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[TheExpressicist]

I still completely disagree with that. Once you have 5 silvers in your deck you do not suddenly start buying wishing wells, lookouts, embargoes, pearl divers, chancellors, thieves, caravans, mines or in fact any alternative card at all. If you want something else than silver then you should generally get it very early before the silver fills the deck. If you are using a lot of silver then keep buying silver. Just what 'any action you could purchase' are you talking about?

The basic premise is that, by the time you have 5 silvers in your deck, your money density will be around $1.5 per card. So any card that offers +1 card automatically offers +$1.5. (You already have a 45-50% chance of replacing the silver with that +1 card).

The second basic premise is that we're operating under the assumption that you haven't already maxed yourself out on terminal cards and can add another terminal.

The third basic premise is that there are most likely alternatives to any crappy card you mention. By the time you have 5 silvers in your deck, you only have a ~5% chance of drawing a hand with exactly 3 coins. And, there is only a ~5% chance that the kingdom has exactly one <=$3 card. So in 1 out of 400 hands, you'll be faced with a dilemma of "Silver vs. One Specific Crappy $3 Card".  The odds are even less for "Silver vs. One specific crappy $4 card".



Wishing Well: It's a cantrip, so $1.5+ Utility. If you haven't done any trashing, you have between a 30-40% chance of hitting Copper if you name it, which adds $.3 to $.4 to its value. So, overall, Wishing Well is worth on average $1.85 per play. So strictly speaking, it's slightly less valuable, $-wise, than silver. But it provides two key utilities: if you do hit Copper, you have filtered a copper out of your next hand which increases your next hand's value. And you have the ability to do the hail-mary Wish (if you have 5 coins and you need one province to win, WW will always be more useful than silver).  .

Embargo:  $2+, Utility, thus more valuable than silver. (As long as it doesn't collide).

Pearl Diver: Cantrip, so $1.5+ Utility. The utility is fairly marginal. You have a ~50% chance of hitting a >=$2 card on the bottom of your deck, thus increasing your next hand's value by ~$.7. So that's ~$.35 worth of utility making it worth ~$1.85. It's highly situational, but yes I would agree that in the 0.25% chance that this is the only <=$3 available, it would likely be preferable to buy silver.

Chancellor: $2 + Utility, thus more valuable than silver (as long as it doesn't collide).

Lookout: The filtering increases the value of your next hand by ~$1, and the trashing and increases the value of all future hands by ~$.20. So it pays for itself in 5 turns. Probably not better than silver, but board-dependent.

Caravan: Cantrip so +$1.5, provides another cantrip so another +$1.5, so $3 of total value.

Thief: Except in extremely specific situations, less useful than silver. But, that's counterbalanced by the fact that there's approximately a .01% (1 in 10000) that you will have exactly four coins, and that this will be the only <=$4 card available. 

Mine: Provides +$1 in value, plus trashing, (the result of which increases the average value of your future hands by $.25). So in four turns it pays for itself. Probably not better than silver.

To answer your question re: "What any action you could purchase?"
1. Any card that provides +$2 (Worth $2+ utility)
2. Any card that provides 2 or more cards, regardless of whether or not there's a discard involved.   (Worth $3+ on average)
3. Any card that provides $1 + 1 card. (Worth $2.5 + utility on average.)

Again, let's all take a step back and remember that this is for the average scenario. Edge cases need not apply. There are definitely individual cards that in almost all circumstances are worse than silver, but the odds of that one card being the only available option range from extremely low to astronomically low.

[TheExpressicist]

Trashing Rules****: If you have a trasher in hand, here are your priorities: 1:Trashing two+ Estates, 2: Trashing three+ copper, 3: Buying an extremely important card (Witch, Mountebank, etc.),  4:Trashing 1 Estate,  5:Trashing 2 Copper.

Ambassador suggests the last two should be reversed. Is there any non-Ambassador situation where you have a choice between trashing 1 Estate and 2 Copper?

I can't think of any non-edge case scenarios where you'd have that choice when it doesn't involve Ambassador.

I based #4 and #5 off the following thread: http://forum.dominionstrategy.com/index.php?topic=9828.msg323096#msg323096 , where the simulation data suggested "that returning 2 coppers over 1 estate is a bad idea." There was a lot of skepticism re: that, so I also looked at it from a statistical perspective. Statistically, removing 2 coppers from your money supply has more of a negative effect on your money density than removing 1 estate, until you have approximately $30 in total money in your deck.

That said, I'm not 100% sold on the order of those two just yet, so I could be convinced to swap them.

[Davio]

Trashing Rules****: If you have a trasher in hand, here are your priorities: 1:Trashing two+ Estates, 2: Trashing three+ copper, 3: Buying an extremely important card (Witch, Mountebank, etc.),  4:Trashing 1 Estate,  5:Trashing 2 Copper.

Ambassador suggests the last two should be reversed. Is there any non-Ambassador situation where you have a choice between trashing 1 Estate and 2 Copper?

Doctor if you happen to know that the next 3 cards are Estate, Copper, Copper?

[DG]

I still think you're completely wrong about the 5 silvers and the logic you are backing it up with shows flaws. Could you please drop the silver thing before you make me completely deconstruct it?

[Stealth Tomato]

Trashing Rules****: If you have a trasher in hand, here are your priorities: 1:Trashing two+ Estates, 2: Trashing three+ copper, 3: Buying an extremely important card (Witch, Mountebank, etc.),  4:Trashing 1 Estate,  5:Trashing 2 Copper.

Ambassador suggests the last two should be reversed. Is there any non-Ambassador situation where you have a choice between trashing 1 Estate and 2 Copper?

I can't think of any non-edge case scenarios where you'd have that choice when it doesn't involve Ambassador.

I based #4 and #5 off the following thread: http://forum.dominionstrategy.com/index.php?topic=9828.msg323096#msg323096 , where the simulation data suggested "that returning 2 coppers over 1 estate is a bad idea." There was a lot of skepticism re: that, so I also looked at it from a statistical perspective. Statistically, removing 2 coppers from your money supply has more of a negative effect on your money density than removing 1 estate, until you have approximately $30 in total money in your deck.

That said, I'm not 100% sold on the order of those two just yet, so I could be convinced to swap them.

Many decks would rather have card density than money density. Mostly this applies to action cards, but I'd also rather have a Silver and an Estate than two Copper, even though they have the same density.

The trade-off between two Copper and one Estate isn't money density--it's that you'll get to the Ambassador faster, and you're also more likely to hit it with two Estates. Particularly notable is the case where you return one Estate twice, vs. two Copper and then two Estates. In the second case, you've returned two extra cards over two turns, and your opponent just gets one Copper instead of one Estate.

[Mic Qsenoch]

Average money density is a crappy way of evaluating your deck, with rare exceptions.

[TheExpressicist]

Many decks would rather have card density than money density. Mostly this applies to action cards, but I'd also rather have a Silver and an Estate than two Copper, even though they have the same density.

Yeah, and these heuristics are definitely not for engine decks. And good points regarding the other aspects of Ambassador. Since those last two only apply to Ambassador anyway, I'm going to change them.

Average money density is a crappy way of evaluating your deck, with rare exceptions.

That's why there are few references to it in the "rules of thumb". It's too complicated of a concept for beginners, and by the time you're good enough to really understand it, it's not really all that applicable in its basic form.

[WanderingWinder]

I still completely disagree with that. Once you have 5 silvers in your deck you do not suddenly start buying wishing wells, lookouts, embargoes, pearl divers, chancellors, thieves, caravans, mines or in fact any alternative card at all. If you want something else than silver then you should generally get it very early before the silver fills the deck. If you are using a lot of silver then keep buying silver. Just what 'any action you could purchase' are you talking about?

The basic premise is that, by the time you have 5 silvers in your deck, your money density will be around $1.5 per card. So any card that offers +1 card automatically offers +$1.5. (You already have a 45-50% chance of replacing the silver with that +1 card).

This is a terrible premise. I basically NEVER get money density that high (in a non-colony game, anyway) unless I'm awful AND playing against someone else who is also awful.

The second basic premise is that we're operating under the assumption that you haven't already maxed yourself out on terminal cards and can add another terminal.

This is another awful assumption. If I'm playing BM, why in the world would I not have bought more terminals by the time I am looking at silver #6? I mean, seriously, I have had many turns of being able to produce any amount of money for which I could have gotten the terminal - why am I waiting until now to do so?

The third basic premise is that there are most likely alternatives to any crappy card you mention. By the time you have 5 silvers in your deck, you only have a ~5% chance of drawing a hand with exactly 3 coins. And, there is only a ~5% chance that the kingdom has exactly one <=$3 card. So in 1 out of 400 hands, you'll be faced with a dilemma of "Silver vs. One Specific Crappy $3 Card".  The odds are even less for "Silver vs. One specific crappy $4 card".

These "odds" (which aren't actually odds) are entirely made up. Who knows what my deck consists of by the time I have five silvers?
Nevertheless, this point actually shows the utter uselessness of the rule. I mean, so it's a first of all bad rule of thumb, because in this situation, you really should be looking long and hard at that silver (and indeed, in BM and slogs, the only reason you don't want more silvers is because you'd rather have other things....). So if you ever need to resort to it, it's just bad advice. Fortunately, it comes up pretty rarely - though more often in some cases (masterpiece, slogs, feodum) - so it won't actually hurt you very much in reality.

Wishing Well: It's a cantrip, so $1.5+ Utility. If you haven't done any trashing, you have between a 30-40% chance of hitting Copper if you name it, which adds $.3 to $.4 to its value. So, overall, Wishing Well is worth on average $1.85 per play. So strictly speaking, it's slightly less valuable, $-wise, than silver. But it provides two key utilities: if you do hit Copper, you have filtered a copper out of your next hand which increases your next hand's value. And you have the ability to do the hail-mary Wish (if you have 5 coins and you need one province to win, WW will always be more useful than silver).  .

Embargo:  $2+, Utility, thus more valuable than silver. (As long as it doesn't collide).

Pearl Diver: Cantrip, so $1.5+ Utility. The utility is fairly marginal. You have a ~50% chance of hitting a >=$2 card on the bottom of your deck, thus increasing your next hand's value by ~$.7. So that's ~$.35 worth of utility making it worth ~$1.85. It's highly situational, but yes I would agree that in the 0.25% chance that this is the only <=$3 available, it would likely be preferable to buy silver.

Chancellor: $2 + Utility, thus more valuable than silver (as long as it doesn't collide).

Lookout: The filtering increases the value of your next hand by ~$1, and the trashing and increases the value of all future hands by ~$.20. So it pays for itself in 5 turns. Probably not better than silver, but board-dependent.

Caravan: Cantrip so +$1.5, provides another cantrip so another +$1.5, so $3 of total value.

Thief: Except in extremely specific situations, less useful than silver. But, that's counterbalanced by the fact that there's approximately a .01% (1 in 10000) that you will have exactly four coins, and that this will be the only <=$4 card available. 

Mine: Provides +$1 in value, plus trashing, (the result of which increases the average value of your future hands by $.25). So in four turns it pays for itself. Probably not better than silver.

To answer your question re: "What any action you could purchase?"
1. Any card that provides +$2 (Worth $2+ utility)
2. Any card that provides 2 or more cards, regardless of whether or not there's a discard involved.   (Worth $3+ on average)
3. Any card that provides $1 + 1 card. (Worth $2.5 + utility on average.)

Again, let's all take a step back and remember that this is for the average scenario. Edge cases need not apply. There are definitely individual cards that in almost all circumstances are worse than silver, but the odds of that one card being the only available option range from extremely low to astronomically low.

Again, you don't actually have $1.5/card. By the time you reach that, you've lost the game.
Your math continues to be really wacky. How do you come up with these numbers? Your chance of hitting the copper is based on how many cards in your deck? Your trashing added-value-to-the-deck-in-the-future is calculated how? How are you taking into account that besides right now, the game will go on for some time longer? How do you know how much longer?

Money density is somewhat reasonable to look at if you're playing a treasure-based deck. Unfortunately, it misses extra "utility", which is often pretty important, timing, which is always important, and most importantly, most treasure-based decks just aren't that good....


Your other rules of thumb are similarly bad (well, silver/silver does get you around a 90% chance of hitting 5, but this is covered (and more precisely)) elsewhere.

I love your enthusiasm, but in terms of strategic advice, this stuff just isn't good.

[dondon151]

As for rules of thumb, having actual speed benchmarks in mind might better. For example,

Can I comfortably get 4 Provinces in less than 15 turns?

That about sums up what you need to know about benchmarking your Province strategy against a big money + X strategy.

Even this rule of thumb has a lot of exceptions.

[Polk5440]

As for rules of thumb, having actual speed benchmarks in mind might better. For example,

Can I comfortably get 4 Provinces in less than 15 turns?

That about sums up what you need to know about benchmarking your Province strategy against a big money + X strategy.

Even this rule of thumb has a lot of exceptions.

I was under the impression that most of the big money + x optimized bots can get 4 provinces in 15 turns on average. Jack may be a couple turns faster. But that's the fastest, right? Is my knowledge of the sims not right?

Now, using the question to determine whether your province strategy is good enough or not, sure. That one question won't tell you whether you have a winning strategy (e.g. "I can't get to 4 Provinces in 15 turns, but there are attacks! The attacks will slow my opponent down, but not me! I'll go with that..."); however, I think it sums up what you need to know about the speed of big money + x when you are also going for a Province strategy, which is what I thought was the stated point of OP.

[TheExpressicist]

The basic premise is that, by the time you have 5 silvers in your deck, your money density will be around $1.5 per card. So any card that offers +1 card automatically offers +$1.5. (You already have a 45-50% chance of replacing the silver with that +1 card).

This is a terrible premise. I basically NEVER get money density that high (in a non-colony game, anyway) unless I'm awful AND playing against someone else who is also awful. [/quote]

"Money density" is a bit of a misnomer in this situation as I'm actually referring more specifically to "Value per Card". E.g. If you Wishing Well and then draw a Smithy. Your literal Money Density may be <$1.5 but the collective impact of the Smithies in your deck means that functionally speaking, every card you draw will, on average, result in $1.5ish.*

edit: The "value per card" is fairly self-evident; if your average value per card was less than 8/5, you would almost never have enough money to buy a Province. Also, this number is going to differ depending on how much gold/treasure/card draw/etc. you have. The actual number is a bit lower but 1.5 is a nice round number.

This is another awful assumption. If I'm playing BM, why in the world would I not have bought more terminals by the time I am looking at silver #6? I mean, seriously, I have had many turns of being able to produce any amount of money for which I could have gotten the terminal - why am I waiting until now to do so?

You should have at least 2-3 terminals by the time you're looking at silver #6. By turn 10 or so, you can afford to add a 3rd-4th terminal to your deck. If you're unlucky enough to be put into a situation where you already have 4 terminals, 5 silvers, and can't afford anything else, then yes, you should probably buy a silver. But the odds of that happening are fairly slim.

These "odds" (which aren't actually odds) are entirely made up. Who knows what my deck consists of by the time I have five silvers?

If you're playing BM you'll probably have 3 terminals and 5 silvers. There are plenty of situations where this isn't the case, but remember, this is not a guide for expert players to reevaluate how they play the game. They're simple rules for beginners to follow to raise the bar of their level of play.

Again, you don't actually have $1.5/card. By the time you reach that, you've lost the game.

See above re: "money density" vs. "value per card".

Your math continues to be really wacky. How do you come up with these numbers? Your chance of hitting the copper is based on how many cards in your deck? Your trashing added-value-to-the-deck-in-the-future is calculated how? How are you taking into account that besides right now, the game will go on for some time longer? How do you know how much longer?

The numbers are just statistical analysis using "average situations" as the assumptions. Chances of hitting copper are based on 7 copper in a deck consisting of between 18-23 cards.  The trashing added-value-to-the-deck-in-the-future is based on statistical simulation of a deck consisting of [X..Y..Z] and the same deck minus Y. As for the length of game, this is assuming that your "silver vs. action" dilemma happens sometime between turns 8 and 13. If it happens beyond that, then it probably doesn't matter at all what you purchase.

[dondon151]

I was under the impression that most of the big money + x optimized bots can get 4 provinces in 15 turns on average. Jack may be a couple turns faster. But that's the fastest, right? Is my knowledge of the sims not right?

I was going for more of the "you can lose 5/3 on Provinces but win 6/2 on Duchies" kind of implication there.

Yeah, you're right, attacks and alt VP confound that rule of thumb even more. Kingdoms containing those cards, in addition to the possibility of winning when down on Provinces, are common enough that at least in my opinion, there are more exceptions to this rule of thumb than there are instances proving it.

[flies]

Many decks would rather have card density than money density.

i always try to build decks with one card per card.  getting your card density higher than that is tricky, but i bet Stef can do it.

[TheExpressicist]

As for rules of thumb, having actual speed benchmarks in mind might better. For example,

Can I comfortably get 4 Provinces in less than 15 turns?

That about sums up what you need to know about benchmarking your Province strategy against a big money + X strategy.

Even this rule of thumb has a lot of exceptions.

I was under the impression that most of the big money + x optimized bots can get 4 provinces in 15 turns on average. Jack may be a couple turns faster. But that's the fastest, right? Is my knowledge of the sims not right?

Now, using the question to determine whether your province strategy is good enough or not, sure. That one question won't tell you whether you have a winning strategy (e.g. "I can't get to 4 Provinces in 15 turns, but there are attacks! The attacks will slow my opponent down, but not me! I'll go with that..."); however, I think it sums up what you need to know about the speed of big money + x when you are also going for a Province strategy, which is what I thought was the stated point of OP.

Yeah as far as I'm aware, 15 is the limit. That's where I got the 14 from; if you can consistently achieve your goal in 14 turns, you can consistently beat BM.
 
Also , after reading the input from everyone re: the un-aptly named "5 silver rule" I'm going to go ahead and recant. The circumstances where it is applicable are few and far between especially if you're following solid greening heuristics. Thus making it not a great "rule of thumb for average cases"

[shark_bait]

Average money density is a crappy way of evaluating your deck, with rare exceptions.

Gotta give my love to Masterpiece/BM.  Average money density works like a gem for this strategy as the vast majority of your hands are 5 card hands that play solely treasure. 

[shark_bait]

Many decks would rather have card density than money density.

i always try to build decks with one card per card.  getting your card density higher than that is tricky, but i bet Stef can do it.

Using Island returns cards to your deck at the end of the game implying that during the game they are not part of your deck.  But if you think of the ratio in terms of cards / cards in deck you in fact can get higher than 1 because you own the cards from Island but they are not in your deck!

[Kirian]

Many decks would rather have card density than money density.

i always try to build decks with one card per card.  getting your card density higher than that is tricky, but i bet Stef can do it.

Celestial Chameleon is able to increase his density to as high as one deck per card.

[silverspawn]

i like seeing posts being picked apart that aren't mine. it's so... peaceful.

[markusin]

i like seeing posts being picked apart that aren't mine. it's so... peaceful.

I'm beginning to suspect it's some sort of secret hazing ceremony.