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[Davio]

If I simulate plain Village vs Farming Village (start both off with 7 Copper, 3 Estate and 9 of its respective card), Farming Village beats regular Village 86% to 9%.

So there is a significant effect going on here and the simulator doesn't even care about managing reshuffles. I doubt whether it's just because the FV shuffles slightly more often.

[ecq]

If I simulate plain Village vs Farming Village (start both off with 7 Copper, 3 Estate and 9 of its respective card), Farming Village beats regular Village 86% to 9%.

So there is a significant effect going on here and the simulator doesn't even care about managing reshuffles. I doubt whether it's just because the FV shuffles slightly more often.

No doubt.  Farming Village helps your current hand.  While Village often draws green, Farming Village will always draw something you can use.  The question is whether it helps your next hand (apart from the effects of non-filtering deck cycling).  The answer is it doesn't.

[Ozle]

If I simulate plain Village vs Farming Village (start both off with 7 Copper, 3 Estate and 9 of its respective card), Farming Village beats regular Village 86% to 9%.

So there is a significant effect going on here and the simulator doesn't even care about managing reshuffles. I doubt whether it's just because the FV shuffles slightly more often.

No doubt.  Farming Village helps your current hand.  While Village often draws green, Farming Village will always draw something you can use.  The question is whether it helps your next hand (apart from the effects of non-filtering deck cycling).  The answer is it doesn't.

Surely the answer is: It can do, but on average it doesnt.

[ecq]

Of course, in the same way Smithy can help your next hand if it coincidentally draws 3 Estates.

[Deadlock39]

Well in a really contrived manner... it helps out your future hands by drawing a better card this hand which allows you to purchase better cards which are then in your future hands.  This still has nothing to do with the expected value of your next hand (unless you reshuffled and drew that better card you just bought).

[Ozle]

Well in a really contrived manner... it helps out your future hands by drawing a better card this hand which allows you to purchase better cards which are then in your future hands.  This still has nothing to do with the expected value of your next hand (unless you reshuffled and drew that better card you just bought).

I think your saying the same thing as me, but for some reason stating it in a different way,

If in the example given, it draws 3 estates and your deck is 20 coppers, 3 estates and 1 smithy. Then the act of playing it has changed your next hand. (You have 4 Coppers and Smithy: Your next expected hand value is about 4.2ish. You play the smith and get three estates. Your next expected hand value is now 5)


On average it doesnt effect your next hand, but sometimes it will, sometimes it wont. Thats why its average.


I am now calling this Schroedingers Draw, because you dont know whether it has changed your next hands expected value until you play it and see what you draw. But you want to know whether it will be good or bad before you play it.

[RiemannZetaJones]

I don't think anyone has proved anything about what might happen at, or after, the reshuffle. Venture and Farming Village are both good at deck-cycling, which presumably has a net positive effect, since you want to cycle your deck more than you want to avoid cycling it. While blueblimp's result on non-improvement of the next hand(s) is fantastic, it only applies away from the reshuffle.

[Chriamon]

On the concept of the "Schroedingers Draw" as Ozle said, perhaps it is possible to use this in the future, as opposed to trying to calculate the benefit of it before you play it. What I mean by this, is say you have perfect memory, and you know exactly every card in your deck, and every card in your discard pile at all times. Is it possible to gain a benefit to cards being in your discard over cards being in your deck. With cards like chancellor, it can be. So the question I would pose, is that in the example that smithy "can help your next hand but on average doesn't," does the smithy actually on average help your next hands when you have cards like chancellor? Intuitively I would say that more sift is helping chancellors, because you have more information about what cards are in your discard, however I would be hard pressed to come up with any math to prove it. (I think it would really be awesome if chancellor had potential applications beyond chancellor/stash).

[ecq]

I don't think anyone has proved anything about what might happen at, or after, the reshuffle. Venture and Farming Village are both good at deck-cycling, which presumably has a net positive effect, since you want to cycle your deck more than you want to avoid cycling it. While blueblimp's result on non-improvement of the next hand(s) is fantastic, it only applies away from the reshuffle.

If you discount the impact of drawing the Venture you played, Venture doesn't do anything to the quality of the hands you get at or after the reshuffle.  If there are 3 cards left in your draw pile at the end of your turn, you're drawing 3 random cards from your deck, then two random cards from your discard pile.  Your discard pile will be a random sampling of your deck, plus whatever you bought since the last reshuffle, plus the Venture we're talking about.  We can't really say anything meaningful about buys, so really the only difference Venture makes to the quality of hands after reshuffle is the fact that it can be drawn, which you could say about any card.

Venture does let you get to the reshuffle faster.  Most of the time, that's a good thing.  Near the end, it's probably a bad thing.

[blueblimp]

I've skipped past most of the arguments that are way over my head (hey, my last math class was on the university 10 years ago!).

What I want to know and was unable to distill for sure is whether the skip ability of Venture is actually helpful or moot.
It's been shown that the money density doesn't change, so playing Ventures doesn't help future hands? Is this correct?

So the only "extra" we get from Venture is the ability to trigger multiple Ventures?


I have a similar question for Farming Village. Is the ability to skip moot or does it actually help the player when there's no interaction between players? Farming Villages find either action or treasure cards, so they're done searching even sooner. The main question is: Is Farming Village only better than a normal Village when e.g. Ghost Ship is in play or are there more benefits to this potential skipping?

I'd summarize results as follows:

• Cards like Venture and Farming Village do not improve (or even change really) your future hands before a reshuffle, unless you know something about the top of your deck (e.g. you got Rabble'd). To visualize this, imagine Venture drawing cards from the bottom of your deck instead of the top. (This intuition doesn't address density, but it's a good start.)
• Nobody knows exactly how these cards interact with reshuffles. But the effect should be minor unless you're reshuffling often.

So, play recommendations:

• If you need a sifter for your next turn, Venture/FV won't do the job. Get a Cartographer instead.
• Although Venture/FV sift your current turn (and that's why FV is better than Village), this is a weak effect compared to sifters like Warehouse & Embassy. For example, if Farming Village draws a Silver and you don't use the extra action, you might as well have just bought a Silver in the first place.
• The cycling effect from these cards is overall a good thing, even if they aren't sifting much, in the same way that Smithy's cycling effect is good.

[WanderingWinder]

I've skipped past most of the arguments that are way over my head (hey, my last math class was on the university 10 years ago!).

What I want to know and was unable to distill for sure is whether the skip ability of Venture is actually helpful or moot.
It's been shown that the money density doesn't change, so playing Ventures doesn't help future hands? Is this correct?

So the only "extra" we get from Venture is the ability to trigger multiple Ventures?


I have a similar question for Farming Village. Is the ability to skip moot or does it actually help the player when there's no interaction between players? Farming Villages find either action or treasure cards, so they're done searching even sooner. The main question is: Is Farming Village only better than a normal Village when e.g. Ghost Ship is in play or are there more benefits to this potential skipping?

I'd summarize results as follows:

• Cards like Venture and Farming Village do not improve (or even change really) your future hands before a reshuffle, unless you know something about the top of your deck (e.g. you got Rabble'd). To visualize this, imagine Venture drawing cards from the bottom of your deck instead of the top. (This intuition doesn't address density, but it's a good start.)
• Nobody knows exactly how these cards interact with reshuffles. But the effect should be minor unless you're reshuffling often.

So, play recommendations:

• If you need a sifter for your next turn, Venture/FV won't do the job. Get a Cartographer instead.
• Although Venture/FV sift your current turn (and that's why FV is better than Village), this is a weak effect compared to sifters like Warehouse & Embassy. For example, if Farming Village draws a Silver and you don't use the extra action, you might as well have just bought a Silver in the first place.
• The cycling effect from these cards is overall a good thing, even if they aren't sifting much, in the same way that Smithy's cycling effect is good.

Except I have to disagree that the reshuffle effect is negligible so much. You are going to reshuffle more often, and, more importantly and moreover, these cards will miss reshuffles significantly more often. Which weakens them. How much more often... well, I don't think it's negligible, but it's really hard to pinpoint.

[blueblimp]

I've skipped past most of the arguments that are way over my head (hey, my last math class was on the university 10 years ago!).

What I want to know and was unable to distill for sure is whether the skip ability of Venture is actually helpful or moot.
It's been shown that the money density doesn't change, so playing Ventures doesn't help future hands? Is this correct?

So the only "extra" we get from Venture is the ability to trigger multiple Ventures?


I have a similar question for Farming Village. Is the ability to skip moot or does it actually help the player when there's no interaction between players? Farming Villages find either action or treasure cards, so they're done searching even sooner. The main question is: Is Farming Village only better than a normal Village when e.g. Ghost Ship is in play or are there more benefits to this potential skipping?

I'd summarize results as follows:

• Cards like Venture and Farming Village do not improve (or even change really) your future hands before a reshuffle, unless you know something about the top of your deck (e.g. you got Rabble'd). To visualize this, imagine Venture drawing cards from the bottom of your deck instead of the top. (This intuition doesn't address density, but it's a good start.)
• Nobody knows exactly how these cards interact with reshuffles. But the effect should be minor unless you're reshuffling often.

So, play recommendations:

• If you need a sifter for your next turn, Venture/FV won't do the job. Get a Cartographer instead.
• Although Venture/FV sift your current turn (and that's why FV is better than Village), this is a weak effect compared to sifters like Warehouse & Embassy. For example, if Farming Village draws a Silver and you don't use the extra action, you might as well have just bought a Silver in the first place.
• The cycling effect from these cards is overall a good thing, even if they aren't sifting much, in the same way that Smithy's cycling effect is good.

Except I have to disagree that the reshuffle effect is negligible so much. You are going to reshuffle more often, and, more importantly and moreover, these cards will miss reshuffles significantly more often. Which weakens them. How much more often... well, I don't think it's negligible, but it's really hard to pinpoint.

Good point. I should have just stuck to saying that nobody knows exactly what's going on with reshuffles, in general.

[timchen]

I finally get to read the proof.

So it is just saying drawing a random card does not change the probabilistic distribution of the remaining deck. This part seems intuitive. However, the less intuitive part is that this fact does not depend on your drawing policy (in which you only get to look at the cards after you draw them, of course.)

For me, probably the problem comes from the fact that this works for all policies, which is anti-intuitive. How can I not come up with some sort of drawing criteria to alter the remaining deck? For example, suppose I have a large enough deck of half copper and half estate. Now I demand I draw until I draw more coppers than estates. How does that not change the remaining distribution of a large enough but finite deck?

I think this is sort of an order-of-limits problem, which always hinders my intuition

[ecq]

It only works for policies that are guaranteed to stop drawing before the draw pile is empty.

[blueblimp]

I agree that it would be really nice to have a proof that is more intuitive, but in general I don't know how to do it except by induction.

[RiemannZetaJones]

I don't think anyone has proved anything about what might happen at, or after, the reshuffle. Venture and Farming Village are both good at deck-cycling, which presumably has a net positive effect, since you want to cycle your deck more than you want to avoid cycling it. While blueblimp's result on non-improvement of the next hand(s) is fantastic, it only applies away from the reshuffle.

If you discount the impact of drawing the Venture you played, Venture doesn't do anything to the quality of the hands you get at or after the reshuffle.  If there are 3 cards left in your draw pile at the end of your turn, you're drawing 3 random cards from your deck, then two random cards from your discard pile.  Your discard pile will be a random sampling of your deck, plus whatever you bought since the last reshuffle, plus the Venture we're talking about.  We can't really say anything meaningful about buys, so really the only difference Venture makes to the quality of hands after reshuffle is the fact that it can be drawn, which you could say about any card.

Venture does let you get to the reshuffle faster.  Most of the time, that's a good thing.  Near the end, it's probably a bad thing.

Venture also changes the residue-class modulo 5 of the number of cards in your draw pile, and consequently changes the number of cards in your draw pile that will miss the reshuffle. I doubt this is ever going to be worth taking into account, but it should affect the quality of hands drawn after the reshuffle.

[blueblimp]

I don't think anyone has proved anything about what might happen at, or after, the reshuffle. Venture and Farming Village are both good at deck-cycling, which presumably has a net positive effect, since you want to cycle your deck more than you want to avoid cycling it. While blueblimp's result on non-improvement of the next hand(s) is fantastic, it only applies away from the reshuffle.

If you discount the impact of drawing the Venture you played, Venture doesn't do anything to the quality of the hands you get at or after the reshuffle.  If there are 3 cards left in your draw pile at the end of your turn, you're drawing 3 random cards from your deck, then two random cards from your discard pile.  Your discard pile will be a random sampling of your deck, plus whatever you bought since the last reshuffle, plus the Venture we're talking about.  We can't really say anything meaningful about buys, so really the only difference Venture makes to the quality of hands after reshuffle is the fact that it can be drawn, which you could say about any card.

Venture does let you get to the reshuffle faster.  Most of the time, that's a good thing.  Near the end, it's probably a bad thing.

Venture also changes the residue-class modulo 5 of the number of cards in your draw pile, and consequently changes the number of cards in your draw pile that will miss the reshuffle. I doubt this is ever going to be worth taking into account, but it should affect the quality of hands drawn after the reshuffle.

Maybe it's significant with a Venture/nothing opening. Usually with a 5/- opener, only 1 card will miss the second reshuffle. If Venture is played on T3 and draws 2 cards (not the most likely case), then 4 cards will miss the reshuffle, and they will all be coppers/estates, so that should be good. On the other hand, if Venture is drawn on T4, especially if you know the one card in the draw deck is an estate, it's less clear to me whether the effect is beneficial.

It seems really difficult to use this effect to advantage in general, since you generally don't know how many cards venture will draw.

[Razzishi]

The assumption a random deck directly implies that the expected value of the density function is constant with respect to place in deck.  There's no need to say anything more.

Would you mind expanding on this? This was my intuition, but I wasn't able to formalize it in the case where the number of drawn cards varies depending on what's drawn (which is the case for Venture). To illustrate the difficulty, observe that the conditional expectation E[d_T(A_end, B_end) | |B_end| = r] is not constant in r! For example, if r=|D|-1, that means Venture drew exactly 1 card, so it was a treasure, which means the density decreased.

So... yeah. Basically I think it is necessary to say more.

The way I see it, the distribution of the deck doesn't change based on making the decision to play Venture, so it doesn't matter what Venture does if it all it does is pick a spot in the deck to start the next hand.  There's still something nagging me like "but how it resolves changes the distribution" and while I think "but it doesn't matter", I definitely am having difficulty formalizing in simple terms why it doesn't matter, and that's what you were doing.  It still seems way too complicated for such an obvious (to me) intuition, but after having spent more time thinking about the problem definitely see why you might do something like this.  I still haven't followed through the logic precisely (because I thought it was pointless) and might try to derive something like this myself independently and see how complicated it actually has to be.

[DG]

The way I see it, the distribution of the deck doesn't change based on making the decision to play Venture, so it doesn't matter what Venture does if it all it does is pick a spot in the deck to start the next hand.

It changes the distribution. If we go back to the 3 estates, 2 coppers example then after you play the venture the expected number of coppers in the deck is now 1, the expected number of estates in the deck is now 2, the expected number of cards in the deck is 3, and the expected card value is unchanged at 2/5.

[RiemannZetaJones]

I don't think anyone has proved anything about what might happen at, or after, the reshuffle. Venture and Farming Village are both good at deck-cycling, which presumably has a net positive effect, since you want to cycle your deck more than you want to avoid cycling it. While blueblimp's result on non-improvement of the next hand(s) is fantastic, it only applies away from the reshuffle.

If you discount the impact of drawing the Venture you played, Venture doesn't do anything to the quality of the hands you get at or after the reshuffle.  If there are 3 cards left in your draw pile at the end of your turn, you're drawing 3 random cards from your deck, then two random cards from your discard pile.  Your discard pile will be a random sampling of your deck, plus whatever you bought since the last reshuffle, plus the Venture we're talking about.  We can't really say anything meaningful about buys, so really the only difference Venture makes to the quality of hands after reshuffle is the fact that it can be drawn, which you could say about any card.

Venture does let you get to the reshuffle faster.  Most of the time, that's a good thing.  Near the end, it's probably a bad thing.

Venture also changes the residue-class modulo 5 of the number of cards in your draw pile, and consequently changes the number of cards in your draw pile that will miss the reshuffle. I doubt this is ever going to be worth taking into account, but it should affect the quality of hands drawn after the reshuffle.

Maybe it's significant with a Venture/nothing opening. Usually with a 5/- opener, only 1 card will miss the second reshuffle. If Venture is played on T3 and draws 2 cards (not the most likely case), then 4 cards will miss the reshuffle, and they will all be coppers/estates, so that should be good. On the other hand, if Venture is drawn on T4, especially if you know the one card in the draw deck is an estate, it's less clear to me whether the effect is beneficial.

It seems really difficult to use this effect to advantage in general, since you generally don't know how many cards venture will draw.

My point was simply to rule out the theorem that the only benefit to your future hands from playing the venture is that it speeds the reshuffle. But I'd be willing to bet that in practice, this is the only benefit you should consider except in really small decks.

[ecq]

The way I see it, the distribution of the deck doesn't change based on making the decision to play Venture, so it doesn't matter what Venture does if it all it does is pick a spot in the deck to start the next hand.

It changes the distribution. If we go back to the 3 estates, 2 coppers example then after you play the venture the expected number of coppers in the deck is now 1, the expected number of estates in the deck is now 2, the expected number of cards in the deck is 3, and the expected card value is unchanged at 2/5.

It doesn't really matter, because one way or the other, you'll draw a 5 card hand next turn.  I'm fairly convinced that the only benefits to the next hand that Venture provides are the normal benefits of reshuffling.  Those can be good (or bad), but they have nothing to do with Venture filtering anything, only cycling.

I just threw together some quick and dirty code to verify that Venture doesn't modify the next hand when it doesn't trigger a reshuffle.  It constructs a draw pile with 2 Golds, 3 Silvers, 5 Coppers, and 5 dead cards (arbitrary numbers that are roughly realistic in a moderately greening deck).  It shuffles the deck, gets the value of the next 5 card hand, simulates a Venture draw, and gets the value of the next 5 card hand again.  It does this a million times and calculates the mean and variance for the Venture / non-Venture hands.

Code: [Select]

trials = 1000000
values_after = []
values_before = []
(1..trials).each do
  cards = [3] * 2 + [2] * 3 + [1] * 5 + [0] * 5
  cards.shuffle!
  values_before << cards.take(5).reduce(:+)
  drawn = []
  drawn << cards.shift while drawn.last.nil? || drawn.last.zero?
  values_after << cards.take(5).reduce(:+)
end

mean_before = values_before.reduce(:+).to_f / trials
mean_after = values_after.reduce(:+).to_f / trials

variance_before = values_before.reduce(0) {|s, v| s + (mean_before - v)**2}.to_f / trials
variance_after = values_after.reduce(0) {|s, v| s + (mean_before - v)**2}.to_f / trials

puts "Average hand value without Venture: #{mean_before}"
puts "Variance without Venture: #{variance_before}"
puts "Average hand value with Venture: #{mean_after}"
puts "Variance with Venture: #{variance_after}"

Output:

Code: [Select]

Average hand value without Venture: 5.664152
Variance without Venture: 3.748250120906077
Average hand value with Venture: 5.665684
Variance with Venture: 3.7451291591780285


[ecq]

My point was simply to rule out the theorem that the only benefit to your future hands from playing the venture is that it speeds the reshuffle. But I'd be willing to bet that in practice, this is the only benefit you should consider except in really small decks.

Fair enough.  I didn't intentionally claim otherwise.  The main idea is that Venture doesn't provide the benefits of a deck filtering card (e.g. Scout), which was a claim made in the original thread.  It, of course, does provide the cycling benefits of a deck drawing card.

[blueblimp]

The easiest way I know to verify this in your mind is to consider the last card in the deck. After you have played your venture, the conditional probability (given what you have seen) that the last card is an estate is the density of estates in whatever cards are left. Thus, before you play your venture, the probability that the last card will be an estate is the expected value of the density afterwards.

I forgot about this post (since I didn't fully understand it the first time). That's a really nice short proof that the expected density doesn't change. In algebra, it'd go something like

Code: [Select]

Original treasure density
  = Pr[the last card of the deck is a treasure]
  = sum_{strict subset A of our draw deck} Pr[last card is a treasure | drew A and stopped] Pr[drew A and stopped]
  = sum_{strict subset A of our draw deck} (the treasure density given we drew A and stopped) Pr[drew A and stopped]
  = E[treasure density after drawing].

The key really being that the conditional probability really does give the density, which is only true because we can't look at cards without drawing them.

[RiemannZetaJones]

My point was simply to rule out the theorem that the only benefit to your future hands from playing the venture is that it speeds the reshuffle. But I'd be willing to bet that in practice, this is the only benefit you should consider except in really small decks.

Fair enough.  I didn't intentionally claim otherwise.  The main idea is that Venture doesn't provide the benefits of a deck filtering card (e.g. Scout), which was a claim made in the original thread.  It, of course, does provide the cycling benefits of a deck drawing card.

No, you didn't claim otherwise at all. I just wondered, once I saw your involution argument, about just quite how strong such a line of thought might be.

[MasterAir]

I think that looking at a single hand is causing people problems here.  If there are 2 decks with only victory and treasure cards that are the same size, and 1 deck contains ventures.  The deck with the ventures would see more treasure per hand (on average) because each time a venture is played it removes some number (sometimes 0, but probably not always) of victory cards that would be dead cards from subsequent hands.

There is no way that this has no effect on the quality of cards drawn.  If I can discard a victory card without putting it in hand, that is almost always a boon.

If you used a spy and only discarded a card if it were a victory card it would improve the average quality of your hand in the same way.  Maybe not if you look at the single next hand, but on average.  I think the theorem is too specific in it's priors to be useful in Dominion games, and it is therefore misleading.

It probably isn't a whitewash, but I'd guess that a big money bot that buys venture over silver on $5 wins heavily over an optimised basic big money bot.  One of the (I'd say main, but the cycling might be more important) differences is that the venture bot sees fewer victory cards in hand than the big money bot.  On average, this means it's effective money density is higher.