The fair Catan board quest

One recurrent topic among regular Catan players is what constitutes a fair (or unfair) initial setup.

But the definition of fairness vary from player to player….

Some people prefer a good repartition of resources and probabilities on the board. Others will desire a setup that allows all players to have good initial settlements placement, with multiple locations offering good resource combinations.

In my previous post, I tried to establish an objective measure of a well balanced Catan Board.

But I did not answer some important questions:

• Are some setups unfairly advantageous to some players?
• Are “balanced boards” also “fair boards”?

Those are the questions I’ll try to answer now.

What to expect today

The goal here is to try to understand what makes a board fair. So I will examine if the order of play benefits some players over the others, depending on the initial board setup.

Does being the first player in Catan offers a real game advantage?

Or is it better to play last?

In today’s post, I will start by examining a simplified version of the fairness problem:

• By considering only the expected card return, we should be able to clearly see if there is an expected payout advantage in playing first.

• I will then add a few strategic considerations to see if the benefits vary according to player play styles.

In a follow-up article,

• I will add resources and harbors considerations to the mix.
• We will have a look at balanced and unbalanced Catan boards to see how they are related to a fair board.

This should offer a way to predict, or measure, which boards are fair or unfair, and see if we can modify the Catan island balance index to add an element of fairness to it!

Are some initial board setups unfair?

Right away, I will make an important distinction.

A balanced Catan board is a board where resources and probabilities are equally distributed on the board and the probabilities equally distributed among resources.

A fair Catan board is a board where all players have an equal chance of selecting good starting positions, no matter in which order they play, consequently offering them similar chances to win the game.

What is a truly fair (or unfair) board, however, is not easy to determine…

The initial game setup

At the start of the game, playing one after the other, the players put one settlement on the island with an attached road segment.

The second settlement-road placement plays similarly but in reverse player order, the players also collect one resource card per tile surrounding this second settlement.

This way of playing directly translate in the following:

• The first player has the most options for the first settlement but may end up being blocked or left with poor choices for the second settlement.

• The last player gets to put his two settlements one right after another, removing all the uncertainties other players have to deal with when deciding where to put their settlements. The last player can be sure of the resource mix they will start the game with, and for which settlement they will receive cards at the beginning.

Is the advantage of choosing first outweigh the advantages of being sure of what we start the game with?

What about the second and third players? Is the game fair for them, or do they lose on both count?

Note: For my analysis, I decided to consider 4 player games, since there is more competition for good spots on the board.

Measuring fairness by payout expectation

When playing Catan, the type of resource cards you get is important, but no more than the rate at which you receive them! Getting less optimal resources can slow down your development, but since you can exchange them against other resources type, it certainly beats receiving no cards!

So to simplify our initial fairness problem, and get a good understanding of how the playing order affects the chances of winning, let’s start by a simplified version of the fairness problem:

• We will only consider the expected payout of our initial settlements in terms of the number of resource cards we expect to receive each turn, ignoring their type.
• We will only consider the first two settlements, thus limiting the fairness question to the initial setup, without consideration of the possibility of expansion.

This way, all hexes are of the same interest, their associated probabilities of paying out being the only thing that differentiates them.

All we need to do now to compare the expected payout for different players is to look at the tiles surrounding their settlements and count the dots under the number we find there.

The dots represent how many times on average this number will be rolled out of 36 rolls of dice. So we can use them as a direct proxy for the expected return of our analysis.

For example, the 6 and the 8 are the most commonly rolled numbers on the Catan board. If you were to roll the dice 36 times, you would expect those numbers to come out 5 times each (on average).

If you truly wanted to know how many cards you can expect each turn, you can divide by 36, and multiply by the number of players (the number of times the dice will be rolled between each of your turns).

A typical starting position will offer you about 21.0 expected return, so:

21 / 36.0 * 4.0 = 2.33

meaning you can expect 2 or 3 new cards each round.

Simulating the initial settlement selection

A few years ago, I began this blog with a series of articles exploring in detail all the possible winning conditions in Catan.

In doing so, I ended up exploring something related to our current endeavor, but not quite the same.

At the time, the goal was to get a general idea of the diminishing return of building new settlements as the good spots disappear through a game. So I ran simulations calculating the average return of first, second, third, and all following settlements for all players.

But now we want to know the expected return for each individual player, depending on their turn order. So I modified my simulation to make it easier to see what is going on.

Each player plays like in a real game, selecting the highest paying number for them. I added a few information to make it easier for us to understand:

• The circled numbers are identifiers for the settlement locations.
• Settlement returns list shows each location associated return
• Grayed-out numbers are for location no longer available.
• Players’ total expected returns are shown separately.

Just to be clear:

• Here the first player is Red.
• The first settlement is put on location #4
• The surrounding tiles have the associated numbers: 8-9-5
• Which have under them the following number of dots: 5-4-4.

This is the first item in the Settlement return list.: #4, which offers an expected return of 13.000 or (5+4+4).

In this particular simulation, the player that ends up ahead is the 3rd player, in white, who will start the game with an expected return of 22.00 cards per 36 rolls of dice. All other players starting at 21.00, as can be seen in the Total player return list.

Simulating the settlement selection

Let’s start with the simplest situation, assuming the players use a simple strategy of pure greed: Each player simply picking the highest-paying available spot for their settlement.

There are no other strategic considerations. If several spots have the same highest payout, the player will simply pick one at random between those.

Let’s see if we can learn something from such a simple approach… Here are the results of simulating 1 million games:

Expected returns for the 2 initial settlements, by player order

Player order	Average return	Standard deviation
1	20.73	0.874
2	20.58	0.901
3	20.56	0.918
4	20.55	0.932

The average return for each player is surprisingly similar, with a small advantage for the first player. The standard deviation, while similar, also indicates a smaller variation on average for the first player, which would point towards a first player advantage.

But such simple numbers do not tell the whole story!

Playing order advantage statistics

Because the previous results were not really exciting, I decided to examine the results a bit more closely… And I was not disappointed!

To check how the number actually plays out, I decided to keep track of how often each player ends up having the best or the worst expected return for its first two settlements (still using our early greedy approach).

I added the chances of getting the best and worst return to the last table:

Player order	Average return	Standard deviation	Best of all return %	Worst of all return %
1	20.714	0.87571	48.43	33.82
2	20.576	0.90499	34.62	38.84
3	20.554	0.92095	33.45	39.24
4	20.558	0.93085	38.29	44.66

The numbers add up to more than 100% because more than one person can have the best or worse return.

In 7.2% of the games, all players ended up with an equal expected return for their two settlements.

For the other cases, everyone with the biggest score was considered to have the best return (up to 3 players). Similarly, everyone with a return equals to the lowest return was considered to have the worst return.

It turns out, there is a real advantage in being the first player! Having the best chances to starts the game with the biggest expected return, and the least chances of starting with the worst expected return.

The last player has a better chance to get the best return than the 2nd and third player, but also has a clearly higher chance of ending up with the worse expected return!

Because more extreme situations exacerbate the feeling of unfairness, I decided to redo the simulation, but only count the times where only one player has the best or the worst return. In order to know the chances of being clearly advantaged or disadvantaged by the initial setup. Here the numbers are the following:

Player order	Best of all return %	Worst of all return %
1	19.01	10.97
2	7.22	9.33
3	6.94	8.81
4	14.27	17.09

Once again, the first player is clearly being advantaged here, with the best return 19.01 % of the time, even if in this case he has a bit higher chances than the second and third players to have the worst return.

The Last player has clearly more of a hit or miss position, with 14.27% of the times ending with the highest return, but a whopping 17.09 percent of chances of ending with the worst.

Having the benefit of selecting both settlements back to back comes with its perks, but one is better to take advantage of this carefully in order to compensate the odds of being left a bit behind from the start…

What the worst situation looks like?

We’ve just seen the statistics about the average case, but maybe the worst case is not that bad, or maybe the worst case is milder for some players than others!

The biggest imbalance I measured between the best and worst score was a difference of 6.0 points in expected returns. Let’s have a look at how such a board looks like:

As you can see, here the player ending the worse off is the first player! The 5 first settlements all get 12.0 as expected return, the highest available score on the board, but after that, the payoff drops rapidly to 9.0, 8.0, and finally 6.0. Making the last player the overall winner!

Compare this to the much fairer exemple below:

In this fair setup, all players end up with the same return. The highest paying spots are less concentrated at the top, allowing far better choices for the player placing the last settlements.

If you are curious, I created a similar simulation for the biggest spread of expected returns favouring each different players, (picked out of a million simulated boards):

Best Lead for player 1

Here the expected returns ends at: 21.0 – 19.0 – 19.0 – 16.0

Favouring the first player by 5.0 points!

Best lead for Player 2

Here the expected returns ends at: 19.0 – 23.0 – 22.0 – 22.0

Favoring the second player by 4.0 points!

Best lead for Player 3

Here the expected returns ends at: 18.0 – 21.0 – 23.0 – 22.0

Favoring the third player by 5.0 points!

We’ve already seen the 4th player winning by 6.0 points with a final score of 18.0 – 20.0 – 21.0 – 24.0 ealier.

(Here I choose to keep the biggest difference between the best and worst score. One could have selected the most unfair board a bit differently, but I think you get the point!)

To compare those setup, I plotted the settlement expected return for each of those different scenarios, it is a bit cryptic, but have a look:

In yellow the scenario benefitting the first player show a steep drop in expected return at the beginning, where the fair scenario in green shows a more distributed returns among the location.

We can surmise that the shape and location of the drop-off in expected returns should inform us of who would be the favored player, but there are many parameters missing from such a simple graph. Results also depend on how close the good spots are of each other on the map!

I thought to include it because it is not totally uninteresting!

Are the board then fair or unfair?

I think the preceding results shows that the simplified boards we’ve seen tends to indicate that there is indeed an element of initial board fairness.

Most boards tend to advantage certain players, but not always the same player. Statistically, you have more chances of starting in a better position if you are the first or the last player, but once the board is set up, we can probably find who will be at an advantage before the players put their first settlement!

This is an interesting finding I think!

What about player strategy?

One of the assumptions I made for the simulation was that players were simply greedy, putting no effort into selecting the settlements beyond direct expected return.

What if players start acting strategically? Does that change the fairness landscape?

Evaluating the strategic values of spots in a game is not straightforward. Which is a good thing, otherwise the game would be probably too simple to be fun!

So once again I made some simplifications.

In Catan, settlements cannot be built on adjacent locations.
You need to leave an empty place in between settlements.

Using this constraint as a base for our player strategies, we will evaluate 2 simple behavior.

• The blockers: All players still select the best paying spot, but pick the one blocking the most expected return from surrounding locations.
• The carebears: All players still selects the best paying spot, but pick the one blocking the least expected return from surrounding locations.

How those two simple strategies affect the fairness of a board ?

In order to demonstrate that this has or not an effect, I decided to find the island which shows the most ends result difference between the two strategies (sum of individual differences:12.0) :

Player order	Blockers’ return	Carebears’ return	Difference
1	21.0	19.0	-2.0
2	21.0	19.0	-2.0
3	19.0	23.0	+4.0
4	19.0	23.0	+4.0
Total	80.0	84.0	12.0 (abs)

One could think that this can only affect positively the first player, but as a counterexample, here a similar table for another game showing almost as big a difference than the previous one (sum of individual changes: 11.0):

Player order	Blocker’s return	Carebear’s return	Difference
1	19.0	22.0	+3.0
2	20.0	22.0	+2.0
3	19.0	21.0	+2.0
4	18.0	22.0	+4.0
Total	76.0	87.0	11.0 (abs)

Here everyone has to gain from being nice to others, but only the second player loses its lead over the others. And one thing to note is that in both cases, there is more return for everyone, and this is just by avoiding blocking others, not reducing the returns explicitly!

For the anecdote, the biggest return difference for a player caused by switching from one strategy to the other was 6.0, so it can make quite a difference for the same board…

Conversely, here is a board that offers the same return for the players, whether they use one strategy or another. The settlements are not the same, but the end result is:

So once again, there is a measurable difference between boards that offers blocking opportunity to those who don’t.

Given that the cards return ends up influencing the speed of a game, since you have to buy a certain amount of things to win, I would be curious to see if we can correlate player aggressiveness to game length (in term of number of turns)!

Who benefits the most from the different blocking strategies?

The previous example showed the difference the blocking strategies can make, but on average, who are the beneficiaries?

I obviously ran the simulation cumulating such statistics and found the following:

Blocker’s game return

Player order	Single Best return %	Single Worst return %	Part of best return group%	Part of worst return group%
1	20.82	9.76	51.01	31.45
2	7.00	9.49	34.24	38.95
3	6.66	9.46	32.31	40.59
4	13.32	17.48	36.70	45.85

Carebears’s game return

Player order	Single Best return %	Single Worst return %	Part of best return group%	Part of worst return group%
1	18.86	10.84	48.27	34.04
2	7.16	9.17	34.62	39.01
3	6.87	8.84	33.53	39.51
4	14.07	16.78	38.38	44.47

So the results here seem to indicate that overall, a blocking behavior will benefit the first player by augmenting the chances of having the best return, and diminishing probabilities of ending with less return than others.

All other players benefit from a nicer approach. But it is not a given that it would be the case if everyone was minimize blocking except the first player…

Is this to say that you should always implement a blocking strategy?

For sure, when selecting your second settlement, there is a lot less to lose at blocking more aggressively, so except in specific cases, the incentives do not really favor being too nice!

A grandmaster playing AI would certainly consider the advantages of not maximizing blocking when selecting its first settlement. Certain board configurations call for it.

But this is a tall order for a human to calculate all returns and predict other player behavior in the time you have to select your starting position.

That being said, it’s always good fun to find situations where being nice to others can ultimately be self-serving!

Because this is just a simplified example, and that it is already a long article, I’ll leave it at that for the moment. But I hope I have sparked some interest in the impact of a blocking or non-blocking strategy. Maybe this is a topic I should revisit in the future!

Is this all there is to fairness?

Going back to the question of Catan island being fair or not.

Even with my fairly modest analysis, I think I have found strong indications of a fairness factor in board initial setups. At least for my simplified version, some boards favoured equal expected returns for all players, but others did quite the opposite.

Will fairness imbalance persist once we add resources to the mix? This remains to be seen as this will make the situation quite a bit more complex.

After all, some strategies can rely on harbours or a specific combo of resources. This may offer a way out for less return privileged players… Or maybe those effects compounds, making for some weird boards to be really unfair. That is to be seen in my next article!

What’s next?

In a real game, not all resources are equally valuable to players. So what happens to the balance of expected returns when we start adding relative values to resources? Or start considering the need for resource diversification?

So, the logical thing to do is to try a similar analysis, but to include resources…

I already started, and even if the fairness results are less crystal clear than what we saw here for the simpler case, there are still very interesting things to be learned!

Finally, there is the question of comparing the fairness to the previously defined balance index.

Is there a clear correlation between balance and fairness, or should I add a fairness component to the balance index?

Stay tuned for the second of part of this analysis:

How to construct a fair Catan board, the resource edition!

If you liked this article, or have suggestions to improve it, let me know in the comments!