A game of *Risk* (photo: A.R.N. Rødner)
The board game *Risk*, though far from being my favourite game (and rated only 5.59/10 on Board Game Geek), nevertheless has some interesting strategic aspects and some interesting mathematical ones.

Combat units in *Risk* (photo: “Tambako The Jaguar”)
A key feature of the game is a combat between a group of *N* attacking units and a group of *M* defending units. The combat involves several steps, in each of which the attacker rolls 3 dice (or *N* if *N* < 3) and the defender rolls 2 dice (or 1 if *M* = 1). The highest value rolled by the attacker is compared against the highest rolled by the defender, and ditto for the second highest values, as shown in the picture below. For each comparison, if the attacker has a higher value, the defender loses a unit, while if the values are tied, or the defender has a higher value, the attacker loses a unit.

Comparing attacker (left) and defender (right) dice in *Risk* (photo: “Val42”)
Working through the 6^{5} possibilities, the attacker will be down 2 units 29.3% of the time, both sides will have equal losses 33.6% of the time, and the attacker will be up 2 units (relative to the defender) 37.2% of the time. On average, the attacker will be up very slightly (0.1582 of a unit). A fairly simple computation (square each of the outcome-mean differences −2.1582, −0.1582, and 1.8418; multiply by the corresponding probabilities 0.293, 0.336, and 0.372 and sum; then take the square root) shows that the standard deviation of the outcomes is 1.6223.

When this basic combat step is repeated multiple times, the result is a random walk. For example, with 10 steps, the mean attacker advantage is 1.582 units, and (by the standard formula for random walks discussed in a previous post) the standard deviation is 1.6223 times the square root of the number of steps, i.e. 5.1302.

The histogram below shows the probability of the various outcomes after 10 steps, ranging from the attacker being 20 units down (0.0005% of the time) to the attacker being 20 units up (0.005% of the time). Superimposed on the plot are a bell curve with the appropriate mean and standard deviation, together with five actual ten-step random walks. While the outcome does indeed favour the attacker, there is considerable random variability here – which makes the game rather unpredictable.

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I remember playing Risk a long time ago and without doing the math at that time I somehow knew that attacking with bigger armies on both sides would improve my chances of winning, i.e. having a better chance (less variance) to achieve an outcome closer to the expectation.

Having a larger numerical advantage at the start certainly improves your chances of winning a combat, but that’s not quite the same thing as reducing the variance. The variance is always 1.6223 times the square root of the number of combat steps. This means that, in the process of winning a combat, one sometimes loses many more troops than expected, and sometimes one loses hardly any.

I think I understand that, but with bigger numbers the variance around the expectation will be relatively small so my chances to get results closer to the expectation improve

The whole point of the post is that that is not actually what happens; the more often you roll the dice, the higher the variance will be. True, victory is virtually certain if you attack with a sufficient numerical advantage, but casualty figures in achieving victory may be range from very low to very high.