Monday, July 6, 2020

Modeling value in Century: Spice Road

When playing Century: Spice Road (or the Golem Edition, I imagine), it's easy to notice some patterns in the values of the cards.

The Merchant cards are less clear, so let me push them aside first. Not all Merchant cards are equal. Some comparisons are hard to make and depend a lot on the situation. For example, is it better to have a card that transforms 2 saffron (red) into 2 cardamom (green), or 3 saffron into 3 cardamom? If you only have two saffron, the first may be better. With three it's likely the latter. With four saffron we're back to likely preferring the first, as we can perform the trade twice. That is, unless we only need three cardamom and need to hold onto a saffron. You could perhaps come up with some metric that attempts to incorporate all of these types of considerations. However, there's a clear counter example. There's one Merchant card that gives you three turmeric (yellow) and another that gives you four. The latter is strictly better than the first. Even if you can't always make use of the fourth turmeric, it is all that the first card is and more. This one example is all we need to show that Merchant cards are not all balanced.

Now let's look at the Point cards.

Figure 1: Points cards

The pattern I noticed when looking at a bunch of these is that it appears that the number of points, $p$, follows a simple formula based on the number of turmeric (yellow), $y$, saffron (red), $r$, cardamom (green), $g$, and cinnamon (brown), $b$.
p = y + 2r + 3g + 4b
That is, the number of points is equal to one point for each turmeric, while each higher level of spice is worth one more point each. This tracks for a large number of the Point cards (24 out of 36). Then you run into a card like the following.

Figure 2: Another Point card

Here we'd predict the card to be worth 12 points.
12 = 1 \cdot 3 + 2 \cdot 1 + 4 \cdot 3 + 4 \cdot 1
However, it's worth two additional points. Why? If you look at this spreadsheet, posted by "GameSnake", it becomes quite clear what the pattern is. There's a column specifying the "Spice cost", matching our analysis up to this point, as well as a "Delta" column. This clearly shows that a Point card is worth an extra point if it requires three different types of spices to claim, and two extra points if it requires four different types of spices. This makes some sense, as it may be more difficult to produce a variety of different spices than just having a good combination to produce exclusively one or two. There's no difference between cards that require only one type of spice and cards that require two types of spice. So our revised equation becomes,
p = y + 2r + 3g + 4b + \max(n-2, 0) ,

where $n$ is the number of distinct spices needed to claim the Point card. Note that $n$ is dependent on $y$, $r$, $g$, and $b$. We could write an equation without this intermediate variable, but it'd be quite a bit more cumbersome. This equation exactly predicts the value of every Point card in the game, which you can check against in Table 1.

Victory points Spice cost Delta
YYRR 6 6 0
YYYRR 7 7 0
RRRR 8 8 0
YYGG 8 8 0
YYRRR 8 8 0
YYYGG 9 9 0
RRGG 10 10 0
RRRRR 10 10 0
YYBB 10 10 0
YYGGG 11 11 0
YYYBB 11 11 0
GGGG 12 12 0
RRBB 12 12 0
RRRGG 12 12 0
RRGGG 13 13 0
GGBB 14 14 0
RRRBB 14 14 0
YYBBB 14 14 0
GGGGG 15 15 0
BBBB 16 16 0
RRBBB 16 16 0
GGGBB 17 17 0
GGBBB 18 18 0
BBBBB 20 20 0
YYRB 9 8 1
RRGB 12 11 1
YGGB 12 11 1
YYRRGG 13 12 1
YYRRBB 15 14 1
YYGGBB 17 16 1
RRGGBB 19 18 1
YRGB 12 10 2
YYYRGB 14 12 2
YRRRGB 16 14 2
YRGGGB 18 16 2
YRGBBB 20 18 2

Table 1: Century: Spice Road Points cards. (data from this spreadsheet)

Note: while I leveraged the table above, I did not read any existing analyses of the game.  I imagine that others have come to similar conclusions.

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