Fair Catan Boards, this time with resources!

So you lost a Catan game despite being the best player? Surely the game was not fair to begin with!

Don’t despair (or start to be afraid!) I’m on a quest to find the fairest islands, boards that will finally allow the true Catan masters to shine!

Well, this may be a bit overselling it… But, given that most Catan games are played on randomly assembled islands, is it a stretch to assume how some initial setups may be unfair?

Let’s find out if we can objectively decide if a Catan island is fair or not!

And it it is impossible, let’s at least look at different boards and try to see what makes a good starting location. This is, after all, one of the fun part of playing Catan.

If you are only looking for examples of fair Catan islands, you can scroll down to the bottom and see a list of them.

This article isn’t the first time I’ve written about fairness in Catan…

(If it is your first time here, before or after reading this, I suggest you have a look at the 102 ways to win at Catan It’s an article that gathered an enthusiastic response from all sort of Catan players)

In The fair Catan Board quest, I started addressing the question of what is a fair Catan board, evaluating if certain island configurations gave an unfair advantage to some players.

By considering the best initial settlement positions available to each player at the start of the game, I was able to detect a certain imbalance in terms of the expected number of resource cards each player would receive each turn, depending on the order of play.

Often, but not always, the first player got a slight advantage.

Can we fix this?

The good thing about being able to measure the fairness of islands is that it allows generating fairer ones (or really unfair ones), where each player gets a chance to start on equally good positions.

The major caveat here is that my initial fairness analysis was done on a simplified problem….

It was only considering the number of resource cards a player gets, but not the type of resource cards. This allowed to clearly analyze the return for each player, but to see if an initial Catan setup is truly fair, we need to consider all the aspects of the game… So let’s revisit the question!

“Exploration is curiosity put into action.”

– Don Walsh (American oceanographer)

What to expect:

Now that we have a fairly high-level view of the article, let’s get into it!

In my last post (What is the strategic value of each Catan resources), I established usable strategic values for resources.

Now it’s time to try finding fair Islands, where all players can get good starting positions.

• First, I will present my new game simulation, now taking into account harbours and assets diversification
• Second I will to update the Catan Balance index with our new finding.
• Finally, I will measure the fairness of millions of boards, and see what does the most fair and unfair boards looks like.

Evaluating the fairness of a board

After I created my Catan Island Balanced Index, (something similar to fairness, but not quite the same) some readers pointed out that some balanced islands were not really fair when it came to some resources-combo locations.

Some otherwise balanced islands offered a single very valuable spot, a mix of sheep–wheat–ore resources with good roll numbers, giving a very strong starting point to the first player, and making those boards not exactly fair for the others.

This shown that having a good distribution, or balance, of numbers and resources over the island was not necessarily the most important aspect when building a board that is fun to play.

Fairness, or the chance for all players to start with good initial settlements, is a more important point, even if a bit more elusive.

In order to improve my evaluation of the board’s fairness, I needed to include the missing aspects of the game, and this is especially true for resources and their relative strategic values.

The new simulation for fairness

Fairness is a characteristic of individual Catan islands. Some islands will be fair, others… less so.

To evaluate if an island is fair, I simulate how 4 players would place their initial settlements, and then compare their positions score to see if there is an unfair advantage for one player.

Quick reminder: The first settlement is placed starting with the first player, the second settlement is placed in reverse player order, giving the following player order for settlements:

Player settlement placement order: 1 – 2 – 3 – 4 – 4 -3 – 2 – 1

We assume that each player will pick the best position they can, so the question is:

How to determine what is the best position?

In the previous fairness simulation, it was simply the number of resource cards the players could expect to receive each turn.

It looked like this:

Now we want to include resources and harbours, to be more like a real game. So I had to revamp the whole display and add a few elements.

This imply modifying the evaluation of the settlement locations, and adding some measure of strategic diversification.

In short:

• Resources now have different relative values (eg: Wheat is worth more than sheep)
• Harbours offer multiplication bonus
• Diversification bonuses alter each player valuation differently.

Without further ado, here is a screenshot of the new interface:

Let me describe the elements of this new simulation.

Ordered list of expected return

On the right side of the simulation GUI, a bit like before, we find a list of every island location, identified by a unique number, and ordered by their associated expected return.

Greyed-out lines indicate positions that are no longer available to settle due to the presence of nearby settlements (the minimum distance between two settlements being 2 road lengths).

Each location score is calculated as the sum of its surrounding hexagons value:

hexagon value = expected card return x resource relative values

Expected Card Return

The expected card return is the same as before, calculated over 36 dice rolls, it is easily found by counting the number of dots at the bottom of each numbered token.

If you are wondering: Why per 36-dice roll ? It simply is the number of different outcome of rolling 2 regular dice, so it simplify the math by making sure we are dealing with whole numbers instead of percents or fractions.

As a side note, in a game of Catan, 50-60 rolls are what you would expect for the whole game, so it is not far off how many times you can expect the numbers to come out!

Resource relative Value

The resource relative value is a new element. It represents the strategic importance of each resource in the game and thus, acts as a modifier when calculating location scores.

Resources relative values are shown in the GUI:

You will notice here they all have a value of 1.0, but we will revisit that in a moment.

Just to give you a concrete example, in my screenshot, the best-valued location is spot #2. With a mix of roll number – resource of:

6-Brick / 9-Wood / 11-sheep

• Number 6 has an expected card return of 5 cards per 36 dice rolls
• Number 9 has an expected card return of 4 cards per 36 dice rolls
• Number 11 has an expected card return of 2 cards per 36 dice rolls.

With all resource having a value of 1.0, the total score for this location is:

5 x 1.0 + 4 x 1.0 + 2 x 1.0 = 11.0

Individual players score breakdown

There are several new elements in the simulation, and most of them can be seen in the individual player score breakdown at the bottom:

Resource breakdown

Since resources are now the center of our evaluation, I wanted to present a clear breakdown of each resource contribution to each player score.

So now each line show how much point per resource type the player’s current settlements are worth.

However, there are now also several important modifiers to the Total score

The harbour multiplication

In Catan, players can exchange resources during the game. However, a player can also exchange resources with the bank, at a rate of 4 cards of one type against 1 card of any type.

Settling a harbour offers the player a much better exchange rate:

• The 2:1 resource harbours allows to exchange 2 cards of the harbour resource type against 1 card of any resource type.
• The 3:1 generic harbours allows to exchange 3 identical resource cards against 1 card of any other resource type.

In order to reflect the strategic value of harbours in the game, I added a resource multiplication factor when players opted to put a settlement next to a harbour.

• 1.4 multiplicator for resource-specific harbors (2:1),
• 1.1 multiplicator for generic harbors (3:1).

With this, I tried to reflect the fact that resource-specific harbours greatly improve the card exchange rates for certain cards, and generic harbours have a more diffuse effect, but it applies to all the resources of the player.

One major side effect of this inclusion is the following:

Locations value are now player dependant!

Some specific examples:

• Brick harbour locations are valued more highly for players previously settled adjacent to brick locations.
• Forest locations are valued more highly for players settled adjacent to a forest harbour.

What that means is that from now on, different locations on the map have different values for each player! This differentiated valuation needed to be reflected in the score breakdown.

In order to keep things simple, I however choose to keep the top right-hand side location list static, showing only the common valuation for all players.

In the GUI, I added small harbour flags beside resource scores to indicate that a harbour bonus was added to the resource score.

Let’s have a look now at other player-specific valuation bonus:

Diversification of roll numbers (Number Bonus)

Diversification of roll numbers is important, placing settlement adjacent to a variety of roll numbers allows the player to spread the statistical risk of the dice and lower the chances of being starved of resources. The more numbers you settle around, the most often you will receive cards during a turn.

To recognize the diversification element of a strategy, I added a Number bonus to the score: Each different roll-number settled by a player gives a bonus of 0.3 (for a maximum of 0.9 per settlement).

The number diversification bonus is shown at the end of the scoreline for each player as the Number Bonus.

Diversification of resources (Resource Bonus)

Resources diversification in Catan is even more important than number diversification, so I choose a value of 0.31 for each individual resource a player settles in the game.

This value is small compared to the typical expected card return of an hexagon, but will influence the selection of the settlement position since it can add up to 0.93 to a score (3 new resources for the player.)

This is shown for each player’s score as the Resource Bonus, and also makes each location slightly differently interesting for each player.

I also choose 0.31 to give it a slim advantage over number diversification.

The robber Tax – An hexagon diversification strategy

Part of the risk when selecting locations in Catan is to concentrate the settlements around the same high-paying position.

Putting two settlements around the same high-paying hexagon increase significantly the risk of becoming the recurrent target of other players’ robbers. A sure way to lose the game.

To avoid this, I added a robber tax to the player score: I subtract half the expected card return if the robber was located on a player’s highest paying hexagon, thus strongly encouraging a hexagon diversification strategy!

One small note, I only calculate the robber tax when placing the second player’s settlements.

What does it look like when animated?

Just for fun, here is a full animation of the player selecting their initial position:

So while this is nice, there is one important thing missing, actually at the core of this whole thing:

Attributing different strategic values to Catan resources

For the resource values to be used in our simulations, we will use the values given by the average expected cost of the top 50 fastest victories. As describe in my previous post: What is the strategic value of each Catan resources?

Those could be different, but I think they are reasonable enough to provide for a good evaluation of board fairness.

Back to the board fairness question

So now that we have established how valuable are resources relative to each other, how can we use that to determine if a board is fair?

By the same way I did in my last board fairness iteration: I generate an island, let fictitious players select the best settlement locations for their starting settlements, and check if their total score are close or very far apart!

So let’s start by a single random simulation using my own resource values, and see where it gets us:

Here it so happen that players ends up with a fairly balanced finale score.

Orange seems well balanced, blue here, while having a strong Ore position is not that well positioned. Red has something going with the wheat harbor, and white could have something similar if it can build on the wood harbor.

How does compare to real players

To give an idea how my algorithm compare to real Catan games, I decided to do a simulation for the board used in the 2020 Canadian championship.

The starting board was the following:

You can watch the whole game here: Video of the Catan Game.

Here is the final settlement positions as selected by the human player in this tournament:

And here are the result of my own simulation on the same board:

I was a bit surprised by the results. While the order of selection is not the same, several positions selected by the players are the same, including the first player pick.

But more than that, if you look at how my algorithm is scoring the human player settlements, the player with the highest starting position score is orange… This player ended up winning the real game! (See below for the final state of the game)

This is only one example, so we should not conclude anything from it, but it definitively made my day!

Does that means I have solved Catan ? Evidently no. There are a lot of issues with my approach, but I do think it is certainly an interesting starting point to evaluate positions.

Now if we go back to the original goal of this post:

Using the simulation to generate fair boards

While I will never claimed that my player simulation and resource weighting is perfect, let’s remember what was the original goal I had in mind:

Filtering unfair boards and generating interesting ones.

To see how all this can lead us to interesting boards, let’s find some extreme outcomes, that way we can choose to exclude those in future generations:

The extreme board of unfairness

To find extreme boards, all that is left to do is to:

• Generate lots of boards
• Run simulations
• Check boards where the players scores are the more spread out.

(Conversely, you can keep the boards with the lowest scores difference and check if they seems fair)

As for the balanced Catan approach, I extracted the four boards causing the most imbalance but each favouring a different player.

I think those are interesting because it takes different type of imbalance to give an advantage to the first player (offer a very strong starting position) compared to one advantaging the last player, often offering a certain amount of strong locations, with a huge drop-off in quality after a certain point, which leaves only scraps for the second location of most players.

The first type of unfairness should be a lot more obvious. But all of it offer interesting insights into Catan board structures…

I will show you first the the empty board, and second my algorithm settlement locations. You can click the initial board to access the complete animation. (If you have a better suggestion for presenting this, by all mean, leave me a comment!)

[Click the starting board images to load the Animated selection gif]

Board giving the biggest First player advantage

And here is the final position board:

As we can see here the board offer a really strong wheat position for the first player, which combined with the wheat harbor on the 8-ore should give a head start to the first player.

Board favouring the second player

Here, we could say that the first player stil get the better position with a very strong first settlement, but that the second player is using the wheat port and my evaluation of it to save the day. The fourth player is limited in it’s choices, at least the player can have some hopes for the longest road!

Board favouring the third player

Here the wheat port strikes again. We can see form the Settlement return that there is a huge drop in scoring from the third to the fourth player. So not much can be expected for the last player.

Board favouring the fourth player

Once again the strongest score comes from a strong wheat port. Maybe this mean that I weight the ports too strongly. But at the same time, it is difficult to deny that such a position offer a lot of flexibility to the player even if no one wants to trade.

Those are ‘extreme boards’ from my scoring point of view, so it is normal for them to display more controversial aspect of the scoring. But I think if we go back to the Canadian tournament board, it shows that it is not as crazy as it seems!

Generating fair board using both the balance and fairness approach

In the Balance Catan board article, I invented the CIBI metric to evaluate board balance. And Now I have presented you a fairness scoring system to evaluate if the initial board settlement has a fair selection for players.

In order to include my new metric, I decided to simplify the CIBI index a bit

I decided to call this new measure the CIBI+ index, that is calculated as follows:

From the old CIBI, I included:

1. Resource clustering
2. Probabilities clustering
3. Harbor Return Balance
4. Resource Probabilities Distribution

This constitue the first half of the measure.

The second half will be the Fairness Measure, a measure of the maximum difference between the top and lowest score on my simulated settlement placement. A score between 0 and 1.0

So the final Cibi + is a simple average of those two score.

This gives more weight to the Fairness measure than the individual balance measures, but I think it is a fair approach. If I ever release this as an interactive app, I’ll make sure to let you choose your own criteria for generating interesting boards!

The final boards

The only thing left to do is to generate millions of board, and keep the ones that are fair and mostly balanced!

Here are the most fair and balanced boards out of a 5 millions I generated at random:

The top 50 boards:

The worse 50 boards:

That’s it! I encourage to have a look at both the best and worse boards, try them out, tell me what you think!

I do not know if they are the more fun, but certainly they offer something interesting to think about.

Maybe you can look at them as a group, and decide which one you want to play!

Conclusion

I think that overall this has been a fun quest to undertake, and certainly a good introspection on how I think about position selection and resources values in the Catan board.

The only thing that would add value to this I can see at the moment (aside from writing a machine learning AI to play), would be to create an app that would allow everyone to enter boards and resource values to play with it.

Maybe this is something I will do, but it may take a bit.

If you have any comments, questions or suggestion, don’t hesitate to reach, or to leave them here!

Thank you for reading me!