@Genghis:
Ok so I tested manually 141 infantry attacking 100 infantry defending and the results was 12 infantry left standing on the attacks vs 0 on the defense. AAcalc gives this result as well.
Shouldn’t the remaining infantry be the same on both sides if the battle is supposed to be exactly 50:50 according to the table/formula?
It gives a 1% avg of 11 Infs defenders winning compared to an avg of 1% 17 Infs attackers wins. If 142 vs 100 for a 51.6% vs 48.6%
If it is 141 vs 100, your odds are 48% vs 52%.
You may get a finner tuning but it is between 1.410… and 1.420 ratio.
50% is a put as 17 @1 vs 12 @2.
Which is 1.41666…
√(P2 / P1) = N1 / N2
√2/1= 1.4142
With more units in play, more accurate the formula will be.
Larry-Marx is the circle of life compared to Vann shoes formula…
@Genghis:
Hey guys, this formula is really useful. I would love to see the proof for this. It’s not that I don’t trust you, I just want to see how you derived it, thanks!
-GK
I think CalvinandHobbesliker made a PDF about this formula:
http://www.axisandallies.org/forums/index.php?topic=39526.msg1642941#msg1642941
I cannot find Kreuzfeld equations but I’m pretty sure he provided these somewhere.
The table above is more accurate and the fractions in the second table are an approximation from the first table.
I tried to provide as small number as possible on both part of fraction.
@Baron:
Extended 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
0.5
1
1.5
2
2.5
3
3.5
4
4, 2hits
5 | 0.5
1.00
0.70
0.58
0.50
0.45
0.41
0.38
0.35
0.22
0.32
| 1
1.41
1.00
0.82
0.70
0.63
0.58
0.53
0.50
0.31
0.45
| 1.5
1.73
1.22
1.00
0.87
0.77
0.70
0.65
0.62
0.38
0.55
| 2
2.00
1.41
1.15
1.00
0.89
0.82
0.76
0.70
0.43
0.63
| 2.5
2.24
1.58
1.29
1.12
1.00
0.91
0.85
0.79
0.50
0.70
| 3
2.45
1.73
1.41
1.22
1.10
1.00
0.93
0.87
0.53
0.77
| 3.5
2.65
1.87
1.53
1.32
1.18
1.08
1.00
0.94
0.58
0.84
| 4
2.83
2.00
1.62
1.41
1.26
1.15
1.07
1.00
0.62
0.89
| 4, 2hits
4.58
3.33
2.64
2.30
2.00
1.87
1.73
1.62
1.00
1.41
| 5
3.16
2.24
1.83
1.58
1.41
1.29
1.20
1.12
0.70
1.00| Avg Power
0.5
1
1.5
2
2.5
3
3.5
4
4, 2hits
5 | 0.5
1:1
12:17
4:7
1:2
4:9
5:12
5:13
5:14
2:9
5:16
| 1
17:12
1:1
9:11
12:17
5:8
4:7
9:17
1:2
3:10
4:9
|
1.5
7:4
11:9
1:1
13:15
7:9
12:17
9:14
5:8
3:8
5:9
| 2
2:1
17:12
15:13
1:1
9:10
9:11
3:4
12:17
3:7
5:8
| 2.5
9:4
8:5
9:7
10:9
1:1
10:11
5:6
4:5
1:2
12:17
| 3
12:5
7:4
17:12
11:9
11:10
1:1
19:20
13:15
9:17
7:9
| 3.5
13:5
17:9
14:9
4:3
6:5
20:19
1:1
20:21
4:7
5:6
| 4
14:5
2:1
8:5
17:12
5:4
15:13
21:20
1:1
5:8
9:10
| 4, 2hits
9:2
10:3
8:3
7:3
2:1
17:9
7:4
8:5
1:1
17:12
| 5
16:5
9:4
9:5
8:5
17:12
9:7
6:5
10:9
12:17
1:1|
|
Awesome Baron, I will go over the math later but does this proof validate the Lanchester table?
Taking the original question in a different direction (strategy instead of math), I noticed recently that I always try to send just enough force to a region that it would be inefficiently expensive for my opponent to challenge me.
For example, as Japan, when invading eastern Russia, how much force do you need to send? If you are much, much stronger than Russia, and you can sack the capital, fine, do that. Not interesting. If you are much, much weaker than Russia, and you are not going to even be able to keep an active beachhead, fine, stay home. Also not interesting. The interesting case is the middle ground, where you are strong enough to be a real pain in Russia’s behind, but not strong enough yet to be seriously planning an attack on Moscow.
In that case, you want to send enough force that it would cost Russia more to repel your forces than to just let you sit in Kazakh and Novosibirsk and Evenki. If you are conquering, e.g., 5 IPCs’ worth of territory, and you expect to hold it for roughly 3 turns before the global strategic situation changes again, then the stakes are a 30 IPC swing (5 IPCs * 3 turns * 2 players), so you need a force that’s large enough that Russia can’t kill it without losing at least 30 IPCs’ worth of troops. Depending on Russia’s force mix, this could be more or less than 30 IPCs’ worth of your units. If Russia has almost nothing but infantry, then even a small Japanese force of, e.g., 4 infantry and 2 tanks (24 IPCs) might be a fair match on defense against 10 attacking Russian infantry. If Russia has plenty of artillery and air support, then even a large Japanese force of, e.g., 4 infantry and 5 tanks (42 IPCs) might get crushed by a Russian counter-attack of 6 infantry, 3 artillery, and 6 fighters – Russia’s not going to lose more than 30 IPCs on that battle, and the planes are not really going to be taken away from their positions, but Russia can still repel your 42-IPC force.
So you want to think about what it would actually cost your enemy to repel your forces, and then make your invasion force just large enough that your enemy will lose more money from building or diverting the counter-attack than it would lose by letting you have the extra territories. That way you set up a win-win situation: if they let you have the territory without a fight, you win because your income goes up and theirs goes down, but if they fight you for the territory, then they lose so many resources that holding the ground winds up being a net economic loss for them.
The reason why you don’t just go in with everything you can afford to build is that you might need some of those forces to set up a similar no-win scenario for your opponents on another front – sticking with Japan, if you send too many assets into Russia, you usually won’t have enough cash left over to make sure that the Pacific islands are too expensive for America to take, and so on. There are exceptions; sometimes you can just steamroll everyone at once – but in those games, you don’t need a strategy guide!
@Genghis:
Awesome Baron, I will go over the math later but does this proof validate the Lanchester table?
I know from various empirical use of AACalc, these table works and Lanchester is working too.
To assess strength of a stack, you need to sum all attack or defense points then multiply by hits instead of adding hits. This work easily for 1 hit units.
the tables and formula (and its proof) work for when all attacking units are the same unit and when all the defending units are the same unit. Otherwise you have to approximate by calculating an average power and average number of units. Example:
on attack: 5 infantry, 4 tanks, 2 fighter.
Total pips: 5+4(3)+2(3) = 23.
Number of units = 5+4+2 = 11
23/11=2
So we can approximate this stack with 11 units hitting at 2, or 11 artillery.
I guess you would use power level 2 and number of units 11 for the purposes of lanchester tables.
@Genghis:
the tables and formula (and its proof) work for when all attacking units are the same unit and when all the defending units are the same unit. Otherwise you have to approximate by calculating an average power and average number of units. Example:
on attack: 5 infantry, 4 tanks, 2 fighter.
Total pips: 5+4(3)+2(3) = 23.
Number of units = 5+4+2 = 1123/11=2
So we can approximate this stack with 11 units hitting at 2, or 11 artillery.
I guess you would use power level 2 and number of units 11 for the purposes of lanchester tables.
Yes, exactly.
This table is made for square number basically.
I extended to half, like 1.5 or 2.5 to make for average power of a mixed stack.
Like 1 A3 and 3 A1 means A6 for 4 hits, hence 1.5 avg.
So, such stack will be about same as 7 D 0.5, for instance 5 AAA with 2 Infs (D4/7 = 0.57)
The table say 50% is 0.58. And table below is an approximate to 4:7 ratio.
Now, if there is a big skew between stack, (not the case in last example because both stack have skew)
I learned from the general formula that you get a 10% meta-power increase. However, such results have to be divided by number of hits to get a better avg of power per unit.
For instance, 1 Tank and 3 Inf gives A64 = 241.1= 26.4 metapower /4 hits = 6.6 for this stack or 1.65 per 1 hit unit.
2 Infs and 5 AAA D4 means 4.4 divided by 7 = 0.63 per 1 hit unit.
Of course, using the whole formula get more accurate result but for fast reference you can use avg and this table.
@Baron:
Extended 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:
P = Average Power of a given stack
N = Number of units of a given stack√(P2 / P1) = N1 / N2
| Avg Power
0.5
1
1.5
2
2.5
3
3.5
4
4, 2hits
5 | 0.5
1.00
0.70
0.58
0.50
0.45
0.41
0.38
0.35
0.22
0.32
| 1
1.41
1.00
0.82
0.70
0.63
0.58
0.53
0.50
0.31
0.45
| 1.5
1.73
1.22
1.00
0.87
0.77
0.70
0.65
0.62
0.38
0.55
| 2
2.00
1.41
1.15
1.00
0.89
0.82
0.76
0.70
0.43
0.63
| 2.5
2.24
1.58
1.29
1.12
1.00
0.91
0.85
0.79
0.50
0.70
| 3
2.45
1.73
1.41
1.22
1.10
1.00
0.93
0.87
0.53
0.77
| 3.5
2.65
1.87
1.53
1.32
1.18
1.08
1.00
0.94
0.58
0.84
| 4
2.83
2.00
1.62
1.41
1.26
1.15
1.07
1.00
0.62
0.89
| 4, 2hits
4.58
3.33
2.64
2.30
2.00
1.87
1.73
1.62
1.00
1.41
| 5
3.16
2.24
1.83
1.58
1.41
1.29
1.20
1.12
0.70
1.00| Avg Power
0.5
1
1.5
2
2.5
3
3.5
4
4, 2hits
5 | 0.5
1:1
12:17
4:7
1:2
4:9
5:12
5:13
5:14
2:9
5:16
| 1
17:12
1:1
9:11
12:17
5:8
4:7
9:17
1:2
3:10
4:9
|
1.5
7:4
11:9
1:1
13:15
7:9
12:17
9:14
5:8
3:8
5:9
| 2
2:1
17:12
15:13
1:1
9:10
9:11
3:4
12:17
3:7
5:8
| 2.5
9:4
8:5
9:7
10:9
1:1
10:11
5:6
4:5
1:2
12:17
| 3
12:5
7:4
17:12
11:9
11:10
1:1
19:20
13:15
9:17
7:9
| 3.5
13:5
17:9
14:9
4:3
6:5
20:19
1:1
20:21
4:7
5:6
| 4
14:5
2:1
8:5
17:12
5:4
15:13
21:20
1:1
5:8
9:10
| 4, 2hits
9:2
10:3
8:3
7:3
2:1
17:9
7:4
8:5
1:1
17:12
| 5
16:5
9:4
9:5
8:5
17:12
9:7
6:5
10:9
12:17
1:1|
|
Taking the original question in a different direction (strategy instead of math), I noticed recently that I always try to send just enough force to a region that it would be inefficiently expensive for my opponent to challenge me.
For example, as Japan, when invading eastern Russia, how much force do you need to send? If you are much, much stronger than Russia, and you can sack the capital, fine, do that. Not interesting. If you are much, much weaker than Russia, and you are not going to even be able to keep an active beachhead, fine, stay home. Also not interesting. The interesting case is the middle ground, where you are strong enough to be a real pain in Russia’s behind, but not strong enough yet to be seriously planning an attack on Moscow.
In that case, you want to send enough force that it would cost Russia more to repel your forces than to just let you sit in Kazakh and Novosibirsk and Evenki. If you are conquering, e.g., 5 IPCs’ worth of territory, and you expect to hold it for roughly 3 turns before the global strategic situation changes again, then the stakes are a 30 IPC swing (5 IPCs * 3 turns * 2 players), so you need a force that’s large enough that Russia can’t kill it without losing at least 30 IPCs’ worth of troops. Depending on Russia’s force mix, this could be more or less than 30 IPCs’ worth of your units. If Russia has almost nothing but infantry, then even a small Japanese force of, e.g., 4 infantry and 2 tanks (24 IPCs) might be a fair match on defense against 10 attacking Russian infantry. If Russia has plenty of artillery and air support, then even a large Japanese force of, e.g., 4 infantry and 5 tanks (42 IPCs) might get crushed by a Russian counter-attack of 6 infantry, 3 artillery, and 6 fighters – Russia’s not going to lose more than 30 IPCs on that battle, and the planes are not really going to be taken away from their positions, but Russia can still repel your 42-IPC force.
So you want to think about what it would actually cost your enemy to repel your forces, and then make your invasion force just large enough that your enemy will lose more money from building or diverting the counter-attack than it would lose by letting you have the extra territories. That way you set up a win-win situation: if they let you have the territory without a fight, you win because your income goes up and theirs goes down, but if they fight you for the territory, then they lose so many resources that holding the ground winds up being a net economic loss for them.
The reason why you don’t just go in with everything you can afford to build is that you might need some of those forces to set up a similar no-win scenario for your opponents on another front – sticking with Japan, if you send too many assets into Russia, you usually won’t have enough cash left over to make sure that the Pacific islands are too expensive for America to take, and so on. There are exceptions; sometimes you can just steamroll everyone at once – but in those games, you don’t need a strategy guide!
Bumped. I don’t want this be burried too fast in past page.
