The math behind Risk

Risk - the game of world domination. My friends and I have started to fall in love with it, but for different reasons. Some like it because it's overtly competitive, some because you get to play with army troops, and me because of the immensity of mathematical probability in the game (what else is new). While I was going to go through the probabilities of each possible roll of the dice and determine a strategy based on numbers of attackers and defenders, what I realized (thankfully) fifteen minutes into the project is that Wikipedia has graciously done this already. What I will do in this post is show some of the most interesting things I found while reading the entry, and give my own interpretation.

Outcome probabilities of one dice roll.

Based on the table above, the defender has an advantage when an equal number of dice are rolled (or if the defense rolls two dice, and the offense one). Otherwise, the advantage does to the offense. But what does this mean in the long run? What if the offense has 26 people, and the defense has 18, which is a real possibility in the later stages of the game? Is that a wise numbers game to play? Luckily, this table offers some insight.

Outcome probabilities of a sequence of dice rolls.

From this table, it does not benefit the offense to attack if 1) the offense has fewer armies than the defense, or 2) the offense and defense have the same number of armies, given that it is four or fewer. So in that 26 on 18 hypothetical situation, the offense can expect to win, and conquer the defending territory. The offense can also expect to carry about eight armies with them, since the 18 will probably cancel out. But what if there are a few territories the offense can attack with only one defender? The last table is a guideline for what you can reasonably attack if all territories in your way have one defender in them.

Want to take out a bunch of one-army territories? Use this table.

A simple rule of thumb if you want to go after a cluster of territories with one army defending is to subtract two from the amount of attackers you have, and divide by two. You can be 90% confident in winning those territories given the amount of offense you have.

None of these tables delineate a specific strategy on how to play Risk; they merely just display the chance involved in each roll of the dice, and the preparation required to unleash a full-blown attack. The decisions are still yours - hopefully this post has helped you in your quest to conquer the world.