Settlers of Catan is a compact game that relies on probability to create compelling play. At times the game can be quite cruel, leaving players stranded with their seemingly solid numbers while their opponents rake in piles of resources. But even when that pesky forest with a number two keeps rolling for your opponent, the game rewards players who bank on probability over a series of occurrences. So that forest might be lucky now, Plot 1 makes it fairly clear that over time that forest will be put in its (probabilistic) place.

For this article, we’re going to use probability to create averaged costs, then take that data to inform how we could most efficiently win the game. We can approach this comparison a few different ways. One method would involve combinatorics to determine how many 10-victory point (VP) possibilities exist, sum their resource costs, and then determine which possibility is most efficient. That method quickly runs into issues. How do we measure development card VPs, of which there are five in a deck of 25? What would be the appropriate cost for such a VP? How do we cost special bonus VPs like Longest Road and Largest Army? These questions need to be resolved before we’re able to proceed. So we’re first going to understand how the development cards might be costed.

### What’s the average cost of a VP development card?

There are five development cards that yield a victory point (VP) out of 25 cards. So P(x) = .2, where x is the likelihood of drawing a VP card. We could say, then, that banking on drawing a VP development card is not a good strategy. But what if we tried to measure the average cost of a VP development card?

The plot below is a hypergeometric distribution that displays the probabilities of several card combinations. It’s worth noting that the distribution is based on solitaire play, so the assumption is that nobody else draws from the deck.

With three card purchases, the probability of obtaining a VP card (red line) is P(x) = ~.5. Therefore, half the time the average cost of a VP development card (our original question) is 3(1 sheep, 1 one, 1 grain).

### Comparing VP Possibilities

We can take the data from our plat above and place it into a table.

We can then take the costs above and apply them to all possible VP purchases.

Settlements are a great value in terms of resources spent for VP. As we already discovered, buying development cards just to draw a VP card is inefficient. Of course this table is missing important context. Each of these options have attendant upsides beyond their VP. There are other constraints missing in the table. Players can’t buy 10 settlements; they can buy five. Cities provide enormous resource production benefits that offset their higher Cost/VP. And development cards provide value outside of strict VP, so the cost of buying a VP development card is also buying other value.

### Strategic Directions

Catan is a game won in inches, which means that at the end of the game the differentiation between players—representative of divergent strategies—isn’t huge. Everyone has to build settlements and usually cities. The main strategic differences emerge in the distribution of those buildings and late-game decision to spend resources on either The Largest Army or The Longest Road.

The VP Possibilities table signals to two synergistic strategies: a city-heavy strategy with The Largest Army bonus, and a settlement-heavy strategy that relies on The Longest Road bonus. Even at its most diverse, the two strategies seem pretty similar, as in the table below.

This build order is based mostly on resource efficiency. We already established that settlements are the most efficient Resource:VP purchase available, so it makes sense that a “settlement heavy” strategy will seek to exploit that. Given that five settlements already requires a heavy investment of brick and lumber, a Longest Road victory complements the strategy. One noticeable weakness, based purely on buildings and their costs, is that brick and road have a short lifespan and do not provide value beyond settlement costs and roads. Because there is a maximum number of settlements, this forces a settlement-heavy strategy to extend into other resources in a way that may be difficult; for example, at least one victory point card may be required, which costs everything except brick and road. This is also unfortunate because development cards don’t complement the rest of the strategy. Plot 3 visualizes the resource commitment of this strategy: it’s a little bit of everything. This is positive news because a heavy dose of settlements can provide a small but diverse set of resources, which is what this strategy requires to succeed.

The city-heavy distribution of settlements:cities is context dependent. One must always have at least as many settlements as there are cities, but one could go 3:3 plus additional VP cards, or 4:4 with no VP card, etc. The city-heavy strategy is a clear contrast to a settlement-heavy strategy in that it requires a significant investment in fewer resources, namely grain, ore and wool. Again, the game’s design creates a complement between this heavy investment and the city’s increased resource gathering ability. A city-heavy strategy also differs in that the VP cards complement the resource investment, giving the strategy a lopsided focus and avoidance of brick and lumber. One weakness is that ore–along with the other mining resource, brick– appears three times on the map, while the other resources have four tiles. Because it’s central to the city strategy, it’s important to nab a productive ore location early in the game. Another weakness is that your two underused resources, brick and lumber, are essential to growth. Getting bottlenecked by these two resources can impede any forward momentum and may end up being problematic in the early and middle stages of the game.

The Settlement Heavy resource requirement is relatively balanced amongst the resources, reflecting the nature of settlements as well as their diverse cost. Unsurprisingly, the City Heavy strategy is skewed heavily to grain and ore. One interesting note is that there isn’t a heavy lumber or brick strategy in the same way as there is a grain and ore strategy. This creates some element of risk for players who invested early in lumber and brick, as they will eventually find that they have little to sink their resources into in the late game.