The Keys to Big Money: Money Density and Opportunity Cost

This is a revised version of a guest article by WanderingWinder, originally posted in the forum: Part I, Part II. WanderingWinder was a semifinalist in the 2011 DominionStrategy.com Championships.

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

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 deck. 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 because it produces $1 extra, you’ve increased your buying power by one. If you could add Peddler to your starting deck, you would 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 terminal Actions, 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 terminal 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 (like Smithy). 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 even if you’re just running a Big Money + terminal card draw strategy, you probably still want multiple copies: with the big exception of Envoy, you probably want two Smithies, Courtyards, etc.  And lots of terminal card draw have ways of mitigating the collision; Vault, Embassy, Courtyard…

So for one to two terminal drawers, this money density look at things is still quite important. 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.

Let’s 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.  As a guesstimate, you’ll have around 3 turns before a reshuffle, and two of those three turns will be 5-card hands and one turn will be 7 cards.  That works out to (roughly) $5.4, $5.4, and $7.5 per hand.  If you now add a second Smithy, 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).  (This seems pretty good, but its hidden cost is discussed in the next section.)

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, and it will be painfully clear to you when you are hit by Militia. Sometimes you want more variance, sometimes you want less.

Opportunity Cost

We may now be left with an interesting little question. The analysis of buying a second Smithy shows that it should be good for our deck pretty early on, right? Like, look at this deck: we’ve got 7 Copper, 1 Smithy, 3 Estates, and 2 Silvers. Our effective money density is around $0.917, we’ve got around a 40% chance of hitting a 7 card hand… adding a second smithy would decrease our effective money density to around $.846, it’s true, but significantly increasing the chances at getting two more cards in our hand is worth it, on the analysis, right?

Well, if the choice were between buying the second Smithy and buying nothing, you’d be right. But it’s not. Any time you can buy a Smithy, you can buy a Silver instead. And if you buy that Smithy, that stops you from buying a Silver. The correct play here is for the Silver, not so much because of the collision problem (though that makes putting a free Smithy in your deck barely worth it in the short term), but more because of opportunity cost, i.e. you have to consider what you buy in terms of what else you could have bought, not in a vacuum.

This is actually an important way of looking at all of Dominion, not just Big Money, but I think it’s easiest to understand in Big Money, because of the money density being available. So if you’re trying to decide whether or not to buy a Market, you can’t just look and see whether that’s good for your deck, you have to see if it’s better for your deck than the alternatives.

One nice little way to look at this is with Potion cards cards. Since whenever you buy a potion, you could’ve gotten a silver, it’s generally true that any time you have $X + a Potion, you could have bought something costing $X+2. For instance, if you buy a Possession, that could have almost always been a Province if you’d gotten Silver instead. Your Alchemists could have been Laboratories (hey, that actually makes a lot of sense), your Familiars could have been Witches, etc. Now, whether or not you should go for Potions for these cards has a lot to do with variance and the usefulness of the potion later in the game, but it’s a tremendous illustration of opportunity cost in action. The opportunity cost of buying a Potion is a silver (or any other $4 or less cost card), and that Silver could have gotten you a Province; instead you have Possession, which is sometimes better than Province early in the game but typically probably worse than just having the 6VP.

Perhaps the simplest, first way many players need to realize the importance of opportunity cost is with the ‘Village idiot’. Villages, like any cantrip, can’t possibly hurt your deck, since they replace themselves, right? This is the thought process a lot of people go through early on. This may be true, but that doesn’t mean you should just gobble up all the Villages you can; if you start of the game buying Village after Village, you’ll have a whole bunch of unused Actions lying around, with no Action cards to use them on.  More importantly, your deck won’t have any buying power, because the opportunity cost of buying a Village is a Silver. Your opponent that buys Silvers will be far ahead of you.

Another great card to look at through the lens of opportunity cost is Hoard. Hoard could have been a Gold. So any time you buy a Duchy with Hoard, you could have bought a gold instead. Now you are gaining that Gold as well, it’s true, but you have to look at when in the game you are… do you want a ‘free Duchy’ in your deck yet? Maybe so, maybe not. To say nothing of the times where the opportunity cost of going Hoard over Gold knocks you down from $8, buying a Province, to $7, settling for a Duchy and another Gold.  Yes, I’m saying that Hoard is overrated and misused.

The concept of opportunity cost extends beyond just once through the deck as well. That Silver that your Village cost you is going to hurt your buying power now, and on the next reshuffle, and on the next reshuffle…. Furthermore, the buying power reduction that you feel now is going to make you have to buy something worse now, which is going to further hurt you on the next reshuffle, and then that further reduction on that reshuffle will hurt your buying power even more on the next reshuffle… this compounding effect is why the early turns are generally more important than the later ones.

And you can further extend this paradigm to what I call implied opportunity cost. If I buy a Chapel on turn one with $3, there are lots of costs to me. I have:

  1. the $2 that I spent for it, which could have been another 2- or 3-cost card (e.g., Silver) instead
  2. the cards that will no longer be in my deck once I use the Chapel
  3. the turns that I’m using Chapel, on which I probably won’t be able to buy things
  4. Chapel eventually being a dead card.

Now, the second one of these, getting rid of the Coppers and Estates you’re trashing, is probably actually a boon more than a cost. And very often, especially in engine decks, it’s a really big boon. But it does take the turn you’re buying the Chapel, plus probably two full turns of trashing things, plus another turn which is partially hampered by trashing, plus all the times you have a worthless Chapel in your deck. Which is all to say, Chapel, strong as it is, is not actually all that great for a pure Big Money deck.

Conclusion

A good way of appreciating the in-game impact of these calculations is with the simulators.  Both Geronimoo’s and rspeer’s simulators provide graphs of the average money generated per turn.  For instance, you can see the significant improvement in average money on the crucial first few turns by comparing Big Money and Big Money/Smithy.  (I use rspeer’s only for ease of linking; Geronimoo’s is generally a stronger “player” and has more cards implemented, but requires a download to run.)

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8 Responses to The Keys to Big Money: Money Density and Opportunity Cost

  1. piemasteruk says:

    Nice article, just one minor comment. When discussing Opportunity Cost at the end you say that:

    If I buy a Chapel on turn one with $3, there are lots of costs to me. I have:

    1.the $2 that I spent for it
    2.the Silver (or other 2- or 3-cost card) I could have bought instead

    #1 is not really an opportunity cost, because money is worthless in itself. The idea of the game is not to accumulate money, it’s to accumulate VP. Of course, money is great for buying you things, but in that case it’s what you would have otherwise bought with the money that is the opportunity cost (i.e. #2) not the money itself.

  2. GwinnR says:

    I think you missed a point with the hoard: Your example with having 5$ is right, but what about other cases?
    I have 2$ and get an estate and a gold istead of a 3$-card.
    I have 8$ and get a province and a gold instead of only a province with 9$.
    I’m buying one or two hoards and after that i buy golds.

    • GigaKnight says:

      This also how I thought about it (that and any time you can buy a non-terminal action with Hoard, get Hoard). WW, I’d really like to hear more of your thoughts on Hoard and when you think one should (or shouldn’t) use it.

    • ftl says:

      I don’t think that was a point that was ‘missed’, it just wasn’t the point of the article. Obviously, those are the cases where the Hoard may have been a good buy. Estate+Gold for $2 is pretty good. Getting a free gold along with your provinces is also good.

      Those are the upsides of Hoard, and they’re the reasons you might want to buy hoard. The point was that those upsides aren’t free – there’s a cost. The cost is that that hoard COULD have been a Gold, usually. It’s a hidden cost, because the Hoard doesn’t have text on it that says “$-1”, that’s just the effect it has relative to what it could have been.

      Look at your first case. Estate+Gold for 2. If that hoard had been a gold instead, you could have just bought a Silver for $3! Estate+Gold is two cards with an average value of 1.5, which is lower than the value of a silver! Of course, in Dominion, it’s often the case that you prefer having one really good card and one bad card over having one middling card, so Estate+Gold is better than Silver, but still. The comparison point is Silver, not a $2-cost card.

      So you have to weight the upsides against the downsides. Whenever you have $X with a hoard in play, you probably could have had $x+1 without the hoard. You also have to take into account how likely the different options are.

      If you draw $2 with a hoard, you can get Estate+Gold, whereas if the Hoard was a Gold, it could have been a silver. This case just seems unlikely – why would you be drawing Hoard alone, with 4 green cards and no other treasure, unless it’s really really late in the game and you might not even see that Gold anyway?
      If you draw $3 or $4 with Hoard, you still have the choice of either Estate+Gold or a card costing up to (3 or 4), whereas if the Hoard had been a Gold, you could have picked up a card costing (4 or 5). Drawing only $3 after you had enough coin to buy a $6-cost card might be rare, but not that rare. Missing on the opportunity to buy a 5-cost card might be pretty bad, 5-costers can be quite strong.
      If you draw $5 with Hoard, you have a choice to get Duchy+Gold, whereas if the Hoard had been a Gold, you could have just bought the Gold. Do you really want a free Duchy in your deck? Maybe. Duchies give points.
      If you draw $6 with a Hoard, then you can do the same things you could have done with Gold, but you could also pick Duchy+Gold if you so choose.
      If you draw $7 with a Hoard, that really sucks, because you could have had a province if only that hoard had been a gold.
      If you draw $8 or more with Hoard, that’s a free gold along with the province you would have gotten anyway. Sweet.

      So there’s both upsides and downsides, and while the upsides are obvious and have big shiny gameplay effects, the downsides are sort of hidden – they’re what you COULD have done if you hadn’t gotten the Hoard.

      Those aren’t to say that Hoard is necessarily a bad buy. In the simulators, the optimized hoard bot beats the optimized Money bot something like 60-33. Hoard also combos well with trash-for-benefit cards (Salvage Golds really early, since you just keep getting more! Or Salvage Duchies! Pick up lots of Bishop points!) and with sifting (Warehouse to draw 3 cards, one of which might be a Gold, discard 3 useless green cards) and with dual-type cards (Great Hall+Gold for $3, Island+Gold for $4, Nobles+Gold for $6!) and possibly with other things.

      But it can be a trap if you don’t think about whether you really want a bunch more golds LATER (with green strings attached) as opposed to one more gold NOW. A while back, Hoard was both on Theory’s ‘best $6-cost cards’ AND his ‘worst $6-cost cards’ lists.

  3. purge says:

    Regarding money density calculations, I tend to think of +Card actions differently. I don’t see Smithy as $2 and 1 Card; I see it as $0 and -2 Cards. Dead actions still complicate things, but I think this is a better way to calculate your density. Certainly if all your actions are Labs, Stables, etc., then it makes a lot of sense.

  4. dontminion says:

    Cantrips are worse than pure virtual because a terminal can draw them, e.g. smithy drawing a village could have drawn a silver instead. So deck size has to be increased above the amount given by counting all cantrips as 0 cards.

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