Risk Board Game — Battle Probability Grid Program

A slight advantage goes to the attacker.

Nature of ‘Risk’ ProbabilityThis made me curious, so I decided to run this tool for the attacking/defending armies with even starting troops for 2–1000, and it gave me some interesting results.

But before we look at the results, let’s think about the nature of battling in Risk by looking at the probability table:Source of image: https://www.


com/how-to-use-math-to-win-at-the-board-game-risk-2013-7This chart illustrates that the attacker’s advantage comes at scale.

When able to roll three dice, the attacker receives a probabilistic advantage that outweighs the defender.

The defending army, on the other hand, can roll a maximum of two dice, but wins ties.

When the number of dice being rolled is comparable between the parties, the advantage swings towards the defender.

To put simply — the bigger the armies, the more advantage goes to the attacker.

It’s not until the attacker has one or two dice that the defender take the advantage back.

It’s important to keep in mind that these tables only represent the probabilities of isolated rolls.

How do the probabilities look after combining both the defender’s early advantage and the attacker’s later advantage?.Or to phase it another way, how long does the early advantage of the defender last as the armies on either side increase?So before I ran the program, I hypothesized that, for even-starting attacking/defending armies, the advantage would be with the defenders for the low numbers.

This advantage would likely decrease up until a certain point.

Then it would be about equal.

I then assumed the advantage would switch to the attacker, and increase with the attacker from that point forward.

Results:Practical Risk Takeaways:12 armies each is essentially flipping a coin.

The larger the battles, the more likely the attackers will win.

The smaller the battles (under 12), the more likely defenders will prevail.

Hope this article adds value to future games.

Use this tool for yourself and experiment with unequal starting armies if you want further insight.




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