Visualizing Resources in Settlers of Catan

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.

PRoll

Plot 1.

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.

CardProbs

Plot 2.

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.

 

Screen Shot 2014-08-17 at 1.07.43 PM

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

 

Screen Shot 2014-08-17 at 1.07.50 PM

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.

Screen Shot 2014-08-17 at 1.19.53 PM

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.

Plot 2.

Plot 3.

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.

Splendor Analysis, Part Two

The last article on game data analysis in Splendor explored some basic descriptive statistics: card cost and level distribution, point:cost ratios, etcetera. One area not fully answered was whether that data informs win strategies. What is the most efficient way to win? This article outlines a few possibilities, starting with one that is unrealistic but highly efficient, then transitioning to something a bit more reasonable.

A useful design principle was uncovered from the exploratory analysis in the last article. Based on our analysis of the card costs and points, the game rewards a high concentration in a single color via point:cost efficiency. Our first strategy proposes a laser-like focus on that principle. As will be detailed, it falls flat because the approach is too narrow to be practically applied. The second strategy takes a slight step back from that principle, and finds a more reasonable balance between the two.

*Note on the data: see the previous article for explanations about data management and processing. The graphic visualizations in this article are mostly in Excel, mostly because I’m still learning to do colored plots in R.

Example Strategy: One-Color Focus

This strategy proposes focusing on purchasing cards that cost a single color (or mostly a single color). This strategy was derived from our prior article, which found that the most efficiently priced cards usually had a heavy single color cost. For this example we’re going to use red as our single color focus.

redcards

The plot above displays cards that cost mostly red and are worth points. The four point card exemplary of a single color strategy. It requires a huge cost in red, but is the most efficiently priced card in the whole game: a player gets 1 victory point for every 1.75 gems. The other two most expensive cards are a 1:2 ratio, respectively. It’s also easy to see how a couple of these cards can easily provide >50% of a player’s win condition (15 points to trigger the game’s end).

The four cards in the graphic represent 14 points. The cost of that route is 25 red and 3 black. That’s by far the most efficient card-to-points and card-to-points ratio possible. That said, this ignores the investment needed to get there, whether buying red bonus cards or hoarding red gems. Regardless, it’s an incredible proposition. But it’s also incredibly risky. This analysis presumes that players have access to these cards. It’s highly doubtful that all of these cards would appear in the same game. I don’t have numbers detailing the average number of cards drawn in a game, but in the four-player sessions I’ve experience the level two and three card stacks don’t receive a huge dent. So if these four cards do not appear, the strategy hugely inefficient as a one-color focused player is forced to buy up less efficient cards. The plot below displays other, less efficient card options available to a player buying heavily in red.

moreredcards

This plot displays cards that feature red in its cost or as its bonus. What this confirms is that a single-color strategy is extremely narrow: the four key cards represent most of the right-most points (ie., the cards that reward a single-color strategy). The rest of the cards are clumped on the left side, so after a few key cards a single-color focus becomes very difficult to pull off.

Summary of Strengths

  • Most efficient victory possible (probably).
  • Few key cards required.
  • Exclusivity of key cards means little competition.

Summary of Weaknesses

  • Key cards are infrequent.
  • After key cards, rest of card pool is inefficient.
  • Failure to see any key cards puts strategy at huge risk.
  • Must obtain large number of red cards before engine works.

When to use

  • Probably never, given the probability of flipping all the key cards.

Example Strategy: Two-Color Focus

Strictly adhering to the efficiency principle (cards with single color costs are more efficient) turns out to be an impractical exercise. The next obvious step would be to look at a two-color focus. And as the prior article intimated, there appear to be some allied colors that play well together in terms of similar bonuses and costs. This strategy focuses on that potential and concentrates on buying cards are heavily priced in one or two colors. Continuing on the red theme, we’re going to look at red paired with black.

redblackcards

The plot above displays cards that contain a black and/or red cost of >50%. It’s immediately apparent that a two-color strategy yields a much deeper bench of useful cards. This shouldn’t be surprising, because now you are able to select from both red and black’s single-color focus cards. In addition, another set of cards become useful because their cost is in red and black combined. Overall that yields 20 cards that mostly cost black and/or red, or ~20% of the 90 total cards. Compared to the four cards for a single color, this strategy seems much more relevant and it feels like we’re hitting on something of practical consideration.

There are drawbacks. It is more resource intensive to work up to a two-color bonus engine. One thing this analysis hasn’t really touched is that a strategy that focuses on a color assumes that a player first buys up cards that contain their color bonus. So there are two phases to this strategy: obtaining cards that provide your color bonuses, and obtaining cards that are paid mostly by those bonuses. The plots below show the costs of cards that provide a black or red bonus. There are a few cards that mostly cost black and/or red and also provide one of those color bonuses, but it’s a mostly even mix. The pie chart, which is an aggregate of those costs, confirms this observation.

redblackbonuscardsredblackbonuscost

The other drawback are the color combinations themselves; not every color combination works together. In fact, there are only five two-color schemes seem realistic. Moreover the color combinations are composed of a ~70% composition of one color and ~30% in another. One indicator is to look at the series of 5-point cards: they each contain a cost of 7 in a major color and 3 in a minor color. Likewise, there are also 2-point cards split along a 5:3 cost ratio. These are plotted below and reveal the color allies: G/r, U/g, R/b, B/w, W/u. The five two-color noble cards also reflect these color affinities.

coloraffinities

Summary of Strengths

  • Takes advantage of cost efficient cards in two colors.
  • Little competition in key cards.
  • Noble card affinity.

Summary of Weaknesses

  • Knowing the allied colors is critical. Investing in white and green will yield no strategic advantages compared to a white and blue strategy, and this differentiation is hardly apparent without diving into the game data.
  • It’s more flexible than a single color strategy, but still narrow. Your important cards are 20% of the entire field, with everything else being basically at-cost because your bonuses don’t provide any efficiency.

When to use

  • Several things, like the right noble card and seeing a strong level-three card in the opening, might move you in that direction.

What’s Next?

The next step would be to examine three and four color strategies, which is a shift from the principle of focusing the most efficient cards to the most advantageous cards on the board. I imagine that Splendor rewards that kind of play most, as it appears to be a game that promotes tactical, incremental gains in efficiency rather than grand strategy. That kind of analysis is a bit more complex because it require probabilities and Monte Carlo simulations to understand how likely it is that a broader strategy will be rewarded with efficiencies given a sample of cards drawn.

Tagged ,