The theory of probability has its roots in gambling theory, and introductory probability courses often draw many examples from gambling applications. Learning to reason probabilistically is a real challenge and it takes a disciplined mind to overcome the human minds tendency to see patterns where their may be only randomness.
I’m not much for gambling, but I do enjoy board games. Settlers of Catan is a very popular game which is a blend of chance and skill. It is based on the players building settlements which border resource sources. At each turn two six-faced die are rolled and if a player has a settlement which borders a resource with a number equal to the sum of the two dice they collect a resource card. The resource cards can be used to build additional settlements, roads, etc.
In my experience Catan games typically feature at least one player who becomes disgruntled at the dice rolls and expresses some level of disbelief at the sequence of dice rolls produced. In this post I will compute some probability measures to investigate the phenomena of the disgruntled Catan player.
Dice Rolls in Catan
To begin lets find out the probability mass function of the sum of the two dice. Let us assume the dice are fair, i.e. that the roll of any number (1,2,3,…6) for each dice is equally likely. Then we have 6×6=36 possible combinations for the two dice values each with a probability of . Therefore, we just need to count the number of combinations of numbers in (1,2,3,4,5,6) which sum to a given value (2,3,..12) to get our Catan probability mass function. This is a standard calculation in the mathematical discipline “combinatorics”-the mathematics of counting things. However, in this case we have a simple visualization of this process:
Where I have written the values of the two dice rolls highlighted in gray and the sums on the two values are written in the interior. Counting up the lengths of the diagonals gives the number of combinations which sum to a given number, and dividing this sum by 36 gives the probability.
The probability mass function for our Catan rolls may be plotted as:
Our analysis tells us the most likely value to be rolled is a 7 with a probability of with 6 and 8 the next most likely values with a probability of . The first thing to note is that this isn’t a very big difference: it is only a difference of difference in likelihood to roll a 7 versus a 6 or 8. We can see the reason in our chart above as each time we move to a less probable number it only costs us one of the entries in the table (or 1/36 in probability of being rolled).
Mathematically, this means our probability mass function falls off in a linear manner from the most probable value. This is counter-intuitive from most of our dealings with probability, where the central limit theorem tells us to expect a gaussian/normal distribution (the proverbial bell curve). For a Gaussian distribution the extreme values fall off in a exponential fashion meaning that the probability of seeing a value very far from the average is much smaller.
This linear fall off in the Catan distribution means that it is almost flat (uniform) and leads to a great degree of variability in the rolls which can be observed in a game.For example, we might expect that building a settlement on an 8 would be much more profitable then building on a 9. After all, the law of large numbers tells us that if the game goes on long enough more 8’s should be rolled then 9’s. However, the law of large numbers requires Large Numbers! If the average Catan game lasts 50 rolls then we may simulate to see the probability that betting on 8 will yield more resource cards then the 9. Surprisingly, we see that in this scenario the bet on the 8 only pays off about 60% of the time- so roughly 40% of the time the player who builds on a 9 will get more cards by the end of the game! If we assume the game lasts 100 turns then this percentage increases to about 68% of the time. Below I plot the odds that a bet on an eight pays off versus a nine as a function of the game length (number of turns).
In my experience, a Catan game usually lasts about 50 turns so we can see that building on an 8 doesn’t provide a huge advantage over building on a 9. So, we see the disgruntled Catan player has a point! You can make what is mathematically the optimal decision and still have a good chance that it doesn’t work out that way. Of course, this is part of the fun of the game.