Resources vs. Rolls
On this note, I will make a claim that I’m up for examining closely and testing later:
Numbers don’t matter (as much) when you spread to a new resource.
This was a statement I made near the end of Spreading vs. Clumping Trends (part 2), and since thinking of it, I knew that the comparison of resources to rolls would be an important one. I’ve been looking forward to investigating the relationship between rolls and resources, and I feel that I finally have sufficient data and reasoning to delve into the discussion.
Resources: When we examined resources last time, we found a less than easy way to order the resources by order of importance. When we looked at the demand for resources throughout the different stages of the game (starting resources, secondary resources, and city development), there didn’t seem to be too much difference. Some resources increased in value later (ore) while others decreased in value later (brick). But generally, there was a fairly set importance structure:
 Wheat
 Wood
 Ore
 Sheep
 Brick
Rolls: By now I’d like to say that we should be fairly comfortable with understanding the roll distribution. It’s the wonderful curve in the graph that tells us the probability of numbers being rolled, and essentially their value as numbers. For example, a 5 has an 11.11% chance of being rolled, whereas an 11 has a 5.56% chance. This means that in comparison, a 5 should be rolled about twice as much as an 11, making it twice as valuable. We can also notice that the difference between adjacent numbers is not constant – a 3 is worth about twice as much as a 2, but a 6 is worth only 25% more than a 5. This has some huge implications when we try to optimize our placing strategy.
So now we’ve got ways of showing importance to rolls and to resources. And here is where I will revisit the claim I began this post with: “Numbers don’t matter (as much) when you spread to a new resource.” Is it true? Why?
Well, first, I’ll explain by analyzing and then show some statistics. Let’s look at a couple of examples. We know that brick is the second least important as well as most useless resource (that is, if you buy into what I’ve been saying). We also know that as the game progresses, brick loses what little value it has. So, let’s say I’m in the middle of the game, and I’m given two options for settling on the coast (that is, on one resource – I’m trying to isolate variables here): a 6 of brick or a 10 of ore. Now, looking at our roll distribution graph, we see that a 6 is twice as likely to be rolled as a 10. So far, the brick is winning. But, when we look at the value of the resources, we see that brick is extremely useless, having a negative value (this is actually an average of it’s small amount of worth near the beginning of the game and it’s large amount of uselessness near the end of the game). Also, we see that ore is relatively useful. So, if it were up to me, I would opt for the 10 of ore. The fact that it’s an ore outweighs the fact that a 10 is only half as productive as a 6.
What about the statistics? Let’s take a look. Here we have the probability of a winner having a settlement or city on any given number. Starting Numbers is the numbers that the player places their first 2 settlements on. If rolls were the MOST important thing in this game, we could assume that the distribution of numbers would look like our roll distribution. We can see from this graph that it doesn’t. Let’s not take this to think that starting numbers don’t have any impact on the game, though, as it is EXTREMELY uncommon for a player who wins to have started on a 2 or a 12. Also, we can look and see some semblance of a normal curve, but it is not fully present. The curve is not uniform (a constant value throughout), but it is not normal (the shape of the roll distribution). This means that there is another factor that is more influential impacting the normal distribution of the rolls. Let’s not stop here though. What happens as the game progresses?
Now we look at the graph of Secondary Numbers. As you can guess, these are the numbers that the player develops to after his first placements. Notice that the end points (2 and 12) do not taper off near as much. Also notice that there seems to be less of a normal distribution shape tot he curve – it almost seems uniform! But no, we can still see that there are some differences in value between numbers, even as we have played longer. But what is causing the numbers to change in worth so much? Resources. Resources directly impact each space’s worth significantly more than the number assigned to the space does. This means that when I am playing, I will make most of my placing decisions, and especially my secondary placing decisions, based not on the probability of the rolls, but on the resources available. I will clamor towards a wheat if I have little wheat and not pay any attention to the 9 of sheep that I could get to.
Does this mean that we should forgo paying attention to numbers completely? No. When we pay attention to Empty Rolls, that is, rolls that a player does not pick up on because they have not settled on them, we see some more insight. Let’s start at the beginning. One of the amazing things that I note of this graph is that if we turn it upside down, it almost carves out the space for a normal distribution. The only problem is the flat section in the middle. What does this mean? Well, 5’s, 6’s, 8’s, and 9’s are all worth about the same amount. 4’s and 10’s are close. After that, 2’s, 3’s, 11’s, and 12’s are pretty worthless to start on. Again, it’s all about the resources. The same thing should happen when we examine the empty rolls at the beginning of the game, right?
Here are the empty rolls at the end of the game. From this, it looks like there are less empty rolls all around. This means that people develop to new numbers. This might give some great insight into the psychology of diversifying the numbers that we pick up on, or it might simply be explained by the fact that the good spots get tapped out quickly. People settle on the good spots, so we need to move elsewhere to lesser quality places for our later settlements. This also explains the fact that 6’s and 8’s are relatively high empty rolls at the end of the game. Basically, if you don’t start on a common number, don’t expect to be able to pick it up later. People will get there first. But that’s fine, because 30% of the time the winner is not on a 6, and 22% of the time they are not on an 8.
The moral of the story?
Numbers don’t matter (as much) when you spread to a new resource.
In fact, numbers don’t matter as much as resources in general.

23/02/2011 at 3:34 pmWhat Now? « Developing Catan