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[Davio]

Actually, the value of drawing another card drops heavily when you draw your entire deck

Only when you can overdraw your deck with spare drawing capacity.

Yeah, a simple case is 4 Colonies and 9 GMs.
If you add a Platinum, you might end up with 4 Colonies and Plat in hand next turn.
If you add a GM, you are guaranteed $20.

[jomini]

Actually, the value of drawing another card drops heavily when you draw your entire deck

Only when you can overdraw your deck with spare drawing capacity.

Yeah, a simple case is 4 Colonies and 9 GMs.
If you add a Platinum, you might end up with 4 Colonies and Plat in hand next turn.
If you add a GM, you are guaranteed $20.

The odds of 4 Col/Plat are pretty low (.05%). This has virtually no impact on expected utility.

The odds of hitting $23 with Plat are 35.7% (No Gm on the bottom of the deck).
Odds of hitting $21 or more coin  are 60.4% (odds of there not being two Gm on the bottom of the deck).
Odds of hitting $19 or more coin (break point for Col + Prov being 18)  are 77%.

In other words, adding in a Gm doesn't do much for you in comparison to plat. Odds are, you will increase your payout in this case by going Plat over Gm, even though you are no longer certain to draw your entire deck. In only 23% of cases will you be able to buy fewer VP next turn than if you got the Gm (assuming we aren't at game end and want to go estate hunting, of course if we are ... why not just buy the province instead of the plat?).

This sort of probability happens a lot, you don't need to have enough draw to be sure of drawing your entire deck for the utility of Gm's +1 card to be less than you'd expect. Playing the odds is often the right bet, particularly when you are hovering around break points like 16, 22, and 13 coin.

So you don't need to be able overdraw (in some cases you don't even need to be able to draw your entire deck - e.g. Spy) for the +1 card on Gm to get swamped by other considerations (the odds of being able to hit 2 colonies). Somewhere near being able to draw your entire deck (exact point varies by deck, and even turn by turn for a given deck), the utility of Gm plummets down to close to that of silver.

[DG]

I've run a few simulator tests on a simple deck with chapel, 3 silvers, grand markets, go green when you have all/enough grand markets. The win rate drops slightly for each platinum it buys, even the first platinum. Perhaps someone can check through Jomini's maths in more detail.

[dondon151]

The point here is not a hypothetical super-trim deck with 9 GMs and 4 Colonies: in the most common kinds of drawing deck that rely on terminal +cards and +2 actions cards, you want cantrip money to increase consistency rather than Treasures.

[jomini]

I've run a few simulator tests on a simple deck with chapel, 3 silvers, grand markets, go green when you have all/enough grand markets. The win rate drops slightly for each platinum it buys, even the first platinum. Perhaps someone can check through Jomini's maths in more detail.

The math is fairly straight-forward. The odds of getting a non-Gm in the first card is 5/14, the odds of the next card then being a non-Gm is then 4/13, so on and so forth. This means that we end up with a 120/240240 chance of drawing a straight dead 5 card hand, this simplifies down to .05%.

So what are the odds that you will draw your entire deck? Consider if the bottom card of the deck is a non-Gm, this means you have 5 cards in hand, only 4 can be non-Gm and everything else in the deck is a Gm - ergo you MUST be able to draw & play the entire deck if the bottom card is a non-Gm. So odds that the bottom card is not a Gm are 5/14 or ~35.7%. Conversely, this means that ~64.3% of the time you DO have a Gm on the bottom so it doesn't get played.

We can now repeat this exercise with one fewer Gm. If we have 8 Gm and a 5 card hand we will play the entire deck if we have a non-Gm on the bottom (5/13). So 5/13ths of the time when we have a Gm on the bottom, we will have a non-Gm second from bottom (thus we only fail to play one Gm this turn). Thus we can so .643 x 5/13 = .247 or 27.4% of the time you will hit 21. Add these two probabilities  (35.7 + 24.7) and you get the odds for hitting $21 or greater. We could continue this for a third iteration and calc the odds that all three of the bottom cards are Gm, but that doesn't seem to be worth the time.

Note: I did the initial calculations in a different manner, it shouldn't have an impact, but I wasn't rigorous with decimals so things may not calc out exactly the same.

Unequivocally, you have a higher expectation value of points you can buy after the plat over the Gm. Unequivocally, the value of adding the Gm to the deck is exactly 2 coin.

So why does DG's simulator have the plat lose? Because we don't care about just one turn's worth of expected cash output. Sure for the first plat, you get a higher expected cash value, but as you buy more dead cards, you have less and less draw. It is only very near the "draw your whole deck, reliably" point where +1 card starts to lose value. His sim starts at the reliable whole deck draw point and promptly moves away from it. After the 3rd colony, he has much better odds of using the +1 card from Gm to chain more Gm (totaling more coin than a Plat). If we are buying 8 or even 4 colonies, this just hammers the Plat.

So why not just forgot about the corner case if it doesn't last that long? In a word: megaturn. If you only buy VP on your final turn or two, it can be far more efficient to do something like smithy/village/plat/plat (total cost 25) than Gm x5 (total cost 30); both leave you with +10 coin, but the latter only gives +1 card and of course the +buy. This becomes particularly acute with things like - being able to get more coin by playing coppers or using Border Village/Smithy to only pay 24. You can get to double colony a half turn sooner and that can be a much bigger deal than +1 card, particularly if you over bought components because of copper concerns.

After all, Plat vs Gm is just a closer set of numbers than Gold vs Market. Somewhere there is a point where the advantages of treasure (e.g. purchase efficiency) outweigh the consistency of +1 card. Gm is closer to plat on those advantages, but it still isn't quite there. Most people don't rave about how +1 card from market just destroys gold in engines, clearly the value of +1 card is fluid and at some point on its relative value, we reach a tipping point where treasure is more effective. For Market vs Gold, this is early on with plenty of use left for the draw. For Gm vs Plat, it is a lot later in the curve, but nonetheless, still there.

Dondon:
Actually, I think that depends on your likelihood of dead draw. A Fishing village/Oracle engine will almost never draw the Gm dead, on the other hand a Hunting grounds/Bandit camp engine is much more likely to make a dead draw. Even with either of those specified, I don't think the math obviously favors Plat or Gm - small changes in deck composition can easily swing it one way or the other.

[dondon151]

Dondon:
Actually, I think that depends on your likelihood of dead draw. A Fishing village/Oracle engine will almost never draw the Gm dead, on the other hand a Hunting grounds/Bandit camp engine is much more likely to make a dead draw. Even with either of those specified, I don't think the math obviously favors Plat or Gm - small changes in deck composition can easily swing it one way or the other.

That's also not my point. I am not talking about the situations in which you blindly play your terminal +cards on a bad draw to fish for Platinum and hope that you don't draw 2-4 villages; those circumstances slightly favor Platinum (however, you should be opting to not play the +cards regardless of Platinum or GM unless you absolutely need the cash to make a VP buy, lest you end up drawing 4 villages and also doom your subsequent hand). When you consider GM's +1 card to Platinum's lack of +1 card during the action phase, that could mean the difference between drawing and not drawing your terminal +cards or your villages.

In summary: GM is always preferable to Plat in deck draw engines except when you can draw your deck beyond some level of consistency (let's say a 95% success rate), or when you need to sacrifice consistency for a couple of extra turns of having $3 extra of value in your deck. (Cue someone complaining about edge cases and kingdom dependency.)

[aaron0013]

The point here is not a hypothetical super-trim deck with 9 GMs and 4 Colonies: in the most common kinds of drawing deck that rely on terminal +cards and +2 actions cards, you want cantrip money to increase consistency rather than Treasures.

With Bishop, anything is possible!

[RisingJaguar]

Actually, the value of drawing another card drops heavily when you draw your entire deck

Only when you can overdraw your deck with spare drawing capacity.

Yeah, a simple case is 4 Colonies and 9 GMs.
If you add a Platinum, you might end up with 4 Colonies and Plat in hand next turn.
If you add a GM, you are guaranteed $20.

The odds of 4 Col/Plat are pretty low (.05%). This has virtually no impact on expected utility.

The odds of hitting $23 with Plat are 35.7% (No Gm on the bottom of the deck).
Odds of hitting $21 or more coin  are 60.4% (odds of there not being two Gm on the bottom of the deck).
Odds of hitting $19 or more coin (break point for Col + Prov being 18)  are 77%.

I did a quick expected $ return similar to yours.  Except it breaks down each case (so odds of $21 would be 24.72% and so forth) and multiplied it by the dollars it was supposed to give you. 

Ironically, (I'm not totally sure why this is...) the expected return comes out as $20 (I double checked and my total probabilities added is equal to 1). 

Now given these two options.  I'm pretty sure GM is better here (as you are guaranteed $20).  Also in general, I would agree with DonDon that cantrip money is usually better than treasure money, as long as you aren't extremely over-drawing your deck CONSISTENTLY (ie. a crossroads deck, would not be consistent imo).

For math purposes, I did 5/14*9/13*8/12*7/11*6/10 for each succession by adding another n/n+4 on top.  Read Jomini's for more understanding.

There may be a flaw in my math, so please feel free to yell at me

[jomini]

Now given these two options.  I'm pretty sure GM is better here (as you are guaranteed $20).  Also in general, I would agree with DonDon that cantrip money is usually better than treasure money, as long as you aren't extremely over-drawing your deck CONSISTENTLY (ie. a crossroads deck, would not be consistent imo).

For math purposes, I did 5/14*9/13*8/12*7/11*6/10 for each succession by adding another n/n+4 on top.  Read Jomini's for more understanding.

There may be a flaw in my math, so please feel free to yell at me

Your math, so far as I can see, is fine. The problem is you are measuring the wrong thing. We don't care about having the same or higher expected coin value, that is really just a proxy for expected VP gain. For the latter, there is a world of difference between having a 1/3 chance of hitting 23 coin (and double colony) vs hitting an assured 20 (colony + prov). You do now have non-zero odds of hitting 17 or fewer coin (colony + duchy or worse), but the double colony gain tends to make that up.

Dondon:
I'm still not convinced there. Plat is space inefficient (compared to Gm), but it is extremely price efficient. Gm is exactly the opposite and I'm not so sure that every draw deck with <95% (or whatever) reliability is going to value space over price, particularly with the copper restriction monkeying around.

[Tables]

However, it's not hard to create a map between coin in hand and victory gain, and we could use that to compare the two (I leave this as an exercise for the reader).

Assuming we're happy to gain Duchies but not Estates we'd have:
$22-23 -> 20
$19-21 -> 16
$16-18 -> 13
$11-15 -> 10
$8-10 -> 6
$5-7 -> 3

From there, you just work out the odds of getting each amount and multiply by the value you're mapping to.

[RisingJaguar]

However, it's not hard to create a map between coin in hand and victory gain, and we could use that to compare the two (I leave this as an exercise for the reader).

Assuming we're happy to gain Duchies but not Estates we'd have:
$22-23 -> 20
$19-21 -> 16
$16-18 -> 13
$11-15 -> 10
$8-10 -> 6
$5-7 -> 3

From there, you just work out the odds of getting each amount and multiply by the value you're mapping to.

If we are honestly going down this route.  Lets say said deck has 4 colonies and 9 GMs, gets $18 dollars... what would you buy? 

The matter is that the previous example makes no sense in reality.  Tell me a time when my deck contents does not include a trasher yet everything is trashed, have 9 GMs and 4 colonies, and i'm deciding with my $18 what to buy?

I was using the situation to proxy what to do, when faced with 4 terminal cards, and a good chunk of GMs, what should you purchase.  I came out with similar numbers for expected return, and chose consistency over extra potential. 

So extrapolating that to an engine, lets say a deck of 5 alchemist and similar situation, barely draw all my cards.  I choose GM.

[Davio]

I really don't know why my example of "4 Col + 9 GM" has gone on this long.

I mean, its only intent was to show that it's possible to be stuck in your drawing chain if you happen to hit 4 Col + Plat, nothing else. Yes it's unlikely, but it was only to show that it was possible.

[jomini]

I really don't know why my example of "4 Col + 9 GM" has gone on this long.

I mean, its only intent was to show that it's possible to be stuck in your drawing chain if you happen to hit 4 Col + Plat, nothing else. Yes it's unlikely, but it was only to show that it was possible.

I ran with it because it illustrates one of my major points - before you deck is 100% reliable to draw everything, plat edges out Gm. Dondon calls it 95%, but there is very clearly a point where playing the odds with plat is better than getting a more reliable Gm. Your example allows for very easy odds calculations, the principle is the same with something like a smithy/village engine; that is just much harder to calc. Gm fits on a weird curve - it is pretty bad in a deck that either dead draws or has low density (e.g. lots of curses, low treasure density, not a lot of cards that have 0 effective space or less), it is really good as your deck gets "denser" (e.g. lots of cards that draw, like Gm itself), and then it gets really bad again when you are drawing most everything. The 2nd turning point is short of "draw your whole deck" and your example gives an easy to understand situation.

[aaron0013]

Tell me a time when my deck contents does not include a trasher yet everything is trashed, have 9 GMs and 4 colonies, and i'm deciding with my $18 what to buy?

With Bishop, anything is possible!

[clb]

Tell me a time when my deck contents does not include a trasher yet everything is trashed, have 9 GMs and 4 colonies, and i'm deciding with my $18 what to buy?

With Bishop, anything is possible!

I think you would have to Hermit-trash the Bishop/other trasher, not buy something to trash the Hermit, gaining a Madman, and then play the Madman to get to a trashed deck with no trasher. So, yes, with Bishop it is possible, but it takes more than that, too.

[jeb56]

You can trash your Bishop to the other guy's Bishop.

Bishop is enough.

[clb]

You can trash your Bishop to the other guy's Bishop.

Bishops is are enough.

Though, I guess since we are quibbling edge-cases, I am sure you could trash all of your cards to an opponent's single Bishop, so I concede.

[PSGarak]

Island. Everyone forgets it because it's not technically trashing, but it's the one thing that will remove both itself and another card. The other card is most likely a more effective trasher.