Attention mathematical geniuses...

[the liquidator]

So let's suppose you're in a completely defensive mode,just trying not to be eliminated--let's say with 12 units left over 4 contiguous territories. What's the best arrangement of those units to have the best odds of survival--should you put 9 on one & 1 on the others? 3 units on each of the 4? Some other combination?

[tahitiwahini]

So let's suppose you're in a completely defensive mode,just trying not to be eliminated--let's say with 12 units left over 4 contiguous territories. What's the best arrangement of those units to have the best odds of survival--should you put 9 on one & 1 on the others? 3 units on each of the 4? Some other combination?

You want a minimum of two armies on each territory because that guarantees that you will throw two dice at least once while defending that territory. The other principle that comes into play is you want to be strongest when your attacker is weakest. The corollary being you want to be your weakest when your attacker is strongest. I'm also assuming you have no other goals to complicate matters, in other words, you're not trying to hold onto a continent bonus or anything like that.

OK, given all that you want to start out with two armies on each territory. That uses up 8 of your 12 available armies. Where do you put your last 4 armies? As far away from your attacker as you can. Ideally, the attacker would have to conquer each of your countries with 2 armies each on them before he got to attack your country with 2+4=6 armies on it.

An example to illustrate from the classic map:

You have 12 armies and you own four countries: Central America, Venezuela, Brazil, and North Africa. Your strongest attacker is located in Egypt. You put 2 armies on North Africa, 2 armies on Brazil, and 2 armies on Venezuela, and 6 armies on Central America.

Let's say the attacker has 15 armies on Egypt, what are his odds of conquering all your territories? Your attacker has a 54.40% chance of eliminating you.

Why is this the optimal defense?

Two reasons. First, it puts you the defender in the position of throwing two dice the maximum number of times possible. Second, it puts you in a position to be throwing two dice when your opponent is at his weakest. The defender deployed as described above is employing a defense in depth strategy. The armies deployed in the first three countries (North Africa, Brazil, and Venezuela) are essentially sacrificing themselves in support of as strong a last-ditch stand as possible in Central America. Even if the attacker loses no men to battle he will still lose men when he advances because he must leave behind one man for every country he advances from. So if everything goes the attacker's way he will still be down 3 armies from the 15 armies strong he started with in Egypt.

Notice that while this is the strongest defense, it is not necessarily the best thing to do overall. By being in such a strong defensive position you are in a bad position to go on the offense. Your troops are for the most part spread out rather than concentrated, and where they are concentrated you are (purposely) far away from the enemy. That caveat aside, it is the optimal defensive deployment.

[oVo]

Another tactic --if one territory is centered and touching all the others-- is to pile all your armies on the one at the center, giving you as much retaliation power possible if your attacker merely takes one country.

[tahitiwahini]

Another tactic --if one territory is centered and touching all the others-- is to pile all your armies on the one at the center, giving you as much retaliation power possible if your attacker merely takes one country.

Yes, that's quite a useful technique. When it's used to keep a continent bonus I've called it the sully defense because that's the person who first brought it to my attention. While it's not as strong a defense against elimination (if there is an actual attack) as the earlier defensive deployment, it does have a number of things going for it. One you've already pointed out is that you are poised for a devastating counter-attack if the initial attack isn't fatal.

Another is that by attacking in that configuration the attacker is exposing himself to the greatest concentration of your forces. There's a psychological reluctance to doing that and so sometimes in the right circumstances the sully defense will actually deter the enemy from making an attack against your forces at all.

So while it may not be statistically superior if there is an attack, a defensive posture that can make an attack less likely to be carried out has a great deal to recommend it. If you can afford to surround your concentration of forces with countries held by two armies each you are effectively combining the best features of a defense in depth with the threat of devastating counter-attack that is the mark of the sully defense. Depending on the size of the attacking force this setup can almost be described as a sort of venus flytrap affair where the attacking army is drawn into battle against the very heart of your defenses. Or instead is perhaps deterred from making an attack in the first place. Either way a very important defensive technique to know.

[Captain Crash]

You want a minimum of two armies on each territory because that guarantees that you will throw two dice at least once while defending that territory. The other principle that comes into play is you want to be strongest when your attacker is weakest. The corollary being you want to be your weakest when your attacker is strongest.

Another tactic --if one territory is centered and touching all the others-- is to pile all your armies on the one at the center, giving you as much retaliation power possible if your attacker merely takes one country.

And that ladies and gents is the difference between tactical/strategic play and statistics/probability.

Good luck to you which ever technique you employ...unless it is against me in which case may I have all the good luck!

[IronE.GLE]

So let's suppose you're in a completely defensive mode,just trying not to be eliminated--let's say with 12 units left over 4 contiguous territories. What's the best arrangement of those units to have the best odds of survival--should you put 9 on one & 1 on the others? 3 units on each of the 4? Some other combination?

You want a minimum of two armies on each territory because that guarantees that you will throw two dice at least once while defending that territory. The other principle that comes into play is you want to be strongest when your attacker is weakest. The corollary being you want to be your weakest when your attacker is strongest. I'm also assuming you have no other goals to complicate matters, in other words, you're not trying to hold onto a continent bonus or anything like that.

OK, given all that you want to start out with two armies on each territory. That uses up 8 of your 12 available armies. Where do you put your last 4 armies? As far away from your attacker as you can. Ideally, the attacker would have to conquer each of your countries with 2 armies each on them before he got to attack your country with 2+4=6 armies on it.

An example to illustrate from the classic map:

You have 12 armies and you own four countries: Central America, Venezuela, Brazil, and North Africa. Your strongest attacker is located in Egypt. You put 2 armies on North Africa, 2 armies on Brazil, and 2 armies on Venezuela, and 6 armies on Central America.

Let's say the attacker has 15 armies on Egypt, what are his odds of conquering all your territories? Your attacker has a 54.40% chance of eliminating you.

Why is this the optimal defense?

Two reasons. First, it puts you the defender in the position of throwing two dice the maximum number of times possible. Second, it puts you in a position to be throwing two dice when your opponent is at his weakest. The defender deployed as described above is employing a defense in depth strategy. The armies deployed in the first three countries (North Africa, Brazil, and Venezuela) are essentially sacrificing themselves in support of as strong a last-ditch stand as possible in Central America. Even if the attacker loses no men to battle he will still lose men when he advances because he must leave behind one man for every country he advances from. So if everything goes the attacker's way he will still be down 3 armies from the 15 armies strong he started with in Egypt.

Notice that while this is the strongest defense, it is not necessarily the best thing to do overall. By being in such a strong defensive position you are in a bad position to go on the offense. Your troops are for the most part spread out rather than concentrated, and where they are concentrated you are (purposely) far away from the enemy. That caveat aside, it is the optimal defensive deployment.

That is one hell of a way to say common sense would dictate 2 armies on each country.

[alster]

Basically, the only difference is between 1 and 2 defensive armies. After that. the defensive dice odds doesn't really go up. So, once you have 2 armies piled up, your odds of survival is as high as it gets.

[chessplaya]

So let's suppose you're in a completely defensive mode,just trying not to be eliminated--let's say with 12 units left over 4 contiguous territories. What's the best arrangement of those units to have the best odds of survival--should you put 9 on one & 1 on the others? 3 units on each of the 4? Some other combination?

You want a minimum of two armies on each territory because that guarantees that you will throw two dice at least once while defending that territory. The other principle that comes into play is you want to be strongest when your attacker is weakest. The corollary being you want to be your weakest when your attacker is strongest. I'm also assuming you have no other goals to complicate matters, in other words, you're not trying to hold onto a continent bonus or anything like that.

OK, given all that you want to start out with two armies on each territory. That uses up 8 of your 12 available armies. Where do you put your last 4 armies? As far away from your attacker as you can. Ideally, the attacker would have to conquer each of your countries with 2 armies each on them before he got to attack your country with 2+4=6 armies on it.

An example to illustrate from the classic map:

You have 12 armies and you own four countries: Central America, Venezuela, Brazil, and North Africa. Your strongest attacker is located in Egypt. You put 2 armies on North Africa, 2 armies on Brazil, and 2 armies on Venezuela, and 6 armies on Central America.

Let's say the attacker has 15 armies on Egypt, what are his odds of conquering all your territories? Your attacker has a 54.40% chance of eliminating you.

Why is this the optimal defense?

Two reasons. First, it puts you the defender in the position of throwing two dice the maximum number of times possible. Second, it puts you in a position to be throwing two dice when your opponent is at his weakest. The defender deployed as described above is employing a defense in depth strategy. The armies deployed in the first three countries (North Africa, Brazil, and Venezuela) are essentially sacrificing themselves in support of as strong a last-ditch stand as possible in Central America. Even if the attacker loses no men to battle he will still lose men when he advances because he must leave behind one man for every country he advances from. So if everything goes the attacker's way he will still be down 3 armies from the 15 armies strong he started with in Egypt.

Notice that while this is the strongest defense, it is not necessarily the best thing to do overall. By being in such a strong defensive position you are in a bad position to go on the offense. Your troops are for the most part spread out rather than concentrated, and where they are concentrated you are (purposely) far away from the enemy. That caveat aside, it is the optimal defensive deployment.

LOL srsly get a job

[Griefor]

Wouldn't it make more sense to put all your troops on the same territory so that there is a nice huge discouraging to attack that particular territory? A territory with 2 troops on it is almost as inviting as one with 1, you want 1 single territory that nobody is interested in because it's too well-defended.

The 2-per territory rule assumes that someone is going to put full effort in killing you off regardless of the situation. And unless there's some quick cards or continents to be earned, the weak player is the last guy I want to attack.

[ParadiceCity9]

probably 3 on each

[ParadiceCity9]

Wouldn't it make more sense to put all your troops on the same territory so that there is a nice huge discouraging to attack that particular territory? A territory with 2 troops on it is almost as inviting as one with 1, you want 1 single territory that nobody is interested in because it's too well-defended.

The 2-per territory rule assumes that someone is going to put full effort in killing you off regardless of the situation. And unless there's some quick cards or continents to be earned, the weak player is the last guy I want to attack.

but i agree with that as well...im doing that in a montreal game..ive got like 18 or so on one country and i have 19 total armies..no1's attacking me

[Hologram]

Another tactic --if one territory is centered and touching all the others-- is to pile all your armies on the one at the center, giving you as much retaliation power possible if your attacker merely takes one country.

Indeed. I used that tactic on classic once where I made some huge mistakes early on and never recovered. I could have died out early, but I consolidated all of my armies on N. Europe (all of my territories were in the area) and I would simply pick off any especially weak territories (namely 1 or 2 versus my pile of 10 or so). I was eventually able to make a counter attack to keep a player from bulldozing me with the 40 or so armies he would have gotten with a pile of cards.

[alster]

Wouldn't it make more sense to put all your troops on the same territory so that there is a nice huge discouraging to attack that particular territory? A territory with 2 troops on it is almost as inviting as one with 1, you want 1 single territory that nobody is interested in because it's too well-defended.

The 2-per territory rule assumes that someone is going to put full effort in killing you off regardless of the situation. And unless there's some quick cards or continents to be earned, the weak player is the last guy I want to attack.

Well. The question was about the mathematical best way to place troops:

So let's suppose you're in a completely defensive mode,just trying not to be eliminated--let's say with 12 units left over 4 contiguous territories. What's the best arrangement of those units to have the best odds of survival--should you put 9 on one & 1 on the others? 3 units on each of the 4? Some other combination?

And mathematical wise, the defense is more likely to succeed with two troops rather than one. But, there's no extra benefit (seeing each attack separately) going from two to three troops etc.

[Georgerx7di]

It makes no difference. People think it does, but it doesn't. either way four of your armies will get attacked as singles and roll one die. On the other hand, some arrangements may have a pscycological impact, ie look stronger, but none will give you better odds with the dice.

[Smurf75]

It makes no difference. People think it does, but it doesn't. either way four of your armies will get attacked as singles and roll one die. On the other hand, some arrangements may have a pscycological impact, ie look stronger, but none will give you better odds with the dice.

Thats what i thought too, and was prepared to question tahitiwahini. Then I realized, attacking 3 armies against 1 has the best odds (thats pretty obvious). And with 2 armies on each you are likely to get a 3-1 attack after the initial 3vs2 attack... which ofcourse is not good. But having just a single army on your countries means theres a 3vs1 from start. With 2vs2 you still have a shot of taking 2 armies from your attacker, and then hope your 2 armies get shot down at once. Heche with tahitiwahini's plan of 2-2-2-6 youll likely to get 1-4 3vs1, with 1-1-1-9 (which was what I before this thread thought was most efficent) will give 3-4 3vs1.
But still, most important for survival is have your strongest troop furthest back since your attacker by then will have the least dice left.

Another great strat for survival is(if possible) to force your attacker to split his big force, having two countries fortified with the same amount of troops and atleast 2 countries in between.

[Griefor]

then hope your 2 armies get shot down at once


Heheh... Now that's an odd thing to hope for.

Even if the consequences of getting a draw (1 dies on each side) are a little worse than usual (you end up with 1 die), I still prefer it over a loss.

The explanation is correct though. 2 armies on a country minimize the chances of getting caught with 1 army on the square, which means your chances of survival are best.

Matter of fact, 2-2-2-6 works (mathematically) better than anything, even something like 3-3-3-3.

Knowing that we want to avoid single-die situations:
1 army gives a 100% chance of getting to single-die
2 armies give single-die if you get a standoff
3+ armies give single-die if you get a standoff at 2 armies AND give single-die if you get a loss at 3.

So 3+ has a higher chance of getting to single-die than 2 does!

[Smurf75]

then hope your 2 armies get shot down at once


Heheh... Now that's an odd thing to hope for.

I want my games to be fair, hence if my attacker looses 2 armies, I would like so myself

[mibi]

i would think that 4 armies are each area would be best. because with 4 armys you guarantee two rolls with both dice, of course 5 on a territory will guarantee two rolls as well, but 6 will guarantee 3 rolls. 2 armies on a territory will only guarantee one. so it seems that a 4-4-4-4 defense better than a 2-2-4-8.

of course i may not know what im talking about, but it seems that even numbers are best, and according to my own internal logic, its better to leave more than 2 on a territory.

[maniacmath17]

If you want to know the numbers... 2 2 2 6 would on average kill 9.4 of the attackers troops. The same can be said for having the 2 2 4 4 set up.

The 3 3 3 3 would only kill about 9.2 attacker troops because if you lose 2 on the first attack you're down to rolling 1 dice which is of course at a huge disadvantage. 1 1 1 9 is even worse at 9.0.

Defending with an even number of troops when deciding between defending with either 1 2 3 or 4 does give a slight advantage. Obviously when the numbers get bigger the difference is virtually nonexistent.

maniacmath17 has spoken so consider this discussion over.

[Skittles!]

.. I liked math, but there's too much here.

[Cachorro]

The best thing you can do to strenghten your defense is to put the 2nd army. The worst you can do is to put the 3rd. Without calculating the asked situation 2-2-2-6 is the best formation. 3-3-3-3 is the worst. The following is my text in some earlier discussion. The quote:

I calculated how many troops the attacker needs in average to wipe out your troops in defense. This assumes that the attackes can always attack with 3 dice.

Def - Att

1 - 0,52
2 - 1,55
3 - 2,31
4 - 3,21
5 - 4,04
6 - 4,90
7 - 5,75
8 - 6,61
9 - 7,46
10 - 8,31
11 - 9,17
12 - 10,02
13 - 10,87
14 - 11,73
15 - 12,58
16 - 13,43
17 - 14,29
18 - 15,14
19 - 15,99
20 - 16,85

From this you can calculate that the best way to use your single unit for defense is to increase the unit amount from one to two as this increases the average loss of the attacker by 1,03 units! Addind the amount from 2 to 3 only increases the average loss by 0,76.

All in all if you just have to defense you'd better put an even number of troops to defense to make it optimal. However, this applies relevant only up to 6 troops as after that the average loss of the attacker is 0,85 if the defense adds one.

Of course, this is just mathematics and the strategy might be sometimes different and you do not want to spread your troops. The only situation this is usable is the one when you have to defense for example Australia and you have 2 extra troops to spread over Siam and Borneo. How is the optimal way - that's some easy thinking for yourselves. Also, in some escalting card games you might want to defense optimally.

[the liquidator]

all very interesting--thanks to all respondents!