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The Key to Big Money Part I: Money Density

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WanderingWinder:
In a big-money kind of deck, there's really two concepts you need to be aware of; the first is money density, the second is opportunity cost.

Money density is the average value in coin production of cards in your deck, i.e. copper produces one, silvers two, estates and such 0. It's important to keep in mind that, based on 5-card hands, you need a money density of 1.6 to buy a province and 2.2 for a colony. You need only 1 for duchies or dukes, and less for things like gardens, islands, tunnels, whatever.

Calculating your money density is very simple if you know what's in your deck: add up all the production values of the money, divide by the total cards in your decks. So for your initial deck, you have 7*1 for the coppers +3*0 for the estates, all divided by the 10 total cards for a money density of 0.7.

Branching out slightly, you probably want to buy at least one card that's not a silver or gold or province or duchy, right? How do other cards fit in to money density? Well, the simplest are cards like Woodcutter. Woodcutter (at least, the first one) provides an obvious benefit over silver in that it gives you a buy. But, for all intents and purposes, it still counts as $2 in your money density.

There's another very simple, very common kind of card to deal with when making your money density calculations: cantrips. (I'm using 'cantrip' here to define any kind of card that always draws at least one card and gives at least one action back to you). Cantrips are what I call, for the purposes of money density calculations, 'virtual cards'. What I mean by that is, because they replace themselves totally in your hand, they don't count toward the total count of cards which you're using as the denominator for your money density calculations. So, if you buy a village and a militia with your two starting buys (not, by the way, a good strategy), you have 7 coppers, 3 estates, 1 village, 1 militia, producing 7, 0, 0, and 2 money respectively and with a total of 7, 3, 0, and 1 cards to count against your deck total. Your total money density is therefore 9/11 = .818181.....

Further expanding on that, if you get a slightly more interesting (in this respect anyway) card, the peddler, into your deck, you've increased your effective deck size by 0 (because it's a cantrip), but as it produces $1 extra, you've increased your buying power by one. Add peddler to your starting deck, and you have $8 total money in 10 effective cards for a density of $0.8.

Okay, once you get that down, you need to think about terminal collision. I think that most of you know that buying only treasures and VP won't get you very far in terms of success (or fun). So you probably want to buy some terminals, and by the end of the game, you probably want to buy more than one. This creates some chance that your terminal actions will collide. The big key to playing big money decks is weighing out the benefits that actions provide you with versus the chances that they collide. Of course, with non-terminals, you don't have to worry about that, but very often, you're better served by taking the risk at some point. Fortunately, calculating the chances for terminals collision isn't too hard in general, you just have to remember to use your effective deck size rather than the actual number of cards in your deck. As for figuring out which benefits are worth it... well, I'll let you guys work that out for yourselves. Just keep in mind that you aren't optimizing your results in a vacuum, you have to beat another player. Which means, generally, that you have to count on yourself getting a little luckier than you should expect to on average, because in those really unlucky cases, you've probably already lost anyway. And the amount you have to count on yourself getting lucky, i.e. the amount of risks you have to take, increases more with the more players you add to the game. Villages will help to ease these wrinkles, but you have to get the village together in the hand that the terminals collide in, which doesn't happen so often as people think.

Of course, this leads us to the very important subject of terminal card-draw. In general, by the time you're mixing multiple terminal draws... you're probably engine building*. And for engine building, things like getting your engine to be able to fire consistently and having a sufficient payload are far more important than the money density concept. But for one to two terminal drawers, this money density look at things is still quite effective. For your first terminal drawer, it's a virtual card to your deck, once more, and then you have a percentage (based on the size of your deck) of having a larger handsize. After all, the reason why average card value is important is so that you can calculate the average value of your hand. So take smithy; if I have 2 silvers, a gold, a smithy, and my starting cards in the deck, that's 13 effective cards, 14 total money, and you've got your chance of getting a 7 card hand rather than a 5. Calculating the exact probability is not as easy as you might think, given how reshuffles work. But you can come up with ways to approximate it. And then for your second (and every subsequent) terminal  card draw, they do count as cards in your deck, with 0 money value. So adding a second smithy to your deck, you have a higher chance of getting your 7 card hand, but your money density has dropped from 14/13 to 14/14 (or $1).

Understanding money density is also helpful in understanding how much your deck will stall out. A deck with 3 gold, 7 silver, 7 copper and 3 estates has a money density of $1.5. A deck with 1 gold, 3 silver, 2 copper, and a chapel has a money density of 11/7, or just over $1.57. But if we add two provinces to both decks... the first deck drops to an average money density of ~$1.364. The second drops to ~$1.222. So we can see that thinner decks generally require more padding, and/or choke more on green cards. Whereas decks rich with big money are much more resilient.

In actuality, things are a little bit more complicated than this model would have you look at, because you don't actually draw average hands. Dominion isn't a game that's continuous; it's discrete. So there's a difference between having two silvers and having a gold and a copper. Sometimes you want more variance, sometimes you want less.

*Actually, this isn't really true for most terminal card draw - envoy being the huge exception; you probably want two smithies even before the end. And lots of terminal card draw have ways of mitigating the collision; vault, embassy, courtyard...

Smartie:
I must say that this is an excellent article! Especially when one is going big money, one often wonder, should I buy a gold to increase money density or should I just province 1st? Of course like any statistics models, it can be difficult to calculate totally, especially with effects like discard effects, curses, attacks etc. Often, one also relies on card draws to boost chances for getting provinces or colonies which can be difficult to predict too! Though simple model, but creative and covers most aspects! Lucky that this game is not entirely based on calculations  :)

Geronimoo:
If this is published, add some graphs from simulations to clarify things. For instance, the explanation how exactly Smithy increases your average $density is not so easy to follow, but the graph clearly shows the turbo injection effect of the Smithy on the average $-production in comparison to a pure treasure deck.

timchen:
Personally I've never found the money density concept useful. Maybe I am horribly wrong, but usually my VP buying rule (for BM-ish deck, anyway) is always something like, ok, I am getting the two golds. Then I am heading for Provinces. If I were to guess my effective money density when I start buying Provinces, I would say it is probably lower than 1.6. This is due to the distribution. The point is, what you want actually is to maximize the chance you have above $8 in a Province game (given the limited time frame), not to have your average money per turn to be $8. Given the starting deck without trashing, I suspect the chance will be quite a bit higher than 50% when you reach the average of $1.6...

WanderingWinder:

--- Quote from: timchen on January 19, 2012, 10:47:06 am ---Personally I've never found the money density concept useful. Maybe I am horribly wrong, but usually my VP buying rule (for BM-ish deck, anyway) is always something like, ok, I am getting the two golds. Then I am heading for Provinces. If I were to guess my effective money density when I start buying Provinces, I would say it is probably lower than 1.6. This is due to the distribution. The point is, what you want actually is to maximize the chance you have above $8 in a Province game (given the limited time frame), not to have your average money per turn to be $8. Given the starting deck without trashing, I suspect the chance will be quite a bit higher than 50% when you reach the average of $1.6...

--- End quote ---


Oh, well you're probably actually waiting longer to green than I am then. I probably ought to make it clear that you don't need the money density of your whole deck to be 1.6 to start greening - that would mean you'd be able to buy province like every turn. Or close to it. Somewhere between 50% and 100%. Anyway, yes, it's not meant to be 'oh my average money per hand is $1.6, it's now safe for me to green'; it's more keeping an eye on how each purchase affects that density.

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