I considered two questions about the board game Risk.
- What are the odds of winning a particular dice roll? (like 3 dice vs. 2 dice, 3 dice vs. 1 die, etc.)
- What are the odds of conquering a territory? (for example, 20 men invading 17 men)
Anyone who has played risk has an intuitive sense of the answers to the first question. Odds of winning a 3 dice vs. 2 dice battle are about 50/50. The invading army gets the advantage of an extra die, but ties go to the defender. It turns out that these advantages roughly balance each other out. Here are the full results.
Question #2 is more subtle. When one army is low on men and forced to roll with fewer dice, the odds change dramatically. For this reason, battles between a large number of soldiers are complicated. It all comes down to who has to roll with a reduced number of dice. (For example, if a large army is cut down early with a string of bad luck, its odds of winning go down much faster.)
Using the results from the table above, I generated a more useful result. This second table gives the odds of winning a whole series of dice rolls and capturing a territory. One is always subject to luck, but with this table at least you can know what you’re getting into. Only the truly dedicated would want to memorize some of this; it is more interesting to look for and internalize patterns. (See below for details on this table.)
Click to enlarge!
About this table:
- Of course there is never a 100% probability of success. I round to 100% when the probability is greater than 99.5%.
- This table goes up to 20 soldiers, but it could easily go higher.
- The most useful information is in the top-left. Notice how 2 vs. 1 is better than 3 vs. 2 but not as good as 4 vs. 3. It’s obvious that more soldiers = better odds. But following along the diagonals reveals more interesting features.
- (Math Warning!) I considered two approaches: a Markov chain and the Monte Carlo method. I believe a Markov chain is doable, but applying it would become messy. Instead, I used the Monte Carlo method, computing 3 800 000 simulated conquests, or 10 000 per entry.
Previous Work. There is a web article by Daniel C. Taflin (2001) that considers Question #1 and explains the underlying mathematics of his approach in detail.
Other important life lessons. Alliances are made to be broken; Asia is weak; never leave Australia unattended.
