*After carefully studying the cheapest way to win, you were determined to blow your opponents out of the water with your newly minted and cost effective strategy.*

*But somehow, your opponents ended up with much more resource cards than you, and won the game… Were the dice simply against you, or is there something we missed in all this?*

## Welcome to the 3rd part of the Catan victories analysis!

While I cannot say that I have exhausted the topic, this article is the last part of three article examining all the different possible victories at Catan.

- First, we examined all the possible ways to win a game of Catan – and their minimum cost.
- Then, we calculated: The expected cost of every possible way to win
**–**based on cards drawing probability - It’s time for the final part, where we will study the
**fastest ways to win at Catan**– looking at settlements and resources income.

It is not absolutely necessary to read the first two parts to understand the current article, but I think everything will make more sense if you read them first!

## What to expect in this post

In the previous post, we examined all the possible victories in Catan, and how to evaluate their costs. However, cost is not everything.

Today, I intend to add a final layer to this analysis by examining how we can use the **minimum cost**, and **expected cost** information in choosing a strategy to win, as fast as we can.

This involve simulating simplified games and understanding how settlements and cities influence resource income during the game.

In the end, we will end up with a graphic like this one, comparing the different victory combinations in term of how fast can we achieve them.

This graph is a bit different than the graphs of the two last posts. So you’ll have to read on to see how we end up with it!

*Note: As before, there is also a complete list of victory, with associated costs and speed, which makes it easier in details than this high-level graph.*

## Why the cheapest way is not also the fastest way to win

Besides the cost of the things you choose to buy and the probability of drawing cards, other elements in Catan affect the speed at which a victory can be attained… There is not much point of going the cheapest route if it is not also the fastest, isn’t it?

In Catan, everything is bought using resources. And resource payout is based on the cities and settlement you build.

By the rules, each settlement and city are built on corners of the hexagonal tiles making the island. This means that all settlement will be associated to between zero and three producing resource tiles, depending on its position on the board (One of the tiles is a desert tile that does not produce anything (shown in yellow). Also *the Robber* can disable any resources paying tile while he is there.

Here is an example of the different places where you can put a settlement, with respectively 3, 2, 1 and zero resources tiles associated with it:

At the beginning of the game, after constructing the island from the hexagonal tiles, each resource tile is given a token with a number between 2 and 12. Those are used for resources production.

Each player starts its turn by rolling a pair of normal 6-faces dice, the sum of which will give a result between 2 and 12. This number indicate what are the resources tiles that produce.

For each **settlement** that a player have around the producing tiles, he will receive 1 resource card of the type indicated by the tile, if it’s a **city**, he will receive 2 such resource cards.

The more cities and settlements a player have on the island, the more likely she is to receive cards each time the dice are rolled. And the corollary to that is: the more you receive cards, the faster you can spend the needed resources cards to win. So more settlements should increase your chances to be the first to reach 10 victory point!

**Let’s examine this!**

The probability of rolling each number is shown in the table below.

I also indicated **how many times** you can expect a particular number to show up **if you roll the dice 36 times**. Why 36?

- It gives a round number, so it easier to understand
- Each number token in Catan has a number of dots printed on it. The number of dots represents the likelihood of this number over 36 rolls (in other words, how many time on average you’ll get this number if you roll the dice 36 times). So I used the same convention.

Dice Roll Result | Probability of roll | 36 rolls expectation |
---|---|---|

2 | 2.78% | 1 |

3 | 5.56% | 2 |

4 | 8.33% | 3 |

5 | 11.11% | 4 |

6 | 13.89% | 5 |

7 | 16.67% | 6 (robber) |

8 | 13.89% | 5 |

9 | 11.11% | 4 |

10 | 8.33% | 3 |

11 | 5.56% | 2 |

12 | 2.78% | 1 |

## So how can we use this to determine the number of turns it takes to win?

Besides directly giving you victory points, cities and settlements determine the resources payout you will receive each turn, and ultimately the speed at which you can attain victory.

The earlier you build your settlements, the more turns they will pay during the game. Not only that, earlier settlements can potentially be built on higher paying grounds since players usually build their settlements on the highest available paying spot, so good ones disappear fast.

To evaluate in how many turns a victory can be attained, I decided to evaluate how many turns it should take, on average, for a player to accumulate the resources needed for each different victories. And lucky for us, we already determined how much resources is needed (** the expected cost**) for each victory in the last post.

Now we only need to determine the **expected payout** of different victory point combinations (some victories involve building more settlements or cities than others)!

The combination of the **expected cost** and the **expected payout** should allow us to have a good idea of the expected speed of any given victory!

**Expected payout:** *How many resources cards a player can expect to receive each turn (on average), based on the cities and settlements he has on the island.*

I made the following assumptions in my evaluation (in this order of importance):

- Players will minimize the amount of road they build in their network, basically building settlements every two road segments.
- Players will build their new settlements on the highest paying spot available to them
- Players will upgrade their settlements to cities as fast as possible
- Players will build their settlement as fast as possible

## How to determine the expected payout of a settlement

If we allow randomization of the initial board, each game you play will have different *expected payout* for the settlements emplacements. But given a particular board, it is easy to determine the *expected payout* of all the possible building site on the island. We simply use the probabilities to determine the expected payout of a settlement, as we did for the cards expected cost.

*But this time, this is much simpler!*

For example, consider a player placing a settlement at the junction of three tiles with the numbers **3**, **4** and **5 **on them, which have a respective payout expectation of **2**, **3** and **4** resources cards for every 36 rolls (as indicated by the number of dots on each number token).

To get the *expected payout* of this settlement, we simply add the expected payout of each tile. This gives us a total expected payout of **2+3+4 so** **9 resources cards for every 36 rolls** for this settlement.

To get the payout for **each dice roll** (instead of 36 dice rolls), you simply divide this number by 36, nothing fancy here:

*9 for every 36 rolls, will give: 0.25 resources on average per dice roll*

**Each player rolls the dice at the beginning of his turn.**

For a 4 player game, this means that a player will have received 4 times his expected payout since his last turn (1 for each of the other player, plus his own dice roll).

Since each dice roll has the same expected payout, in our example, the player will add to his hand the following expected number of cards:

*For a 4 players game, a settlement scored at 9 for every 36 rolls will give *

*9 / 36 * 4 = 1 resource card on average for each time it’s the player turn to play.*

Using the same process, we can calculate how much each settlements will add to a player expected payout.

*Note: If there is a Robber on one tile, this particular tile won’t pay out any resources, but this is game dependant, so we will ignore it for today.*

## Going from the expected payout of a specific settlement to the expected payout of any settlement

As I was saying earlier, determining the expected payout of a given settlement is easy. But how to we generalize if we want to consider the payout of all the settlements and cities that will typically be involved in a randomized game?

*I decided to try simulating a few hundred thousand games and find out!*

Well, those are not full fledge game simulations. I simply took my assumptions for how players construct their settlements, randomized the initial game board for each different trial, simulated 4 players building settlements one after another (round-robin style), and averaged the return for each of the player settlements, in their build order.

First, I started with some basic assumption:

- Players will build their first settlements on the highest paying spot.
- They will always augment their network going after the next best spots.
- Player only consider the closest spot available (distance of 2)

I know this is not exactly how a game plays out. In a real game, a player would also evaluate at least the following points:

- A player could build 3 part road network or more between settlements to reach higher paying spots.
- A player could build a settlement or road at a non-optimal place to block the development of an opponent.
- Going for the longest road may restrict how the network is built, adding compromises to be made on where to build.
- Around the edge of the map, there are different harbors, allowing players to have better ratio a trading certain goods. This can be helpful in obtaining missing resources cards at reasonable prices and could change the value assigned to a given spot.

But I nonetheless think that my approach gives a pretty good estimate on how the high paying positions disappear first during a game. And this should at least be considered a strong baseline in evaluating such returns!

Here is an example of how such a 4 players simulation plays out:

It gave me the following average return (expressed in expected payout for 36-rolls):

Average return for each player settlements in a four player game First : 11.951125 Second: 9.14635 Third: 6.72839 Fourth: 5.27427 Fifth: 4.06856 Sixth: 2.99628 Seventh: 2.13481 Eight: 1.67107 Ninth: 1.40315

*Note: Here 11.95 the average expected payout for your first settlement. This is true for the first 4 settlement on the board, so those numbers are independent of player order.*

The same goes for a 3 players simulation (Shown here because it’s fun to watch):

For three players, the numbers are slightly different, but not that much…

Average return for each player settlements in a three player game First : 12.36953 Second: 10.0588 Third: 7.9837 Fourth: 6.7231 Fifth: 5.5394 Sixth: 4.5396 Seventh: 3.6368 Eight: 2.7425 Ninth: 2.0064

## Determining the *speed* of a victory

When I talk about the speed of a victory, I refer to game speed, which in this case would be measured in term of game turns.

All the information we need was determined in the earlier part of this series.

- In Part One, we determined the composition of each possible victory. Including
**how many settlements and cities**need to be built by the player. - In Part Two, we determined
**how much resources each specific victory is expected to really cost**. - In the above section, we determined what are the
**expected resources return of any number of player settlements**.

To determine the speed of a victory, we can take the above information and determine how many turns it should take to receive all the resources needed for a victory, based on the number of cities and settlements involved in said victories.

The only caveat here is that since resources are needed to build settlements and cities, a player will slowly build those as the game advance, and will gradually augment his expected resource payout during the game.

So I wrote a simple simulator to calculate how fast a player could expect to attain each different victory if he built all his needed cities and settlements as fast as he could, maximizing his expected payout along the way.

In Short

- Each turn the player receive his expected payout in resource cards.
- The player spend his resources toward his goal (upgrading and building settlement first)
- As he build settlements and cities, his payout increase
- From there I calculated how long before he receives all the resources needed to win.

This gave me a list of how many turns a player can be expected to take to win the game, for each strategy!

*Note: One important thing to note is that for all the evaluations I consider all resources cards to be equal, and do not take into account the types of the resources cards the player need to spend. This has the advantage of comparing the victories more easily. Otherwise, we would need to consider different lists depending on expected resource types.*

## Simulating a victory turn-by-turn

You can expand the following box for an example of how this would play out on a turn-by-turn basis, for a specific victory:

### + Victory spending and income simulation

## What are the fastest victories

For a 4 player game, the fastest victory takes **6.34 turns:**

- Longest Road (2 vp)
- 2 development card points (Biggest Army or Vp cards)
- 3 Cities (6 vp)

A lot of the fast victories involve having the longest road, 3 or 4 buildings, and scoring a few points with development cards. So it’s nothing earth shattering. But it is interesting to note that a balanced approach is faster than focusing only on development cards, or on settlement building. This indicates at least that the game is well balanced!

However, one thing that was made very clear, is that **building settlements and upgrading city as fast as you can** really give you an edge toward victory. It’s the Catan equivalent of compounding interests. Building settlement increase resources income that allows building settlement faster, which will increase the income even more! It may makes sense to delay other purchases (such as development cards), even if this mean you end up not doing anything during a few turns.

In my experience, a game usually involves 60 to 80 rolls of dice, which for a 4 player game means a 15 to 20 turns for each player. Settlements built early (or city upgrade) will make a world of difference in the game as a whole.

For a 3 player game, the fastest victory takes **8.17 turns:**

- Longest Road (2 vp)
- 3 development card points (Biggest Army or Vp cards)
- 2 Cities (4 vp)
- 1 settlement (1 vp)

Interestingly, the order of victory change slightly between 3 and 4 players games. With 3 payers it takes more turns, because there is less roll of dice per turn, but it takes less roll of dice in total.

**At four players: **4 * 6.34 =** 25.36 dice roll**

**At three players: **3 * 8.17** = 24.51 dice roll**

Both effects are due to the fact that with 3 players, the settlements and cities have a higher expected payout on average. You can have a look at both speed of victory lists (for 3 and 4 players) down below. But both list are close enough that the difference are mostly anecdotal.

## What are the slowest victories

The slowest victory, at **18.14 turns**:

- Longest Road (2 vp)
- 7 points from development cards (Biggest Army + 5 Victory point cards)
- 2 Settlements (2 vp)

It’s an 11 point strategy, the corresponding 10 point victory, with only 6 victory point from development cards, is 14.63, but that’s partly left to the chance of drawing cards.

But that still means that aiming for a victory only using victory points by development cards is more than twice as slow as the fastest victory.

## The power of upgrading to cities

One thing to note is that even victory depending on drawing a lot of cards can be made much faster by upgrading settlements to cities first.

For example, the victory involving **2 cities**, and **6 victory points** from development cards takes **9.83 turns**. Faster than all other victory using 6 or 7 development cards points.

This is due to ignoring everything else and concentrating on buying cards (after upgrading the settlements you get at the beginning). No need to build other road or settlements. The expected Cost is high, but the total expected payout compensate!

One way to convince ourselves of this is by looking at the settlement expected return simulation of earlier. **Doubling the payout of the highest paying settlement by upgrading it to a city is as much rewarding than building the 3rd and 4th settlement… combined! **

At a cost of **5 resources** for a city upgrade versus a cost of **12 resources** for 2 settlements and 2 roads, we can see why the former can be the most interesting option!

*As a side note, you may consider not upgrading your highest paying settlement to a city first, but maybe your second one, spreading the impact (and the attractiveness) of an opponent placing the robber on your highest paying resource. Also, for similar reasons, it makes sense to avoid to building many settlements around the same tile if you can help it!*

## The list of fastest victories

You can **expand the following box** to see a list of all victories with their expected speed. I encourage you to play with the column ordering to get a sense of how much each element affect speed and cost!

### + Expected turns to win for the 143-ways of winning at Catan for 4 players game

### + Expected turns to win for the 143-ways of winning at Catan - 3 players games

## Real wins take longer

If you played Catan before, it will be apparent to you that games usually take longer than the ideal times shown here.

This correlates well with the fact that resources we receive are not the ones we want, and that we often end up trading at a disadvantage to get them. So you’ll need a bit more resources to win than the expected cost. Also, the player that looks like he is in advance is more likely to be hindered by other players, refusing to trade, or putting the robber on his highest paying lands.

For this reason, strategies that involve building near trading harbors, or settlements with the right mix of resource producing tiles should not be forgotten.

But all thing being equals, faster strategies should still give an edge in the game!

## What does it look like on a Graph?

Let’s look at the composition of all victories, in a 4 player match:

In order to be consistent with the two preceding post, I wanted to find a way to represent each victory on a graph, sorted by their speed to win.

I decided to keep the breakdown cost of each victory, but to scale each cost by the speed the victory can be attained.

This has the advantage of preserving the readability of the victories. You can still see what the victory is made off by looking at the bar representing it.

But faster victories shrink the bar height according to the speed, and this does not indicate anything about the cost of the victory itself.

## Final thought and other open questions

Ouf! This whole thing took far longer than I expected! But I’m happy with the results. I have a feeling that I will revisit this work to improve upon it and use it as a basis to analyze other games.

One thing I’ll try to aim for is shorter and more frequent posts. But this is not always easy since that, even starting with a clear plan, results and inquiries tend to have a life of their own.

That being said, here is a list of questions I did not develop in this post, but that I think deserve some inquiry:

**Is it best to develop new settlements first, or to upgrade your existing settlements to cities?**

Expanding your network early will tend to block others in their development. And you’ll have access, in theory, to higher paying spots on the island.

At the opposite, since your first settlements will be on the highest paying spots, upgrading them will give you the higher increase in expected payout.

Which one should you choose?

I’m seriously considering simulating this in the near future, with competing players using different strategy to see how it influences, or hinder the fastest ways to win. But this also falls into a category of evaluation that will vary from game to game. Sometimes you’ll have room for expansion that is not threatened by other players, so you are in no hurry to develop, and could go for the more lucrative strategy of upgrading to a city first.

### Should we diversify payout number when placing settlements?

When selecting new settlements, is it better to select the higher expected return, or is it better to diversify the numbers that will generate resources. Thus trading *higher expectation* for a *reduced variance*, on resources distribution and lowering your exposure to a bad series of dice rolls. As I said… life is made of choices!

Just to be clear in case jargon lost you: If you diversify the different numbers around which your settlements are built, you are less likely to be “bad lucky” and not receive sufficient resources to do anything for several turns (if your numbers never show up on dice rolls).Selecting the highest paying spot, without regards to number diversification will maximize your chance to receive resources.

Aiming to diversify the numbers for which you receive resources will lower your chance of spending several turns without receiving anything, and maybe avoiding to be left behind early in the game.

### How the early game resources selection affect the speed to win?

One thing I now really want to do is take this list and redo the analysis considering the resources you choose for your first 2 settlements.

Since different victories rely on different resources count, we could give a resource conversion factor, and re-evaluate the speed of each victory according to this. Thus making a custom victory approach for different starting resources!

We could even combine the list and see which resources bring you faster victories!

### Evaluating the real cost of resources

Finally, going a bit further on the resource front, access to trading port can change drastically the real cost of resources you need during the game. This should be factored somehow in the victory speed list, in order to evaluate how fast a victory can be attained, with or without access to the optimal harbor! Something I’ll think about!

**If you have other open questions to add to this list, let me know. I may add some, as a kind of to-do list for future analysis!**

For all the articles that refer to the Catan board game, check Analyzing Settlers of Catan. A post that serves as a kind of *Table of Content* that will be updated as I add new content related to this game.

*Ok, I’m done for today.*

*I hope you had some fun and learned a thing or two about Catan, or the math involved in evaluating the best strategies.*

*Expect a come back about other aspects of Catan in the future, but I’ll write about other game first. **Is there something you would like to know I did not mention?*

*If you spot any mistake, let me know, it will be a pleasure to correct any error in this. I greatly appreciate your contributions in this regard.*

Very nice!

Your specific turn-by-turn example only builds 4 roads (the 2 you start with plus 2 needed for extra settlements) yet it requires the longest road for its victory condition. If this oversight exists for all of your calculations, then each victory condition that includes the longest road and fewer than 5 buildings will not have built the necessary roads.

Good catch!

Fortunately, this is only an error in the explanation of the simulation.

The actual program simulate all the turns until all the needed settlements and cities are built. Keeping track of the costs.

Then it look at the remaining cost for the victory and check how much turns it will take to get the remaining resources once the player has reached his maximum resource payout.

I added by hand the last two turns to make it more explicit (but did not think about the road!).

I’ll correct it to remove the confusion!

Thanks a lot!

I don’t have much to add, since your analysis went well beyond what I would be able to do. However I am grateful you have shared this as I have learned a lot !

Where would the concept of nash equilibrium fit into the analysis? Can this entire analysis be considered “game theory”?

As far as I understand, a Nash equilibrium happen when each player ends up having an optimal strategy to follow, and there is no point for anyone to change their strategy.

This is something most often considered for pure strategy game.

I think in a game like Catan, if such strategy exist they would highly depends on the current board. And luck based card drawing or resource distribution would change the game each turn. So anything looking like a Nash equilibrium could only be determined at a precise point in time. Not something you would establish for an entire game.

I did not study game theory beside casual reading, so it’s my interpretation of things.

But everything described here could certainly be a part of a game theory analysis of Catan, since it calculate cost and probabilities that can be used to make informed decision during the game, which is pretty much the basis of game theory analysis.