AntiDredge Strategy  from a Dredge Expert

@evouga And you were the one that brought up Shops. The point that was being made is that while it is normally correct to play 4 Leylines, it is not always so. Just like when you said that "almost all Vintage decks run either 0 or 4 Workshops", the "almost" is the whole entire point of the conversation and basically what I have tried to do is explain circumstances in which it might make sense to play less than 4 Leylines. So your metaphor is basically restating the argument and I don't know what you point is. Which, granted, is not an accurate use of the phrase moving the goalposts but I was in the middle of match and used it incorrectly.
@IamActuallyLvL1 I was tired of getting Chaliced and Leyline of Sancitied out of certain Dredge builds. I agree in general that Tormod's Crypt is more synergistic.

@ChubbyRain ajfirecracker presented a (wellreasoned) argument against the "0 or 4 Leylines" advice, based on a hypothetical format where 5 Leylines are legal, and offered three potential resolutions to the paradox he constructed. Search for "There are a limited number of ways to avoid that conclusion." to find his post.
I was responding to this argument. I'm not claiming that it's never correct to run 2 Leylines, nor am I claiming that no Vintage deck runs 2 Workshops, but these rare corner cases are not especially germane to analyzing ajfirecracker's argument that the "0 or 4" advice is generically unsound.

Re: Leyline at 4 and 0, I've actually never heard an argument for "4 Leylines or None" that made sense to me, or at least any argument that wouldn't equally apply to any magic card. Sure, you want it in your opening hand, but that's true of all dredge hate. Chubbyrain's "1 Leyline after I've added my tutor targets" plan makes total sense to me (except for the part where there's 5 slots, but that's another topic entirely).
It's the total opposite of a card like Workshop that pushes your deck in a particular direction and works in multiples. Leyline is worse in multiples, and has uniquely low deckbuilding costs, as you see it in lists that have no way of casting it. Dredge hate in particular has an added pressure to be diverse, as it's hard for a Dredge deck to pack answers to multiple card types, making a card like Leyline even worse in multiples than something like Moat is against aggro decks.
It's true that you want to see Leyline on turn one of not at all, but that doesn't encourage any particular count. The likelyhood of drawing a card in your opening hand and drawing it later are linked. It's maybe true that you want some of the benefits of running 4 and some of the benefits of running 0, but you can't actually do that, and running 4 doesn't get you any closer to that than 3 does.
I sort of feel like somebody did some bad math back in Guildpact, and everyone just rolled with it for 10 years.

I think the 0 or 4 folks are assuming you have to mulligan for Leyline or it's a wasted slot, and the "13 is sometimes reasonable" folks are imagining that you don't particularly mulligan for Leyline, and accept that sometimes you start with it and sometimes you don't

As someone who plays a lot of Dredge, but also dabbles in every other archetype, I must say that the most important thing to choosing a hate piece is really how it synergies with your deck. Some decks even want to run these cards main deck, and not just leave them to the sideboard. This usually leads decks to play a 4of since most decks have 1 hate piece that is just superior for their deck list. For those that don't pick antihate that synergies with their deck, they run the risk of jeopardizing their own gameplan and thus playing into the Dredge players' benefit of both of you having worse decks in game 2/3.
For example, just playing Leyline because it theoretically can hurt Dredge the most is not a good reason to run Leyline. If you're on a Xerox deck that wants to filter through its library and now you have a bunch of bad cards in there. If you're on a Outcome deck that wants to activate its mox opals and cast outcome to bounce as many permanents its also not a great choice because Grafdigger's, Tormod's and others are artifacts and are better at being bounced. If you're on an aggro deck, the damage difference between a swinging Yixlid Jailer / Containment Priest and a Leyline of the Void can also not be palatable.

I noticed Pitch Dredge jumped back and forth between Leyline of the Void and Ravenous Trap for the same reasons listed by everyone else. Leyline of the Void is solid in Dredge because it greatly reduces the number of creatures the opponent can use to break Bridge from Below. However, there were lists for a while that used Ravenous Trap since you could still make use of it if you drew it for the turn or off of a Bazaar of Baghdad activation. Both have their merits, but it seems Leyline of the Void is winning out more often.

@brassman The two properties that make leyline special amongst other hate pieces are:

it is probably best hate piece IF you have it in your opening hand.
It is uncounterable, it is free, it is assymetrical, the effect is very potent , it is down on turn 1 no matter if you start on the play or on the draw. 
it is probably one of the worst hate pieces if you draw it later in the game.
If you would draw Leyline, the card is horrible, a lot of decks would have to use extensive resources (crack lotus, use dark ritual etc.) to be able to cast it, or not beeing able to cast it at all and to make matters worse it wont exile anything that is already in your opponents graveyard. However: this downside is so very greatly reduced if you already have a leyline in play from your opening hand and are just drawing your second one. In that case, it most likely doesnt really matter that you have drawn it, since your opponent is already very much desperate to handle your first one which will give you enough time and resrouces to either win, or if you really have no other option cast your second one.
I think these attributes make Leyline very special compared to your peers. In my eyes the whole reason to play Leyline is to have it in your opening hand, if not its just so much worse than other options.
In very extreme terms: Leyline is possibly game winning in your opening hand and absolutely useless outside of it. Logically it should follow that either you want to maximize the possiblity of having it in your opening hand, or just eliminate it all together and go for other options.@ajfirecracker said in AntiDredge Strategy  from a Dredge Expert:
I think the 0 or 4 folks are assuming you have to mulligan for Leyline or it's a wasted slot, and the "13 is sometimes reasonable" folks are imagining that you don't particularly mulligan for Leyline, and accept that sometimes you start with it and sometimes you don't
"sometimes you start with it, and simetimes you dont" is not a 50/50 thing, it can be modified by playing more or less leylines, and thats the whole point of playing 4 of them. This is like saying "you sometimes win the lottery and sometimes you dont" which is completely ignoring that there is a propability underlying those events.
Also as i explained over and over again in the posts above it is not about always mulliganing to a leyline, it is mulliganing to a keepable hand, while keepable hands are either very broken and fast, or nearly all hands with a leyline in them. Leylines just up the count of keepable hand by a ton. All of this still in the context of playing DPS and following the 'broken or hate' strategy. 

I agree that if your mulligan strategy involves finding Leyline of the Void in your opening hand you probably want to run as many as possible
On the other hand if your mulligan strategy involves finding some piece of Dredge hate and you have (for example) 4 Cage and 2 Leyline, then you have 6 cards that can count as a success. The fact that the Cages aren't literally Leylines doesn't matter if you're mulliganing for them like Leylines

@baishuu said in AntiDredge Strategy  from a Dredge Expert:
I noticed Pitch Dredge jumped back and forth between Leyline of the Void and Ravenous Trap for the same reasons listed by everyone else. Leyline of the Void is solid in Dredge because it greatly reduces the number of creatures the opponent can use to break Bridge from Below. However, there were lists for a while that used Ravenous Trap since you could still make use of it if you drew it for the turn or off of a Bazaar of Baghdad activation. Both have their merits, but it seems Leyline of the Void is winning out more often.
I personally think Ravenous Trap was always just wrong in that deck. You don't actually draw that many cards (because you are using all your draws to dredge), and it's much weaker in that matchup than Leyline.

A more rigorous take on this debate:
Suppose you are allowed to run as many Leylines as you want. Let p[n] be the probability of winning a tournament with optimal deck construction given the constraint that you must run exactly n Leyline’s.
For any integer n, let S(n) be the following assertion:
max(p[0], p[n]) > max(p[1], p[2], ..., p[n1])
The “0 or 4” crowd is then asserting that S(4) is true and can be deduced from first principles. Let’s assume they are right.
Now if Leyline is playable, then S(75) is clearly false.
This implies that if you consider the statements S(4), S(5), S(6), ..., S(75), then there is some magic k for which S(k) can be proven to be true from first principles, while S(k+1) cannot.
What is this magic k, and what is so special about it, that allows you to make a fromfirstprinciples argument for S(k) but not for S(k+1)?
If no such k can be identified, then the “0 or 4” crowd must be wrong. It may indeed be the case that 0 or 4 is better than 1, 2, or 3, but that fact cannot be deduced from first principles.

One consideration with respect to the possibility of the “magic” k is that there are diminishing returns in terms of the the probability of seeing a leyline in your opener when you add more leylines to your deck. The magic k may be the point at which the increasing likelihood of drawing 2 outweighs the increasing likelihood of getting 1 in your opener.

Returns diminish monotonically absent selfsynergy
Even if you accept that you should Mulligan until you hit a Leyline, the most valuable Leyline is the 1st one you put in your deck  each subsequent Leyline (starting already with the 2nd) suffers from diminishing returns

@ajfirecracker it seems like you clearly do not understand the basic concept of probability

@aelien said in AntiDredge Strategy  from a Dredge Expert:
@ajfirecracker it seems like you clearly do not understand the basic concept of probability
If you explain how you think I'm wrong it's a conversation
If you just tell me I don't understand it's an insult

@ajfirecracker said in AntiDredge Strategy  from a Dredge Expert:
Even if you accept that you should Mulligan until you hit a Leyline, the most valuable Leyline is the 1st one you put in your deck  each subsequent Leyline (starting already with the 2nd) suffers from diminishing returns
Each Leyline above the first ups the probability of you having it in your opening hand, making your strategy of mulliganing to the first one incredibly bad if you only play 1 and way better if you play the maximum amount of leylines.
I really dont get why this is even an argument at all... You play 4 bazaars, not because you always want a second bazaar but because you really want to have a bazaar in your opening hand, you should be very familiar with this concept. 
I used a hypergeometric calculator to put some actual numbers to it.
If you have X Leylines and mulligan until you have at least one in your opening hand, your odds of success are below:
 0%
 38.6%
 62.6%
 77.4%
 86.5%
So, yes, more Leylines does indeed translate into a higher chance of success.
However, the most impactful Leyline is the first one, and if we subtract the prior odds of success to see the marginal impact of adding 1 more Leyline, we see that returns diminish smoothly:
 38.6%
 24.0%
 14.8%
 9.1%
So the most valuable Leyline is the first one, and each one after that helps increase the probability of "success", but not as much as the one before.

This post is deleted! 
@dshin said in AntiDredge Strategy  from a Dredge Expert:
A more rigorous take on this debate:
Suppose you are allowed to run as many Leylines as you want. Let p[n] be the probability of winning a tournament with optimal deck construction given the constraint that you must run exactly n Leyline’s.
For any integer n, let S(n) be the following assertion:
max(p[0], p[n]) > max(p[1], p[2], ..., p[n1])
The “0 or 4” crowd is then asserting that S(4) is true and can be deduced from first principles. Let’s assume they are right.
Now if Leyline is playable, then S(75) is clearly false.
This implies that if you consider the statements S(4), S(5), S(6), ..., S(75), then there is some magic k for which S(k) can be proven to be true from first principles, while S(k+1) cannot.
What is this magic k, and what is so special about it, that allows you to make a fromfirstprinciples argument for S(k) but not for S(k+1)?
If no such k can be identified, then the “0 or 4” crowd must be wrong. It may indeed be the case that 0 or 4 is better than 1, 2, or 3, but that fact cannot be deduced from first principles.
If your premise is
 you must mulligan until you find a Leyline (even if you have zero Leylines in your deck) and you cannot attempt any other deck construction or mulligan strategy
then you and @ajfirecracker are entirely correct. The first Leyline most increases your chance of success, and each subsequent Leyline is weaker than the first (with 75 Leylines obviously useless, unless you're on the drawgountiltheydeckthemselves plan).
I don't understand why people are so hung up on the (obviously absurd) premise I boxed above. The alternative to running 1 Leyline and mulliganing until you find it is not running 0 Leylines and mulliganing until you find it. It's building a completely different deck with a different strategy for defeating Dredge (such as cantripping into multiple copies of Priest or Rest in Peace.) In this sense a "Leyline mulligan" strategy with 4 Leylines can be more effective than a completely different "cantrip into hate" strategy with zero Leylines which is in turn more effective than a "Leyline mulligan" strategy with 3 Leylines.
This is the argument I and other "4 or 0" people are advancing. Not some strawman (and incorrect) simplistic assertion about hypergeometric distributions. @dshin's calculations are completely valid but irrelevant to the argument I'm actually advancing.
I again suggest thinking of Workshops as a second view on the same phenomenon. A blue deck with zero nonMoxen artifacts is strictly worse if you add a random Workshop. Once you add the first Workshop and start adding expensive artifacts, you are pursuing a new strategy and now adding three more Workshops is optimal (with diminishing returns as you add more Workshops, with the tipping point well above the maximum allowed 4 copies). The vast majority of Vintage decks want 0 or 4 Workshops. This has nothing to do with p[n].

@evouga I think you make a very good point  if you are oriented around some particular strategy, the cards that support that strategy are good and should generally be maximized, but there might be some other strategy that is better or comparable, but you can't necessarily take half and half and get a good result
However, I think you are defining "strategy" in too narrow a way if your choices are some number of a given named card. "Mulligan until you hit a piece of graveyard hate" is a fine strategy for decks that can protect hate and/or have some way of winning fairly quickly without using a large number of cards. This strategy can be executed with 2 Leyline and 4 Grafdigger's Cage just as well as 4 Leyline 2 Cage. Typically we see Workshops decks lean on Cage and Leyline as their main hate cards, and I don't see why they shouldn't prioritize Cage from that selection in order to enhance other matchups like Oath of Druids.

@evouga said in AntiDredge Strategy  from a Dredge Expert:
I again suggest thinking of Workshops as a second view on the same phenomenon. A blue deck with zero nonMoxen artifacts is strictly worse if you add a random Workshop. Once you add the first Workshop and start adding expensive artifacts, you are pursuing a new strategy and now adding three more Workshops is optimal (with diminishing returns as you add more Workshops, with the tipping point well above the maximum allowed 4 copies). The vast majority of Vintage decks want 0 or 4 Workshops. This has nothing to do with p[n].
Agreed, there are good generic arguments that for Magic card X you should run either 0 or the maximum legal number of copies, and only if you have some special reason should you run some number in between (the most common special reason being you actually tested it and the diminishing returns make 1 or 2 or whatever the right number).
With Workshops I would guess the optimal number is somewhere in the 1012 range, but of course there's no way to get there from first principles  in theory 3 could be the optimal number, but we all know from experience that 3 is silly and you'd prefer to have more of that particular effect. The same applies here  there's no way to get from first principles to the optimal number being 4 or greater.
Moreover, the discussion has all been in the context of a particular deck. Your 0 or 4 division is generated by having different decks with different strategies. I interpret the "0 or 4" claim as applied to a particular deck with a particular set of 71 other cards at its disposal. I agree that generally you would want 0 or 4 because either Leyline meshes well into your strategy or some other card does, and you want the better of the two. However, I don't see any reason at all why Leyline should be an exception to the same principles that frequently push people to play 1, 2, or 3 copies of a card.