as for what could be most useful for the topic of this thread, i think Dromar asked a very poignant question: what further analysis could be convincing to someone who doesn't want to take as proof a) the experience of pros and b) these good arguments? what further analysis could make it feel more proven? maybe very interesting analysis could be done that hasn't been brought up yet, i dont' know?
(at any rate, i don't think a single person in this thread actually thinks anymore than playing 41 cards is correct, at least not for 99% of actual Limited games!)
@Phyrre:
regardless of if i think you're correct or incorrect in your arguments about this topic, i really appreciate what you wrote here. for me, it humanizes the person behind your words, and it makes it more relatable/understandable why you'd be so upset by arguments for 41 card decks that seem like only mere "yes, but.." objections, when there seems to be solid evidence about what is correct (ie evidence from a) good (if not "proof") arguments and b) years of corroboration from pros (ie because they would be motivated to find the advantage of 41 card decks, if it's there) ).
your analogy to the anti-vaccination camp put the feeling of being upset about misinformation based not on evidence or truth-finding methods, but hypotheticals or anecdotal evidence.
@Velxa:
So do it, if you really teach math, that a) shouldn't be a problem and b) you should know is only way to show something is proven.
i, too, am curious about Puddle's math, but don't forget that Puddle also said that we would likely not understand their math without them over-simplifying it.
that, and it takes effort (and patience!) to lay out the math, and who wants to do that in an antagonistic environment where people will likely try to find fault with the effort? i can't blame Puddle if they don't want to share their math here. (pause) though i must say i am nevertheless still curious about it
I NEVER said, I run 41-card decks or that it's correct to do so. I just said, that there are many people who say it is proven that it is wrong and I'm saying that there ISN'T actual proof anywhere here - just good evidence that it might be true.
i think that Puddle might be the only one who has used words like "proven", and that mua1000, myself, magicmerl [?], and Axelrod would all agree completely with what you wrote. (and about Puddle, i'm not even 100% sure i am reading them correctly -- i /think/ they're saying that it's provable, but i might be misunderstanding them).
About thought experiments: can you imagine nonland card X, such that decks 17L/23X and 18L/22X has the same chance to win in Karstens' experiment?
i wonder if i'm understanding your thought experiment here. are you saying "throw in a Wrath of God to make a 41 card deck" or "throw in an Adaptive Automaton to make a 41 card deck"? because if that is your suggested experiment, isn't that experiment flawed because obviously the Wrath of God / Adaptive Automaton is not the worst card (and, the point of 40 card decks is that you cut out the worst card)?
@Axelrod:
i think you articulated perfectly what is still lingering in my curiousity
i mean, Phyrre does have a point: the pros, if anyone, would be most motivated to investigate.
but it does strongly bring up the question Dromar brought up: what kind of demonstration / though experiment / proof WOULD satisfy my urge to still defend 41 card decks as POSSIBLY correct? what would it take?
(i already believe 41 card decks aren't correct from Phyrre's one example, but that only uses my intuition. something like Frank's experiment is more solid evidence for me, except i don't believe it's comprehensive enough. "the pros have thought of this" isn't satisfying emotionally to me, even though i agree it's a good point. so what kind of demonstration would satisfy me more?)
@Dromar:
What additional tests would need to be done to convince you? A deck that has X lands, Y 2/2s for 2 and Z 3/3s for 3 vs a deck that has X+1 lands for mana ratio reasons? Or more complex than that?
good question!!
hm. i know how to write simple computer programs, though i dont' know any statistics. maybe i should tinker around a little bit on this. your idea of doing Frank's experiment, modified using "vanilla creatures with a curve" sounds like something possible to try.
As far as the other factors you mention, I think they have been considered but overall the side effect of drawing your best cards less frequently outweighs any advantage of extra synergy or having more specific answers available in your deck. Also these additional synergies are hurt by having a larger deck now anyways as you are statistically less likely to draw the cards together than if you just cut a non-synergistic card for whatever 24th card you wanted for that purpose. There's certainly something of an argument to be made for sacrificing power for the sake of synergy, but sacrificing power and consistency for the sake of synergy is just not worth it to me.
i agree with all of this that you wrote but i agree with it only through using my intuition (and possibly knowing that others have had many, many trials of in-game experimentation to corroborate it).
i wonder if there is any way to demonstrate what you wrote in a more "concrete" or "stronger" way than using my intuition. (i have trouble coming up with a way. maybe it would help (??) if someone can come up with a card deck list that uses exactly the factors that i am interested in (ie the two worst cards give a better curve, better synergy, and better flier-protection), and show that /even then/ it's better to draw your good cards a turn earlier?)
2. The last card you add to your deck is the worst. At the very least, if you're anguishing over exact valuation, it's worse than the average.
3. For any 41 cards deck in its exact shuffled configuration, there is a corresponding 40 cards deck with that card cut off.
4. When you'd draw that card, you drawing a card that is always (if you really chose the worst card) worse than any non-land cards in your deck. At the very least, it's worst than the great majority of cards and at best about the same power as the other in its bad lot.
5. Your deck contains fewer lands than non-land, so in the majority of cases, the card you would draw instead of that 41st card is better. The better you are at card valuation, the more the alternative card is better.
6. In the case where the 41st card is in the early top of your deck, even drawing a land is better than drawing that card.
That's it. Points 5 and 6 are the important bits: they show that just taking relative value of cards, the great majority of times, you're better off not having that 41st card.
The argument of 41-carders is basically to refute #5 by saying it's better drawing a very bad spell than a land. They forget point #6 and that even late in the game, having extra land is not bad: you can play many spells per turn, have mana sinks, keep trick mana up, etc. But that ignores that better spells form more than 50% of your deck!
Sorry if my last post was stressing you out, silph.
This has been mentioned before, but if a card gives you a better curve, better synergy, and better protection, then it probably isn't your worst card. Maybe you could post an actual decklist example and we can figure out a better cut?
This has been mentioned before, but if a card gives you a better curve, better synergy, and better protection, then it probably isn't your worst card. Maybe you could post an actual decklist example and we can figure out a better cut?
This is a good point and demonstrates why theory can sometimes be a wasted effort. If you can't present an example of the situation beyond a hypothetical, maybe that situation does not exist, or exists so rarely as to be not worth the effort investigating it.
We have discussions this like with modal cards sometimes. Take Gudul Lurker for example. Imagine a world where Gudul Lurker didn't have a mana cost. The only way to get it into play would be as a Morph. While there are situations where playing Gudul Lurker face up is correct, they are rare. It's extremely likely that by simply having a mana cost, Gudul Lurker has been misplayed face-up more than it's been correctly played face-up. The option makes the card strictly better but in practice has made it worse due to user error.
The same may hold true for 41 card decks. Even if we can invest much time and effort into crafting the perfect theory of when to play 41 cards -- that may be a negative compared to pretending you're not even allowed to play 41 cards and focusing your thought in other areas. Even at the individual level, I don't necessarily think the theory could be designed in such a way that it produced more benefit than harm.
Maybe your interest in the topic is purely academic so you don't really care if the 41 card deck can be applied well -- you just want to know it exists. I have trouble empathizing with that mindset because to me, if I'm going to ponder something purely academic, the theoretical application of 41 card decks that cannot be productively put into practice is not something I'd ever want to spend my time on. What's the point? There have to be a thousand more interesting ways to spend your time thinking about Magic (or any other field of academia) that may have better applications or rewards.
2. The last card you add to your deck is the worst. At the very least, if you're anguishing over exact valuation, it's worse than the average.
3. For any 41 cards deck in its exact shuffled configuration, there is a corresponding 40 cards deck with that card cut off.
4. When you'd draw that card, you drawing a card that is always (if you really chose the worst card) worse than any non-land cards in your deck. At the very least, it's worst than the great majority of cards and at best about the same power as the other in its bad lot.
5. Your deck contains fewer lands than non-land, so in the majority of cases, the card you would draw instead of that 41st card is better. The better you are at card valuation, the more the alternative card is better.
6. In the case where the 41st card is in the early top of your deck, even drawing a land is better than drawing that card.
That's it. Points 5 and 6 are the important bits: they show that just taking relative value of cards, the great majority of times, you're better off not having that 41st card.
The argument of 41-carders is basically to refute #5 by saying it's better drawing a very bad spell than a land. They forget point #6 and that even late in the game, having extra land is not bad: you can play many spells per turn, have mana sinks, keep trick mana up, etc. But that ignores that better spells form more than 50% of your deck!
All right, I take this, but the conceptual problem I have is that some, if not most, of the cards in your deck are situational. Your point #1 is actually not nearly as clear cut as you would like it to be. A Mana Leak, to use a simple example, is good when you have it early, and terrible when you draw it late. Re: point #4, the "best" card in your deck - depending on its mana cost, isn't always going to be your "best" draw. If your gameplan relies on an Ugin to win, but you draw him when you kept a two lander on the play that's not a "better than average" draw.
That seems to be the problem with simply asserting that "Card X is better than card Y. Period. And therefore the math says that drawing card X is better than drawing card Y. Period." Except we know that's not always true, for almost any card you can think of.
So are we back to, "well, it's usually better to draw card X, so from a practical standpoint it doesn't change the argument..."
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Ambush Krotiq makes me laugh so much. I keep rereading the card and it keeps not having Flash. In what sense is this an ambush again? I just have visions of this huge Krotiq poorly concealed in some bushes, feeling slightly sad that his carefully planned ambushes never seem to work.
For most limited decks, almost all of the cards are not situational. For a typical deck, you'll have ~15 creatures, ~4 removal, ~5 tricks (pump, bounce, draw, equipment, etc.). I'd say every deck easily has 20 cards that are not situational at all, unless you're ready to stretch the meaning of the word.
We've already read enough mind-bending justifications without resorting to now argue that card evaluation is too hard to build a deck?
You cannot fix having N situational cards (and I agree with pierrebai, N should be small) by having N+1 situational cards, because 50% of the the time (with N=1) you will draw them in the wrong order.
I think we can all admit that on an intuitive level, it really seems like 41 cards should be OK. It's such a small thing. It's a decision made with good intentions. But it might still be very wrong in the same sense that a person thinking the world is flat is very wrong regardless of their intentions. And generally it takes decades or centuries to make the transition from widely accepted theory to provable truth because we don't always have the technology to do so.
Ironically, the vast majority of people - even authority figures [read: pros] - claimed for centuries that the world was flat. Anyone trying to say otherwise were ridiculed and even killed.
As someone else mentioned, 15-20 years ago it was an unwritten rule that you played 1/3 lands in your deck. If you at that time would argue that control decks should play 26-27 lands, people would probably laugh in your face.
Look, I'm not saying 41-card decks are better than 40-card decks. What I'm saying is that we should question these unwritten rules.
Look, I'm not saying 41-card decks are better than 40-card decks. What I'm saying is that we should question these unwritten rules.
Questioning is great. But we looked at both sides of it (mana consistency vs. likelihood of drawing your best cards, with most other considerations identified as too minor to compare to these two effects) and it's pretty clear that you're more likely to miss drawing your best cards vs. the subtle difference in mana ratio having a positive impact.
I've yet to see anything approaching a cohesive argument against this. Everything since then has been "But that's not proof" which is a waste of breath. There will never be proof. The system is too complex. By the time we have the technology to prove such a thing, Magic will be considered a trivial relic of the past. Everyone concerned with proof in this demonstrates a fundamental lack of understanding of the complexity involved here.
I could make a statement like "The middle class, statistically speaking, always live more comfortably than homeless people." Is it true? Yes. Is it proven? Yes. Could you conceivably come up with a situation that temporarily seems to be an exception? Maybe, but who cares? That doesn't change what the fact is.
If the current limited rules were rewritten to allow 35 card decks, wouldn't you play 35 cards? If your answer is an automatic yes, why do you feel differently about 40 vs. 41 cards?
What additional tests would need to be done to convince you? A deck that has X lands, Y 2/2s for 2 and Z 3/3s for 3 vs a deck that has X+1 lands for mana ratio reasons? Or more complex than that?
good question!!
hm. i know how to write simple computer programs, though i dont' know any statistics. maybe i should tinker around a little bit on this. your idea of doing Frank's experiment, modified using "vanilla creatures with a curve" sounds like something possible to try.
As far as the other factors you mention, I think they have been considered but overall the side effect of drawing your best cards less frequently outweighs any advantage of extra synergy or having more specific answers available in your deck. Also these additional synergies are hurt by having a larger deck now anyways as you are statistically less likely to draw the cards together than if you just cut a non-synergistic card for whatever 24th card you wanted for that purpose. There's certainly something of an argument to be made for sacrificing power for the sake of synergy, but sacrificing power and consistency for the sake of synergy is just not worth it to me.
i agree with all of this that you wrote but i agree with it only through using my intuition (and possibly knowing that others have had many, many trials of in-game experimentation to corroborate it).
i wonder if there is any way to demonstrate what you wrote in a more "concrete" or "stronger" way than using my intuition. (i have trouble coming up with a way. maybe it would help (??) if someone can come up with a card deck list that uses exactly the factors that i am interested in (ie the two worst cards give a better curve, better synergy, and better flier-protection), and show that /even then/ it's better to draw your good cards a turn earlier?)
Maybe if someone asks Frank really nicely he'll revisit that specific part of his article with more complex/specific examples haha.
As far as the last part, a lot of that is really hard to prove as it is all seems very specific. Like you'd not only need to take into account your whole decklist for things like curve considerations, but also how likely you are to face the specific cards your extra synergy/answers care about (like if you're arguing to run Plummet as the 41st card for example...well we need to know how prevalent flyers are in the example format). The only easily provable things are that:
a) you'll draw your best cards less often running a 41 card deck as opposed to a 40 card deck (clearly no rational person would argue this point), and
b) playing 18/41 can give you a better mana ratio than 18/40 if you know you want to maximize your potential to hit x lands drops by turn y while minimizing your potential to flood out (too lazy to look up the specific numbers here).
That's again why I think an example like Frank's is a good argument for 40 cards being usually (if not always) better than 41, not factoring in things like anti-mill or Lost in the Woods.dec. You take out a), the simplest, provable argument against running 41 cards and are left only with b), the simplest provable argument in favor of 41 cards...and the result is 40 cards still out performs it. It's very simplified of course but that combined with as you mention, intuition and my own experiences from playing and watching better players than myself play limited makes me content to side with 40 cards being usually better. If I'm wrong a small percent of the time that's acceptable to me as I think you'd need to find very complex and/or convoluted examples where 41 cards would in fact actually give you a better deck. And when you're in the midst of a limited tournament you probably don't have the time to go in-depth to determine those necessary calculations anyways.
If you said "you can't get wrong with playing 40 cards", "about 99,9% situations 40 cards is correct", "it usually isn't good idea to play 41 cards" or "I never play 41 cards because I think it's not worthy", I wouldn't have any problem with that. But you ignore possible "temporary exceptions" in your posts and say it is fact they don't exist, which is misleading and is even worse if you do that as mathematician.
Let me try to explain something about statistics using a simpler example:
Let's say you flip a coin. The odds of getting heads is always 50%. It is never going to be anything but 50%. The words always and never in these sentences don't imply anything about the final outcome after it happens, but rather about our ability to predict the outcome before it happens. Another good example is that there is always a negative expected value for buying lottery tickets. Maybe you win the lottery and it seems like the greatest idea you ever had, after the fact, but that doesn't mean it was a statistically valid way to spend your money beforehand.
In the same way, it might be true that you lose a game as a direct consequence of playing 40 cards instead of 41, once in a while, but those are statistically irrelevant because a person's inability to predict the quality of their draws will always favor them playing 40 cards. If you could see the future, then yeah, there would be times when playing 41 cards would be correct, because then you'd be relying on predictable cause and effect, rather than statistics. But if you can't predict the future, and you're stuck with statistics as your tool for making best guesses, then 40 cards is always best. That is what I mean when I talk about something "seeming" to be a temporary exception. I don't mean it actually *is* a temporary exception, just that people are really bad at judging the quality of a statistical judgment if they already know what the outcome was going to be.
*Statistically speaking* none of those statements you offered up are any good except the first, which is just expressed as opinion anyway. "In 100% of situations, 40 cards is statistically correct", "It is never a good idea to play 41 cards unless you're cheating and the cards are not in random order", and "I never play 41 cards because it is not worthy" are all fine. I'm not saying this to try to get under your skin at all, please understand. I just want you to be clear on what statistics does for arguments like this: it makes it possible to make value judgments about inherently uncertain circumstances.
magicmerl a few posts up gave a good summary of their understanding:
To the people who basically don't want to be forced to decide between the 23rd and 24th card in their decklist, imagine that your game is going to take 8 turns on average, withou card drawing. That means you will draw exactly 15 cards from your deck. Let's assume that you have one bomb in your deck. By running a 41 card deck you are reducing your odds of drawing your bomb in any given game from 15/40 to 15/41. So in roughly 1% of your games you would have drawn your bomb, but you don't because of your suboptimal deckbuilding decision. In any given FNM you might play 9 games. So you'll never see that tiny mistake writ large in your life. But it is still the (wrong) decision to accept a worse result in your deck.
By refusing to choose between the worst cards in your deck you are giving yourself options on both of them, but those options come at the expense of all of the other (better) cards in your deck. The latter outweighs the former.
but i still see this argument as appealing to intuition, much like your own argument that convinced me. merl says that "The latter outweighs the former", which is an assertion i agree with, but my understanding is that it still appeals to my intuition, not to anything close to being unambiguous or provable; intuition says that the benefits from keeping both worst cards are comparatively smaller than drawing yoru best card -- but this isn't proven. maybe intuition is incorrect here. (not that i believe this is likely the case, but it's not been proven).
I don't know what sort of thing you are looking for that would constitute proof.
Imagine that the 41st card you are thinking of is a Naturalize. This is a card that is almost universally regarded as being a 'sideboard card', in that when it has a good target it is worth playing, but otherwise not. Let's say that you decide to play it, and lo and behold, you run into someone with a bomb that you can target with it. Yay, right? Not really. Even though there is a situation where it helps you, on balance it hurts you more by being dead in your other matches.
Here's another way to look at this penomenom: Imagine a bag with 7 red balls and 3 green balls. You need to predict what the colour of the ball to be drawn out is. You can choose red, and a whopping 30% of the time, you will be right. But it's still WRONG to choose red, when green is a better percentage play.
Choosing to play 41 cards might not always hurt you. It might even be beneficial like in the Naturalize example above. But it's still not the correct percentage play because you lower your chances of drawing your best cards in any given game.
the only thing i'm "holding onto" and wanting to be shown wrong about, is the assertion that this (probably correct) conclusion is so "factually, provably correct". (and if it has been proved, i missed it, and any gentle redirection of my attention would be appreciated!)
as for what could be most useful for the topic of this thread, i think Dromar asked a very poignant question: what further analysis could be convincing to someone who doesn't want to take as proof a) the experience of pros and b) these good arguments? what further analysis could make it feel more proven? maybe very interesting analysis could be done that hasn't been brought up yet, i dont' know?
(at any rate, i don't think a single person in this thread actually thinks anymore than playing 41 cards is correct, at least not for 99% of actual Limited games!)
Let's step back a minute. What is the practical application for the knowledge in this thread? Surely it's to help make better 40 card decks, which has to be done within some sort of timeframe, right? If the only way that you can work out whether 41 lands is correct is via some in depth analysis, then it's almost certainly incorrect to do so. We're arguing in a hypothetical vacuum here, but in reality we're going to be making this sort of decision while deckbuilding in 5-10 minutes TOPS. There's no way this sort of knowledge is going to be useful in a practical sense other than to convince us to always stick to 40 cards.
In the same way, it might be true that you lose a game as a direct consequence of playing 40 cards instead of 41, once in a while, but those are statistically irrelevant because a person's inability to predict the quality of their draws will always favor them playing 40 cards.
See, this is the kind of blanket statement that I (and I believe veXlaMtg and others) have an issue with; you postulate that 40-card decks are always the correct decision. But what if - with a given pool of cards, in a given meta - the 41-card deck would on average be better than the 40-card deck? It would, statistically speaking, be correct to play the 41-card deck.
Imagine that the 41st card you are thinking of is a Naturalize. This is a card that is almost universally regarded as being a 'sideboard card', in that when it has a good target it is worth playing, but otherwise not. Let's say that you decide to play it, and lo and behold, you run into someone with a bomb that you can target with it. Yay, right? Not really. Even though there is a situation where it helps you, on balance it hurts you more by being dead in your other matches.
[...]
Choosing to play 41 cards might not always hurt you. It might even be beneficial like in the Naturalize example above. But it's still not the correct percentage play because you lower your chances of drawing your best cards in any given game.
This is essentially the same straw man argument that Puddle presented above; you postulate that the 41-card deck is inferior to the 40-card deck (on average), then present a scenario where the 41st card actually wins you the game, then goes on to show that you were actually wrong playing the 41-card deck because on average the 40-card deck is better.
Let's step back a minute. What is the practical application for the knowledge in this thread? Surely it's to help make better 40 card decks, which has to be done within some sort of timeframe, right? If the only way that you can work out whether 41 lands is correct is via some in depth analysis, then it's almost certainly incorrect to do so. We're arguing in a hypothetical vacuum here, but in reality we're going to be making this sort of decision while deckbuilding in 5-10 minutes TOPS. There's no way this sort of knowledge is going to be useful in a practical sense other than to convince us to always stick to 40 cards.
I agree. The 41-card deck is a thought exercise with minimal impact in practice.
I'm pretty sure it's impossible to (mathematically) prove that the 40-card deck always is a superior choice to the 40+ decks, because I believe that would require running infinite game simulations with all possible card pool permutations. On the other hand, the statement is proven false once (if) a single counter example is found.
I am going to chip in here in a reaction to mu1000.
See, this is the kind of blanket statement that I (and I believe veXlaMtg and others) have an issue with; you postulate that 40-card decks are always the correct decision. But what if - with a given pool of cards, in a given meta - the 41-card deck would on average be better than the 40-card deck? It would, statistically speaking, be correct to play the 41-card deck
This thread has gone in two different directions and it all boils down to this:
1)playing 41 cards will make you draw your bombs less
2)playing 41 cards gives you a better mana ratio, making sure every land counts and upping the average value of your deck by upping the value of the 18th land (since it does not cost you a card, but a 1/40-1/41 or 0,00061 (0,06%) of the value of every other card). 0,00061*40 still is not a whole card though so I can still see this deck win out based on the average value.
The fallacy here is that the pro-41 and pro-40 is in effect the same argument. You are looking for what gives you the best possible card sequence. IE the best possible 17/40 or 17/41 cards (on average most draft formats will not last much longer than 7-8 turns). The difference between those two depends on the average expected value of your cards and is only slightly influence by the few bombs you'll get, but it is also influenced by your expected number of lands drawn. Land value drops significantly if you get nothing to cast with it.
In extremum, the arguments heard here are like this:
you play 41 cards, so I will play my bomb more vs you play a less optimal land ratio, so I will curve out more.
Both arguments never take into account that you will never see your entire deck. As I have shown here, you will draw your bomb 0,06% less. Someone else has calculated mana ratio's and behold, the difference is also rather small
The 0.06% number is wrong, it only relates to a single card drawn out of a 40 vs 41 card deck. As you draw more cards the odds of you finding your bomb increases in the 40-card deck's favor. Eg, as you draw card #40 the odds are 40/40=100% that you draw your bomb in the 40-deck vs 40/41 in your 41-deck, ie about 2.44% difference.
Yes, this is exactly what I have problem with. This is how math works, statements like "I lost to bad deck because my opponent played more than 40 cards" is exactly what Puddle is indirectly saying here and that statement can clearly be false, because it was NEVER proven. That 41-card deck might be the best thing construable from that cardpool (even if it almost everytime isn't).
Here's my question though -- why do you care?! PuddleJumper has made it abundantly clear that he's interested in practical knowledge and practical application of theory. In a practical world, he's right.
Why do you feel the need to prove him wrong when the only issue at hand is semantics? (Practical "proven" vs. Scientific "proven") There's no reason to come at him repeatedly the way you have been. This isn't a contest. This isn't philosophy, it's a game, and it's perfectly reasonable for people to focus on the practical side of a game.
Like if someone says to you "I had the best time at FNM last week!" do you say "Really, the best time? You had literally the most optimal period of time...pff I doubt it!"
The problem of proving that 40-cards deck are better has nothing to do with proofs. We've provided multiple different kind of proof, from statistics, comparison arguments, showing that the last additional card is by definition of lower value than all the others.
The problem rest entirely that the deniers:
1. Never give a definition of a proof that would satisfy them. Every time someone offers a proof, they go in denial that it constitutes a proof.
2. Never bother to give a good argument for the 41-cards deck.
That's a quasi-definition of a troll.
Furthermore, the whole argument of better mana with a 41-cards deck is laughable. Between the exact mana curve of a particular deck, the number of colors it runs, how many mana fixing it has, how many mana rocks, how many card draw spells, the exact distribution of colored mana pips among cards along the curve... you're telling me you can say that 16, 17, 18, or any land count is mathematically correct? What. A. Joke. It would takes thousand of simulation run with cunning software that can know the nature of each cards, when they should be played along the curve, etc.
One one hand they argue that the advantage of drawing your top-5 best spells more often in a 40-cards deck, which is easily and obviously provable, is inconsequential. That it's not a proof. On the other they argue about fractional percentage of mana balance when exact perfect mana composition is actually impossible to be known in the first place.
Is there a deck out there where 41-cards is correct? Well, yes, just take lost-in-wood.dek with 42 cards and be happy.
For the rest of the times, the statistical possibility that such elusive deck exists, is extremely small if not zero.
TL;DR: Given that a correct mana base is actually deck-specific and impossibly hard to determine scientifically, stick with the factor (chance of drawing good spells) that is easily provably better. Play 40 cards.
I agree, impractical thought exercise, but if it makes people think about it, then still useful. I am also not sure if it is provable, but statistically many proofs of "unexistence of something" are among the hardest problems math study. (And still, Puddle says he proved it and I am really interested in the proof, because it will, probably, be very interesting. Even if I slowly begin to lose hope that he at least tries to make that.)
I thought of a, hopefully, suitably simple way to present this.
The difference in success between a 40 and a 41 card deck containing one kind of card is zero. The draws are identical.
Now imagine that we ratchet up the number of different cards by one so that we have deck consisting of two kinds of cards. This is what Frank Karsten's article examined. So I'll assume that in this particular narrow case, you'll believe me when I say that the 40 card version of such a deck is the better performer.
The reason why the 2-types versions weight toward the 40-card version where the 1-type versions do not is because of the effect of the extra card on exactly one term: the variance. Variance is essentially defined as a measure of how far you can get from the average case, and still be within a certain level of probability - in other words, if you increase the variance, you draw the average number of any specific kind of card less often, and you get more extreme cases more often. Or even more bluntly, you get mana screwed/flooded more.
Now, a proof by induction requires the following: a base case, a case where you keep everything exactly the same except to increase one variable by exactly one (in this case the number of different cards), and a predictable effect of that increase (in this case, an increase in variance weighted toward the 41-card deck). I.e. we start at no difference in variance, and then every time we add a different kind of card to the deck, variance goes up more in the 41 card deck than in the 40-card deck.
I haven't laid this out using weird symbols or anything, and I've taken the shortcut of using Frank's article as a given rather than actually demonstrating the n+1 case. But if you do accept that as a given (and I don't think anyone here does not), then proof by induction is a valid way of demonstrating that variance in a 41-card deck is always higher than in a 40-card deck. You will always get mana screwed/flooded more often.
On the subject of land percentages...isn't the optimum land ratio gonna be curve dependent anyways? How do you even accurately gauge the theoretical impact of having lands at 17/41 vs 17/40 or 18/41 vs 18/40 on different decks with different curves?
I'd much rather go with the guaranteed higher chance of drawing my bombs or removals in a 40 card deck.
You don't only change the variance, though, you change the probabilities of drawing specific cards too; they are not equal for the 40 and 41-card decks. If you had the same ratio of card #1 to card #2, the "average draw" would be the same, but the variance would be different. It's not possible to get the same ratio in 40 and 41 card decks, though. The purpose of adding the 41st card would obviously be to change the probabilities to draw specific cards, which is exactly what you get.
I, like Karsten himself, am not convinced that the findings in the article can be extrapolated like that. It's an indication, though, and probably the best "evidence" I've seen.
I, like Karsten himself, am not convinced that the findings in the article can be extrapolated like that. It's an indication, though, and probably the best "evidence" I've seen.
Wow you did exactly what pierrebai predicted you'd do FIVE POSTS AGO. You didn't define what would actually constitute proof, you denied what was presented, and you failed to make any coherent argument about why the contrary might actually be right. (Well, you said it "changes the probabilities to draw specific cards" but that was debunked as useless a while ago.)
Regardless of intent this is now indistinguishable from trolling, so we should put this conversation to rest.
@Puddle:
you didn't stress me out at all with that last post. atually it made me wonder what i was looking for in a stronger proof or demonstration.
your thought that "maybe it isn't your worst card after all, if you feel it gives you benefits" got me thnking, when i was thinking to make a sample decklist.
@Phyrre:
at first i was going to say that i rather like the purely academic pursuit, even though i can understand why you would not be interested.
but then i found myself too tired to follow this conversation further!
i was half-thinking of investigating this by writing a computer simulation and then asking for feedback, but i all of a sudden have lost interest in that.. .
i was watching the latest LSV draft (where he did go 41 cards! ... accidentally ;-) ), and during sideboarding, it occurred to me: "who actually /really/ knows the correct way to sideboard? is there any real proof?". no, there isn't. Magic is too complex a game!
there are strong arguments and considerations. i feel that there exist demonstrations that would feel stronger and maybe more insightful to me (ie probably just concrete decklist examples of ideas already discussed), but i /asked/ this Spike community for their ideas, and i /got/ good ideas. i think i'm satisfied with that, rather than pushing for more analysis.
----
of course if others want to continue the discussion, go ahead! challenges and new ideas might still be interesting for me to casually look over -- but i would hope only the parties who are still enjoying actively creating such discussion would further it. i hope not too much bad feelings developed here..
so yeah, i'm still intrested in the discussion f(for those who actually /want/ to further it), but please dont' think it rude if i don't repsonde or actively participate.
(at any rate, i don't think a single person in this thread actually thinks anymore than playing 41 cards is correct, at least not for 99% of actual Limited games!)
@Phyrre:
regardless of if i think you're correct or incorrect in your arguments about this topic, i really appreciate what you wrote here. for me, it humanizes the person behind your words, and it makes it more relatable/understandable why you'd be so upset by arguments for 41 card decks that seem like only mere "yes, but.." objections, when there seems to be solid evidence about what is correct (ie evidence from a) good (if not "proof") arguments and b) years of corroboration from pros (ie because they would be motivated to find the advantage of 41 card decks, if it's there) ).
your analogy to the anti-vaccination camp put the feeling of being upset about misinformation based not on evidence or truth-finding methods, but hypotheticals or anecdotal evidence.
@Velxa:
i, too, am curious about Puddle's math, but don't forget that Puddle also said that we would likely not understand their math without them over-simplifying it.
that, and it takes effort (and patience!) to lay out the math, and who wants to do that in an antagonistic environment where people will likely try to find fault with the effort? i can't blame Puddle if they don't want to share their math here. (pause) though i must say i am nevertheless still curious about it
i think that Puddle might be the only one who has used words like "proven", and that mua1000, myself, magicmerl [?], and Axelrod would all agree completely with what you wrote. (and about Puddle, i'm not even 100% sure i am reading them correctly -- i /think/ they're saying that it's provable, but i might be misunderstanding them).
i wonder if i'm understanding your thought experiment here. are you saying "throw in a Wrath of God to make a 41 card deck" or "throw in an Adaptive Automaton to make a 41 card deck"? because if that is your suggested experiment, isn't that experiment flawed because obviously the Wrath of God / Adaptive Automaton is not the worst card (and, the point of 40 card decks is that you cut out the worst card)?
@Axelrod:
i think you articulated perfectly what is still lingering in my curiousity
i mean, Phyrre does have a point: the pros, if anyone, would be most motivated to investigate.
but it does strongly bring up the question Dromar brought up: what kind of demonstration / though experiment / proof WOULD satisfy my urge to still defend 41 card decks as POSSIBLY correct? what would it take?
(i already believe 41 card decks aren't correct from Phyrre's one example, but that only uses my intuition. something like Frank's experiment is more solid evidence for me, except i don't believe it's comprehensive enough. "the pros have thought of this" isn't satisfying emotionally to me, even though i agree it's a good point. so what kind of demonstration would satisfy me more?)
@Dromar:
good question!!
hm. i know how to write simple computer programs, though i dont' know any statistics. maybe i should tinker around a little bit on this. your idea of doing Frank's experiment, modified using "vanilla creatures with a curve" sounds like something possible to try.
i agree with all of this that you wrote but i agree with it only through using my intuition (and possibly knowing that others have had many, many trials of in-game experimentation to corroborate it).
i wonder if there is any way to demonstrate what you wrote in a more "concrete" or "stronger" way than using my intuition. (i have trouble coming up with a way. maybe it would help (??) if someone can come up with a card deck list that uses exactly the factors that i am interested in (ie the two worst cards give a better curve, better synergy, and better flier-protection), and show that /even then/ it's better to draw your good cards a turn earlier?)
Goblins have poor impulse control. Don't click this link!!
some of my favourite flavour text:
Wayward Soul
"no home no heart no hope"
—Stronghold graffito
Raging Goblin
He raged at the world, at his family, at his life. But mostly he just raged.
1. Some cards are better than others.
2. The last card you add to your deck is the worst. At the very least, if you're anguishing over exact valuation, it's worse than the average.
3. For any 41 cards deck in its exact shuffled configuration, there is a corresponding 40 cards deck with that card cut off.
4. When you'd draw that card, you drawing a card that is always (if you really chose the worst card) worse than any non-land cards in your deck. At the very least, it's worst than the great majority of cards and at best about the same power as the other in its bad lot.
5. Your deck contains fewer lands than non-land, so in the majority of cases, the card you would draw instead of that 41st card is better. The better you are at card valuation, the more the alternative card is better.
6. In the case where the 41st card is in the early top of your deck, even drawing a land is better than drawing that card.
That's it. Points 5 and 6 are the important bits: they show that just taking relative value of cards, the great majority of times, you're better off not having that 41st card.
The argument of 41-carders is basically to refute #5 by saying it's better drawing a very bad spell than a land. They forget point #6 and that even late in the game, having extra land is not bad: you can play many spells per turn, have mana sinks, keep trick mana up, etc. But that ignores that better spells form more than 50% of your deck!
This has been mentioned before, but if a card gives you a better curve, better synergy, and better protection, then it probably isn't your worst card. Maybe you could post an actual decklist example and we can figure out a better cut?
This is a good point and demonstrates why theory can sometimes be a wasted effort. If you can't present an example of the situation beyond a hypothetical, maybe that situation does not exist, or exists so rarely as to be not worth the effort investigating it.
We have discussions this like with modal cards sometimes. Take Gudul Lurker for example. Imagine a world where Gudul Lurker didn't have a mana cost. The only way to get it into play would be as a Morph. While there are situations where playing Gudul Lurker face up is correct, they are rare. It's extremely likely that by simply having a mana cost, Gudul Lurker has been misplayed face-up more than it's been correctly played face-up. The option makes the card strictly better but in practice has made it worse due to user error.
The same may hold true for 41 card decks. Even if we can invest much time and effort into crafting the perfect theory of when to play 41 cards -- that may be a negative compared to pretending you're not even allowed to play 41 cards and focusing your thought in other areas. Even at the individual level, I don't necessarily think the theory could be designed in such a way that it produced more benefit than harm.
Maybe your interest in the topic is purely academic so you don't really care if the 41 card deck can be applied well -- you just want to know it exists. I have trouble empathizing with that mindset because to me, if I'm going to ponder something purely academic, the theoretical application of 41 card decks that cannot be productively put into practice is not something I'd ever want to spend my time on. What's the point? There have to be a thousand more interesting ways to spend your time thinking about Magic (or any other field of academia) that may have better applications or rewards.
All right, I take this, but the conceptual problem I have is that some, if not most, of the cards in your deck are situational. Your point #1 is actually not nearly as clear cut as you would like it to be. A Mana Leak, to use a simple example, is good when you have it early, and terrible when you draw it late. Re: point #4, the "best" card in your deck - depending on its mana cost, isn't always going to be your "best" draw. If your gameplan relies on an Ugin to win, but you draw him when you kept a two lander on the play that's not a "better than average" draw.
That seems to be the problem with simply asserting that "Card X is better than card Y. Period. And therefore the math says that drawing card X is better than drawing card Y. Period." Except we know that's not always true, for almost any card you can think of.
So are we back to, "well, it's usually better to draw card X, so from a practical standpoint it doesn't change the argument..."
For most limited decks, almost all of the cards are not situational. For a typical deck, you'll have ~15 creatures, ~4 removal, ~5 tricks (pump, bounce, draw, equipment, etc.). I'd say every deck easily has 20 cards that are not situational at all, unless you're ready to stretch the meaning of the word.
We've already read enough mind-bending justifications without resorting to now argue that card evaluation is too hard to build a deck?
Ironically, the vast majority of people - even authority figures [read: pros] - claimed for centuries that the world was flat. Anyone trying to say otherwise were ridiculed and even killed.
As someone else mentioned, 15-20 years ago it was an unwritten rule that you played 1/3 lands in your deck. If you at that time would argue that control decks should play 26-27 lands, people would probably laugh in your face.
Look, I'm not saying 41-card decks are better than 40-card decks. What I'm saying is that we should question these unwritten rules.
Questioning is great. But we looked at both sides of it (mana consistency vs. likelihood of drawing your best cards, with most other considerations identified as too minor to compare to these two effects) and it's pretty clear that you're more likely to miss drawing your best cards vs. the subtle difference in mana ratio having a positive impact.
I've yet to see anything approaching a cohesive argument against this. Everything since then has been "But that's not proof" which is a waste of breath. There will never be proof. The system is too complex. By the time we have the technology to prove such a thing, Magic will be considered a trivial relic of the past. Everyone concerned with proof in this demonstrates a fundamental lack of understanding of the complexity involved here.
Maybe if someone asks Frank really nicely he'll revisit that specific part of his article with more complex/specific examples haha.
As far as the last part, a lot of that is really hard to prove as it is all seems very specific. Like you'd not only need to take into account your whole decklist for things like curve considerations, but also how likely you are to face the specific cards your extra synergy/answers care about (like if you're arguing to run Plummet as the 41st card for example...well we need to know how prevalent flyers are in the example format). The only easily provable things are that:
a) you'll draw your best cards less often running a 41 card deck as opposed to a 40 card deck (clearly no rational person would argue this point), and
b) playing 18/41 can give you a better mana ratio than 18/40 if you know you want to maximize your potential to hit x lands drops by turn y while minimizing your potential to flood out (too lazy to look up the specific numbers here).
That's again why I think an example like Frank's is a good argument for 40 cards being usually (if not always) better than 41, not factoring in things like anti-mill or Lost in the Woods.dec. You take out a), the simplest, provable argument against running 41 cards and are left only with b), the simplest provable argument in favor of 41 cards...and the result is 40 cards still out performs it. It's very simplified of course but that combined with as you mention, intuition and my own experiences from playing and watching better players than myself play limited makes me content to side with 40 cards being usually better. If I'm wrong a small percent of the time that's acceptable to me as I think you'd need to find very complex and/or convoluted examples where 41 cards would in fact actually give you a better deck. And when you're in the midst of a limited tournament you probably don't have the time to go in-depth to determine those necessary calculations anyways.
Let me try to explain something about statistics using a simpler example:
Let's say you flip a coin. The odds of getting heads is always 50%. It is never going to be anything but 50%. The words always and never in these sentences don't imply anything about the final outcome after it happens, but rather about our ability to predict the outcome before it happens. Another good example is that there is always a negative expected value for buying lottery tickets. Maybe you win the lottery and it seems like the greatest idea you ever had, after the fact, but that doesn't mean it was a statistically valid way to spend your money beforehand.
In the same way, it might be true that you lose a game as a direct consequence of playing 40 cards instead of 41, once in a while, but those are statistically irrelevant because a person's inability to predict the quality of their draws will always favor them playing 40 cards. If you could see the future, then yeah, there would be times when playing 41 cards would be correct, because then you'd be relying on predictable cause and effect, rather than statistics. But if you can't predict the future, and you're stuck with statistics as your tool for making best guesses, then 40 cards is always best. That is what I mean when I talk about something "seeming" to be a temporary exception. I don't mean it actually *is* a temporary exception, just that people are really bad at judging the quality of a statistical judgment if they already know what the outcome was going to be.
*Statistically speaking* none of those statements you offered up are any good except the first, which is just expressed as opinion anyway. "In 100% of situations, 40 cards is statistically correct", "It is never a good idea to play 41 cards unless you're cheating and the cards are not in random order", and "I never play 41 cards because it is not worthy" are all fine. I'm not saying this to try to get under your skin at all, please understand. I just want you to be clear on what statistics does for arguments like this: it makes it possible to make value judgments about inherently uncertain circumstances.
I don't know what sort of thing you are looking for that would constitute proof.
Imagine that the 41st card you are thinking of is a Naturalize. This is a card that is almost universally regarded as being a 'sideboard card', in that when it has a good target it is worth playing, but otherwise not. Let's say that you decide to play it, and lo and behold, you run into someone with a bomb that you can target with it. Yay, right? Not really. Even though there is a situation where it helps you, on balance it hurts you more by being dead in your other matches.
Here's another way to look at this penomenom: Imagine a bag with 7 red balls and 3 green balls. You need to predict what the colour of the ball to be drawn out is. You can choose red, and a whopping 30% of the time, you will be right. But it's still WRONG to choose red, when green is a better percentage play.
Choosing to play 41 cards might not always hurt you. It might even be beneficial like in the Naturalize example above. But it's still not the correct percentage play because you lower your chances of drawing your best cards in any given game.
Does the above help?
Let's step back a minute. What is the practical application for the knowledge in this thread? Surely it's to help make better 40 card decks, which has to be done within some sort of timeframe, right? If the only way that you can work out whether 41 lands is correct is via some in depth analysis, then it's almost certainly incorrect to do so. We're arguing in a hypothetical vacuum here, but in reality we're going to be making this sort of decision while deckbuilding in 5-10 minutes TOPS. There's no way this sort of knowledge is going to be useful in a practical sense other than to convince us to always stick to 40 cards.
See, this is the kind of blanket statement that I (and I believe veXlaMtg and others) have an issue with; you postulate that 40-card decks are always the correct decision. But what if - with a given pool of cards, in a given meta - the 41-card deck would on average be better than the 40-card deck? It would, statistically speaking, be correct to play the 41-card deck.
This is essentially the same straw man argument that Puddle presented above; you postulate that the 41-card deck is inferior to the 40-card deck (on average), then present a scenario where the 41st card actually wins you the game, then goes on to show that you were actually wrong playing the 41-card deck because on average the 40-card deck is better.
I agree. The 41-card deck is a thought exercise with minimal impact in practice.
I'm pretty sure it's impossible to (mathematically) prove that the 40-card deck always is a superior choice to the 40+ decks, because I believe that would require running infinite game simulations with all possible card pool permutations. On the other hand, the statement is proven false once (if) a single counter example is found.
The 0.06% number is wrong, it only relates to a single card drawn out of a 40 vs 41 card deck. As you draw more cards the odds of you finding your bomb increases in the 40-card deck's favor. Eg, as you draw card #40 the odds are 40/40=100% that you draw your bomb in the 40-deck vs 40/41 in your 41-deck, ie about 2.44% difference.
Here's my question though -- why do you care?! PuddleJumper has made it abundantly clear that he's interested in practical knowledge and practical application of theory. In a practical world, he's right.
Why do you feel the need to prove him wrong when the only issue at hand is semantics? (Practical "proven" vs. Scientific "proven") There's no reason to come at him repeatedly the way you have been. This isn't a contest. This isn't philosophy, it's a game, and it's perfectly reasonable for people to focus on the practical side of a game.
Like if someone says to you "I had the best time at FNM last week!" do you say "Really, the best time? You had literally the most optimal period of time...pff I doubt it!"
The problem rest entirely that the deniers:
1. Never give a definition of a proof that would satisfy them. Every time someone offers a proof, they go in denial that it constitutes a proof.
2. Never bother to give a good argument for the 41-cards deck.
That's a quasi-definition of a troll.
Furthermore, the whole argument of better mana with a 41-cards deck is laughable. Between the exact mana curve of a particular deck, the number of colors it runs, how many mana fixing it has, how many mana rocks, how many card draw spells, the exact distribution of colored mana pips among cards along the curve... you're telling me you can say that 16, 17, 18, or any land count is mathematically correct? What. A. Joke. It would takes thousand of simulation run with cunning software that can know the nature of each cards, when they should be played along the curve, etc.
One one hand they argue that the advantage of drawing your top-5 best spells more often in a 40-cards deck, which is easily and obviously provable, is inconsequential. That it's not a proof. On the other they argue about fractional percentage of mana balance when exact perfect mana composition is actually impossible to be known in the first place.
Is there a deck out there where 41-cards is correct? Well, yes, just take lost-in-wood.dek with 42 cards and be happy.
For the rest of the times, the statistical possibility that such elusive deck exists, is extremely small if not zero.
TL;DR: Given that a correct mana base is actually deck-specific and impossibly hard to determine scientifically, stick with the factor (chance of drawing good spells) that is easily provably better. Play 40 cards.
I thought of a, hopefully, suitably simple way to present this.
The difference in success between a 40 and a 41 card deck containing one kind of card is zero. The draws are identical.
Now imagine that we ratchet up the number of different cards by one so that we have deck consisting of two kinds of cards. This is what Frank Karsten's article examined. So I'll assume that in this particular narrow case, you'll believe me when I say that the 40 card version of such a deck is the better performer.
The reason why the 2-types versions weight toward the 40-card version where the 1-type versions do not is because of the effect of the extra card on exactly one term: the variance. Variance is essentially defined as a measure of how far you can get from the average case, and still be within a certain level of probability - in other words, if you increase the variance, you draw the average number of any specific kind of card less often, and you get more extreme cases more often. Or even more bluntly, you get mana screwed/flooded more.
Now, a proof by induction requires the following: a base case, a case where you keep everything exactly the same except to increase one variable by exactly one (in this case the number of different cards), and a predictable effect of that increase (in this case, an increase in variance weighted toward the 41-card deck). I.e. we start at no difference in variance, and then every time we add a different kind of card to the deck, variance goes up more in the 41 card deck than in the 40-card deck.
I haven't laid this out using weird symbols or anything, and I've taken the shortcut of using Frank's article as a given rather than actually demonstrating the n+1 case. But if you do accept that as a given (and I don't think anyone here does not), then proof by induction is a valid way of demonstrating that variance in a 41-card deck is always higher than in a 40-card deck. You will always get mana screwed/flooded more often.
I'd much rather go with the guaranteed higher chance of drawing my bombs or removals in a 40 card deck.
Wow you did exactly what pierrebai predicted you'd do FIVE POSTS AGO. You didn't define what would actually constitute proof, you denied what was presented, and you failed to make any coherent argument about why the contrary might actually be right. (Well, you said it "changes the probabilities to draw specific cards" but that was debunked as useless a while ago.)
Regardless of intent this is now indistinguishable from trolling, so we should put this conversation to rest.
you didn't stress me out at all with that last post. atually it made me wonder what i was looking for in a stronger proof or demonstration.
your thought that "maybe it isn't your worst card after all, if you feel it gives you benefits" got me thnking, when i was thinking to make a sample decklist.
@Phyrre:
at first i was going to say that i rather like the purely academic pursuit, even though i can understand why you would not be interested.
but then i found myself too tired to follow this conversation further!
i was half-thinking of investigating this by writing a computer simulation and then asking for feedback, but i all of a sudden have lost interest in that.. .
i was watching the latest LSV draft (where he did go 41 cards! ... accidentally ;-) ), and during sideboarding, it occurred to me: "who actually /really/ knows the correct way to sideboard? is there any real proof?". no, there isn't. Magic is too complex a game!
there are strong arguments and considerations. i feel that there exist demonstrations that would feel stronger and maybe more insightful to me (ie probably just concrete decklist examples of ideas already discussed), but i /asked/ this Spike community for their ideas, and i /got/ good ideas. i think i'm satisfied with that, rather than pushing for more analysis.
----
of course if others want to continue the discussion, go ahead! challenges and new ideas might still be interesting for me to casually look over -- but i would hope only the parties who are still enjoying actively creating such discussion would further it. i hope not too much bad feelings developed here..
so yeah, i'm still intrested in the discussion f(for those who actually /want/ to further it), but please dont' think it rude if i don't repsonde or actively participate.
Goblins have poor impulse control. Don't click this link!!
some of my favourite flavour text:
Wayward Soul
"no home no heart no hope"
—Stronghold graffito
Raging Goblin
He raged at the world, at his family, at his life. But mostly he just raged.