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  • Writer's pictureAlex Cates

Are 6's and 8's really that great?

I received an email the other day with a contrarian idea from Sjors:


"Hi Alex, ... I've found that any prime spots of course get blocked more, so 6's and 8's get blocked more. Especially if you play with less experienced players it is always 6 and 8 that are the target ... I counted blocks for 6 games, and the result was that a 5/9 had almost the same resulting pips as a 6/8, being 3.8 instead of their 4/5. How does this blocking bias filter through into your stats?"

It's an interesting question; and considering I have largely ignored the role of the robber, seemed like a great excuse to explore the robber's role. Are the 6's and 8's too obvious such that 5's and 9's offer as good or even better settlement locations?


I can see the argument. After all, I personally normally do put the robber on the 6’s and 8’s, especially in the early game where I don’t have a more targeted strategy for how to weaken my opponents.  While my expectation is that not enough resources end up being blocked for this to really matter across lots of games, I don't like leaving it at a gut answer when we have the data!


So today, let's look at the value of different numbers and maybe see if there is a more precise measure than the theoretical pips.


The Data

For this analysis, I will be using a sample of ~400 games collected from Colonist.io. As I have previously described, I don’t know what the board looks like, but I do know how many resources are collected and the distribution of dice. Given resource distribution and each players build history we can infer their starting numbers and create some win probabilities based on where you start. Note if you would like to contribute to the dataset, you can do so at colonist_data (anvil.app). It really helps.



Route 1: Effective Pips

One initial route can be to create an effective pips metric. Essentially, how many resources are produced by a tile with that number on a given turn.  First, we calculate how many resources each number produced along with how many times each number was rolled. Next, we can calculate the how many resources are expected to be produced whenever a number is rolled. Finally, we can incorporate the true odds of a number being rolled and the number of tiles with that number on the board (remember these are all base catan games so 1 tile with a 2 and 12, 2 tiles of the rest) to create an average number of resources gained per turn by a tile having a given number and see the results below. Note: while some people question how random the dice on colonist are, I have previously shown that are random and conform to the theoretical distribution with enough rolls.

Bar plot of the amount of resources gained with each roll of the dice by a tile with a given number.

 

Well that was anticlimactic. That really looks like it matches the theoretical pips. Unfortunately, there is likely a bias on where settlements and cities get built that does affect the results here. People are more likely to settle on a 6 or an 8 and to upgrade those settlements to cities (and vice versa with 2 and 12). Such build patterns may be artificially inflating the value of 6's and 8's, canceling out any punishment from the robber. Given that, let's try a different approach.



Route 2: Should you be changing where you settle at the start to avoid the robber?

Focusing on Sjors email, should you change what numbers you start with to avoid the robber? We can examine this directly by looking at the odds of winning the game when your starting settlement borders a particular number. For this we will focus on games without any bots (152 total) to try to maximize player quality. We will also count any starting settlement on a single number, regardless of what other numbers came with that starting settlement (so there are likely some correlations between numbers, but that's for another post).



Point plot of the odds of winning the game based on settling on a tile with a given starting number. error bars represent 95% standard error.


A couple of interesting insights right off the bat. First, the 2 and 12, they don't happen as much and have massive variation. my guess is the happen with 6-5-2 situation and the odds are driven by the other tiles. That said, the takeaway is to not be afraid of the 2 and 12 if the rest of the spot is good enough. Second is that there is not much of a difference across the other numbers except maybe 8 and 9. I am not sure that supports the original theory that started this whole investigation but is interesting. I think the larger takeaway is how similar the win rates are from different number, suggesting that it's the resources that matter more.



Limitations

The most obvious limitation is that I never actually tracked the robber! Unfortunately, Colonist only recently added a note of resources being blocked by the robber so I don't have enough games with the robber information to meaningfully use it. Instead, we have to infer its effect. Additionally, a lot of the arguments made are from indirect inferences. Even a player's starting numbers is dependent upon them receiving a resource from that number before they build a third settlement, hardly definitive, but the best we can do with the data available. So while the evidence we have doesn't suggest a significant difference in effective vs theoretical pips, more research could certainly be done.


Conclusions.

So finally, returning to Sjor's question:


Are 6's and 8's really that great?


Yes, yes they are.


Overall, I don't think there is enough of an impact from the robber to warrant choosing different starting locations. That does not mean that controlling the robber isn't a valuable strategy or that you shouldn't be blocking your opponents. Just that you should not dictate your starting placements out of fear of being blocked by the robber. This may change based on your opponents, but as a whole, you are better off maximizing your resources and pips than worrying about "effective pips" or whether you are setting yourself up to be robbed.


Questions? Comments? Let me know at ac@alexcates.com. Want to read more breakdowns like this? Sign up for my newsletter. Finally, like what I do? Consider supporting me on buy me a coffee.

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