Thursday, January 3, 2013
Exponential Growth in Settlers of Catan
Mathematically, this means that the number of resources you collect as the game progresses follows an exponential curve, shown to the below. This is a very common curve for growth, and is especially common in places where how much you grow depends on how much you have. Other common places of exponential growth is financial (interest on money grows depending on how much you have invested) and populations (the more people or animals there are, the more they are making babies).
Setting up the game, you start off with a settlement and a city, which allows you to touch or collect resources from up to six different numbered hexes. To begin, you need to build a road and a settlement faster than anyone else. This requires two wood, two brick, a wheat and a sheep. If you can collect these before any of your opponents, you have significantly increased your odds of winning. While I have not been nerdy enough to record these statistics during all the games I've played, I feel confident saying that more than half the times I've played and managed to build the first settlement, I've won.
Why is this strategy so important? Let me illustrate with a simple choice, early on in the game. There is a rule that says you must build your first settlement two roads away. Suppose instead you decide to take time to build three roads away -- figuring it will help you to branch out a little first. This will likely take an additional 3 rolls at the beginning of the game, because you only average 2 cards per 3 rolls early on. This seemingly insignificant choice actually puts you significantly behind. If you now change strategies and try to build as aggressively as your opponent did from the get-go, you'll find yourself just falling further and further behind. In this final graph - the red curve shows aggressive building from the beginning and the black curve growth delayed by just one choice - waiting to build a settlement after an extra road. Notice how the two curves are getting further and further spread out as the game progresses.
Now I'll admit this is a simplified graph - and there are times where building farther away might be advantageous. For example, maybe building in one location over another allows you to create a much steeper rate of return than another location -- but in the grand scheme of things, it would have to be perhaps twice as good to make up for the extra turn it takes to build there. If it only causes you to touch a 6 instead of 5 -- it's probably not going to catch up with someone who has already built and started walking along the red curve.
It also suggests making efforts to obtain the important cards that cause you to build quickly. Wood and bricks (to make roads and settlements) are crucial at the beginning -- the others can be obtained in time, which you'll have if you are able to build a settlement first and always be on a steeper curve than your opponents.