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 ,

Game Data Analysis: Splendor

Splendor has become a very popular game recently, likely due to its Spiel des Jahres nomination. Splendor is a card game about using limited resources to build an efficient engine. There are five colored gems, and playing only a few times it might feel like the costs and distribution of the colors are the same. In other words, the colors are different currencies with no other differentiating qualities. If that were true, we’d be able to add up all the costs and find that the colors are evenly distributed. But even with a few plays of Splendor that assumption is difficult to maintain. The rules are simple and there are no exception-based cards, so it’s a great game to analyze by number crunching. I’ve set out some research questions to help guide the analysis. Some of them are basic, but I think they’ll open up other directions for exploring the data.

Research Questions

RQ1: are color bonuses evenly distributed?

RQ2: are color costs evenly distributed?

RQ3: what is the average point:cost ratio for a level one, two, and three card?

RQ4: what is the percentage of “on color” cards? (Cards that give a bonus for a color that is also in the cost).

Process

CardRules

Figure 1. Card Description.

It’s always useful to communication the analysis process, but if you’re interested strictly in results then scroll down to the Results section. The process first involved developing a taxonomy to store the attributes, entering and cleaning the data, then validating the data. Some of this may seem excessive for a small project like this, but there are always unexpected events that a formal procedure can mitigate.

To operate on the card data I input it into Excel. The data entered would consist of the cards and all their attributes, including the bonus, level and cost (see Figure 1). Noble cards would also be input. There was some taxonomic thinking that went into the process: how should I manage the terms used? For example, instead of writing out the word “black,” it’s easier to input “b,” but there’s also “blue” then term management needs to standardize that “blue” becomes “u” instead.

Data validity was also an issue because I was manually uploading the information; I didn’t perform a second check nor have others do it, so I had no reference to determine whether the uploaded data was correct. I did do a card count to ensure I had input the correct number of cards, but nothing beyond that. An auto-incrementing CardID attribute was created to ensure I could individually track each card.

Analysis and data visualization was performed in R Studio.

Results

Table1

RQ1: are color bonuses evenly distributed?

Yes. See Table 1.

RQ2: are color costs evenly distributed?

No. If you sum all the card’s costs, the colors are relatively equal. But when slicing those costs by level, the distribution becomes uneven.

Table 2 charts the frequency of color cost. That means it counts occurrences of color cost. So the Black row under Level One indicates that White and Blue occur as a cost in four level one black cards. Note that this table does not track intensity of those costs (that’s for later).

Table2

In terms of frequency, on-color costs are infrequent (few cards with a color bonus also have that same colored cost). There are some other odd frequency dynamics: level two blue cards feature a blue cost five times, with other colors appearing two times. So each color has a unique frequency of cost, as seen in the figures below.

Rplot02

Figure 2. White costs by card color.

Rplot03

Figure 3. Red costs by card color.

These figures displays the distributions of White and Red costs among the cards. Because the two figures are different, we can conclude that certain card colors require varying degrees of red and white costs. Because the figures are box plots we can also discern the expected white or red cost depending on the card color. A white card will have a higher red cost on average when compared to a green, red, or blue card.

While each color has a unique frequency of cost, those patterns do not occur from level to level. In other words, investing in white bonuses to buy red cards at level one will not be a reasonable strategy at level two or three because the white cost frequency changes.

Table 3 charts the intensity of color cost, which is where the uneven distribution really comes into perspective. The level three cards are representative of this trend. Like a lot of other games, the expensive cards reward a vertical strategy of investing a lot in a few colors. For example, the sum total of white costs in level three blue cards is 23, with all other colors totaling less than 8. As a proportion of the total costs of all level three blue cards, that’s 70%. So there are clear color affinities, but they change at each level, so white costs won’t always be strongly tied to blue bonuses.

Table3

RQ3: what is the average point:cost ratio for a level one, two, and three card?

For cards with point values, the average cost of a point goes down as the card level increases. At level three, the average cost a point is 2.7, level two is 3.7, and level one it is 4 (every level one card with a point costs 4). This isn’t too surprising; a lot of games reward big investments by making them more efficient. The trade off, of course, is that they require long-term planning to efficiently buy them, while low-level cards require little.

Rplot01

Figure 4. Card Points and Card Costs.

After a few plays of Splendor you’ll probably notice that some cards cost more for the same point value. Indeed, Figure 4 shows that some cards are just strictly more efficient. However, the cards that cost the most (14)  spread the cost among 3-4 colors, while the highest value cards (5 points) are concentrated on high costs of one or two colors. So the design of the costs provides two clear paths to efficient point gain.

Table4RQ4: what is the percentage of “on color” cards (Cards that give a bonus for a color also in the cost)?

My playgroup assumed these cards are high value because players have a tendency to collect things they already own. They occur infrequently in level one, but in card levels two and three they’re at least 50% of all cards.

Conclusions

One important aspect of this analysis is that it’s dealing with static information. This data tells us about the set information of the game, but it doesn’t describe any behaviors. We can infer some behaviors based on the incentives provided by the set rules of the cards, but we should be conservative in those conclusions.

That said, the frequency and intensity of the color costs does validate the potential for a vertical, one or two color strategy. But it seems like those strategies require careful planning, as by level 3 each color is essentially “locked” into buying up one particular color. So if that color is available to buy, well, you might have some problems. At the same time, it may be a viable strategy if every other player is competing for a wider strategy with nobles.

That said, I remain skeptical of a strategy concerned with concentrating on level three high value cards, and remain convinced that Splendor is very much a game of opportunistic, incremental victories that rewards turn-by-turn efficiency rather than grand strategies. Consider for example that the average point value of a level three card is four. To win, you would need to purchase three plus something else or four. That’s a tall order, especially considering the bonus return is the same as a level one or two card. Instead, it seems much more likely that–like all other parts of the game–level three cards are purchases of opportunity. If a player is able to efficiently purchase a level three card, it is because they just happen to have the correct bonuses and gems at hand, rather than a “from turn one” plan. That said, without behavioral data to acknowledge that, I can only speculate.

Tagged ,