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Author Topic: The Key to Big Money Part I: Money Density  (Read 10919 times)

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WanderingWinder

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The Key to Big Money Part I: Money Density
« on: January 19, 2012, 08:53:14 am »
+4

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

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Re: The Key to Big Money Part I: Money Density
« Reply #1 on: January 19, 2012, 09:11:50 am »
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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  :)
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Geronimoo

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Re: The Key to Big Money Part I: Money Density
« Reply #2 on: January 19, 2012, 09:30:19 am »
+1

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.
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timchen

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Re: The Key to Big Money Part I: Money Density
« Reply #3 on: January 19, 2012, 10:47:06 am »
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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...
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WanderingWinder

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Re: The Key to Big Money Part I: Money Density
« Reply #4 on: January 19, 2012, 10:53:53 am »
0

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...


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.

DG

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Re: The Key to Big Money Part I: Money Density
« Reply #5 on: January 19, 2012, 11:08:44 am »
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I (predictably) apply some slightly different maths to look at drawing effects. When adding a treasure card with value t to a deck with T total treasure and C cards in deck, the average card value moves from T/C to (T+t)/C+1. When adding a drawing card to a deck however I would approximate this by subtracting the number of cards drawn from the denominator. So adding a single smithy would change the average card value from T/C to T/(C-2). The smithy itself adds a card to the deck but then draws three, and one minus three equals minus two.

This method is a very broad and inaccurate approximation that I'm not going to justify beyond it's simplicity. Nevertheless this approach does allow some conclusions to be drawn. The first thing you can do is set up an (inaccurate) equation for when buying one smithy increases your hand values more than buying a silver. This is (T+2)/(C+1) < T/(C-2). A little bit of algebra gives you a result of buying the smithy when T/C > 2/3 - 4/3C. This suggest that the drawing card has less impact the more cards you have in your deck (true) and that the first smithy is almost always better than a silver, unless you are drawing a very poor deck (true).
« Last Edit: January 19, 2012, 11:18:10 am by DG »
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WanderingWinder

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Re: The Key to Big Money Part I: Money Density
« Reply #6 on: January 19, 2012, 11:17:44 am »
0

I (predictably) apply some slightly different maths to look at drawing effects. When adding a treasure card with value t to a deck with T total treasure and C cards in deck, the average card value moves from T/C to (T+t)/C+1. When adding a drawing card to a deck however I would approximate this by subtracting the number of cards drawn from the denominator. So adding a single smithy would change the average card value from T/C to T/(C-2). The smithy itself adds a card to the deck but then draws three, and one minus three equals minus two.

This method is a very broad and inaccurate approximation that I'm not going to justify beyond it's simplicity. Nevertheless this approach does allow some conclusions to be drawn. The first thing you can do is set up an (inaccurate) equation for when buying one smithy increases your hand values more than buying a silver. This is (T+2)/(C+1) < T/(C-2). A little bit of algebra gives you a result of buying the smithy when T/C > 1/2 - 2/C. This suggest that the drawing card has less impact the more cards you have in your deck (true) and that the first smithy is almost always better than a silver, even if you are drawing a very poor deck (true).


The reason I don't like doing things this way, apart from the nonsensical answers it gives you when you have really few cards in your deck, has to do with the point I make at the end, that dominion is discrete rather than continuous. Smithy doesn't at all help hands without smithy in it, and in those hands WITH smithy, the bump is significantly larger. So (to oversimplify) it's half the hands 40% better and half the hands 10% worse (as compared to silver), rather than an across-the-board kind of thing. Of course that's also true of the silver itself, but it's felt more when the power is concentrated in one hand, like smithy does for you.
But yeah, you can do it that way.

ecq

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Re: The Key to Big Money Part I: Money Density
« Reply #7 on: January 19, 2012, 11:21:06 am »
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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.

I've been thinking about this recently (and really wishing I had paid more attention in stat class).  If your average money per hand is $7 with no variance, you'll never buy a province.  If it's $7 with moderate variance, you may be able to afford a province every couple of hands.  With high variance, you may have several hands with too little money followed by a hand with far too much.

I'd be really interested to hear someone with better math skills than me shed some light onto how variance impacts things, and how to assess the variance of a deck.
« Last Edit: January 19, 2012, 11:23:41 am by ecq »
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DG

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Re: The Key to Big Money Part I: Money Density
« Reply #8 on: January 19, 2012, 11:31:45 am »
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I initially got the maths wrong. I don't do much algebra these days.

The maths can give some useful answers sometimes. You generally get a mix of the obvious - a smithy is worthy more than a silver if your average card value is more than 2/3 - plus some extra factor that shows how card power changes with the size of the deck or treasure. If you put reasonable limits on the maths, such as you're using your smithy with at least 7 treasures in your deck, then it no longer gives nonsensical answers.
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DStu

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Re: The Key to Big Money Part I: Money Density
« Reply #9 on: January 19, 2012, 11:36:11 am »
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I'd be really interested to hear someone with better math skills shed some light onto how variance impacts things, and how to assess the variance of a deck.

First is that variance alone does not tell you much. You can have mean $7 with high variance because you have $6 with probability 1/2 and $8 with probability 1/2, or you can have $7 in 90% of the cases, but $0 in say 5% and $14 in 5%, that will also give you a high variance, whereas the possibility to buy a Province with it is quite low.
So what you really would need to say something is the distribution. But this in general is a too complicated thing, and most decks locks kind of the same (decks with 90% $7 hands, but 5% $0 or $14 hands is not something you usually see), so you take the variance as an indicator of how the distribution will look like.
And then it tells you basically: The larger the variance is, the more likely it is to get hands that are significantly better (and worse) than the average hand. But I don't think there is a good way to really quantify this (beside simulations), as first you have to few cards in your deck compared to your hand that you could say you just apply some Gaussian limit, and also the deck (and it's variance) of course changes all the time. Also your ability to influence it is quite limited, without harming your mean to much (buying Coppers/Curses instead of Silvers of course would increase the variance of most deck, but you don't really want to do this) There are some strategies where you can say you rely on variance (Treasure map), but that is more to rely on the variance of the strategy (sometimes it is really good, sometimes it is really bad), than to rely on a deck whose hand have high variance.

Hmm, written a lot of stuff, but basically said nothing...
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timchen

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Re: The Key to Big Money Part I: Money Density
« Reply #10 on: January 19, 2012, 12:12:39 pm »
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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...


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.

Precisely my point! So, what should one do with his eyes on the money density? For me it is just a rough feeling; do I have bought a few silvers that I can afford Duchy-dancing? something like that. It would be much more useful, to have some quantitative criteria.

And it is really important not to mislead people to think that they should somehow get their money density to 1.6 in a Province game...
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ecq

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Re: The Key to Big Money Part I: Money Density
« Reply #11 on: January 19, 2012, 03:02:00 pm »
+1

Maybe variance is a loaded word, because it has such a specific meaning in the stat world.  The question is how the quality of your money affects your expected outcome, all other things being equal.

I just ran a couple of sims.  Let's say it's turn 9, and we're both just buying money.  I've bought 2 Golds, 4 Silvers, 2 Coppers (yeah, hypothetical).  You've bought 8 Silvers.  We both have the same total money, the same total cards, and therefore the same average money.  We both start buying Provinces and only Provinces.  I win 54%/25%.

Things change when we add more cards to the mix.  At 16 cards purchased (me with 6 Golds, 4 Silvers, 6 Coppers and you with 16 Silvers), you're winning.  It seems the reason you win is that you can grab the first couple of Provinces far more consistently than I can.  Once things get greener, the higher-variance buys relatively more Provinces, but not enough to catch up.

The conclusion seems to be:
  1. When things get green, it's better to have a few Golds, even with the same total money in-deck.
  2. If things don't get very green, consistency (lots of Silver) wins.  (See: Double-Jack)

Can anyone speak to this from a stat perspective?  (I'd love to know the math)
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HiveMindEmulator

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Re: The Key to Big Money Part I: Money Density
« Reply #12 on: January 19, 2012, 03:10:46 pm »
+1

This a very nice idea for an article with some good information in it, but it could definitely use some cleaning up. As it is currently formatted, it seems it would be very easy for one to get lost in the math and not get the big picture out of it. It's nice to have some big bullet points or conclusions for the reader to take home. How is this money density important? Do I really need to know the difference between $1.6 and $1.57? Or is "relatively high" vs "relatively low" enough? Or is actually important to keep track of total money vs total cards so you can think about the derivative of money density with purchases (in some sense)? There are basically 2 major decisions you have to make in big money decks: when to add more actions, and when to buy duchies. Is there some sort of rule of thumb for how to make these decisions?
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Asklepios

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Re: The Key to Big Money Part I: Money Density
« Reply #13 on: January 21, 2012, 07:17:50 pm »
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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.....

Not sure if it was mentioned somewhere within this article (if so, I may have missed it), but I think its worth noting that cantrips DO effect your money density calculations on the final draw.

So, say you play a deck that has four golds, four silvers, two coppers, ten Schemes and one Smithy. Before the Smithy is played, the Schemes are "virtual cards" as you say. But when you play the Smithy, suddenly they come into play again, as if you draw them on the Smithy draw, they're $0 cards, not virtual cards, so you need to factor in the number of Schemes left in the deck.

I guess the way to put it is that if you are using your last action to draw, then the virtual cards need to be counted again.

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WanderingWinder

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Re: The Key to Big Money Part I: Money Density
« Reply #14 on: January 21, 2012, 11:10:00 pm »
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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.....

Not sure if it was mentioned somewhere within this article (if so, I may have missed it), but I think its worth noting that cantrips DO effect your money density calculations on the final draw.

So, say you play a deck that has four golds, four silvers, two coppers, ten Schemes and one Smithy. Before the Smithy is played, the Schemes are "virtual cards" as you say. But when you play the Smithy, suddenly they come into play again, as if you draw them on the Smithy draw, they're $0 cards, not virtual cards, so you need to factor in the number of Schemes left in the deck.

I guess the way to put it is that if you are using your last action to draw, then the virtual cards need to be counted again.


Actually, all that matters is terminal card draw. That's the only way a non-terminal's gonna hit you. And I imply it I think, but probably need to make it more explicit.

I plan on making some revisions either tomorrow or Monday.

dondon151

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Re: The Key to Big Money Part I: Money Density
« Reply #15 on: January 22, 2012, 05:44:52 am »
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Can anyone speak to this from a stat perspective?  (I'd love to know the math)

This is not going to be extremely rigorous math-wise, but let's take your provided example of a deck (player A) with 2 Golds, 4 Silvers, 9 Coppers, 3 Estates against a deck (player B) with 8 Silvers, 7 Coppers, and 3 Estates. Then let's add 3 Provinces to each deck (21 cards total).

Let it be given that player A draws a Gold as 1 of the 5 cards in his hand. He can get to $8 in the following ways:

No Victory cards in hand - 14C4 / 20C4 = 20.66%
All combinations get a Province - 100%
Cumulative: 20.66% * 100% = 20.66%

1 Victory card in hand - 6C1 * 14C3 / 20C4 = 45.08%
All combinations with exactly 0 Copper get a Province - 5C3 / 14C3 = 2.75%
All combinations with exactly 1 Copper get a Province - 5C2 * 9C1 / 14C3 = 24.73%
Only 1 Gold with 2 Copper get a Province - 1C1 * 9C2 / 14C3 = 9.89%
Cumulative: 45.08% * 37.36% = 16.84%

2 Victory cards in hand - 6C2 * 14C2 / 20C4 = 28.17%
Must draw 1 Gold and 1 Silver - 1C1 * 4C1 / 14C2 = 4.40%
Cumulative: 28.17% * 4.40% = 1.24%

Chance to buy Province given that Gold is in hand: 38.74%

Player B's treasure is more homogeneous, so we don't have to do conditional analysis to figure out how he can get to $8:

4 Silvers in hand - 8C4 * 13C1 / 21C4 = 15.20%

3 Silvers, 2 Coppers in hand - 8C3 * 7C2 / 21C4 = 19.65%

Chance to buy Province: 34.85%

How about we add 4 Provinces apiece instead in order to figure out the chances of each deck closing out the game with a 5/3 Province split?

Player A's chance to buy Province given that Gold is in hand: 37.08%
Player B's chance to buy Province: 29.47%

Now, OK, it's not quite fair to always assume that player A has a Gold in hand and directly compare the probabilities of buying a Province. But player A will draw a Gold into his hand at some point. In a deck bloated with Victory cards, player B has trouble closing out a win when he has a 4/3 Province advantage, but player A doesn't have nearly as much trouble as long as he finds his Golds. That can probably explain your lopsided win ratios.
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Asklepios

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Re: The Key to Big Money Part I: Money Density
« Reply #16 on: January 22, 2012, 12:01:24 pm »
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On the topic of Money Density, what do people think of Cache?

Obviously if there are good $5 actions out there its pretty weak, but say you're playing 2 Smithys and Big Money towards Provinces, and you hit $5. Is Cache a good call or is Silver better?

To me, I think it may well be but I find it hard to explain why in statistical terms. Gold and two coppers is only $1.666 average, which makes it worse than buying  single silver, but in experience Cache seems to have worked out for me more often than not. Has anyone simulated 2 Smithy + Big Money + Caches on $5 vs 2 Smithy + Big Money?
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Thisisnotasmile

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Re: The Key to Big Money Part I: Money Density
« Reply #17 on: January 22, 2012, 12:07:15 pm »
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Completely unoptimized it's about 59-33 to the Cache player.
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HiveMindEmulator

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Re: The Key to Big Money Part I: Money Density
« Reply #18 on: January 22, 2012, 12:14:44 pm »
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Cache is better than silver in basically any non-trimmed deck that relies on gold to buy provinces. When money density is even reasonably close, gold-heavy is generally better than silver-heavy.
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DG

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Re: The Key to Big Money Part I: Money Density
« Reply #19 on: January 22, 2012, 12:47:26 pm »
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Another simple situation is woodcutter + treasure, where you might get the option to buy an gold+copper or silver+silver instead of a gold. In both cases you should buy the one gold if you're trying to buy provinces. If you are not trying to raise your average hand value much above 5, perhaps to buy duchies and dukes, then more/lower treasures can become superior.
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Asklepios

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Re: The Key to Big Money Part I: Money Density
« Reply #20 on: January 23, 2012, 03:48:19 am »
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Completely unoptimized it's about 59-33 to the Cache player.

That was what I expected. Can someone explain why that should be the case statistically though, given that Cache has a lower average money per card than silver? I suspect its to do with distributions of probabilities rather than averages.

That is, in statistical terms I think it must be more important to know p($8+) than the $mean when you are analysing money density.
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DStu

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Re: The Key to Big Money Part I: Money Density
« Reply #21 on: January 23, 2012, 04:42:22 am »
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Completely unoptimized it's about 59-33 to the Cache player.

That was what I expected. Can someone explain why that should be the case statistically though, given that Cache has a lower average money per card than silver? I suspect its to do with distributions of probabilities rather than averages.

I think you can also argue for Cache without even going to the probabilities, and I think the article even gives all you need. It's not a clear call, I think I could argue in the other direction also if the facts where different, but anyway, here is what is in Caches favour in BM-decks:
1) $5/3 is more than you need on average to get to Provinces, even without Smithies. It's also more than your average hand usually is in BM-decks when you go for Provinces.
2) Silver is even higher, but it's only one card. Cache is three cards. Three cards have an higher influence on the deck than one. And you are more resilent to clogging with a larger deck. Extreme case: You have a deck consistent of only 2 victory cards, and the alternative of gaining a Cache or a Silver. Cache brings you deck to average $1, Silver to average 2/3.
I don't believe there is really a situation where Cache overtakes Silver in this way, as the effect gets smaller with both more money in the deck and more cards in the deck, just to illustrate that gaining 3 cards for 5/3 must be valued stronger in many cases than gaining just one of it.
3) Two Smithies are already pretty much for BM, might be optimal but I think it was close, as they tend to collide. Adding 3 cards to your deck reduced the risk of collisions more than adding one.
4) Variance should also help the Cache.
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Asklepios

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Re: The Key to Big Money Part I: Money Density
« Reply #22 on: January 23, 2012, 08:16:45 am »
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Yes, I agree.

I think as another extreme circumstance the following example might indicate why mean$ is less  important than p($8).

First deck: $1 x 8, $2 x2 has got a mean $ of $1.2, but zero chance of making $8.
Second deck: $1 x 10, $2 x1, $3 x1 $ of $1.15, but has a fair chance of making $8.

By the same count, I think Cache likely increases your chance of reaching $6 faster than Silver does, so Cache is a quicker than silver as a route to gold.

Does this make sense? I mention this only because I think its a trap to think of money density purely in terms of mean $ per card.

That is, the second from last paragraph in the article is probably the most important one. Gold + Copper =/= Silver + Silver

Or to put it in Cache terms:

Gold + Copper + Copper > Silver.
« Last Edit: January 23, 2012, 08:19:19 am by Asklepios »
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HiveMindEmulator

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Re: The Key to Big Money Part I: Money Density
« Reply #23 on: January 23, 2012, 03:30:37 pm »
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To add to the cache vs silver thing, think about what hands that buy provinces look like. Unless you're going for a really heavy silver strat like trader or double jack, you basically need either a gold (or cache) or a +cards card to hit $8. And if you have 2, you're basically guaranteed a province. Given reasonable money density, gold (+ cache) density actually becomes more important than money density. Similarly with buying golds. Pre-greening, a gold (or cache) in hand basically allows you to buy another gold almost regardless of what you draw it with.
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GendoIkari

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Re: The Key to Big Money Part I: Money Density
« Reply #24 on: January 23, 2012, 04:00:35 pm »
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To add to the cache vs silver thing, think about what hands that buy provinces look like. Unless you're going for a really heavy silver strat like trader or double jack, you basically need either a gold (or cache) or a +cards card to hit $8. And if you have 2, you're basically guaranteed a province. Given reasonable money density, gold (+ cache) density actually becomes more important than money density. Similarly with buying golds. Pre-greening, a gold (or cache) in hand basically allows you to buy another gold almost regardless of what you draw it with.

So I assume that this means also that in a hand of Mine + Copper + Silver, Silver -> Gold is always better than Copper -> Silver?
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