## Friday, May 11, 2012

### Dominion: Optimal "Big Money" strategy?

In a previous post I gave a quick overview of the rules of the board/card game Dominion.
Because there are so many different and attractive actions cards to purchase, they can be very tempting to purchase, especially those allowing you to draw and play even more action cards. However, a very simple yet efficient strategy at Dominion is called "Big Money" and essentially ignores all the action cards.

Big Money strategies

"Big Money" consists in only purchasing treasure cards and buying Provinces. Assuming a two-player game with 8 Provinces in play, I will look at how many turns it takes to buy 4 Provinces. As a hand consists of 5 cards and players only start with copper, the maximum hand value is 5, so before the first Province can be bought, silvers and golds will have to be bought.
The algorithm for "Big Money" can be written as:
• if hand value >= 8, buy Province
• otherwise, if hand value >= 6, buy gold
• otherwise, if hand value >= 3, buy silver
• otherwise, do nothing
But some variations exist. Indeed, in Dominion it is usually important to have a high money density, to maximise the value of a 5-hand card. So although coppers are worth 1 and cost nothing to buy, it would be foolish to gain as much of these as possible as the values of your hand will be capped at 5, and make higher purchases impossible. So going back to the variations, we ideally want as many gold as possible to raise the average value of a 5-card hand. But what about silvers? We need to buy at least one silver in order to buy a gold (4 coppers + 1 silver = 6, cost of a gold). But if the player has multiple turns with a hand of 3, 4 or 5, should silvers always be bought, or are they going to bring the average hand value down? Wouldn't it be worth skipping those turns and wait to buy gold instead?
I therefore created variations of Big Money, depending on variable k which is the maximum number of silvers the player will buy. If the player already has k silvers and has a turn with 3, 4 or 5 in money, the player will not buy a new silver:
• if hand value >= 8, buy Province
• otherwise, if hand value >= 6, buy gold
• otherwise, if hand value >= 3 and [less than k silvers in deck + hand + discard pile], buy silver
• otherwise, do nothing
Analysis results

And now for the long awaited results. Let us consider 9 different "Big Money" variants, respectively capping silvers at 1, 2, ...7, 8 and no capping, I've displayed the number of turns it took to buy 4 Provinces based, on 100,000 simulations in each case.
Apparently, capping is not a good idea, especially for very small values. For larger caps (7, 8), it is unlikely the limit will even be reached! But just to confirm let's take a closer look excluding the first two caps:
Looking at the mean number of turns in each situation:
• When Capping at 1 silver purchase, the average number of turns required to purchase 4 Provinces was 31.14.
• Capping at 2: 21.63 turns on average
• Capping at 3: 19.47  turns on average
• Capping at 4: 18.25  turns on average
• Capping at 5: 17.55  turns on average
• Capping at 6: 17.13  turns on average
• Capping at 7: 16.91  turns on average
• Capping at 8: 16.83  turns on average
• No capping: 16.80  turns on average
So, when applying "Big Money", don't think twice go ahead buy the most expensive treasure you can!
In the next post we will take a look at some characteristics of "Big Money". How much Gold will you end up with? How much money in total? On which turns can you expect to purchase your first three Provinces?

There is much literature about "big Money" and my objective is not to repeat what can easily be found elsewhere, but to use "Big Money" as a simple benchmark for other strategies I would like to model, explore and compare.