ajfirecracker


« on: November 15, 2015, 07:14:09 pm » 

I've been one of relatively few Dredge players running Library of Alexandria for the past year. Today I got my hands on the Python code that was used a while back to calculate Bazaar of Baghdad probabilities. This program is an easy way to do the whole combinatorial tree, so it is an exact answer. It is not an approximation from repeated simulation. I wanted to investigate the odds of being able to keep a Library of Alexandria as a substitute for Bazaar of Baghdad. I calculated that, ignoring for the moment the odds of using Library as a bait for Wasteland or as a draw engine postboard. I calculated some values initially, ignoring Library. This confirmed that my results were consistent with the earlier results others have calculated. Probability of drawing Bazaar any time in the mulligan process: 0.94168 Probability of playing Bazaar by turn 1, on the draw (under old mulligan rule): 0.94610 Probability of playing Bazaar by turn 1, on the draw (includes new mulligan rule): 0.95026 My new results: Probability of keeping a hand with a specialty land (Library is only kept on 7): 0.95116 Probability of activating a carddraw land Turn 1, on the draw (includes new mulligan rule): 0.95831 So if we are willing to keep Library on the play (but only on a 7card hand) we eliminate 16.26% of our mulligans to oblivion. (This ignores play/draw and only looks at the hand we keep) If we demand a specialty land activation on the first turn we eliminate 16.18% of mulligans to oblivion on the draw, therefore about 7.5% of our mulligans to oblivion total. (Mulligans to oblivion happen more often on the play than on the draw once we include scry and the firstturn draw) Without Library, you will mulligan to oblivion in 1 in 17 games on the play and 1/20 games on the draw. Keeping Library in your 7card hand on the play you will mulligan to oblivion in 1/20 games. On the draw, Library cuts mulligans to oblivion to 1 in 24. My revised Python code is here: https://github.com/ajfirecracker/VintageDredge/blob/patch1/MulligansWithLibraryThe original Python code is here: https://github.com/Dritte/VintageDredge/blob/master/mulliganToBazaar.pycredit for original code to Eugenio Fortanely, I believe


« Last Edit: November 15, 2015, 07:28:39 pm by ajfirecracker »

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kingneckbeard on MTGO
"I fully believe that if Dredge could play a transformational sideboard it would just win all the tournaments yet it just doesn’t have one because there is just nothing that it can play. It’d be awesome to completely ignore all those very specific hate cards people bring against you but how are you going to do that?"  Paulo Vitor Damo da Rosa



jcb193


« Reply #1 on: November 16, 2015, 08:00:42 pm » 

I'm not going to pretend to understand the math you mentioned, and I've rarely played dredge, but I did enjoy reading your post. Didn't want it to go unrecognized. I like the concept of playing LoA in dredge. Thanks for explaining it.



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gkraigher
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« Reply #2 on: November 16, 2015, 10:08:23 pm » 

I've wanted to know this math for a long time, thanks for posting it.
I've considered playing LoA and I'd still consider it in the right build.
One take away that I never realized in theory crafting is that: LoA on the play is the same as it is on the draw.



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ajfirecracker


« Reply #3 on: November 16, 2015, 10:12:58 pm » 

The odds of hitting Library in a hand of 7 are the same on the play/draw
On the play, you can't activate Library on Turn 1



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kingneckbeard on MTGO
"I fully believe that if Dredge could play a transformational sideboard it would just win all the tournaments yet it just doesn’t have one because there is just nothing that it can play. It’d be awesome to completely ignore all those very specific hate cards people bring against you but how are you going to do that?"  Paulo Vitor Damo da Rosa



vaughnbros


« Reply #4 on: November 16, 2015, 11:40:49 pm » 

The number 16.18% is extremely deceptive. In reality you are talking about 0.008 percentage point change or 1 in 125 games.
This is also a false dichotomy. You've changed what was a dichomous get bazaar or mull to oblivion into a trichotomy of get bazaar, library, or mull to oblivion. I think this is about 4.5% of games, but you should be able to calculate the exact number using the code. So instead the distribution looks like: 91% of games with Bazaar, 4.5% of games with Library, 4.5% mull to oblivion. You undoubtedly will have a lower win % in games you open with library instead of bazaar since not only does it dredge slower (by an entire draw per turn), but it also doesnt immediately pitch a card nor can you cast any of your disruption or play any lands from your hand without turning the library off. Compared to the original distribution of 95% bazaar, 5% mull to oblivion your library hands need to be able to win at least 90% of the games Bazaar would be able to win in order to break even. I am highly doubtful that this will be the case.
This all being said I think library in dredge is decent for games 2/3 given that it's a single card draw engine that requires no other investment. The 10% chance of opening with it on 7 seems that it should at least be worth consideration in most lists.



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gkraigher
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« Reply #5 on: November 17, 2015, 12:55:03 am » 

The odds of hitting Library in a hand of 7 are the same on the play/draw
On the play, you can't activate Library on Turn 1
Yes I get that. But saying "library, go" on turn one on the play is the same thing as saying "draw, library, go" on the draw. With the exception of wasteland/strip mine. It's very much akin to your opponent playing actual "manaless dredge" in legacy and you choosing to draw first, even though you want to play first, because they are actually the same thing except one nets you a card.


« Last Edit: November 17, 2015, 01:05:45 am by gkraigher »

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vaughnbros


« Reply #6 on: November 17, 2015, 01:00:32 am » 

The odds of hitting Library in a hand of 7 are the same on the play/draw
On the play, you can't activate Library on Turn 1
Yes I get that. But saying "library, go" on turn one on the play is the same thing as saying "draw, library, go" on the draw. With the exception of wasteland/strip mine. It's very much akin to your opponent playing actual "manaless dredge" in legacy and you choosing to draw first, even though you want to play first, because they are actually the same thing except one nets you a card. The problem is you aren't dredging until turn 3 if you are on the play with library. When we turn ourselves into a turn 4/5 combo deck with little to no disruption we become very vunerable.



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gkraigher
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« Reply #7 on: November 17, 2015, 01:06:04 am » 

Vaughnbros, I understand how you could find it deceptive. I don't. I find it follows its premises to an exact and calculates the math. It doesn't say shit about winning percentages, only the percentages of an event occurringin this case drawing a land that fuels dredge.
I don't think he is making any deceptive claims at all. He's simply presenting the actual probabilities of events and letting everyone in the forums draw their own conclusions about whether or not this suites their build of dredge.
If anything, I feel like you don't get his point and are polarizing it by calling it deceptive.
Sometimes forums are about information, not arguments.



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ajfirecracker


« Reply #8 on: November 17, 2015, 01:11:21 am » 

@vaughnbros I'm going to rerun a bunch of numbers to address your post, but off the top of my head: in a 4x Serum Powder deck the odds of Library in a 7card opener is actually around 18% (if you Powder for it... in practice you'll keep some Powderable hands and the real percentage will be somewhere between 10 and 18)
To be precise, you will get a hand of 7 with your 1of Library 18.475% of the time, assuming you Powder any hand without it. Postboard, I think it's reasonable to assume something like a 15% Library rate.
On the draw, with the extra scry, your odds of finding Bazaar on a mulligan to 6 or less are 89.659%. This is after any 7card Powders are taken, so this is the figure to beat for a 7card hand with Library and no Powder. In essence, if a 7card hand with Library and no Powder wins in 89.659% or more of the situations in which a Bazaar hand (of any sort, including 12 card hands and including scrying into the Bazaar from 1 card on your first turn) would win the game, then you should keep. If your Library hand wins less than 89% of the games in which a random Bazaar hand would win the game, you should mulligan to 6 (also assuming the Library hand doesn't win any games where the random Bazaar hand loses). Once you start including variables like disruption in hand or Petrified Field, I think you can easily make the argument that some particular Library hand does or does not pass that 89% threshold. Another point to make is that you may have gained information by this point, most notably from an opponent's Serum Powders, if any. (Since you are ostensibly on the draw if you're seriously considering Library)
Expanding my original scope slightly: You will keep a hand of 7 with Bazaar 54.127% of the time. 10.169% of the time you will have Library in that same hand of 7, if you run it. Multiplied together (the ~10% is conditional on the Bazaar odds, so you can multiply these) you get a rate of 5.504% to have both Library and Bazaar in a given hand of 7, after any Powders.
A related point I just realized is that if you keep your 7card Library hand you get an extra draw to find Bazaar; a Turn 2 Bazaar following up a Library activation is basically as good as a Turn 1 Bazaar, except in the most absurdly fast matchups. You don't start dredging until your third draw, because the Library draw replaces it for discard purposes without having the option to Dredge. Assuming zero Powders have been used (worstcase for the proLibrary side) you have a 14.66% chance to hit a Bazaar in those two draws. In the first draw step alone you're 7.55% to hit, in which case you're no worse off than if you had mulliganed for Bazaar successfully (and probably better off, since you have 8 cards and the option to play around Wasteland, but are not committed to being any slower than you otherwise would have been). As a result, the Library by itself only needs to win 75% or 83% of the games that a Bazaar would. The extra percentage is made up by "lucky" Bazaar topdecks. As before, this interacts with information about your opponent and your hand. You can mulligan Library as if it's worthless against known fast combo opponents and Dredge mirrors. You can keep Library aggressively against known slow decks, for the "guaranteed" win. To push the case even further, consider an "average" number of Serum Powders of around 0.8. You will exile an extra 6 cards and be 8.51% to hit Bazaar on your first draw and 16.47% to hit the Bazaar before you start dredging (i.e. on either the first draw or the Library draw).
An interesting point you raised was what the odds are of keeping a hand of 7 with Library and no Bazaar, given that you were mulliganing for Bazaar but will always keep a 7card hand with Library. The odds of keeping a 7card Bazaar hand (totally ignoring Library) are 54.127%, as I mentioned above. The odds of keeping a 7card Bazaar/Library hand are 61.878%. This is one of the few cases where we can simply subtract  7.751% of the time you'll keep a Libraryonly hand. I think this mulligan strategy is overly simplistic, but of course so is the Bazaar strategy, as people divert given sufficiently good nonBazaar options.


« Last Edit: November 17, 2015, 02:26:19 am by ajfirecracker »

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kingneckbeard on MTGO
"I fully believe that if Dredge could play a transformational sideboard it would just win all the tournaments yet it just doesn’t have one because there is just nothing that it can play. It’d be awesome to completely ignore all those very specific hate cards people bring against you but how are you going to do that?"  Paulo Vitor Damo da Rosa



bactgudz


« Reply #9 on: November 17, 2015, 10:43:45 am » 

Interesting. You may want to expand the state space to count dredgers and reevaluate for keeping only in those states where you have library+dredger, I'm not sure you'd keep library alone.



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vaughnbros


« Reply #10 on: November 17, 2015, 11:40:37 am » 

The full distribution of hand and library sizes would be nice.
What scenarios are you deviating from the strategy? I can count on one hand the number of times I've decided to not powder and I've never mulled a a bazaar. It seems like the percentage of situations that will cause deviation are few and far between and shouldn't effect the end resulting probabilities much.
At 7.751% we are now talking about needing to win a larger number of games with library. Top deck bazaar is roughly 1(14/53)*(14/52)=0.148 of those games leaving us with 6.615%. So we need to win 5.810% of games with Library opener in addition to break even, or in total 87.78% of the games we would have won with Bazaar. I'm not sure how you got your 75% or 83% figures. Regardless this is a high number of wins that I am still doubtful that library could win. Remember that library is also vunerable to cabal therapies, unmasks, thoughtseizes, and other discard effects that can take you off it. This is in addition to the difficulties of not being able to use your hand much or quickly dump your hand into the yard.


« Last Edit: November 17, 2015, 11:49:06 am by vaughnbros »

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ajfirecracker


« Reply #11 on: November 17, 2015, 12:21:38 pm » 

Yeah, I agree. You can finetune your keeping decisions based on the number of Powders you have used. I don't think I'm going to be able to generate that, but I might try.
The .148 you mentioned is a probability, not a percent. We're talking about 14.8%, which is why it knocks things from ~89 to ~75.
The decision I'm investigating is:
P(Win  Keep Library) > P(Win  Mull to 6)
We can break this down to:
P(Win  Library only)*P(Library only) + P(Win  Library > Bazaar)*P(Library > Bazaar) > P(Find Bazaar)*P(Win  Find Bazaar) + P(Don't find Bazaar)* P(Win  Don't find Bazaar)
Assuming you never win on a mull to nothing, and assuming Library > Bazaar is exactly as good as finding Bazaar in mulligans, and substituting approximate known values:
P(Win  Library Only)*0.85 + P(Win  Find Bazaar)*0.15 > P(Win  Find Bazaar)*0.89 + 0.11*0
P(Win  Library Only)*0.85 > P(Win  Find Bazaar)*0.74
P(Win  Library Only) > P (Win  Find Bazaar) * 0.87
Huh. I guess you're right. I was trying to structure it in a way that would let me avoid having to downweight the library only option (in which case you get that 0.74 or 0.75 figure) but I don't think there's a way to frame it like that.
As to the scenarios where you deviate from the Bazaar strategy, the canonical two are Ancestral + Blue source in 6+ cards (if you run such things) or heavy disruption in 24 cards. Obviously Ancestral is worse now that Mental Misstep is everywhere, but I think it's fair to say that you would at least consider a keep there.
I think in the past year I've probably lost one game with Library where Bazaar would have won, and that was to a very fast Mentor start. I've also won several games where I think a random Bazaar hand would be likely to lose, due to keeping Petrified Field and disruption. I know for sure that I've had a few keeps where I went Library > find Bazaar and that hand won where Bazaar only would have lost. I suspect that my Library winrate is probably actually higher (i.e. surpassing 100% when the figure to beat is 87%) than my "random 6orfewer card Bazaar" hand winrate, because I don't keep every 7card Library hand, I look for Dredgers and/or disruption and/or resilience. Now, I have lost some games with Library (surprisingly few) but I think for the most part those are games where Bazaar also would have lost. Overwhelmingly we're talking Wastelands, fast combo, T2 win on the play from the Dredge mirror, that kind of stuff.
To put it another way, if you take a step back and say that your Bazaar hand wins 80% of the time, your Libraryonly hand only has to win .8*.87 = .696 = 70% of the time. Just a reminder to keep the translation into actual winrate in mind.
Another thing to consider are hands with Library + Petrified Field, which will routinely assemble Bazaar by Turn 3. The 3turn sequence for Bazaar on the draw is: 3x Draw, 3x Discard, 2x Dredge, 3x Discard, 1x Dredge, 2x Dredge, 3x Discard, 1x Dredge The 3turn sequence for Library > Petrified Field > Bazaar on the draw is: 2x Draw, 1x Discard, 2x Dredge, 1x Discard, 2x Dredge, 2x Dredge, 3x Discard So overall you have the same number of Dredges by Turn 3 and have simply delayed one of them from Turn 2 to Turn 3. You've missed out on 4 discards. If you hit a Bazaar before the end of Turn 2 (i.e. in the first 812 cards), you can use Petrified Field on your own turn to get an extra discard. Otherwise, you can dredge up to 24 cards trying to hit a Bazaar. You also missed out on a single card draw into hand. If you hit Bazaar before the 4th Dredge (whether on Turn 2 or Turn 3) you can also move your Turn 3 Library draw to after the Bazaar discard. If you hit on the 2nd Turn and get the extra discard, you play Bazaar, tap Library holding priority, and tap Bazaar. If you hit on the 3rd Turn, you simply play and tap Bazaar then tap Library.
Unfortunately the numbers take us only so far. As you can see, we can hardly get into the discussion at all before we start hitting things like "I don't think you can win 87% of the games where Bazaar would win" or "I almost never lose where Bazaar would have won". I have a year of anecdotal, personal, biased experience. It's worth something, but it's fundamentally subjective and nonnumeric. I'd simply encourage you to give the card a try, see if it works for you, and keep these numbers in mind as a guide rather than possibly being definitive as to whether you should run Library.


« Last Edit: November 17, 2015, 12:46:49 pm by ajfirecracker »

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kingneckbeard on MTGO
"I fully believe that if Dredge could play a transformational sideboard it would just win all the tournaments yet it just doesn’t have one because there is just nothing that it can play. It’d be awesome to completely ignore all those very specific hate cards people bring against you but how are you going to do that?"  Paulo Vitor Damo da Rosa



vaughnbros


« Reply #12 on: November 17, 2015, 02:22:47 pm » 

Yeah, I agree. You can finetune your keeping decisions based on the number of Powders you have used. I don't think I'm going to be able to generate that, but I might try.
Yeah I'm not familiar enough with Python so it would take me a few hours to figure this out. I'm assuming there is a final denominator (deck size), and final number of draws (hand size) that could be output from each iteration of the simulation. Then we can just run a crosstab frequency on these to find the distribution. Adding the trichotomy of Bazaar, or library, or mull to oblivion adds more complications, but if we have the final distributions of both strategies we can compute the trichotomy from there. I think when we start considering only certain Library hands the math becomes irrelevant to a certain degree. The only thing we can conclude is that by playing library, and being willing to stop on a 7 with library, we reduce our mull to oblivion by a maximum of 1 in every 125 games we play. Considering this is a maximum, and it is still less than 1% it seems that the effect of adding library to our deck on reducing mulls to oblivion is negligible. If library allows us to keep more strong 7's, and therefore increases our win % in that fashion, I think that is a different story entirely as it becomes more of a statistics problem less of a probability problem.



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ajfirecracker


« Reply #13 on: November 17, 2015, 03:17:58 pm » 

It's not a simulation  the code breaks out all the combinatorial probabilities. It builds a big probability tree exactly once and gets you a precise answer.
I.e. the odds of drawing 01234 Serum Powder in your opening hand, and the odds of having Bazaar in each of those, and the odds of finding Bazaar on subsequent Powders/mulligans (and so on and so on)
I think 1 game in 125 is actually fine for a single card. A 5th Bazaar would be a lot higher impact, around 1 game in 40. Unfortunately, we can't run a 5th Bazaar. I'm curious as to what card(s) would provide more win percent than the Library  probably a fatter Dread Return package is the closest competitor.


« Last Edit: November 17, 2015, 03:29:34 pm by ajfirecracker »

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kingneckbeard on MTGO
"I fully believe that if Dredge could play a transformational sideboard it would just win all the tournaments yet it just doesn’t have one because there is just nothing that it can play. It’d be awesome to completely ignore all those very specific hate cards people bring against you but how are you going to do that?"  Paulo Vitor Damo da Rosa



vaughnbros


« Reply #14 on: November 17, 2015, 03:24:29 pm » 

It's not a simulation  the code breaks out all the combinatorial probabilities. It builds a big probability tree exactly once and gets you a precise answer.
I.e. the odds of drawing 01234 Serum Powder in your opening hand, and the odds of having Bazaar in each of those, and the odds of finding Bazaar on subsequent Powders/mulligans
Oh is there a way to output the probability of each branch then? Maybe make an empty 6x28 array with a spot for each possibility and then assign the probabilities to it there. EDIT: Alternatively, it looks like I could output the given probabilities directly from the code right now. ie. P(Find Bazaar  Deck=32,...,60; Hand=1,..,7). What I'm looking for is P(Deck=32,...,60; Hand=1,..,7  Found Bazaar). So I could use Bayes rule if its easier to find the overall P(Deck=32,...,60; Hand=1,..,7).


« Last Edit: November 17, 2015, 04:42:14 pm by vaughnbros »

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ajfirecracker


« Reply #15 on: November 17, 2015, 04:41:19 pm » 

I'll manually do the cases that are relevant to Library. Even on 7 cards, the probability tree grows quite a bit.
Odds of hitting a Bazaar mulling to 6 or fewer cards:
Given 0 Powders on 7: 88.658% Instead of mulliganing, if you keep a Library here you're 14.659% to draw Bazaar on your first draw step or Library activation You need Library by itself to win at least 86.710% of the games that Bazaar would win in order to keep Library rather than mulligan here Given 1 Powders on 7: 90.717% 16.812% to draw Bazaar via Library 88.841% relative wins to keep Library Given 2 Powders on 7: 92.693% 19.703% to draw Bazaar via Library 90.900% relative wins to keep Library Given 3 Powders on 7: 94.529% 23.790% to draw Bazaar via Library 92.821% relative wins to keep Library Given 4 Powders on 7: 96.159% 30.000% to draw Bazaar via Library 94.513% relative wins to keep Library Given 1 Powders activations that eat 2 Powders: 88.795% 16.812% to draw Bazaar via Library 86.531% relative wins to keep Library Given 1 Powders activations that eat 3 Powders: 86.726% 16.812% to draw Bazaar via Library 84.043% relative wins to keep Library Given 1 Powders activations that eat 4 Powders: 84.541% 16.812% to draw Bazaar via Library 81.417% relative wins to keep Library Given 2 Powders activations that eat 3 Powders: 90.777% 19.703% to draw Bazaar via Library 88.514% relative wins to keep Library Given 2 Powders activations that eat 4 Powders: 88.658% 19.703% to draw Bazaar via Library 85.875% relative wins to keep Library Given 3 Powders activations that eat 4 Powders: 92.646% 23.790% to draw Bazaar via Library 90.350% relative wins to keep Library
Odds of hitting a Bazaar when Powdering 7 cards:
Given 0 prior Powders: 96.136% Instead of Powdering, if you keep a Library here you're 14.659% to draw Bazaar on your first draw step or Library activation You need Library by itself to win at least 95.472% of the games that Bazaar would win in order to keep Library rather than Powder here Given 1 prior Powders: 97.124% 16.812% to draw Bazaar via Library 96.543% relative wins to keep Library Given 2 prior Powders: 97.967% 19.703% to draw Bazaar via Library 97.468% relative wins to keep Library Given 3 prior Powders: 98.649% 23.790% to draw Bazaar via Library 98.227% relative wins to keep Library Given 0 prior Powders, 1 spare Powders exiled: 94.794% 14.659% to draw Bazaar via Library 93.900% relative wins to keep Library Given 0 prior Powders, 2 spare Powders exiled: 93.204% 14.659% to draw Bazaar via Library 92.037% relative wins to keep Library Given 0 prior Powders, 3 spare Powders exiled: 91.385% 14.659% to draw Bazaar via Library 89.905% relative wins to keep Library Given 1 prior Powders, 1 spare Powders exiled: 95.856% 16.812% to draw Bazaar via Library 95.019% relative wins to keep a Library Given 1 prior Powders, 2 spare Powders exiled: 94.283% 16.812% to draw Bazaar via Library 93.128% relative wins to keep a Library Given 2 prior Powders, 1 spare Powders exiled: 96.785% 19.703% to draw Bazaar via Library 95.996% relative wins to keep a Library
"spare Powders exiled" includes the current hand as well as previous exiles
One thing worth pointing out is that it's better to Powder 4 of your Powders in a single activation than to mulligan that hand. I think that's the worstcase in terms of Powder usage, so it should always be correct to use the Serum Powder (if your only goal is to find Bazaar)
Regarding framing of the raw mulligan capability, I agree that "1 in 125 games" sounds pretty unimpressive. I think "cuts your mulligans to oblivion from 6 in 125 games to 5 in 125 games" sounds a lot more impressive and conveys the same information. I do tend to agree that Library has more value than just guarding against mulligans to oblivion, but I think getting almost 1% winrate from a single card is actually pretty good value. The best cards in your deck obviously add a lot more % wins than that (ex: boosting from 3 Bazaar to 4 Bazaar is worth well over 1%) but I don't know what card you're going to run over Library that provides 1% or more winrate from that slot. (And of course Library probably provides a few percentage points due to the other modes)


« Last Edit: November 17, 2015, 06:44:30 pm by ajfirecracker »

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kingneckbeard on MTGO
"I fully believe that if Dredge could play a transformational sideboard it would just win all the tournaments yet it just doesn’t have one because there is just nothing that it can play. It’d be awesome to completely ignore all those very specific hate cards people bring against you but how are you going to do that?"  Paulo Vitor Damo da Rosa



gkraigher
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« Reply #16 on: November 17, 2015, 09:42:07 pm » 

I don't know what card you're going to run over Library that provides 1% or more winrate from that slot. there isn't one. And pushing the "event rate" up 1% when it's close to 100% is even more difficult.


« Last Edit: November 18, 2015, 01:08:06 pm by gkraigher »

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vaughnbros


« Reply #17 on: November 17, 2015, 10:22:16 pm » 

I don't know what card you're going to run over Library that provides 1% or more winrate from that slot. there isn't one. And pushing the "winrate" up 1% when it's close to 100% is even more difficult. A maximum of 1 in 125 reduced mull to oblivion is not an expected return of 1% to your win %. I've won a number of games on a mull to oblivion, and there is no guarantee of victory by just not mulling to oblivion. I think there are certainly cards that could alter your win % by a lot more than 1%. We only play 60+15 cards after all. I think you are looking for hard proofs in an area where the reality is there is a lot of uncertainty. Optimal magic theory is a problem of statistics for which we don't have much data and our data is highly confounded by a number of unrecorded variables. It is not a probability problem with perfect knowledge, and absolute truths. Even if we were to assume that 1% is indeed the number why would we choose to play a card that boosts our game 1 percentage by 1% when we know we can get a lot more percentage points from playing cards to boost our game 2/3 percents. I agree its certainly a lot more difficult to go from 80% to 81% than from 40% to 41%. This is an arguement for cards that excel in game 2/3 not game 1 though. Especially considering we play more game 2/3s than game 1s. This all being said I think library is a fine card post board. It can single handedly win the card advantage war, which is something dredge usually can never do.



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