Risk simulation: conquering a territory

In previous posts I've showed the probabilities of succeeding in a single attack. In today's post, we'll see how difficult is to conquer an enemy territory. Just as a reminder, you can download the game instructions here. For this simulation, you can download my Python script here.

The simulation will tell you the chances of conquering a territory given the total amount of attackers and defenders. The defenders will be the number of armies in the enemy territory. The attackers will be a number of armies the attacking player wish to risk. So, if you have 7 armies in a territory that is going to attack an enemy territory but you want to make sure that you leave always 4 armies in your territory, you would set the attackers argument to 3 (i.e. 7 – 4 = 3).

During the simulation, I will assume that the attacker will attack with the maximum amount of armies he's willing to risk. The defender will act in the same way (we've concluded in previous posts that it's better to attack using as many dice as possible).

As I did in a previous post, here the simulation will be repeated a number of times to return more accurate data.

Let's prove our intuition. The probability of succeeding in a single attack with one dice against one defender is 41.6667%. So, if I run the program with one attacker and one defender I should conquer the territory a 41.6667% percent of the times or so.

As you can see, the conquest rate is 41.682% (very close to 41.6667%) in a population of one million simulations. However, we already knew that (the result was predicted). Let's try more complex scenarios:

As one should expect, the more armies defending a territory, the harder is going to be the conquest. I've been puzzled for a minute with the results on the losses. In the second to last simulation on the image above, the probability of conquering the territory is around 29.813%, but the defender expects more losses than the attacker. After thinking about it, I find it logical, because even thought the probability of conquest is below 50%, it is taken into account more cases in which the defender losses more armies (i.e. in case of conquest the defender will lose more armies and even if the attacker doesn't conquer the territory, there is also a case in which the defender loses 6 armies, more than the attacker).

We can also conclude that more armies involved in the attempt of conquest will be translated in more casualties (in both sides).

Finally, we already know that attacking with 3 dices will be a likely victory (whether the defenders uses 1 or 2 dices). In the past, I've experienced situations in which two players put many armies in only one territory menacing the other player but never attacking and expending many reinforcements in the business... They were wrong. Be the first one to attack. Let's see it with big numbers:

It took my computer a while to finish this one! Conclusion: as we wanted to prove, even if the enemy outnumbers (not by a huge amount of armies) you, it's better to start the attack.