Introduction

The goal of this article is to quantify some aspects of Catan so that we can make better decisions and optimize our strategy accordingly. Notice that we do not discuss any strategies in this article

This article is created by the knitr package of R. Since this is written for gamers but not programmers, I hide all the codes. Comments and suggestions are welcome and can be sent to wanshunwong at gmail dot com.

Development Cards

Expected Economic Value of Knight Cards

On average we will roll a 7 once every 6 times. Therefore after using a Knight Card, the Robber is expected to affect 5 Resource Productions before a 7 is rolled and the Robber is moved again. During those 5 Resource Productions, the expected number of resource cards prevented by the Robber per Settlement is shown below. It is clear that the expected number depends on the number token that the Robber sits on.

Let’s do an example. Suppose the robber is on a number 8 token (with 5 dots), and you, player A, and player B each has one Settlement adjacent to the corresponding Terrain Hex. You use a Knight Card, and move the Robber to a number 6 token (again with 5 dots) where players A, B and C each has a Settlement adjacent to the corresponding Terrain Hex. There is no net effect on players A and B. On the other hand, you start receiving resource cards while player C’s production is now hurt. This is a swing of 1.67 cards on average. Together with the card you rob the expected economic value of this Knight Card is 2.67 cards, which is actually less than the cost of a Development Card.

The expected economic value of Knight Cards in other scenario can be easily computed in the same manner. The following plot is made to facilitate those computations. Since we rarely put a Robber on a number token with 2 dots or less, we exclude those cases from the following plot.

This plot also helps us to decide where to put the Robber. For example, it is better to put the Robber on a 4 dots number token with “4 adjacent Settlements”, i.e. 2 Settlements plus 1 City, or 2 Cities, than on a 5 dots number token with only 3 adjacent Settlements.

In the late-game phase where most Settlements are upgraded into Cities, we see that the expected economic value of a Knight Card can be much higher than its cost. Notice that, however, the above calculation is based on the assumption that after you use a Knight Card, the Robber remains stationary until a 7 is rolled. If another player uses a Knight Card after your turn, then obviously the above calculation does not apply.

Largest Army

Since there are 14 Knight Cards out of 25 Development Cards, the probability of drawing one is 56 percent. This means on average you need around 5.36 Development Cards to get 3 Knight Cards for the Largest Army. Combining with the rules that you can play at most one Development Card per turn, and you cannot play the Development Card you bought during the same turn, you need to plan early if you want to get the Largest Army. Obviously you need to draw more Development Cards if there are other players competing the Largest Army with you.

An interesting fact is that as 20 percent of the Development Cards are Victory Point Cards, you are expected to draw (at least) one on your way to the Largest Army. Therefore you only need 7 victory points from your Settlements and Cities, although building more will help your resource production and won’t hurt.

Resource Cards

Building Costs

We know that there are 5 types of Resource Cards, and we also know that they are not identical. In this section we want to know how many Resource Cards of each type we need in order to get to 10 victory points. The following computations are based on using the minimum number of Road segments required. The cost of the initial 2 Settlements and 2 Road segments is zero. There are 3 main ways to obtain 10 victory points:

  1. Getting 10 victory points just from Settlements and Cities. There are 2 combinations on the number of Settlements and Cities to do so.

  2. Getting 8 victory points from Settlements and Cities, and having the Longest Road. There are 3 such combinations. In the following calculation we assume 8 Road segments (in addition to the inital 2) have to be built to get the Longest Road. We choose length 10 as a benchmark because a lot of the combinations to get 10 victory points need 6 or more Settlements/Cities. And if they all lie on a single road then that road is at least of length 10.

  3. Getting 7 victory points from Settlements and Cities, and having the Largest Army and 1 Victory Point Card. The following computation assumes buying 6 Development Cards will give us the Largest Army and also 1 Victory Point Card.

The labels used in the X-axis come from the initials of the building items. For example “2C 4S 10R” means 2 Cities, 4 Settlements, and 10 Road segments. The first observation is that unless we are going the Largest Army route, the demand for Wool is very low. On the other hand, the demands for Brick and Lumber are fairly consistent, except for the extreme case “3C 1S 6D”, whether we are trying to get the Longest Road or not. This is because the distance rule in building Settlements requires us to build many Road segments. Finally, all the above combinations require a substantial amount of Grain and Ore, since Settlements alone cannot get us to 10 victory points and we need to upgrade some of them to Cities.

The next observation is that the Building Costs of the 3 combinations with the Longest Road are the lowest. Therefore if you think you have a high chance of getting the Longest Road at around length 10 you can consider going for that (provided you have a decent Brick and Lumber production).

Another thing to keep in mind is that if there are other players competing with you for the Longest Road or the Largest Army, the buiding costs can go up a lot.

To conclude this section we remark that the above chart does not consider the economic value of Development Cards nor the resource production capabilities of Settlements and Cities. Depending on the map and your resource production capabilities certain combinations with high building costs might actually be your best option.

Production Capabilities

There are 18 Number Tokens on the whole map, and the total number of dots on them is 58. Hence if a resource has a total of 10 or less dots, its production capability is low. On the other hand if a resource has a total of 13 or more dots, its production capability is high. The difference in production capabilities among different recources will become more and more significant as the game proceeds and players start expanding.

Harbors

It is very hard to determine the value of Harbors because it depends on how willing your opponents are to trade with you, players’ resource production capabilities, the availability of different Resource Cards, and much more (e.g. having a relevant Harbor will give you more bargaining power in negotiation). In this section we focus on one specific scenario so that we are able to do some concrete computation.

Suppose we are going to build our last Settlement, and later on we will only upgrade our Settlements into Cities. Hence we don’t need Bricks anymore. Now our task is to decide the location of our last Settlement: Whether we build it at a 2:1 Brick Harbor, or at a location with higher resource production. We assume the extra Terrain Hexes produce resources that we need, say Ore. Our choice depends on our current Brick production capability and also the extra Terrain Hexes’ number tokens, therefore we have the following graph.

First notice that we can always trade 4:1, thus all the lines above have positive slopes. We see that 2:1 Harbor outperform 2 and 3 extra dots when we have more than 8 and 12 Brick dots respectively, which roughly correspond to having 2 and 3 Settlements adjacent to Hills. We remark that the flexibility of a Harbor is not accounted above. For example in the above scenario we can use our Harbor to get Grain after gathering sufficient Ore.

For completeness we also provide a similar graph for 3:1 Harbor. We see that most of the time having extra Terrain Hexes will give a higher effective production capability. Therefore building at a 3:1 Harbor is a better choice only if other factors like flexibility and bargaining power in negotiation are important.