Please login or register.
1 Hour 1 Day 1 Week 1 Month Forever
Login with username, password and session length

This section allows you to view all posts made by this member. Note that you can only see posts made in areas you currently have access to.

101

Dominion Articles / Re: A theorem about drawing and density

« on: March 12, 2012, 12:55:52 pm »

The code I posted earlier in the thread looked at variance as well.  It showed that, over a million trials, Venture does not impact the average coin value of your next hand or the variance of your next hand for the draw pile I tested.  Is there some other metric we should look at apart from mean and variance?

I just re-ran the simulation with a deck containing 6 coppers and 1000 dead cards.  It's the same story.  No change to average coin value, and no change to variance after playing Venture.

We have to be a little careful considering decks with fewer than 6 treasures, since there are cases where the next hand could cause a reshuffle.  It's not really worth considering what happens after the reshuffle, because at that point you're looking at the benefits of cycling not of filtering.

Venture will nearly always change the next hand that you draw in one way or another.  It does so unpredictably, though, (when played blindly) and on average it won't change the quality of the next hand.

102

Dominion Articles / Re: A theorem about drawing and density

« on: March 12, 2012, 12:09:03 pm »

Well, sort of. It's unchanged from before you play the ventures to after you play the ventures, on average, assuming you don't have other ventures or banks or special treasures in your draw deck, assuming you don't reshuffle. Okay, all of these are are actually significant assumptions (that make it unclear as to whether playing the ventures when you don't need the money is good or not), but it's still something.

The main point is that you shouldn't automatically play Venture in an attempt to improve future hands.  Assuming you have no use for additional treasure (so, no HoP or Loan) or the $1 from Venture, the effect of Venture is deck cycling.  The decision of whether or not to play Venture in that case is similar to the decision of whether or not to play Smithy when it can't draw anything to help your current hand.

103

Dominion General Discussion / Re: Homage to the Best Card

« on: March 10, 2012, 10:03:16 am »

104

Dominion Articles / Re: A theorem about drawing and density

« on: March 10, 2012, 12:58:57 am »

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.

105

Dominion Articles / Re: A theorem about drawing and density

« on: March 10, 2012, 12:45:47 am »

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


106

Dominion General Discussion / Re: Sea Hag or Young Witch with weak Bane?

« on: March 08, 2012, 10:50:27 pm »

My money is on Young Witch.  Sea Hag's main advantage is that it top-decks the Curse, but Young Witch weakens that.  Generally, Young Witch's filtering/cycling is pretty nice in cursing games.

107

Dominion Articles / Re: A theorem about drawing and density

« on: March 08, 2012, 10:41:49 pm »

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

108

Dominion General Discussion / Re: The Bureaucrat pin

« on: March 08, 2012, 06:05:58 pm »

KC, KC, Scheme, Scheme, Bureaucrat, KC, Scheme, Bureaucrat, Bureaucrat

109

Puzzles and Challenges / Re: Emptying a pile as fast as possible.

« on: March 08, 2012, 10:11:41 am »

I guess I hadn't thought about 5 players just emptying piles on their two buys. But you give me an idea to do it with 4P.

Turn 1
P1 opens Noble Brigand.
P2,3,4 open Ambassador.

Turn 2
P1 buys estate.
P2,3,4 Ambassador estate returning none, buying an estate. That drains 4 estates per player. More than enough for the 12 in the supply.

Fun fact: Council Room lists over 30 Noble Brigand openings that are worse than Noble Brigand / Estate

110

Dominion Articles / Re: A theorem about drawing and density

« on: March 07, 2012, 02:12:20 pm »

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.

111

Dominion Articles / Re: A theorem about drawing and density

« on: March 07, 2012, 10:36:34 am »

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

112

Dominion Articles / Re: A theorem about drawing and density

« on: March 07, 2012, 10:13:44 am »

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.

113

Dominion Articles / Re: A theorem about drawing and density

« on: March 07, 2012, 09:39:10 am »

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?

When you calculate an average (mean) card value for a draw deck then you're considering something like T/C, T= treasure and C= number of cards. When you play a farming village T remains unchanged but you might see C decrease when green cards are discarded. Therefore T/C increases.

T decreases by 1 (assuming Farming Village draws a treasure, not an action).  It's (T - 1) / (C - discarded_cards - 1).  If your draw pile is 5 Estates and a Copper, and the Copper was on top of the deck, you go from 1 / 6 to 0 / 5, so the second term can decrease.

For any given play, the money density will probably change slightly.  On average, though, the money density will remain equal to T / C.

114

Dominion Articles / Re: When to Venture

« on: March 06, 2012, 09:11:33 pm »

ECQ: But you won't have 15 otherwise identical cards most of the time in an actual game, that is my point. I get the basic math, adding a duchy decreases money density of whichever stack it is in. That being said, the stacks are rarely equal. If one of them tends to contain more money, then you want to play that deck more. When you've bought a province, you know you have 8 coin, and very likely 8 dense coin, in the discard. Unless you have a low variance BM strategy going on (e.g. BM/envoy), you will find that cycling can be a net benefit with venture.

If I understand your question rightly, I have something around a 30-40 card deck and I have a province buying hand of gold/gold/chancellor. In that case I am going to want to know the composition of the deck, but generally in BM decks 8 coin should be at the upper end of your variance distribution. In that case I have a better than average hand going into the discard and average hands already in the discard and average hands in the draw deck - chancellor all the way. Now for something where I have low variance, sure don't chancellor. The easiest example is a hand of

I get that venture doesn't help your next hand inately, however it gets you closer to a reshuffle. Think about it this way, if you've just seen all four of your golds go into a province buy, would you rather wait one turn or two turns to see it again?

It's easy to construct situations where cycling late in the game is a good thing.  Yes, if you just watched all your Gold go by, you probably want to get through your deck ASAP.  But on average, when greening, it's a bad thing. 

Specifically, I'm saying that if you have a Chancellor in your deck, no Stashes, and you're at the stage where you're buying Victory cards often, it will be better treat it as a terminal Silver and skip the deck discard more often than not.  If you're greening, then what's going into your discard is, on average, worse than what's in your draw pile.  So, again on average, you do not want to shuffle it into your draw pile.

Maybe this comes down to differing definitions of "greening?"  Two features of greening: (a) you're choosing dead victory cards instead of treasure or more useful actions on nearly every turn (b) $8 hands are becoming more sporadic because of said dead victory cards.  It seems obvious to me that you'd rarely want to put the cards from (a) in your draw pile, because they'll just make (b) worse.

Venture is a great card.  It's also great at resisting greening, but only because of its money-seeking ability, not because of its deck-cycling ability or non-existent filtering ability.  The deck cycling is generally a late-game drawback, but it's easily outweighed by the free $1 with replacement treasure in your current hand.

115

Dominion Articles / Re: When to Venture

« on: March 06, 2012, 07:09:49 pm »

Wouldn't it depends on how you cycle? If you're blowing past everything bad and getting to play your good cards, then cycling is great. But that's cycling plus sifting. If you're just cycling past cards fast, regardless of whether they're good for you or not, then cycling while greening is bad.

I was talking about the average case.  On average, it's bad to cycle while greening.  If you're buying nothing but victory cards and you draw a Chancellor, you should choose not to reshuffle most of the time (Stash combos aside), because most of the time the expected value of a hand from your entire deck will be worse than the expected value of a hand from just your draw pile.  Sometimes that won't hold, but holds more often than not.

Venture cycling does give you some selectivity though - it only cycles past green cards, so it does help you play the treasures more often. So that's definitely in Venture's favor.

Venture is only worth what it draws.  If you play it blindly and it doesn't cause you to reshuffle, it does nothing, on average, to your next hand.  It might skip 3 Duchies, and yes, that's good.  It might also hit a Gold immediately, which is good for your current hand but not for your next.  All the scenarios average out.  Think about what Venture draws.  It draws an off-the-cuff poor approximation of your money density: a single sample of the number of non-treasure cards you have to encounter before finding a treasure.

116

Dominion Articles / Re: When to Venture

« on: March 06, 2012, 12:11:46 pm »

If you are greening, cycling tends to be a good thing, just not as good as when you are buying value adding cards. Think about it this way, you have enough cash in hand to buy a province. Is your deck or your discard more likely to have higher coin density? In most cases, the discard is the correct answer - we KNOW that there is at least 8 coin in it and it is more likely to be the dense coin. This is why if you are playing BM/Chancellor and you hit gold/gold/chancellor you normally want to reshuffle immediately - you are unlikely to have too many more golds and the money density is going to drop precipitiously in the rest of the deck (for instance a deck of 7 copper, 3 estates, 4 silvers, chancellor, 2 golds, and a province before this buy will have a money density of 1.15 while the discard has a money density of 1.33).

This is not the case.  Cycling is, on average, very bad when greening.  Sure, it can be good, but it's not typically so.  Let's say I offer you one of two decks to draw a hand from.  I tell you that they each have 15 cards that are identical, but the second deck has 3 extra Duchies mixed in.  Which one would you rather draw from?

In your example, you've played a hand or two, though, so I take 10 cards off the top of each of my two piles and set them aside.  Did that change your decision?

117

Dominion Articles / Re: A theorem about drawing and density

« on: March 06, 2012, 10:13:49 am »

I think having fewer than 5 cards in the draw pile is confusing things.  Let's assume you have 10 cards in the draw pile: 3 Estates, 7 Coppers.

Before playing Venture, the average value of a card is $1 * 7 / 10 = $0.7

Venture will draw 0-3 Estates followed by 1 Copper.  The number of shuffles placing a given number of Estates on top of the deck is: (number of ways to pick the number of estates from a set of 3) * (number of ways to pick a single Copper from a set of 7) * (the number of ways to shuffle the remaining cards)

For the 4 possible Venture draws, we need to calculate the number of shuffles that produce the given draw:

• C -  1 * 7 * 9!
• EC - 3 * 7 * 8!
• EEC - 6 * 7 * 7!
• EEEC - 6 * 7 * 6!

Sanity check:  There are 10! different ways to shuffle a deck of 10 cards.
1 * 7 * 9! + 3 * 7 * 8! + 6 * 7 * 7! + 6 * 7 * 6! = 3628800 = 10!

So, we've accounted for all the different shuffles producing any given venture draw.

Drawing C leaves an average coin value of $(6/9) in the draw pile.  EC leaves $(6/8).  EEC leaves $(6/7).  EEEC leaves $(6/6).

Multiply those values by the number of shuffles they appear in to get the sum of all possible coin values:

(6/9) * 1 * 7 * 9! + (6/8) * 3 * 7 * 8! + (6/7) * 6 * 7 * 7! + (6/6) * 6 * 7 * 6! = 2540160

Divide by the total number of shuffles to get the average coin value:
2540160 / 10! = $0.7

Whoa, exactly where we started.  Venture didn't change the average coin value in our draw pile at all.

There are at least 6 cards remaining in the draw pile and you'll only draw 5 next turn.  We can go through and calculate it, but I'm confident that the expected value of the next hand hasn't changed by any metric.

118

Dominion Articles / Re: When to Venture

« on: March 05, 2012, 07:19:08 pm »

Played blindly, without triggering a shuffle, Venture doesn't help or hurt your next hand at all, on average, regardless of the amount it green in  your deck.

Imagine if your draw pile is Estate, Estate, Copper, Estate, Estate, Copper, etc.  Play Venture.  Yes, it skips 2 Estates, but what did it do to change your next hand?  It didn't do anything to modify your draw pile composition beyond the Copper.

119

Dominion Articles / Re: When to Venture

« on: March 05, 2012, 12:57:49 pm »

Long story short, in a large enough deck ventures reduce the effective space that VP cards take up. This gives you a higher money density and allows you to green out sooner as you will simply play fewer greens each shuffle.

I don't think this is right.  You'll draw fewer greens each shuffle, but you'll also shuffle more.  If you're greening, shuffling more means you're seeing more greens per hand on average, not fewer.

Venture's power has little to do with skipping cards.  It's powerful because it adds $1 to the current hand while doing as little as possible to sacrifice the quality of the hand.  It can't be drawn dead.  It doesn't make you wish you bought a Silver, because it draws you a treasure card to replace itself.

120

Simulation / Re: Dominiate: a Dominion simulator that runs on the Web

« on: March 05, 2012, 12:03:08 pm »

I've been considering how to test the simulators and had kind of a neat idea this weekend.  We could grab a couple of days worth of Isotropic logs, reject games with unimplemented cards and/or Masquerade, and exercise the code using the remaining game logs.  We'd parse the logs, feed hands and decisions from the game through the simulator, and ensure that everything made sense.

Obviously this is imperfect because Isotropic doesn't tell you precisely what was drawn or discarded, but it could serve as baseline sanity checks for the game and card implementations, making sure we're counting money, buys, actions, and VP correctly.

I think it'd be hard to implement, but maybe not as hard as writing the equivalent set of tests manually, and probably less error-prone.  Obviously we'd still need tests for AI decisions, strange edge cases, and things that aren't available from scanning logs.

I'm betting Geronimoo's simulator could benefit from this idea as well, but I know Dominiate is actively looking for tests.

121

Dominion Articles / Re: When to Venture

« on: March 05, 2012, 10:13:44 am »

Consider a draw deck of three estates and two coppers. When you play a venture you will discard the estates before the first copper, and leave behind the estates after the first copper and after the second copper. The distribution of the draw deck will be changed to have (on average) twice as many estates remaining as coppers. Extending this method to look at larger decks shows that a venture/loan/adventurer/golem/scrying pool will nearly always make a small change to the distribution of the remaining draw deck.

The remaining draw deck, yes.  It's clear that Venture must make a small change, since you can't make a 3/5 ratio from fewer than 5 cards.  It's not obvious to me that Venture, on average, makes any appreciable changes to the next hand, though (discounting the effects of cycling, because Venture does make you reshuffle faster which can be good or bad depending).

Let's say Venture discarded an Estate and drew/played a Copper in your example, leaving behind 2 Estates and a Copper.  The Estate and Copper Venture drew go into the discard pile at the end of your turn.  When you draw your next hand, you draw 2 Estates, a Copper, then you reshuffle and draw 2 average cards from a pile that's missing 2 Estates and a Copper.  So, you're not necessarily drawing Estates 2:1 with Coppers.

If you assume that your discard, hand, and draw pile have roughly the same composition (again, I know that in practice they don't), would Venture mess with the distribution of the next hand?  Intuitively, it doesn't seem like it would or at least not in any perceptible way, but I haven't sat down to do the math.

122

Dominion Articles / Re: When to Venture

« on: March 04, 2012, 10:17:45 pm »

I think blueblimp is right about the cycling thing.  Venture doesn't do much of anything to your next hand on average.  Assuming Venture doesn't trigger a reshuffle, your next hand will contain 5 random cards from your draw pile one way or the other.  Yes, Venture might skip some green.  It also might not skip them, or it might skip your Smithy.  It all evens out and is pretty much a wash.

The one thing it does do is move you closer to the next shuffle.  If you're actively buying green, that's a bad thing.

Venture is only as good as the card it draws, plus $1.  Where Venture really shines and green-proofs your deck is when it's likely to hit another Venture or a high-value treasure (Gold, Bank, Platinum).  If Venture hits a Gold, it's worth $4, which is really nice.  If it hits a few Ventures before the Gold, $$$.  If you can get it to do that consistently, you don't need to draw much treasure to consistently buy Provinces.  Obviously the card loves Mint.

123

Dominion General Discussion / Re: Is there a card you always buy?

« on: March 03, 2012, 09:52:03 am »

I was going to say Hunting Party as well.  There are very very few decks where it doesn't add something.  Getting one early is great for accelerating the game.

124

Puzzles and Challenges / Re: 7 copper and 3 estate vs. 50 _____?

« on: March 02, 2012, 11:47:28 am »

Loan seems almost balanced, winning and losing in a wide range of circumstances.

1) Loan
2) Kingdoms with strong non-terminals, alternate VP, or treasures costing up to $5.  IGG, Hunting Party (though be careful about reshuffles), Duke, maybe Venture, Laboratory, or Cache
3) Kingdoms where there's a strong engine, good big money support (Smithy), or compelling $6+ cards

125

Variants and Fan Cards / Re: A slightly different card idea... Bluff.

« on: March 02, 2012, 10:20:52 am »

I love the idea for the card.  I'd make two tweaks, though:

1) Force the named card to be a card present in the current game, otherwise you end up in situations where someone plays double-Tactician and you're stuck either letting them play a platinum during their action phase or letting them play arbitrary cards from outside the game.

2) Have the bluff-caller draw a card *after* you play the revealed card, otherwise you could, say, bluff King's Court using a Militia and negate most of the drawback of bluffing.