomogeneous compositions. I highly doubt there’s a closed form solution there. Also, that calculation you’re definitely not doing in your head!
OP asumes forces of equal strength where this is not the case in A&A, it only works if you are attacking with (for example )art vs inf because both have attack/defence of 2 ( there are other combinations but they are all unrealistic )
Not sure if you’re referring to my post here, but I’m certainly not assuming homogeneous compositions or forces of equal strength. I start with a simple scenario just to show that my derivation isn’t coming out of thin air, but I go on to show how heterogeneous compositions work, and I include examples…
“What is the minimum ‘power’ needed to win an attack”.
You could use metapower here for this, if you calculate theirs and see how much you’d need to send in. One issue is that metapowers don’t add together, so they’re clunky to tinker with. This is sort of the nature of the problem though  2 forces combined really do better than the sum of their parts.
I haven’t done any analysis to see if you can determine a % victory rather than an avg number of units; I somewhat suspect that you’d need a simulator for that. I find for large battles it tends to be “everything I have” just because each additional unit to a close and large battle results in a huge TUV swing.
Interesting difference between your ideas is that HolKann claims the number of units is more important than the power; while your calculations show metapower is a multiplication of power and the number of units so they are symmetrically important.
I think my results support HolKann’s. Given that metapower = units * power = units^2 * avg(power), I think it’s safe to say that the number of units is more important metapower than power is, and in fact, this shows you how much more important it is. And also how skew doesn’t add much
I like the term skew much more than distribution; I spent forever trying to come up with a better word and this eluded me!
For F2F games the best thing you could do is.
Count number of hitpoints
Count nummer of attacking/defending point.If your at the table you dont want to spend 30 minutes with a pen/paper and a calculator to determine your odds you want to do a quick check.
The 2 things you suggest counting are the 2 parts of the metapower. Multiplying them together shouldn’t take 30 minutes and a pencil and paper
If you have more units and more power, you’ll likely win. If you have fewer units and less power you’ll likely lose. It’s the cases inbetween where things get interesting and metapower can help there.
That said, I have no doubt that with the number of games you’ve played that the heuristics/gut you have does fine as well, and with less math!
And of course, it goes without saying that odds and numbers don’t win games. If a battle is your best path to victory, the odds don’t matter!

Thing is that the force composition is very important. But it is not in your calculation that i can see.
Consider 10 inf + 10 tanks vs 20 inf.
20 units 40 to hit. So metapower is identical.
But the 10 inf + 10 tanks will win most of the time ( dice can be fickle )Now with 10 inf + 10 art vs 20 inf it is basicaly a coinflip, most likely mutual annihilation or close.
But still 20 units and 40 to hit. So metapower is identical to the first example.In the first example the defender loses power quicker then the attacker in the 2nd example the attacker and defender lose power at exactly the same rate.

@zergxies:
“What is the minimum ‘power’ needed to win an attack”.
You could use metapower here for this, if you calculate theirs and see how much you’d need to send in. One issue is that metapowers don’t add together, so they’re clunky to tinker with.� This is sort of the nature of the problem though  2 forces combined really do better than the sum of their parts.
I haven’t done any analysis to see if you can determine a % victory rather than an avg number of units; I somewhat suspect that you’d need a simulator for that. I find for large battles it tends to be “everything I have” just because each additional unit to a close and large battle results in a huge TUV swing.
Naw, no need for that. My numbers come from the solving of some simple equations, and you can add some bonus for distribution. The beginning is quite simple. Lets assume that the attacker has X Units with A chance of hitting. The defender has Y Units and B chance of hitting. (where A and B is some number between 0 and 1). This gives you the equations[ (1, A) ; (B,1)] * [X_0;Y_0] = [X_1;Y_1] for the development of the system.
Solving this system gives the eigenvalues 1 +/ sqrt(A/B), with the eigenvectors [+/sqrt(B/A);1]. Discarding the negative eigenvector and the positive eigenvalue (ie, we don’t have an increasing number of units). We can easily see that SQRT(AverageOfAttacker/AverageofDefender) will show you how many units you need to bring to the battle.
No need for the Lanchester laws, however, his estimates comes from simliar types of analysis.

Thing is that the force composition is very important. But it is not in your calculation that i can see.
Consider 10 inf + 10 tanks vs 20 inf.
20 units 40 to hit. So metapower is identical.
But the 10 inf + 10 tanks will win most of the time ( dice can be fickle )Now with 10 inf + 10 art vs 20 inf it is basicaly a coinflip, most likely mutual annihilation or close.
But still 20 units and 40 to hit. So metapower is identical to the first example.In the first example the defender loses power quicker then the attacker in the 2nd example the attacker and defender lose power at exactly the same rate.
It is in the calculation. He does give a 510& increase. However, this is an interesting point. Most players overestimate the value of forcesomposition and hiters, as compared to fodder. This is why they buy so many tanks, an no where near enough mechs as germany. This is also why they don’t build about 40% Subs in their battlefleet in the pacific (assuming you are strong enough against a force without DDs). Ususally, subs is the most costeffective unit to buy for defence.
Did some simulations. Attacked with 200 Art against 50% Ftr/ 50%AAA to see what the maximum advantage was. Did also 200 inf against 25% Ftr and 75% AAA. (to get the correct average for the stacks).
About equal odds where
200 Art against 87 AAA 87 Ftrs (87% of attacking force)
200 infs against 117 AAA 39 Ftrs (78% of attacking force).These examples are the most extreme possible. Since none will ever be as loopsided as the ftr/aa combo, we can safely assume that 10% Addition in the metapower will be absolutely maximum. You will rarely have more than 5% extra in metapower.
Edit, wrote Art, when i meant AAA

200 infs against 117 art 39 Ftrs (78% of attacking force).
These examples are the most extreme possible. Since none will ever be as loopsided as the ftr/aa combo, we can safely assume that 10% Addition in the metapower will be absolutely maximum. You will rarely have more than 5% extra in metapower.
Please explain that one… 200 infantry @1 attacking versus 117 artillery @2 and 39 fighters @4? I don’t understand the learning point. 200 pips against 390.

Something is wrong on that 117 art / 39 fighter problem. I see it balanced when 200 inf attack 76 art + 39 fighters (200 offense against 308 defense with 200 attackers and 115 defenders).
The calculations are so simple when it is one type of unit attacking another. For example 2829 infantry are required to attack 20 defending infantry. Simply take the square root of the relative defense/attack strength, 1.41 in this case. When I have mixed units in a big land battle like Moscow and no access to a battle calculator, I give 5% extra to the side with significantly more cannon fodder unless it is a ridiculous amount of fodder in which case I add an extra 10%. The cannon fodder benefits can be much more significant in big naval battles.

Thing is that the force composition is very important. But it is not in your calculation that i can see.
It is in the calculation. He does give a 510& increase. However, this is an interesting point. Most players overestimate the value of forcesomposition and hiters, as compared to fodder. This is why they buy so many tanks, an no where near enough mechs as germany. This is also why they don’t build about 40% Subs in their battlefleet in the pacific (assuming you are strong enough against a force without DDs). Ususally, subs is the most costeffective unit to buy for defence.
Thanks, ya that was only a small section in the otherwise lengthy post. Mostly for 2 reasons  I had no real explanation for the numbers, other than that I modeled tens of thousands of battles and these had the “best fit”. Your 2 examples show basically a similar result. Secondly, the impact was so relatively small, when skews are similar you can honestly ignore it most of the time.
Most players overestimate the value of forcesomposition and hiters, as compared to fodder. This is why they buy so many tanks, an no where near enough mechs as germany. This is also why they don’t build about 40% Subs in their battlefleet in the pacific (assuming you are strong enough against a force without DDs). Ususally, subs is the most costeffective unit to buy for defense.
I think this is the most interesting part. It’s very counterintuitive to me and I really have to think about this during the buy phase.

@zergxies:
Interesting difference between your ideas is that HolKann claims the number of units is more important than the power; while your calculations show metapower is a multiplication of power and the number of units so they are symmetrically important.
I think my results support HolKann’s.� Given that metapower = units * power = units^2 * avg(power), I think it’s safe to say that the number of units is more important metapower than power is, and in fact, this shows you how much more important it is.� And also how skew doesn’t add much
I like the term skew much more than distribution; I spent forever trying to come up with a better word and this eluded me!
Pft why didn’t I think of that (numbers^2). Thanks, that’s it.

Ususally, subs is the most costeffective unit to buy for defence.
I’m not following this. Suppose I have 48 IPCs and want to buy a defensive fleet.
I buy 8 subs, receive metapower=881=64.
I buy 6 destroyers, receive metapower=662=72.Aren’t destroyers the units with a higher metapower and as such better as defensive units?

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 convoyraiding Allied subs. The Allies can primarily focus on subs with just enough destroyers for ship blocking purposes.

I have solved the case of the homogeneous battle, where every unit on each side has the same hit probability. I found closed forms for the length of the battle in number of combat rounds and for the number of units left on the winning side. I typed it up in LaTeX to make the math look prettier. I have attacked a pdf.

200 infs against 117 art 39 Ftrs (78% of attacking force).
These examples are the most extreme possible. Since none will ever be as loopsided as the ftr/aa combo, we can safely assume that 10% Addition in the metapower will be absolutely maximum. You will rarely have more than 5% extra in metapower.
Please explain that one… 200 infantry @1 attacking versus 117 artillery @2 and 39 fighters @4? I don’t understand the learning point. 200 pips against 390.
not Art, AAA (0 die, but 1 hp)
so, then it is 117 AAA (AAguns), and 39 ftrs, which will be 200 pips against 0117 + 394 = 156 pips

Ususally, subs is the most costeffective unit to buy for defence.
I’m not following this. Suppose I have 48 IPCs and want to buy a defensive fleet.
I buy 8 subs, receive metapower=881=64.
I buy 6 destroyers, receive metapower=662=72.Aren’t destroyers the units with a higher metapower and as such better as defensive units?
I think I also said that it was dependent on what other ships you had available. Subs isn’t best if you only have subs. I would expect the optimal ratio for subs vs other ships would be between 40 and 60 % of your fleet.
if you have no fleet, then you would be correct.
lets ssume you have 2 CV + 4 ftr, + 2 DDs + 2 subs. I will count the CV as 2 units. For your fleet so far, you have 26 pips and 12 hp. if you buy 6 subs, you will have 38 pips and 18 hp. if you buy 10 subs, you will have 34 pips hand 20 hp.
with sub
342020 = 13600
with DD
381818 = 12312So, as you can see, the sub will be better for your metapower. It will also give you a better lossdistribution.
But, I think your point is interessting. I think this formula can be used to figure out what you want to buy. Just calculate your metapower and hp, and figure out what units you should add. This can be extremely nice russia, germany, japan and USA.

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.


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.
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.


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.
This makes sense  if you are playing conservatively. If you’ve been having bad luck or have been getting outplayed, a riskier strategy is usually correct.
I normally just round down, but I find I’m rounding down a lot I might round up in a subsequent battle round. Especially if the exact same remainder keeps “lingering”.
The big thing about rounding IMO is to adjust the next round. If your combat strength is, eg, 45, that’s 7.5 hits. if you round down to 7, and the rounded off 3 points to your calculated combat strength next iteration.

Ususally, subs is the most costeffective unit to buy for defence.
I’m not following this. Suppose I have 48 IPCs and want to buy a defensive fleet.
I buy 8 subs, receive metapower=881=64.
I buy 6 destroyers, receive metapower=662=72.Aren’t destroyers the units with a higher metapower and as such better as defensive units?
I think I also said that it was dependent on what other ships you had available. Subs isn’t best if you only have subs. I would expect the optimal ratio for subs vs other ships would be between 40 and 60 % of your fleet.
if you have no fleet, then you would be correct.
lets ssume you have 2 CV + 4 ftr, + 2 DDs + 2 subs. I will count the CV as 2 units. For your fleet so far, you have 26 pips and 12 hp. if you buy 6 subs, you will have 38 pips and 18 hp. if you buy 10 subs, you will have 34 pips hand 20 hp.
with sub
342020 = 13600
with DD
381818 = 12312So, as you can see, the sub will be better for your metapower. It will also give you a better lossdistribution.
But, I think your point is interessting. I think this formula can be used to figure out what you want to buy. Just calculate your metapower and hp, and figure out what units you should add. This can be extremely nice russia, germany, japan and USA.
This need a few Edits to be correct:
lets assume you have 2 CV + 4 ftr, + 2 DDs + 2 subs. I will count the CV as 2 units.
For your fleet so far, you have 26 pips and 12 hp.
If you buy 6 Destroyers (D2= +12Def pips), you will have 38 pips and 18 hp.
If you buy 8 subs (D1= +8Def pips), you will have 34 pips and 20 hp.with Submarines:
342020 = 13600with Destroyers:
381818 = 12312But applying avg pips of stack to follow the formula almost to the letter, this gives:
with Submarines:
(34/18)2020 = 34*20 = 680with Destroyers:
(38/16)1818 = 38*18 = 684And, considering Carrier being 1 unit for 2 hits, to apply this stack formula to the letter:
with Submarines:
(34/18)2020 = 755.6with Destroyers:
(38/16)1818 = 769.5So, in lasts 2 cases, you get a better defense with Destroyers compared to Subs purchase.
This contradict your initial thesis.This rise a question: how consider the number of units compared to hits in this formula?
75 Cruisers Attack 3 vs 40 Battleships D4, 2 hits
Cruisers 225 pips, 75 hits vs BBs 160 pips and 80 hits
Cruisers
3* 75^2= 16875Battleships
4* 80^2 = 25600, too high!!!
Or
4* 40^2 = 6400, too low!!!But AACalc gives a pretty even match still win by Cruisers:
Overall %*: A. survives: 53.4% D. survives: 46.2% No one survives: 0.4%Maybe the formula need this addition (1.618034) on hits or rolls for all 2 hits units
Battleships
*4 [(401.618034)^2] = 16755**
So, this seems to work.
However, IDK the derivative formula…So, the complete Stack formula would be :
**Metapower = units^2 * powerN1 is nb of 1 hit units
N2 is nb of 2 hits units
(N1+ N2*1.618034)^2 * avg power [total power/(N1+N2 units)] = Metapower**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 = 698.92 CV + 4 ftr, + 8 DDs + 2 subs
(14 hits + 2*1.618034)^2 * 38 pips/16 units = 705.6And this confirmed that adding DDs brings more defense metapower.
But not that much.
And if you take into account the skew effect, which is higher with Sub fodder, then it is the 10 Subs fleet which becomes a bit better.But there is also the tricky Subs, DDs and planes triangle.
If attacker brings a lot of planes, Subs but no DDs, defending Subs will not protect the fleet core from air attack. 
I think you’d all be better off if you just stuck with the VANN FORMULAS.
Uh Baron, you didn’t actually think my comment was serious did you?