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.
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 = 12312
So, 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 = 13600
with Destroyers:
381818 = 12312
But applying avg pips of stack to follow the formula almost to the letter, this gives:
with Submarines:
(34/18)2020 = 34*20 = 680
with Destroyers:
(38/16)1818 = 38*18 = 684
And, considering Carrier being 1 unit for 2 hits, to apply this stack formula to the letter:
with Submarines:
(34/18)2020 = 755.6
with Destroyers:
(38/16)1818 = 769.5
So, 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= 16875
Battleships
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 * power
N1 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 = 698.9
2 CV + 4 ftr, + 8 DDs + 2 subs
(14 hits + 2*1.618034)^2 * 38 pips/16 units = 705.6
And 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?
No kidding?
Vann formula, even revamped into a Second Edition formula, only help calibrate cost and strength for judging OOB units and HRs units.
The one above is far more useful for combat eval as said in title.
EDIT: Written ironic comments are difficult to judge without any tone from real voice.
It did not do bad if I answered straight forward.
Especially going in the same direction as the real ironic intent.
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.
Anything that uses “formulas” other than
HP vs HP
ATT POWER vs DEFENSE POWER
6 / 36 / 108 style statistics for d6is too complex to be used during any game, or too abstract to counsel any action. No battle calcs are allowed at the tourney, which is fine, because they are totally unnecessary. Counting and addition are all that is necessary.
? 108 ?
6+36= 42
636= 216
3636= 1296
Any link which explains this Punch formula?
Cell phone are not allowed?
Before judging a formula is too difficult for mental calculations, you have to ask how accurate is a formula.
Then, you can tell which is too cumbersome but accurate compared to inaccurate but come in handy.
People used clock and compass, or sun and moss but now GPS.
There is many ways to Rome.
Some faster but costlier, other longer but funnier.
While looking for Punch formula, I found the first occurrence of the OP stack formula based on Lanchester’s Law posted by akreider2 in 2007:
@akreider2:
I’m not sure if I ever mentioned this before, so I apologize if I did.
The formula comes from my experience playing the old version, where I used buy almost entirely infantry (and no tanks). I ran several simulations and came up with this formula. Frankly, it’s really amazing that it works and that I just figured it out inductively.
My sister actually proved this formula - but I don’t recall the proof.
The Formula
Power= (number of units) (number of units) (number you need to hit)**
IF attacker power = defender power THEN you have a 50% chance of winning.
This only works if you are using units of the same level of strength (both attacker and defender - eg the defender can use a mixture of artillery and infantry as they both defend on ‘2’).So this formula is useful for calculating your odds of winning.
For instance
A: 3 inf - power = 331 = 9
D: 2inf - power= 222 =8
So the attacker has an advantage.Frood Says
A. survives: 50.7% D. survives: 45.5% No one survives: 3.7%A: 40 inf - power = 1600 (40401)
D: 20 fig - power = 1600 (20204)
A. survives: 48.2% D. survives: 51.4% No one survives: 0.4%Now using larger numbers helps:
A: 80 inf
D: 40 fig
A. survives: 49.3% D. survives: 50.5% No one survives: 0.1%As you move to infinity units, it approaches 50% to 50%. There is some kind of limit/calculus going on. It seems to be approaching 50% from below (for the attacker, eg the attacker odds start at 48% and increase to 50% as the number of units increases) which is weird. If I didn’t get timeouts on Frood when using 10,000 simulations, and if it allowed me to use more than 100 units, then I might have better luck proving whether the approach is random (and purely due to standard deviation) or from below.
I think it’d be possible to create a formula for calculating odds in general, but I’m not a complete math genius.
When are people going to learn that no single calculation is going to win them the game. There is no silver bullet.
Further - many battles I fight are losing propositions(in terms of calculations, but strategic and necessary evils, because guess what folks - winning the war requires sacrifice
Lanchester’s Table for Axis and Allies 2nd Edition
I made it on AACalc then I revised numbers by applying this formula derived from above Stack formula:
√(P2 / P1) = N1 / N2
This might be another way to estimate outcomes…
@Baron:
It is my first shot working with Table feature of the Forum.
If someone can do better, I will appreciate.
With this small 6 x 6 table, you get in a glimpse what is the 50%-50% break even according to the ratio of units and the average power of a given stack.
This table may also be written in decimal instead of a ratio, below. But ratio are better to understand how each ratio is paired to another one which is simply reversed.
A more complete table would include average power: 1.5, 2.5 and 3.5
But, in F-2-F, you can always round up the average enemy’s power, so you do a safer battle to be sure you are above break even ratio.
X means the ratio is 1:1. I did not want to overcharge this table with obvious infos.
| Power
1
2
3
4
4, 2hits | 1
X
12:17
11:19
1:2
7:23
| 2
17:12
X
9:11
12:17
4:9
| 3
19:11
11:9
X
13:15
10:19
| 4
2:1
17:12
15:13
X
5:8
| 4, 2hits
23:7
9:4
19:10
8:5
X
| Power
1
2
3
4
4, 2hits | 1
X
0.70
0.58
0.50
0.30
| 2
1.42
X
0.82
0.70
0.43
| 3
1.73
1.22
X
0.87
0.53
| 4
2.00
1.42
1.15
X
0.63 | 4, 2hits
3.33
2.30
1.87
1.60
X
This table can also be memorized with the main 5 basic ratios and you can reverse at will, or 6 if we include 1:1 ratio, the @4 with 2 hits for BBs might not be very relevant in F-2-F:
1:2 (4 vs 1), 11:19 (3 vs 1), 12:17 (4 vs 2 or 2 vs 1), 9:11 (3 vs 2), 13:15 (4 vs 3)
OR same order but from reverse ratio:
2:1 (1 vs 4), 19:11 (1 vs 3), 17:12 (1 vs 2 or 2 vs 4), 11:9 (2 vs 3), 15:13 (3 vs 4)
For instance, reading from left row to the right, if you have an average power of 1.3 for 17 units and the defender has 2.2 average power for 10 units.
You may cross-referenced the 1 row with the 2 column, saying you need 17: 12 ratio, or 1.42 more units than defender.
So, your 17:12 ratio or 1.42 more units give an above 50-50% odds of winning.
Of course, during battle, ratio of units and average power may changes, especially when fodders are done.
So, you may decide at critical moment to recheck your odds of success.
For example, the defender may have only 3 Tanks left, while attacker only 4 Infs and 1 Tank.
This give a 5: 3 units ratio. And the 1 (A7 / 5= 1.4) row compared to 3 column, says: 19 to 11 or 1.73.
If looking the 2 row, it says 11:9 or 1.22.
So, assuming this rounding up or down, it reveals you are above or below the break even point.
Thus, it is still near 50%-50%.
In fact, AACalc says:
Overall %*: A. survives: 70.6% D. survives: 24% No one survives: 5.5%
But, Punch formula might be better to anticipate results?
A7 for 5 hits vs A9 for 3 hits.
7+5 =12 or 9+3 =12, so still even…?
Stack formula is more revealing IMO:
251.4= 35 vs 93= 27
Clearly the 5 attackers units are on winning side. |
|
Why all these charts, calculations,and complications?
What’s wrong with:
Attack Power divided by 6 = X Hits
vs
Defense power divided by 6 = X hits,
and then seeing where that will take each side hit points after X rounds.
This is really simple stuff guys.
Why all these charts, calculations,and complications?
What’s wrong with:
Attack Power divided by 6 = X Hits
vs
Defense power divided by 6 = Y hits,and then seeing where that will take each side hit points after Z rounds.
This is really simple stuff guys.
What is the name of this method? Pips substraction?
Is it different from Punch formula?
Once all methods identified, it can be easier to compare relevance and accuracy.
One thing, I saw with multiple divisions and substractions is that it overloads short term memory and you have to repeat for defender the process, at each step.
Stack formula use almost only multiplications and may avg power more intuitively.
You can call it whatever you want .
How about common sense or axis and Allies 101.
No one’s brain is getting overloaded with that method, as opposed to the calculus phone books some of the guys posted earlier.
I thought that since it is an old ways of predicting outcomes, it would have been named or nicknamed by someone.
Common Sense Formulas.
@SS:
Common Sense Formulas.
We should add a .com and make this a website
