The Robber Strikes Again

If you’ve played Settlers of Catan, you are most likely familiar with the following scenario: Say you picked a great spot for one of your settlements at the beginning of the game and the resources have been pouring in so much that you haven’t been able to use them all. Another player rolls a seven and BOOM you’ve lost half your hand.

What you experienced is actually a clever game mechanic based on probability to keep jerks like you (in this hypothetical scenario) from hoarding the game’s finite resources.

What is probability? It’s a measurement, between 0 and 1, of the likelihood of something occurring. When you flip a coin, for example, there’s a 50% chance you’ll get heads or tails. (That’s 0.5 if you’re new to this and confused about the between 0 and 1 part.) That’s because there are only two outcomes to flipping a coin. In Settlers of Catan, however, distribution of resources is determined by rolling a pair of dice. One roll can produce any number between 2 and 12. How does that increase the likelihood of rolling a 7 and activating the robber?

This guys is the worst

To demonstrate this nifty trick, let’s simulate five games of Catan with 100 dice rolls each possibly the number of times players will roll in a long game of Catan, but don’t quote me on that.

If you’re not interested in the statistical programming language R, you can skip this code chunk.

set.seed(500)
game1 <- sample(1:6, 100, replace=T) + sample(1:6, 100, replace=T) 
game2 <- sample(1:6, 100, replace=T) + sample(1:6, 100, replace=T) 
game3 <- sample(1:6, 100, replace=T) + sample(1:6, 100, replace=T) 
game4 <- sample(1:6, 100, replace=T) + sample(1:6, 100, replace=T) 
game5 <- sample(1:6, 100, replace=T) + sample(1:6, 100, replace=T) 

Now let’s look at the outcomes of each simulation.

Outcomes of 100 Dice Rolls for Five Games of Catan
2 3 4 5 6 7 8 9 10 11 12
Game 1 4 5 13 10 11 18 12 8 7 10 2
Game 2 7 4 4 7 15 23 10 16 9 3 2
Game 3 3 7 7 14 12 20 10 15 9 2 1
Game 4 1 3 9 9 14 20 12 15 8 8 1
Game 5 9 6 14 16 16 10 11 7 6 5 4

If you’re more of a visual learner, here are the distributions of the outcomes for four of the games (because you can’t make a lovely, even graph with five).

fig1-robber-strikes-back

What the table and plots above are showing is that for each game, 7 was rolled the most. You can see in the four plots that they are not quite what you would call in statistics a normal distribution. For example, in game 1, 4 was rolled more than 6 or 8, which have higher probabilities of being rolled. How do I know that?

Take rolling a 2 or 12 as an example. There is only one way to roll a 2, both dice showing 1 or [1,1] and one way to roll a 12, both dice showing 6 or [6,6]. There are 36 possible outcomes from rolling dice (6 outcomes on one die x 6 outcomes on the other). If there is only one way to get a 2 or 12 out of 36 outcomes, the probability of rolling one of those numbers is \(\frac{1}{36}\) or 2.8%.

Rolling a 7, on the other hand, is a lot simpler. The possible combinations are [1,6],[2,5],[3,4],[4,3],[5,2] and [6,1]. That’s 6 out of 36 combinations, or 16.7%.

Let’s revisit the five games of Catan above. Seven came up 18%, 23%, 20%, 20%, and 10% of the time for each game.

Probability doesn’t tell us that 7 will be rolled 16.7% of the time in every game, only the likelihood of it being rolled. Probability is not a fortune teller. But let’s say we simulate the world’s longest Catan game with \(10^4\) rolls (that’s 10,000 rolls). Something beautiful happens

fig2-robber-strikes-back

Now that’s a normal distribution! How many times did the players roll 7 in what must be the worlds longest game of Catan? 17.12% of the time. That’s pretty darn close to 16.7%. And if we simulated even more rolls, we’d get even closer.

The larger the number of events, the closer the outcomes will come to their mathematical expectation. Like in an infinite number of coin flips, heads and tails will always come up 50/50 (it actually takes less coin flips than that, but stick with me on this). It’s called the Law of Large Numbers. The makers of Catan know this, which is why the robber is activated when rolling a 7. There are only so many resource cards in the package, and if one player is allowed to hoard them all, it’s not going to be much fun for the other players (like Monopoly, really).

So if you’re settling down for a really long game of Catan, it’s probably not a bad idea to build next to resources with 5,6,8, and 9 on them and spend those resources every chance you get. That doesn’t prevent you from getting robbed, but it’s a lot better than the alternative.