Mathematical Probability Is it better to have one attack say 58 (65%) vs 37 (85%) or three attacks of 20-18-20 against the original 37.

The real game scenario was: Ava Bomber against the Russian Capitol

Actual losses were

Def    Att

16 to 20 (all)

9  to 18 (all)

10 to 8

With the defender left with 2 attacker 12.  Territory untaken.

I know that in one attack the territory would be taken more  than half the time. But, is it the same odds for three attacks? Would the losses been the same? My gut says no, but my basic math skills comes out about the same. If someone could prove it that would be awesome.

I think the Attack would have been the same...but not positive on the Loss (Defending), without thinking through the math (or probably even knowing it :P ) I would tend to lean towards you may lose more with 3 attacks than a single attack...The Defending Units got to roll 3 times with 85% kill rate rather than a single roll of 85% kill rate.

Should be interesting thread...dunno if we've had this specific conversation before (I would be surprised if there wasn't one).

I agree, an average attack would yield the same kills whether you attack all at once or in groupings.  However, your losses will be greater with multiple attacks since the defender rolls multiple times.

So, for example, taking 40 vs 20 (50% v 50%)

One attack:

40 v 20, attacker loses 10, defender loses 20

Multiple (4 territories at 10 each attacking single territory)

10 v 20, attacker loses 10, defender loses 5

10 v 15, attacker loses 7.5, defender loses 5

10 v 10, attacler loses 5, defneder loses 5

10 v 5, attacker loses 2.5, defender loses 5

Net - attacker loses 25, defender loses 20

The results won't always be this severe, so in some cases it may still make sense to attack from 2 territories, but you will always lose more by splitting attacks against a single territory.

Here is a good thread to read

http://www.wargear.net/forum/showthread/1703p1/Simulgear_Questions

SquintGnome wrote:

I agree, an average attack would yield the same kills whether you attack all at once or in groupings.  However, your losses will be greater with multiple attacks since the defender rolls multiple times.

So, for example, taking 40 vs 20 (50% v 50%)

One attack:

40 v 20, attacker loses 10, defender loses 20

Multiple (4 territories at 10 each attacking single territory)

10 v 20, attacker loses 10, defender loses 5

10 v 15, attacker loses 7.5, defender loses 5

10 v 10, attacler loses 5, defneder loses 5

10 v 5, attacker loses 2.5, defender loses 5

Net - attacker loses 25, defender loses 20

 

The results won't always be this severe, so in some cases it may still make sense to attack from 2 territories, but you will always lose more by splitting attacks against a single territory.

 

Cool, great example. I should have thought of using simpler numbers.  So yes, in this 4 attack scenario the losses were 2.5 times a single strike. That would definitely affect the probability of taking the territory. How much is a good question.

My gut and Yertle's was right. I wonder if there is a formula to calculate the probability of taking the territory vs # of attacks. Or if there is a ratio for # of attack vs the increase loss percentage.

Any mathematicians in the house?

CK66 wrote: Here is a good thread to read

http://www.wargear.net/forum/showthread/1703p1/Simulgear_Questions

Cool. I totally forgot about that thread.

In the AvA Bombers map Russia now has no surrounding territory larger than 20. So it's ability to defend is stronger than I had previously had thought.  If loaded with 45 troops it can take a bomber attack and three full country attacks of 20 and possibly live. The question is what % of the time can it do that...

Your probability of taking the territory is, I think, constant regardless of how you distribute the attacking units across attacking territories. Definitely your expected value is constant; I don't quite know the math for determining the actual probability of taking the territory (it seems to involve nested binomials), but I ran a simulation with some simple numbers and it seemed to work out evenly.

That said, you always want to try to attack with a single large stack instead of multiple smaller stacks to minimize your own losses. If you have to split the attack due to territory maximums (as in AnA), it's generally best to maximize the biggest stack you can and attack with it last. Sometimes the tactical positions of territories and attack timing will offer incentives to do otherwise, but I think it's a good rule of thumb.

So for your AnA example, say that Germany was attacking Russia with 58 from Archangel, West Russia, and Caucasus. You'd want to max out Caucasus at 30, max out Archangel/W. Russia at 20, and then put the remaining 8 on the other. You would attack 8, 20, 30 in that order to minimize defender losses.

Note: I didn't notice the difference between the Bomber scenario territory maxes and the original board when I wrote that.

The probability will remain the same.  At work I have a spreadsheet with probabilities for different attacking and defending units, I will bring it home if I can remember.

Kjeld wrote:

Your probability of taking the territory is, I think, constant regardless of how you distribute the attacking units across attacking territories. Definitely your expected value is constant; I don't quite know the math for determining the actual probability of taking the territory (it seems to involve nested binomials), but I ran a simulation with some simple numbers and it seemed to work out evenly.

That said, you always want to try to attack with a single large stack instead of multiple smaller stacks to minimize your own losses. If you have to split the attack due to territory maximums (as in AnA), it's generally best to maximize the biggest stack you can and attack with it last. Sometimes the tactical positions of territories and attack timing will offer incentives to do otherwise, but I think it's a good rule of thumb.

So for your AnA example, say that Germany was attacking Russia with 58 from Archangel, West Russia, and Caucasus. You'd want to max out Caucasus at 30, max out Archangel/W. Russia at 20, and then put the remaining 8 on the other. You would attack 8, 20, 30 in that order to minimize defender losses.

Note: I didn't notice the difference between the Bomber scenario territory maxes and the original board when I wrote that.

Cool, thanks Kjeld.

I'm guessing you meant minimize "attacker" losses.  Good to know that it doesn't change the capture probability, just how many you burn through. 

Yeah, Thingol put Orel-Kursk in between (not a historically good territory for Germany), so now Russia is surrounded by 20, 20, 20, 15, 15, 15. 

So with the Capitol bonus, if Russia is full, a three state attack with bombers is only a little better than a coin flip, and if Russia also has bombers loaded to hit one of the three you need 4 to be sure.

Caucuses are now a little more exposed, but cannot be used as a knockout punch on Moscow.  So now Russia either has to mess up by using up it's troops, or get hit hard a couple turns in a row for the capitol to fall. 

SquintGnome wrote:

The probability will remain the same.  At work I have a spreadsheet with probabilities for different attacking and defending units, I will bring it home if I can remember.

Cool. I also noticed no one answered your last comment on the other thread about unfilled orders. Did that ever get answered?

Kjeld wrote:

You would attack 8, 20, 30 in that order to minimize defender losses.

If you're the attacker then shouldn't you attack 30, 20, 8 to minimize losses?

In the 50% A and 50% D scenario if going against 20 units...

30 v 20, attacker loses 10, defender loses 15

20 v 5, attacker loses 2.5, defender loses 5

Net - attacker loses 12.5, defender loses 20

Compared to...

8 v 20, attacker loses 8, defender loses 4

20 v 16, attacker loses 8, defender loses 10

30 v 6, attacker loses 3, defender loses 6

Net - attacker loses 19, defender loses 20

Right?

itsnotatumor wrote:

SquintGnome wrote:

The probability will remain the same.  At work I have a spreadsheet with probabilities for different attacking and defending units, I will bring it home if I can remember.

Cool. I also noticed no one answered your last comment on the other thread about unfilled orders. Did that ever get answered?

I think they were answered in that thread right? Also with the question How are orders executed by the system? here: http://www.wargear.net/help/display/FAQ#SGOrders

Yertle wrote:

itsnotatumor wrote:

SquintGnome wrote:

The probability will remain the same.  At work I have a spreadsheet with probabilities for different attacking and defending units, I will bring it home if I can remember.

Cool. I also noticed no one answered your last comment on the other thread about unfilled orders. Did that ever get answered?

I think they were answered in that thread right? Also with the question How are orders executed by the system? here: http://www.wargear.net/help/display/FAQ#SGOrders

Ahah! But, you have to click on page two, like I'd notice page 2. I don't listen to B-sides either.

Wait, there's a help menu? Look a giant bird! (Turns and walks away quickly)...

Yertle, it depends on what your goal is. If your goal is to minimize loss, then, yes, you would want to attack 30, 20, 8.

If your goal is to capture the territory and have the most units there - for defense or to further your attack next turn - you are better off attacking smallest to largest. Although, in the stated example, you would do best by attacking 20, 30, 8. That would give you, on average, 33 units in the target territory.

Yertle wrote:

Kjeld wrote:

You would attack 8, 20, 30 in that order to minimize defender losses.

If you're the attacker then shouldn't you attack 30, 20, 8 to minimize losses?

In the 50% A and 50% D scenario if going against 20 units...

30 v 20, attacker loses 10, defender loses 15

20 v 5, attacker loses 2.5, defender loses 5

Net - attacker loses 12.5, defender loses 20

Compared to...

8 v 20, attacker loses 8, defender loses 4

20 v 16, attacker loses 8, defender loses 10

30 v 6, attacker loses 3, defender loses 6

Net - attacker loses 19, defender loses 20

Right?

Serious again, attack order is dependent on the number of defenders, hit percentage, (ignoring possibility of friendly fortify), etc. The original premise was attacking 37 with 58, with defender hitting at 85% attacker 65&.

In that scenario

30 v 37, attacker loses 30 (all), defender loses 19.5

20 v 17, attacker loses 14.45, defender loses 13

6 v 4, attacker loses 3.4, defender loses 3.9 (all)

Net - attacker loses 47.85, defender loses 37

8 v 37, attacker loses 8, defender loses 5.2

20 v 32, attacker loses 20 (all), defender loses 13

30 v 19, attacker loses 16.5, defender loses 19.5 (All)

Net - attacker loses 44.5, defender loses 37

If I did the math right.

Hitting with the 20 first is right for my scenario too…

20 v 37, attacker loses 20 (All), defender loses 13

30 v 24, attacker loses 20.4, defender loses 19.5

8 v 4, attacker loses 3.4, defender loses 4 (All)

Net - attacker loses 43.8, defender loses 37

If I did the math right.

itsnotatumor wrote:

Yertle wrote:

itsnotatumor wrote:

SquintGnome wrote:

The probability will remain the same.  At work I have a spreadsheet with probabilities for different attacking and defending units, I will bring it home if I can remember.

Cool. I also noticed no one answered your last comment on the other thread about unfilled orders. Did that ever get answered?

I think they were answered in that thread right? Also with the question How are orders executed by the system? here: http://www.wargear.net/help/display/FAQ#SGOrders

Ahah! But, you have to click on page two, like I'd notice page 2. I don't listen to B-sides either.

I don't even have a Page 2...you have different Settings :P