Strategic question: should I attack or defend?

[ORBOTRON]

I would be curious as to how the situation would unfold if you were able to attack using only one die, risking one army, then leaving two to defend once you lost that one army....

 

[Robinette]

I would be curious as to how the situation would unfold if you were able to attack using only one die, risking one army, then leaving two to defend once you lost that one army....

In the original RISK board game, you could choose to attack with one or two dice, and you would be risking that same number of armies. As a side note, when you would conquer a territory, you HAD to move in no less than the number of dice that you had rolled. Another interesting twist,, the defender could choose how many dice to use as well. Although, it is always better to defend with full force.

But to answer your question...

If you attacked with 1 dice, and the enemy defends with 2 dice... you will lose 75% of the time.
If you attack with 1 dice, and the enemy defended with 1 dice... you will lose 58% of the time.

So with 3 troops, your greatest strength will always be as a defender. Same with 2 troops.

 

[ORBOTRON]

yep, the original board rules were what I was thinking of when I asked

 

[Evan]

How do things change with 4+ troops? I'm pretty sure I've read elsewhere that you have overall betters odds attacking??

 

[Robinette]

How do things change with 4+ troops? I'm pretty sure I've read elsewhere that you have overall betters odds attacking??

Yes, rolling 3 dice vs 2 dice provides a slight advantage, even with the ties going to the defender.

37% chance you will win 2
34% chance you'll each lose 1
29% chance you will lose 2

So if you did this 100 times, the odds are such that you would end with 8 more men if you were the one attacking

 

[giuppi]

In a nutshell, the question was what will yield more damage to your opponent in the following situation:
You have 3 isolated troops being targeted by an enemy and you have decided not to commit any resources there: so to hurt your opponent the most, should you attack with the 3 troops, or defend.

I replied with the following:

if you do a single attack with your 3 troops, you have a 23% chance of killing 2
a 32% chance that you will each lose 1
and a 45% chance you will lose 2

if you wait to defend,
the 1st attack has a 29% chance of 2 dead attackers,
a 34% chance that you will each lose 1,
and a 37% chance that you will lose 2

The odds are against you either way, but defending provides you a greater advantage.
If you ran this same scenario 100 times, you would have 14 more men by defending than by attacking.

If we generalize your reasoning to as many attacks as possible in the specific configuration, we get this:

Case 1 - Attack with 3 troops against 4
Chances of killing 0 defending troops: 44.83%
Chances of killing 1 defending troop: 24.16%
Chances of killing 2 defending troops: 16.35%
Chances of killing 3 defending troops: 5.53%
Chances of killing 4 defending troops: 9.13%

For a total damage of 1.10 defending troops killed


Case 2 - Defend with 3 troops against 4
Chances of killing 0 attacking troops: 24.52%
Chances of killing 1 attacking troop: 14.96%
Chances of killing 2 attacking troops: 7.55%
Chances of killing 3 attacking troops: 52.98%
Chances of killing 4 attacking troops: 0% (impossible because the last remaining troop cannot attack)

For a total damage of 1.89 attacking troops killed


Therefore you cause more damage by defending.
But what if your enemy decides to attack with more than 4 troops? Well, interestingly, in that case he will likely lose EVEN MORE troops (2.07 out of 5, 2.18 out of 6, 2.29 out of 7, and so on) although with a decreasing marginal increment.

So, if we consider one 'wave' of attack, it's clear that the best strategy is to defend. So far so good.
But what if we consider a second wave of attack? In case 1 (you attack with 3) there are 90.87% chances that you don't conquer the territory, therefore your enemy can counterattack with more troops against whatever is left of yours (1 troop). As we saw before, this is a chance to inflict even more damage.
In case 2 (you defended 3 troops against 4) there are 52.97% chances that you are still alive by the end of the attack, so your enemy again will be back with more troops. Also in this case, more chances to inflict further damage. How much this 'second wave' potential damage is, I haven't figured out yet, but I suspect not enough to tip the scale in favor of attack vs defend.
And then by generalization there is a third wave, a fourth wave and so on. Anybody wants to calculate it?

Finally, just for fun, I've calculated the odds for a strategy made up by one-strike attack followed by defense.
So, you start with a 3vs4 attack, which can result in either 3-2 (best case), 2-3 or 1-4 (worst case). Whatever the result, you stop attacking, and just wait for the enemy to attack you (without reinforces). In this case, the total damage you can inflict to the opponent is 0.93 troops, which is worse than both pure defense and pure attack strategies.

 

[Incandenza]

Well, robin already busted out the maths, but generally speaking, the ONLY time you'd ever want to attack with fewer than 3 dice (i.e. with a 2 or a 3) is to A) break an opponent's bonus, B) get a card in an escalating game, or C) to kill someone, and even then it's much better attacking against a 1 than a 2+. Attacking with 3's one of the early things that can separate the men from the boys, strategically speaking.

 

[Evan]

Well, robin already busted out the maths, but generally speaking, the ONLY time you'd ever want to attack with fewer than 3 dice (i.e. with a 2 or a 3) is to A) break an opponent's bonus, B) get a card in an escalating game, or C) to kill someone, and even then it's much better attacking against a 1 than a 2+. Attacking with 3's one of the early things that can separate the men from the boys, strategically speaking.

ah ok excellent thanks. Maybe this is why some people didn't understand the nature of the question... because it is so rudimentary to the game that people with experience might think this is an obvious thing. Thanks inca.

 

[Incandenza]

The TL;DR version:

Attacking with less than 3 dice (i.e. with a 3 or a 2) is almost always a bad idea, unless you're trying to A) break an opponent's bonus, B) get a card in an escalating game, or C) to kill someone, and even then it's much better attacking against a 1 than a 2+.

Personally I've never quite had the math to really parse out the whole "expected damage" concept, tho I get the general idea in theory.

For further exploration and experimentation, check out this site, which provides attack odds.

 

[giuppi]

True, but some of the game wisdom is based on the idea that you want to conquer territories. Evan's question was interesting in that his goal was to maximize the enemy losses only, without conquering new territories. If my math is not wrong, I've shown that what you say still holds true, but it was worth double checking it. In fact, it also shows something that was unexpected, and that is that the more troops you attack with, the more losses you incur in (as I wrote above).
There are many other situations that I would love to explore, where common wisdom might fall short.
Also, as Robinette says, asking questions is interesting not only to find the right answer, but also to reveal what the people in the community think.


PS: yeah, that's the site I've used for the probabilities.

 

[Robinette]

Hate to say it, but I don't like where you are going with this giuppi...
I will highlight in RED what doesn't look right, and make comments in GREEN

If we generalize your reasoning to as many attacks as possible in the specific configuration, we get this:

Case 1 - Attack with 3 troops against 4
Chances of killing 0 defending troops: 44.83%
Chances of killing 1 defending troop: 24.16%
Chances of killing 2 defending troops: 16.35%
Chances of killing 3 defending troops: 5.53%
Chances of killing 4 defending troops: 9.13% Clearly, something is wrong here... inverted perhaps? This just seems to jump off the page at me, and yet I have no real interest in checking your math inasmuch as the broad result of this scenario will always be somewhat of a suicidal play.

For a total damage of 1.10 defending troops killed


Case 2 - Defend with 3 troops against 4
Chances of killing 0 attacking troops: 24.52%
Chances of killing 1 attacking troop: 14.96%
Chances of killing 2 attacking troops: 7.55%
Chances of killing 3 attacking troops: 52.98% This must be WAY off... Maybe the decimal is in the wrong place? Speaking of decimals, I don't think we really need to be calc'ing these to the nearest one hundredth of one percent, do we? Please say no, lol.
Chances of killing 4 attacking troops: 0% (impossible because the last remaining troop cannot attack)

For a total damage of 1.89 attacking troops killed


Therefore you cause more damage by defending.
But what if your enemy decides to attack with more than 4 troops? Well, interestingly, in that case he will likely lose EVEN MORE troops (2.07 out of 5, 2.18 out of 6, 2.29 out of 7, and so on) although with a decreasing marginal increment.
lol, it almost sounds like you are saying the only way to win in war is to not play the game
Click this to see That Scene From War Games that made that line famous
Seriously though... this is such an erroneous thing to say... if i took it to the extreme, I could say that I risk losing 49 troops if I attack 50 vs 1... while technically true, it implies that attacking with more men increases the risk of losing to a smaller force.


So, if we consider one 'wave' of attack, it's clear that the best strategy is to defend. So far so good.
But what if we consider a second wave of attack? In case 1 (you attack with 3) there are 90.87% chances that you don't conquer the territory, therefore your enemy can counterattack with more troops against whatever is left of yours (1 troop). As we saw before, this is a chance to inflict even more damage.
................

In case 2 (you defended 3 troops against 4) there are 52.97% chances that you are still alive by the end of the attack, so your enemy again will be back with more troops. Also in this case, more chances to inflict further damage. How much this 'second wave' potential damage is, I haven't figured out yet, but I suspect not enough to tip the scale in favor of attack vs defend.
And then by generalization there is a third wave, a fourth wave and so on. Anybody wants to calculate it?
Ohhh Nooo... I've already used up my Face Palm... *sigh* Ok, Just kill me now!

Please don't take offense... I really do appreciate all the time you put into this, and I really do appreciate that you actually worked out some of the math,,,, and I also understand that you were simply mathing out more facets of the op's question about inflicting the most damage... it's just that you said a few phrases that I just couldn't wrap head around, in fact, my cute little head just wanted to explode, lol... So how about a nice game of chess instead...lol

Did you click the YouTube link above? You really do need to watch it... and learn...
Click this to see That Scene From War Games

 

[giuppi]

So the facepalm is because the self-proclaimed queen of math doesn't get math anymore?

I'm not offended, Robinette, but I don't understand if you really believe what you write or you just like to provoke people.

So, hoping it's the former, I offer some explanation here.

Case 1 - Attack with 3 troops against 4
Chances of killing 0 defending troops: 44.83%
Chances of killing 1 defending troop: 24.16%
Chances of killing 2 defending troops: 16.35%
Chances of killing 3 defending troops: 5.53%
Chances of killing 4 defending troops: 9.13%

These figures are correct. Fine you don't want to do the math, so I'll help you out: killing 4 defending troops is the same as conquering a territory. Use the link that Inca posted and you will find out that 9.13% is exactly the chance you have to conquer a territory in a 3vs 4 attack. Consequently, the chance of NOT conquering the territory must be 90.87%, which exactly the sum of the remaining events (killing 0, 1, 2 and 3 troops). And it's intuitive that those remaining odds are distributed in decreasing order.

Same thing for the Case 2. 52.98% is the chance of killing 3 attacking troops in a 4vs3 attack, which is also the chance of NOT conquering the territory. Again, check Inca's link if you are not convinced. [actually I checked myself, it should be 52.97% there...] And the remaining events (killing 0, 1 and 2 troops) have a total chance of 47.02%, distributed as above.

One more point, War Games doesn't apply here. It is not true that "the only way to win in war is to not play the game". In fact, you still have more chances of conquering a territory in a 5vs3 attack than you have in a 4vs3 attack. You just risk losing (marginally) more troops. This result seems counterintuitive, but only until you realize that, for instance, you risk losing more troops in a 10vs20 attack than you do in a 5vs20 attack. Does it make sense? In both cases (5vs3 and 10vs20), you have more chances of conquering a territory (than 4vs3 and 5vs20 respectively) but also more chances of losing more troops while trying it.
The difference is that in the original case where you defend with 3 troops, I suspect there will be a value of attacking troops n, with n>7, such that the attack (n+1)vs3 will not cause an increase on the likely damage suffered by the attacker. I'd calculate n now if I knew the formula. I can only get to it by simulation on a spreadsheet and I'm too tired for that now.

So, have we saved your cute little head from explosion?

 

[Robinette]

1st things 1st... can we drop the one hundredth of a percent thing please.. these are just probabilities afterall...

and the 2nd thing... I am not, nor have I ever been, the self-proclaimed queen of math... this is a title that has been bestowed on me by others...


okay, now that I got that off my chest, let's talk numbers...
and maybe you will see why I had a problem with your post. I'll start with this:

Case 1 - Attack with 3 troops against 4
Chances of killing 3 defending troops: 5.53%
Chances of killing 4 defending troops: 9.13%

while this is technically correct, it implies that you have a greater chance of killing 4, than of killing only 3. In reality, you have a 4 or 5% chance of winning with 2 men left, and a 4 or 5% chance of winning with just 1 man left (feel free to calc this to the hundredth of a % if you like) Combined these give us that 9% chance you are referring too...

and of course... this leads us to the other part of your post that showed the more an enemy risks to take out your little stack of 3, the more men they are putting in harms way... once again, technically correct, but implies the opposite of what will normally occur.

okay... i'd love to chat more... but I've gotta run now...