The Mathematics of Risk Battles

Risk is a strategy game governed primarily by dice battles. The choice of whether to attack or not is an important decision to make, so understanding the probabilities involved can help you to make an informed decision and maximise your chance of success!

A couple of years ago I made a version of Risk to be played in Excel, using VBA code to write the program.

RiskExample

This is was a really fun project for me and in the absence of physical dice, I had to replicate the probabilities of each possible outcome. I’ll start by explaining how attacking in Risk works.

The Process of Attacking

One territory attacks an adjacent territory. Each territory has a number of troops in it (anything from 1 up to 50 is quite normal).

  1. The attacker chooses to attack with 1-3 troops (usually 3 are used if available), and rolls that number of dice. He must always leave one troop sitting in his current territory.
  2. The defender rolls the same number of dice as he has troops, up to a maximum of 2.
  3. The highest dice roll of the attacker is compared to the highest of the defender, and the person with a lower score loses a troop. Likewise for the 2nd highest roll if applicable. The defender wins ties.
  4. The attacker can choose to repeat steps 1-3 or to stop attacking.

The Possible Battle Types

Here are the 6 individual battle types and the number of dice that each person uses. In the headings are the number of troops present in the territories. In the result cells the format is “Number of dice for attacker – Number of dice for defender”.

Battle Types

The first thing to note is that the attacker generally gets more dice than the defender, giving him an advantage. However, the defender wins ties, which gives them an advantage. Here are the probabilities of each outcome for each of the 6 individual battle types. These can be worked out with fairly simple probability.

BattleProbs

These probabilities can immediately give you essential information about whether you should go ahead with a single attack.

  • Attacking against just one defence troop is relatively easy with 2 or 3 troops, but getting the defence down to that stage may require many 3-troop attacks.
  • A 1-2 attack should never be considered unless there’s a very large advantage to be gained by taking the territory – you would have to win with 25% probability and then also win a 1-1 battle with 42%, giving just a 11% chance of taking the territory.

Attacking Multiple Times

Each roll of the dice results in a maximum of 2 deaths, but we know that you don’t just want to weaken the enemy – you want to destroy them! As explained earlier, the attacker can choose to repeat his attack, in order to try and eliminate the enemy and take control of the territory. Of course the other final outcome is that the attacker goes down to 1 troop and he can no longer attack. The above probabilities tell you what will happen each time you choose to attack, but we can take that further and work out the probability of successfully eliminating the enemy by repeatedly attacking.

The chance of victory or defeat in the overall battle for the territory can be modelled as a Markov chain using a transition matrix with the number of troops on each side as the possible states. All states with 0 troops on either side are absorption states. Unfortunately there is no simple formula to calculate the probability of victory, but I’ll provide some graphs and figures.

Here’s a graph showing the probability of the attacker claiming victory if the sides have an equal number of starting troops.

When a battle starts with lots of troops on both sides, most of the individual battles will be 3 dice against 2 – this will remain true until either the attacker is down to 3 troops or the defender down to 1 troop. As the probability table above shows, the attacker has a slight advantage, with a 37% chance of killing 2 defence troops compared to a 29% chance of losing 2 troops. As demonstrated in the graph, this means that as the number of troops on both sides gets bigger, the chance of the attacker successfully taking the territory increases.

What you’re probably more interested in knowing is how to feel safe about attacking. Here’s a graph showing you the number of troops you need in your territory to start with in order to have a 50%, 90% or 98% chance of overall victory in the battle. You can hover over the lines to see the values.

If these figures aren’t enough to keep you happy, you can download the full victory probabilities for up to 250 troops on each side here: Download RiskProbabilitiesFull

Now you should have a better idea of the mathematics of Risk battles, so I hope that once you make informed and logical decisions based on the probabilities, the dice gods don’t take their wrath out on you! In other words, good luck in your next game!

Comments are closed.

  1. MDW says:

    Hi Daniel. I have a risk-related question. There is a box which says “do not open, ever”. Statistically speaking, should I open it?

    • kinch2002 says:

      Haha, I assume this is Risk Legacy. I’m actually working through the game with my friends at the moment and we haven’t opened it yet. I suspect we’ll open it near the end though. The only probabilistic advice I can offer is that if you Google it you are ~99% likely to spoil the fun of it!