Fluctuating Incomes and Cool Graphs
So this post is going to be a better explanation and answering of the questions posed in “Spreading Vs. Clumping” (part1 and part 2). I asked a lot of questions in these, and even I wasn’t satisfied with the answers that I gave. I think it’s time to attack the beast, with a little help from Paul Gebel. Here’s the story:
When I began my look into Settlers of Catan through my math-lenses, I tweeted about it. Paul somehow found it, and has been a great encouragement since then. The other day, he tweeted something about the “resource curse”, which sounded extremely Catanish. Upon reading it, I found that it was exactly relating to how I wanted to describe fluctuating income.
So, that brings us here. I’m going to follow the format I did last time with introducing my analysis with one of my tweets:
So, there’s a couple of jobs to do. First, the definition of “resource curse” needs to be applied to a Catan setting. Then, sine curves need to be used to help model incomes in Catan. Let’s start with “resource curse”.
The definition, as quoted from Investopedia (which is where Paul found it):
A paradoxical situation in which countries with an abundance of non-renewable resources experience stagnant growth or even economic contraction. The resource curse occurs as a country begins to focus all of its energies on a single industry, such as mining, and neglects other major sectors. As a result, the nation becomes overly dependent on the price of commodities, and overall gross domestic product becomes extremely volatile.
So basically, it’s an over-dependence on one income source. The easiest way to examine this is to examine an over-dependence on numbers, and here’s why: there are only 5 options for resource dependence, while there are 10 options for numerical dependence. This means that it is much more likely to be bullied or forced into a setting where you have to punt a certain resource (that is, go without it). It’s not a huge deal, although in my statistical analysis, no player ever won on only 3 resources (and I’ve tried a lot, even setting myself up with 3 ports, but to no avail).
Anyways, let’s assume we’re comparing a player clumped on a few numbers to a player spread out across more numbers.
Have you seen a graphical representation of the business cycle? Probably. It’s a regular sine (or cosine really) curve. We can use this model to represent personal income as the business cycle moves. Let’s use it to model the number of cards a players hand as the game moves forward. We can see in the picture below that the number of cards in our hands fluctuates, bottoming out at a hand of zero cards for a time being.
We can use this graph, and play with it to show the difference between spreading and clumping. For example, if a player holds too many cards in their hand, an upper limit is imposed (when a 7 is rolled). Also, a player who spreads his numbers may not have as many occurrences where they end up not picking up cards on a roll. So, we need to change our models.
In the below graph, two curves are shown on the same set of axes. The red curve represents a player who has spread out their settled numbers, while the blue curve represents a player who has built around similar numbers.
Ok, let’s figure out what’s going on. The most noticeable thing should be the awkward shape of some of the oscillations in the blue curve. This represents a 7 being rolled at the height of the curve. In this situation, it seems that blue was over the limit, while red was not, and therefore blue had to cut its supply of cards in half. This doesn’t happen to red in this case, although even people who spread out will be forced to discard. This is just to show a point – if you rely on getting income in spurts, keeping your supply of cards at or under the limit of 7 cards will be hard. Another noticeable facet of the blue curve is the extended “dry spell” – the bottom of the curve is much more linear than normal. This is all based on the roll distribution: if you rely on a smaller amount of numbers, you will undoubtedly have time periods where your numbers are not rolled. The advantage is that when they do, they will provide more income. This is shown by the blue curve’s maximum being higher than the red curve’s (although it probably could be exaggerated more). The red curve has some interesting characteristics as well. For instance, it never reaches zero. This is because with more options available, players will be picking up cards more often, and will not likely have “dry spells”. They are picking up less cards though. Also, the period of the red curve is longer than that of the blue curve. Again, this is due to a more balanced spread on the numbers.
So, which is best? My answer is the red curve, because of a few different observations.
- Clumping your income leaves you very vulnerable to having to split your hand more often, leaving you with a less net income.
- Adaptability is huge in Settlers, as strategy can change instantly. Having a steady supply means that every turn strategy can be assessed and actively changed. If I have to wait a couple of turns to really bring in all of my resources at once, I’ll most likely lose any race to a port, or won’t be able to switch from concentrating on city development to defending the longest road.
- Picking up from a variety of different sources means that if one of my hexes gets blocked, it has less of an overall impact on me.
There really aren’t numbers or stats to back up my claims here, so I’m open to discussion in the comments below. What do you think?