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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....
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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....
How do things change with 4+ troops? I'm pretty sure I've read elsewhere that you have overall betters odds attacking??
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.
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.
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!
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...Case 1 - Attack with 3 troops against 4
Chances of killing 3 defending troops: 5.53%
Chances of killing 4 defending troops: 9.13%