People like winning games. It’s the American way. Some games are purely chance, some are mostly skill, and most are a mixture of the two. Fortunately, in many games math can help you figure out how to win. In fact, the field of probability was developed after some gamblers asked mathematicians such as Fermat and Pascal to help them out. Math does more than just dice and cards; sabermetrics, baseball enthusiasts’ favorite branch of statistics, analyzes players’ performances and helps steer management in a direction of which players to play when, and who they should sign in free agency.
I am writing this article because I lost a game, and I did some math to try to break that losing streak. The game in question is the card game Dominion: Intrigue. For those of you unfamiliar with the game, the point is to accumulate money to buy as much land (or victory points) as possible (and therefore increase the size of your domain). There are many strategies that one can use to reach this goal, as there are many combinations of cards that can lead to increased money or land. The strategy that I was using was a combination of two cards, a Duchy card (worth three victory points) and a Duke card (worth one victory point for each Duchy in one’s possession).
Obviously one would want to buy 3 Duchy cards before purchasing any Dukes (as Dukes have no intrinsic value unless one already owns Duchies). But after one has 3 Duchy cards, what is the Duchy/Duke ratio that maximizes points?
The mathematical relationship between them is as follows:
Victory Points = 3*(# of Duchies) + (# of Duchies)*(# of Dukes). For the sake of nice math, I’m going to let Victory Points be represented by z, Duchies by x, and Dukes by y. We now get the equation:
Which happens to describe a hyperbolic paraboloid, or our favorite saddle function from multivariable calculus (plotted below for only positive numbers):
At this point I realized things were more complicated than I thought they were going to be. But this is the Berkeley Science Review, so I don’t feel too bad about doing calculus on here. In order to optimize this, we need a boundary condition. There aren’t infinite cards, so I set the total number of Duke + Duchy cards that one can have to some constant, k.
This is now a quadratic, which is much easier to solve. Basically where the derivative of quadratic is zero (no slope) that is the maxima of z or total victory points one can get (as the quadratic term is negative).
we can now go back and plug in x to our boundary condition:
That’s a pretty manageable equation (remember x= Duchies, y= Dukes, and k is the total number of Duchies and Dukes). The equation has a discontinuity at 3, which makes sense because we want to buy three Duchies before ever buying a Duke, anyway. If the total number of cards you can buy is 10, you will want 1.85 Duchies to every Duke, which comes out to around 6.5 Duchies and 3.5 Dukes (a 7:3 or 6:4 Duchy:Duke ratio results in the same number of victory points). If for some reason you get lucky and can buy lots of cards, the ratio of dukes to duchies will asymptote to 1 at infinitely many cards. However, there are only a total of 12 Dukes and 12 Duchy cards available, so if you are able to nab most of them, you should buy with a slight favor toward the Duchies, for instance if you can buy 18 cards, you would want 10 Duchies, and 8 Dukes.
The point of this article is that even for people who might say they “don’t like math”, it certainly helps to figure it out if you want to win. At the end of the day, winners like math.