Tiles, Miles, and (hopefully) Smiles
Welcome back! Today I will be talking a little bit more about strategy in Catan. Last time we talked a little about hexes, their associated numbers, and how that affects where we might want to place the robber. This left us with several unanswered questions about other phases of game play. What is the value of a resource? Where do we place our settlements? And what is the value of a Development Card?
Most of these questions do not have clear cut answers, but I will try to share some ideas that seem to make sense to me, and hopefully lend themselves to a reasonable strategy. If I have the time and energy, I might try to code up a bot that plays Catan (we’ll call him Stevie the Settler), and see what strategy he comes up with. If Queen Victoria keeps me from watching this season of the Bachelor long enough, I might just make it happen…
For now though, let’s stick to simpler, “human” strategies.
Development Cards
What are Development Cards, and are they worth going for? At any point in the game, a player can spend 1 Ore, 1 Wool, and 1 Grain to buy a Development Card. The game has 25 Development Cards, comprised of 14 Knights, 5 Victory Point, 2 Road Building, 2 Year of Plenty, and 2 Monopoly Cards.
This may actually be a good time to explain Victory Points (VP), which is exactly what they sound like, points you need for Victory. The first player to 10 Victory Points wins the game. You can get 1 victory point for each settlement, 2 for each city, 2 for having the Longest Road (we’ll talk about that later), 2 for having the Largest Army, and of course, 1 for each VP card. Then, it makes sense for us to compute the expected value of a development card in terms of Victory Points. Indeed, every decision we make in the game (building settlements, upgrading to cities) should be viewed as a tool for gaining more Victory Points.
For VP Cards, we can easily assign a value: 1 VP. Each Road Building Card allows us to build 2 roads. Since the Longest Road is worth 2 VP, we assign the value 2/L where L denotes the length of road required to have the longest road. For example, if the longest road is currently 6 edges long, and we have a road with 4 edges, then L = 3. This valuation in some cases overvalues the Longest Road, as other players may be holding Longest Road cards without revealing them. Similarly, the length required for the Longest Road will increase throughout the game, so you might also want to discount this value a little initially. On the other hand, expanding your empire and building roads at the start of the game could also be very beneficial.
Similarly to Road Building, Knight cards let us work towards Largest Army. We will ignore the effects of placing the robber for now. When we discuss resources and tile placement, maybe there will be a good way to include this factor. Like Road Building, if S is the number of knights required to take the largest army, we assign Knight cards a value of 2/S Victory Points.
Year of Plenty and Monopoly are a little hard to quantify, and completely depend on the current state of the game. Let’s say you had some Brick and Lumber (in addition to whatever you spent on the Development Card), then Year of Plenty would be worth at least 1 VP, since you can build a settlement. On the other hand, if you were tied for the Longest Road, then Year of Plenty would allow you to seize that lead, and thus be worth 2 VP. As we can see, the value of a Year of Plenty (and similarly the Monopoly Card), depends heavily on what options are available to you at a given point in the game, so let us just give them arbitrary values Y and M for now. (Yes, this is quite the cop-out non-answer but we’re going with it.) You can think of Y and M as how many Points you would receive if you used these cards as effectively as possible. For another example, if you had a steady supply of Brick, Lumber, Wheat, and Grain, calling Monopoly on Ore would be an incredibly powerful move. After all, without Ore, a player can only get 7 VPs (5 settlements and Longest Road), making winning impossible.
Anyways, with these valuations in hand, we know exactly the value of a Development Card! Initially, we would have,
Of course, as the game progresses and players reveal cards, the probabilities of each card being drawn will change, but the same calculation will still work.
Resource Evaluation
Again, it’s a little hard to quantify the value of a certain type of Resource. This powerpoint takes a stab at it, and apparently when it comes to bricks, Catan and basketball have a lot in common. In fact, in 13% of games (although I’m not entirely sure where their statistical data comes from), the winning player never settles on Brick. As we saw in the previous post, the value of a resource is weighted according to its rarity.
Anyways, as it’s a little unclear how to value a certain resource, we will temporarily avoid this question. Indeed, the value of a resource changes throughout the game, again depending on what is available to you. For example, at the start of the game, the priority is to build roads and settlements and take up as much land as possible. For this, you need Brick and Lumber. Furthermore, a settlement early in the game is worth more than a settlement late in the game: both are worth 1 VP but one will produce more resources than the other.
Later in the game, you might be concerned with upgrading to cities, or hoping to draw a couple VP cards from the Development Card deck, so that Ore and Grain grow in importance. Then, depending on how you value each resource, we can determine the value of each tile. When Stevie is built and trained, I will come back to this and see how he determines the worth of a resource.
Tile Placement
Given resource valuations, it is then pretty straightforward to determine the value of a tile. In particular, a tile has two attributes, the number k and its resource type. Then, for any settlement location, we can simply add up the value of its adjacent tiles as follows,
In particular, this P tells us exactly the expected production of a given tile in a single turn. Generally, we would like to place settlements/cities on tiles that have the greatest production. However, later on in the game, we may want to ignore this and settle a harbor in order to make favorable trades. For example, if we are producing a lot of Wool, settling a 2:1 Wool Harbor would be great way to produce whatever resource we need.
Decision Making
Equipped with our analysis, we can now decide what to do on a given turn. Again, for simplicity, we will ignore resource trading for now. (Including trading just means giving yourself more options, so you can evaluate all your options). This leaves us with several options: building settlements, cities, or buying a development card. You will also need to build roads, to give yourself room to expand and build more settlements. Especially at the beginning of the game, expansion is key as you can always upgrade a settlement or buy a Development Card (usually), but without expansion, another player could easily cut you off and seize a large portion of the island.
For a Development Card, we already know how to determine its value in terms of Victory Points. We will also say that a settlement is worth 1 VP, while a city is worth 1 VP (since it replaces the settlement). Dividing each of these by their cost gives us the value of each action. For example, given the value of a Development Card from above, we can find the value of buying a card as,
For a settlement, the cost is a little more complicated, as a settlement produces resources in the future. The earlier in the game, the greater the return on a settlement. Thus, we can add this to the value of a settlement to get,
Again, it’s a little unclear what exactly f should be: should it decrease linearly, exponentially, in some other way? Without a clear analytical answer, this is another question that will require lots of simulation and empirical results to give us some insight.
Building a settlement can also have many other advantages in addition to simply grabbing more land. In rare cases, a settlement could break another player’s road, either solidifying your lead in having the Longest Road or making significant progress in claiming the title for yourself in the future.
Anyways, with these tools, this should help us make some better decisions in our turn, whether to settle, upgrade, or buy a development card. In the case of settling and upgrading, this should also give some insight on where to settle and upgrade. Unfortunately, in some ways, this post has raised more questions than it has provided answers, and I have more or less convinced myself to at least get started on trying to program a bot that learns to play the game. Whether I finish (and maybe more importantly, whether it works) will be something for a later post.
Hope you enjoyed this second post! Let me know if you have any thoughts or ideas in the comments!
