by Grzegorz Kotkowski

During the Christmas and New Year’s vacation I got introduced to Catan board game (Settlers of Catan). It is a popular (classic) game – far more complex and entertaining than simple dices-throwers like for example “Monopoly”. I would like to share with you my opinion and strategies about it.

First of all as a statistician and a person who likes to win I always try to play the optimal strategy. Some people would call me a soulless player who mercilessly steals the joy of the game from other players. Well, in my opinion the purpose of the game is to give fun from winning, crossing the plans of other players, feeling the adrenaline when the opponents are just behind you and much, much more.

Basic rules

In Catan two dices are rolled for each players turn. From this reason the numbers from range 2-12 are not going to be drown with equal frequency. For example occupying the tiles with numbers 2,3 and 11 is equivalent to single tile with number 8 (from statistical point of view). Below I present the barplot of the probabilities for throwing particular numbers on the two dices.

The probability of drawing for example number 8 is 5/36=13.9%. However it does not mean that in every 36 draws there is going to be exactly 5 numbers 8. Or for example number 9 is chosen with probability 4/36=11.1% but given 50 rolls of dices with probability 24.4% number 9 occurs more often than 8.

From the Law of Large Numbers as the number of draws goes to infinity the fraction of observed events converges in probability to the theoretical value. In simpler words having only a few draws the frequencies could be a little different from the theoretical one so we never can depend fully on the drawing mechanism as it would be deterministic. Below I present the comparison between the theoretical frequencies and the one obtained during one of my games.

As we see on the graph above during my game numbers 5,6 and 8 appeared less frequently than they theoretically should. However the randomness of appearing numbers is a very important part of the game although it could seem unfair that sometimes particular numbers just don’t want to be thrown on the dices.

Average Revenue

I wanted to calculate some index of welfare that would summarize the economical situation of the players. The most basic is the average number of resources that the player gets during the turn (average revenue). It is the sum of all the probabilities for drawing the player’s tiles (the particular tiles are summed a few times if the player has multiple settlements or cities respectively).

The index was calculated in every turn throughout the game for each player. The resulting time series are presented below.

For the presented game Player 3 was victorious. This player started with the best index and quickly built the new settlement. The strong economy lies in foundation of the successful game, however the Player 3 in order to win collected also the longest road and 3 points from the development deck.

The average revenue index is a helpful device for controlling the economical situation among the players. On the other hand calculation of the index could be time consuming and irritating for the others. Finally as I played more games I conclude that the strongest economy is not always the crucial aspect of winning.

The key strategies

The most important is the placement of the first two settlements. My main strategy would be to occupy a good position with bricks and wood. This enables a quick development at the beginning of the game, building roads and new settlements. At the beginning bricks and wood are in shortage among players so it should be easy to trade them for other resources.

After building 3 settlements you should also end up with the longest road (7 points in total). At this point it could be difficult to obtain any ore and other player might not want to trade it as you might seem to be the most promising player to win. The last 3 points are therefore the most difficult to gather. It is important to have an elastic harbor for 3:1 trade. I’d recommend to collect as much of development cards as possible in order to collect the points, because building the town might be too expansive at this point.

The second strategy is to monopolize on some particular resource and have an access to a special harbor that trades the resource 2:1. However this strategy works only if the given resource is in abundance – there is no sense in building the economy on tiles with low probability of being drown.