In Risk, a classic board game, the best strategy is to attack tooth and nail, according to mathematics
In this episode of one of our favorite channels, numberphile they interview the no less admired Marcus du Satoy about the mathematical strategies of Risk, one of the archetypal board games in the strategy category. TL;DR: attacks to death.
It is interesting that James Grime, the interviewer, has never played Risk, being forty. I remember the afternoons and evenings of entertainment discovering curious things like where I was Kamchatka and how to calculate probabilities with the dice to fine-tune attacks, apart from some other mythical comeback.
In any case, at the beginning of the video there is a brief summary of what Risk consists of and how the important thing to see the most appreciated countries and continents is the topological scheme of the map. There is no mention of its cultural importance – the references are many – but, well, it is a video of mates.
One of the details that affects the most is that two strategies are usually used in Risk: there are those that play defense and that they play to attack. In rolls of up to three dice in which the one who obtains greater results than the other person wins, comparing dice by dice and ordering them. That is why for a long time it was believed that “playing defense” was a superior strategy, because a tie between dice benefits the one who defends.
Du Satoy explains why this is not the case, given that this comparison is influenced by the ordering of the resulting values. It is a classic problem of combinatorics and probability, which turns the game into a mathematical beauty of topology, with Markov chains and basic probability. The best strategy is to attack everything you can.
Researching this I found a very old and probably outdated Risk FAQ where they talk about the rules, probabilities, mathematical hope and some tricks for beginners (attack with large groups, start in the Southern Hemisphere, try to get Australia – although it pays little in each round – and don’t get obsessed with the missions. About this too There are some standard and not so standard variants (“house rules” and people’s inventions) with attractive names such as Tactical Nuclear Risk, Martian Risk or Multi-Earth Risk. To investigate if you are interested in this type of game.