is it better to..

[mibi]

Ok here is a game play problem.

Lets say you have South America and north africa and an enemy has egypt. you have 4 armies to defend SA with. is better to put 3 armies on brazil and 1 in north africa or to put 2 on brazil and 2 in north africa guaranteeing turn rolls of 2 dice.

i suppose a more general question is do people use two armies on a country as a way to soften an approaching enemy or do they wait until the enemy hits a wall of your armies?

[tahitiwahini]

is better to put 3 armies on brazil and 1 in north africa or to put 2 on brazil and 2 in north africa guaranteeing turn rolls of 2 dice.

Good question, but I think you hit upon the answer already.

It's always better to place 2 in Brazil and 2 in North Africa, than 1 in Brazil and 3 in North Africa.

The reason is the one you gave already, a 2,2 defense guarantees you will get a chance to roll two defender dice twice, while you have no such guarantee with a 3,1 defense.

To quantify it:

Let's say there are 6 attacking armies on Egypt. The plan is to attack North Africa and then Brazil.

Brazil(2) <----- North Africa(2) <----- Egypt(6)
The attacker has a 51% chance of conquering both countries (North Africa and Brazil) if the defender deploys a 2,2 defense.

Brazil(3) <----- North Africa(1) <----- Egypt(6)
The attacker has a 55% chance of conquering both countries if the defender deploys a 1,3 defense (1 up front, 3 in the rearl).

Brazil(1) <----- North Africa(3) <----- Egypt(6)
The attacker has a 60% chance of conquering both countries if the defender deploys a 3,1 defense (3 up front, 1 in the rear).

So the best thing for the defender is to put 2 in each country, the next best thing is to put 1 in front and 3 in the rear, and the worst thing is for the defender to put 3 up front and 1 in the rear.

This is true regardless of how many attacking armies are in Egypt. The differences in probabilities of the attacker's success among the three defense strategies will narrow as the size of the attacking force grows, but it is always the case that 2,2 beats 1,3 which beats 3,1.

Let's say there are 11 attacking armies on Egypt. The plan is to attack North Africa and then Brazil.

Brazil(2) <----- North Africa(2) <----- Egypt(11)
The attacker has a 94.28% chance of conquering both countries (North Africa and Brazil) if the defender deploys a 2,2 defense.

Brazil(3) <----- North Africa(1) <----- Egypt(11)
The attacker has a 95.16% chance of conquering both countries if the defender deploys a 1,3 defense (1 up front, 3 in the rear).

Brazil(1) <----- North Africa(3) <----- Egypt(11)
The attacker has a 95.78% chance of conquering both countries if the defender deploys a 3,1 defense (3 up front, 1 in the rear).

[mibi]

thanks for the great reply.

i get the 2-2 and 3-1 defense, but what if you have more armies.

say, is there a difference between an 18-2 defense and a 10-10 defense? or does it not matter unitl it gets down to 1 army to through the percentages off.

[alex_white101]

is better to put 3 armies on brazil and 1 in north africa or to put 2 on brazil and 2 in north africa guaranteeing turn rolls of 2 dice.

Good question, but I think you hit upon the answer already.

It's always better to place 2 in Brazil and 2 in North Africa, than 1 in Brazil and 3 in North Africa.

The reason is the one you gave already, a 2,2 defense guarantees you will get a chance to roll two defender dice twice, while you have no such guarantee with a 3,1 defense.

To quantify it:

Let's say there are 6 attacking armies on Egypt. The plan is to attack North Africa and then Brazil.

Brazil(2) <----- North Africa(2) <----- Egypt(6)
The attacker has a 51% chance of conquering both countries (North Africa and Brazil) if the defender deploys a 2,2 defense.

Brazil(3) <----- North Africa(1) <----- Egypt(6)
The attacker has a 55% chance of conquering both countries if the defender deploys a 1,3 defense (1 up front, 3 in the rear).

Brazil(1) <----- North Africa(3) <----- Egypt(6)
The attacker has a 60% chance of conquering both countries if the defender deploys a 3,1 defense (3 up front, 1 in the rear).

So the best thing for the defender is to put 2 in each country, the next best thing is to put 1 in front and 3 in the rear, and the worst thing is for the defender to put 3 up front and 1 in the rear.

This is true regardless of how many attacking armies are in Egypt. The differences in probabilities of the attacker's success among the three defense strategies will narrow as the size of the attacking force grows, but it is always the case that 2,2 beats 1,3 which beats 3,1.

Let's say there are 11 attacking armies on Egypt. The plan is to attack North Africa and then Brazil.

Brazil(2) <----- North Africa(2) <----- Egypt(11)
The attacker has a 94.28% chance of conquering both countries (North Africa and Brazil) if the defender deploys a 2,2 defense.

Brazil(3) <----- North Africa(1) <----- Egypt(11)
The attacker has a 95.16% chance of conquering both countries if the defender deploys a 1,3 defense (1 up front, 3 in the rear).

Brazil(1) <----- North Africa(3) <----- Egypt(11)
The attacker has a 95.78% chance of conquering both countries if the defender deploys a 3,1 defense (3 up front, 1 in the rear).

well i took this all very seriously, until i reached the large bit, which to my immature sense of humour made me laugh for quite a while.....

[reverend_kyle]

is better to put 3 armies on brazil and 1 in north africa or to put 2 on brazil and 2 in north africa guaranteeing turn rolls of 2 dice.

Good question, but I think you hit upon the answer already.

It's always better to place 2 in Brazil and 2 in North Africa, than 1 in Brazil and 3 in North Africa.

The reason is the one you gave already, a 2,2 defense guarantees you will get a chance to roll two defender dice twice, while you have no such guarantee with a 3,1 defense.

To quantify it:

Let's say there are 6 attacking armies on Egypt. The plan is to attack North Africa and then Brazil.

Brazil(2) <----- North Africa(2) <----- Egypt(6)
The attacker has a 51% chance of conquering both countries (North Africa and Brazil) if the defender deploys a 2,2 defense.

Brazil(3) <----- North Africa(1) <----- Egypt(6)
The attacker has a 55% chance of conquering both countries if the defender deploys a 1,3 defense (1 up front, 3 in the rearl).

Brazil(1) <----- North Africa(3) <----- Egypt(6)
The attacker has a 60% chance of conquering both countries if the defender deploys a 3,1 defense (3 up front, 1 in the rear).

So the best thing for the defender is to put 2 in each country, the next best thing is to put 1 in front and 3 in the rear, and the worst thing is for the defender to put 3 up front and 1 in the rear.

This is true regardless of how many attacking armies are in Egypt. The differences in probabilities of the attacker's success among the three defense strategies will narrow as the size of the attacking force grows, but it is always the case that 2,2 beats 1,3 which beats 3,1.

Let's say there are 11 attacking armies on Egypt. The plan is to attack North Africa and then Brazil.

Brazil(2) <----- North Africa(2) <----- Egypt(11)
The attacker has a 94.28% chance of conquering both countries (North Africa and Brazil) if the defender deploys a 2,2 defense.

Brazil(3) <----- North Africa(1) <----- Egypt(11)
The attacker has a 95.16% chance of conquering both countries if the defender deploys a 1,3 defense (1 up front, 3 in the rear).

Brazil(1) <----- North Africa(3) <----- Egypt(11)
The attacker has a 95.78% chance of conquering both countries if the defender deploys a 3,1 defense (3 up front, 1 in the rear).

Everyone knows.. ALWAYS 2 up front and 1 in the rear.

[reverend_kyle]

fastposted sort of.

[alex_white101]

by almost an hour?

[Molacole]

thanks for the great reply.

i get the 2-2 and 3-1 defense, but what if you have more armies.

say, is there a difference between an 18-2 defense and a 10-10 defense? or does it not matter unitl it gets down to 1 army to through the percentages off.

it comes down to what you're trying to accomplish.

If you're trying to hold onto africa for a amount of territories bonus then fort it up heavy or even if you want to just prevent them from getting a card.

If you're trying to protect your bonus then keep the numbers even like 2-4, 2-6, 4-10 or even 2-16 with the smaller number on north africa. It's dependant on how many troops they have in attacking position.

I personally like to put all my men in their face and would leave brazil with 2 troops on it at the most. If they attack and have a bad run at the dice they'll get destroyed. If they get lucky dice then it wouldn't matter how your troops were positioned. You want all the troops you want to be able to attack with in north africa so if you see an oportunity open up you'll be ready to take advantage over it. Like eliminating another player...

I would never split my troops in half like a 10-10 because you're just opening the door for attacks against you. If I wanted them to attack me in hopes they make themselves weaker I would leave something like a 16-4 with 4 on north africa or maybe even 14-6. You don't want to handicap yourself. If you split your troops 10-10 all you're doing is giving them persmission to test out the dice on your army. If they get good rolls they can continue and if they don't get good rolls then they can stop and you wont have as much to attack them with from north africa until you can fort into attacking position from brazil.

[St John]

So the best thing for the defender is to put 2 in each country, the next best thing is to put 1 in front and 3 in the rear, and the worst thing is for the defender to put 3 up front and 1 in the rear.

I get that 2 by 2 deployment is the better. But why is it better to deploy 1 in North AFrica and 3 in Brazil than vice versa? The Egyptian attacker has to overcome 1 army and 3 armies in two consecutive battles, what difference does it make in which order they are fought?

And by placing 3 in North Africa you also make it harder for the opponent to gain any land at all, so why place only 1 there?

[tahitiwahini]

So the best thing for the defender is to put 2 in each country, the next best thing is to put 1 in front and 3 in the rear, and the worst thing is for the defender to put 3 up front and 1 in the rear.

I get that 2 by 2 deployment is the better. But why is it better to deploy 1 in North AFrica and 3 in Brazil than vice versa? The Egyptian attacker has to overcome 1 army and 3 armies in two consecutive battles, what difference does it make in which order they are fought?

And by placing 3 in North Africa you also make it harder for the opponent to gain any land at all, so why place only 1 there?

Good question.

It's all about the dice.

We understand why 2,2 is a better defense so I won't go into that again.

Why is 1,3 (front,rear) a better defense than 3,1?

There are several important thresholds when attacking. For attackers these thresholds are:

>3 armies: attack with 3 dice
3 armies: attack with 2 dice
2 armies: attack with 1 die
1 army: can't attack anymore

For the defender they are:

>1 army: defend with 2 dice
1 army: defend with 1 die

Let's see what happens in a 6 army attack against a 1,3 defense:

The first attack is a 6 v 1.

These are the worst possible odds for the defender. Since it's the first attack the attacker is at the zenith of his strength (he will never be stronger). In our scenario you will have to endure one of these 3 v 1 dice attacks at some point. Why do you want to take it up front when the attacker is at his strongest? Because essentially the attacker's strength at this point is really overkill. In other words, the attacker's odds in a 6 v 1 attack are the same as his odds in a 5 v 1, which are the same as a 4 v 1 (it's not until 3 v 1 that the attacker's odds drop due to the loss of the third die).

Look what happens as a result of the first attack. Either you lose a defender or the attacker loses an attacker.

Case 1: you lose a defender. In the first case, you suffer a loss, but at these odds you really expect to. What happens next? Well the attacker advances all but one of his armies into the conquered country. So even though he "won" his strength has just been reduced by 1 army.

Case 2: attacker loses an attacker. Good for you, the attacker's strength has been reduced by 1 and now he has to go through another attack. The best the attacker can hope for on the second attack is Case 1. We know from case 1 that the attacker will lose a further army in the advance into the conquered country. So best case, the attacker will lose at least 2 armies, possibly more.

So the effect of the 3,1 defense is that even in the worst case scenario you have reduced the attacker's strength before he has to face your strongest defense. When I say strongest defense I'm not just thinking in terms of number of armies but how far you as a defender are away from crossing your thresholds. When I say the attacker's strength has been reduced I'm not just thinking in terms of number of armies but how much closer he is as an attacker to crossing his thresholds.

So, best case in conquering the first country the attacker loses an army (in the advance). Now let's consider the second country.

Here we have a 5 v 3 attack.

There are 3 cases for the outcome of the attack.

Case 1a: attacker loses 2 armies. Great for you because the attacker has just passed through one of his thresholds, he will lose one of his dice on the next attack.

Case 2a: attacker loses 1 army, defender loses 1 army. Leaves attacker with 4 armies and you with 2 armies. If the attacker loses an army in the next attack, he will lose one of his dice on the next attack.

Case 3a: defender loses 2 armies. Bad for you, now you lose a die for the next attack.

Remember if we were in a 3,1 defense the attacker would be facing these outcomes with an additional army cushion (i.e., he would be making this attack against our strongest defense with 6 armies instead of 5). That additional army cushion keeps the attacker away from crossing his thresholds and that's a bad thing for us, we want our attacker to be closer to crossing his thresholds -- we don't like the additional army cushion! Instead of the attacker being one army loss away from crossing his threshold in Case 2a he would be two army losses away from losing one of his attacking dice. Not a good thing!

The point is this: As the defender you want to put the attacker in a position where he is starting to cross his thresholds before you do. By putting one army up front (essentially a sacrifice) you whittle down the attacker's strength before he faces your strongest defense (the 3 in the rear).

In the 3,1 (front, rear) defense you allow the defender to attack your strongest defense at a point when his strength is the strongest. What that means is that he will be more likely to stay above his thresholds (which allow him to throw more dice than you) for the series of attacks against your strongest defense.

Finally, your last point is correct, a 3,1 defense is your best shot at not losing any territory, if that's what's really important to you. But presumably the overriding concern is to prevent you from losing your South America bonus. You've already lost your Africa bonus (if you even had it to begin with) because you're being attacked from Egypt.

[tahitiwahini]

thanks for the great reply.

i get the 2-2 and 3-1 defense, but what if you have more armies.

say, is there a difference between an 18-2 defense and a 10-10 defense? or does it not matter unitl it gets down to 1 army to through the percentages off.

Here's the rule: put 2 armies on each territory and then the remainder available in the rear.

Purely from a mathematical point of view it is always better in defense to have your greatest strength in the rear, rather than in the front. There may be other strategic concerns that outweigh this rule, but all other things being equal (that is, in the absence of a compelling counter-reason), the rule is the optimal way to deploy your defenses.

Let's take the example of a 21 army attack against 20 defenders spread between 2 countries.

Following the rule we would:

Brazil(18) <--- North Africa(2) <-- Egypt(21)
The attacker would conquer both countries 58.67% of the time.

Brazil(10) <--- North Africa(10) <-- Egypt(21)
In an intermediate deployment the attacker would conquer both countries 59.03% of the time.

Brazil(2) <--- North Africa(18) <-- Egypt(21)
In a front-loaded deployment the attacker would conquer both countries 59.71% of the time.

In the most foolish deployment:
Brazil(1) <--- North Africa(19) <-- Egypt(21)
The attacker would conquer both countries 62.33% of the time. Foolish both because it is front-loaded and because it doesn't put 2 defenders on each country.

[St John]

Thanks, tahitiwahini!

Your reasoning is sound, I actually think I will change my defensive strategies in such scenarios accordingly.

P.S.
Did you see that dugcarr1 called you tinniwinni in gamechat 323797? If I should happen to call you that in a future post, please don't take offence. I find that "tinniwinni" is an amusing name, especially coming from our infamous (and probably fast typing) friend dugcarr1.

[tahitiwahini]

Did you see that dugcarr1 called you tinniwinni in gamechat 323797? If I should happen to call you that in a future post, please don't take offence. I find that "tinniwinni" is an amusing name, especially coming from our infamous (and probably fast typing) friend dugcarr1.

Yes, I rather liked it. Sort of struck me as a diminutive of my full name; you know kind of a term of endearment.

I never took offense at anything Dug said (although I could do without the profanity which I find tiresome), I only took offense at what he did. Specifically the harm he did to other players and to the site which I have a great deal of affection for.

[yeti_c]

Well bugger me - Tinniwinni - you've come up with one of the most interesting posts I've read on this site...

Great bit of insight here!!

C.

[detlef]

That is some very nice work. However, considering the very slim defensive advantage in all situations but when all players have very few armies, I don't think it's worth it considering the degree to which it handicaps your ability to attack.

In the case where we are talking about upwards of 20 armies, that is a whole lot to devote to nothing more than protect a 2 pt bonus. Your options are extremely limited when you bury such large armies behind your front lines.

More than a slight percentage gain in successful defense, I think the decision should be made based on what type of message you are trying to send or what you're trying to accomplish. If it appears that everyone is beating up on each other and you want to try to stay out of the fray, then burying your armies behind N Africa is a great way to make yourself look very unattractive and somewhat harmless. Of course, you may miss on the opportunity to eliminate a player for his cards if you have to waste a turn moving your men into attack position.

Quite rarely are you fortunate enough to play against someone foolish enough to attack that front line and expose themselves to your big army.

[tahitiwahini]

That is some very nice work. However, considering the very slim defensive advantage in all situations but when all players have very few armies, I don't think it's worth it considering the degree to which it handicaps your ability to attack.

In the case where we are talking about upwards of 20 armies, that is a whole lot to devote to nothing more than protect a 2 pt bonus. Your options are extremely limited when you bury such large armies behind your front lines.

More than a slight percentage gain in successful defense, I think the decision should be made based on what type of message you are trying to send or what you're trying to accomplish. If it appears that everyone is beating up on each other and you want to try to stay out of the fray, then burying your armies behind N Africa is a great way to make yourself look very unattractive and somewhat harmless. Of course, you may miss on the opportunity to eliminate a player for his cards if you have to waste a turn moving your men into attack position.

Quite rarely are you fortunate enough to play against someone foolish enough to attack that front line and expose themselves to your big army.

All very good points. The defensive deployment does matter more when you're dealing with fewer armies because you are then playing closer to the thresholds. The larger the number of armies the less important the defensive deployment becomes.

The question I was looking at was: what is the theoretically best defense given the deployment of armies over two countries? To answer that question I was looking at the defense in isolation. In an actual game you can never consider defense in isolation. You must be concerned about offense as well. So your point is valid, the small tactical advantage of a defensive deployment may be far outweighed by the negative consequences the defensive deployment has on attacks going forward, especially the opportunity cost of not being able to eliminate an opponent as you pointed out.

Psychological factors are extremely important as well. How you are perceived by your opponent may mean you needn't be concerned with the best chance of surviving an attack because through psychological manipulation you may prevent the attack from happening at all. It's best to achieve your goal without battle since battle is costly even to the victor (I'm sure some famous strategist said this in a much more elegant way, but I can't remember who and what he said exactly).

Your last point just emphasizes the effectiveness of the rear-loaded defense. Those front line troops are in effect protected by the rear echelon. If your opponent passes on making an attack, you've successfully repelled a threat. If your opponent attacks the front-line troops and is unable to conquer the rear guard then the rear guard can be used to launch a successful counter-attack.

The fun in the game is balancing the competing interests of attack and defense, daring and restraint, so that you position yourself for the most successful outcome possible under the circumstances.

[tahitiwahini]

The same principle applies to attacking as well, especially to attacks leading to a player's elimination.

Back to the classic map for an example:

For the purpose of this example assume you want to conquer both countries (in this case because it will lead to the elimination of an opponent).

You are the attacker with 12 armies on North Africa. Your opponent is down to his last two countries (2 on Brazil, and 7 on Egypt). Which do you attack first and why?

You attack the 7 on Egypt first because you always want to attack where the enemy is strongest when you are strongest.

Let's look at the probabilities of successfully conquering both countries:

According to our rule we should attack Egypt first then Brazil. Our probability of successfully conquering both countries in a 12 v 7,2 attack is 67.6%.

If we were instead to attack Brazil first then Egypt, a 12 v 2,7 attack, our probability of success would be 66.4%.

So we get a 1.2% increase in our probability of success by attacking the stronger enemy position first.

[mibi]

So lets say a player has to go through 3 countries to get to your bonus.

and it looks like this Bonus-country-country-country-enemy.
and you have 10 armies
which is best from a defense standpoint

2-2-2-4
1-3-3-3
7-1-1-1
3-2-3-2

?

[tahitiwahini]

So lets say a player has to go through 3 countries to get to your bonus.

and it looks like this Bonus-country-country-country-enemy.
and you have 10 armies
which is best from a defense standpoint

2-2-2-4
1-3-3-3
7-1-1-1
3-2-3-2

?

Here's the rule: put 2 armies on each territory and then the remainder available in the rear.

So given bonus-country-country-country-enemy, the optimal defensive deployment of armies is 4-2-2-2 for the reasons given in my earlier posts in this thread.

For fun, here are the probabilities of the attacker being successful against each defensive deployment assuming the attacker has 12 armies:

2-2-2-4: 43.59%
1-3-3-3: 50.47
7-1-1-1: 47.26
3-2-3-2: 43.79

and the optimal deployment

4-2-2-2: 41.88%

and the worst possible deployment

1-1-1-7: 52.49%

So the optimal is better than the worst deployment by 10.61%.

The fewer the attacking armies the more important the deployment, the greater the attacking armies the less difference there will be between the best and worst deployments.

[mibi]

thanks!... your good with those numbers.