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Thread: Cantrip Math regarding 3-vs-4 of's.

  1. #21
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    Re: Cantrip Math regarding 3-vs-4 of's.

    Quote Originally Posted by wonderPreaux View Post
    Still, if you're at the point where you'd play over 4 cantrips to offset dropping 1 of a card, i cant help but wonder why you wouldnt just play 4 of the original thing in the first place
    It's more the idea of "You have to run Brainstorm in blue" and I wanted to check the math to see if this assumption made sense in the case of my very-quadlazer build. My bad math seems to imply it was and I was so excited by that fact I had to share it. Now that I'm wrong but it looks like I could be right with a better cantrip (or with a 5th cantrip?) it seems plausible enough to keep looking into it. I guess at this point it's an engineering challenge.

    If we solve the parity problem and you have access to blue; you can start really considering every card slot more critically from a statistician point of view. Maybe you could break it down to Matchups, your consideration of how much you expect to face each, and most-appropriately weigh cards to tailor the last 4-8 slots of your deck best.

    Notes:
    -If we want a "pure draw 3" then Preordain is nice and simple to avoid the "shuffle" problem.
    -"Preordain" as above with multiples would find it roughly 42-42.5% of the time (give or take .2 let's say)
    -I think Ponder will break parity, but probably only barely.

    Final thought until we can get some better math, I think it'd be best/easiest to write a program for this; as due to diminishing returns I'm also curious of 2-of vs. 3-of., 1-of vs. 2-of.


    EDIT: I mean, brainstorm being ~42% vs. 45% is pretty minor for all the other benefits it provides, but the idea that maybe that 5th cantrip breaks parity or what have you could allow for "truly right/wrong" choices in deckbuilding. Not running Brainstorm in blue seems like an egregious sin, but having evidence as to the "why" or having an idea of relative weight of a 4-of in a blue deck vs. a 4-of in a non-blue deck is telling of consistency problems. As a mostly non-blue player I'm very interested by the idea that a 4-of is like a 5-of in another deck, or a 3-of is like a 4-of in another deck. It's hard to wrap my head around the non-number side of "Well, I *really* want to see this card a lot" and then go cutting card slots for cantrips based on anecdotes.
    Quote Originally Posted by Nestalim View Post
    Wrong. Gideon Emblem protect you from losing and you can even open your binder and slam some cards on the board, not even the HJ can DQ you now.

  2. #22

    Re: Cantrip Math regarding 3-vs-4 of's.

    Ponder is slightly stronger in terms of finding individual cards than Brainstorm since you can see up to four cards with it.

    It's not that hard do this sort of thing with a spreadsheet. Assuming one dig for 3 cards, I get the following:

    Code:
    	Dig-3s											
    n-of	0                       1                       2                       3                       4                       5                       6                       7                       8
    1	0.13333333333333	0.14011299435028	0.14607442041691	0.15130374152799	0.15587939750018	0.15987269725773	0.16334834704671	0.16636494875035	0.16897546945542
    2	0.25084745762712	0.26253652834600	0.27279009915203	0.28176197360731	0.28959197313192	0.29640697271815	0.30232187801941	0.30744054606857	0.31185665183648
    3	0.35417884278200	0.36925569329519	0.38244793749423	0.39396116879521	0.40398194418681	0.41267922094178	0.42020571044127	0.42669915236241	0.43228351241458
    4	0.44482040870733	0.46206076266060	0.47710688974708	0.49020259295199	0.50156867497888	0.51140470750216	0.51989069634577	0.52718864675128	0.53344403281314
    5	0.52413177889200	0.54256359777292	0.55860610679891	0.57252979387808	0.58457913846581	0.59497465144347	0.60391479260425	0.61157777074206	0.61812323123478
    6	0.59334897468953	0.61221289354422	0.62858535141810	0.64275382457819	0.65497760534376	0.66549005680215	0.67450072948077	0.68219734572709	0.68874765742608
    7	0.65359357103182	0.67230833140500	0.68850379711256	0.70247635576222	0.71449275620093	0.72479252800553	0.73359024975530	0.74107767252106	0.74742570486594
    8	0.70588133389494	0.72401370359230	0.73965731666454	0.75311082390666	0.76464240154277	0.77449229077361	0.78287517522539	0.78998240334755	0.79598406265071
    Edit:
    For similar assumptions, 3 + 10 cantrips is comparable to a 4 of.

  3. #23

    Re: Cantrip Math regarding 3-vs-4 of's.

    Quote Originally Posted by rufus View Post
    Ponder is slightly stronger in terms of finding individual cards than Brainstorm since you can see up to four cards with it.

    It's not that hard do this sort of thing with a spreadsheet. Assuming one dig for 3 cards, I get the following:

    Code:
    	Dig-3s											
    n-of	0                       1                       2                       3                       4                       5                       6                       7                       8
    1	0.13333333333333	0.14011299435028	0.14607442041691	0.15130374152799	0.15587939750018	0.15987269725773	0.16334834704671	0.16636494875035	0.16897546945542
    2	0.25084745762712	0.26253652834600	0.27279009915203	0.28176197360731	0.28959197313192	0.29640697271815	0.30232187801941	0.30744054606857	0.31185665183648
    3	0.35417884278200	0.36925569329519	0.38244793749423	0.39396116879521	0.40398194418681	0.41267922094178	0.42020571044127	0.42669915236241	0.43228351241458
    4	0.44482040870733	0.46206076266060	0.47710688974708	0.49020259295199	0.50156867497888	0.51140470750216	0.51989069634577	0.52718864675128	0.53344403281314
    5	0.52413177889200	0.54256359777292	0.55860610679891	0.57252979387808	0.58457913846581	0.59497465144347	0.60391479260425	0.61157777074206	0.61812323123478
    6	0.59334897468953	0.61221289354422	0.62858535141810	0.64275382457819	0.65497760534376	0.66549005680215	0.67450072948077	0.68219734572709	0.68874765742608
    7	0.65359357103182	0.67230833140500	0.68850379711256	0.70247635576222	0.71449275620093	0.72479252800553	0.73359024975530	0.74107767252106	0.74742570486594
    8	0.70588133389494	0.72401370359230	0.73965731666454	0.75311082390666	0.76464240154277	0.77449229077361	0.78287517522539	0.78998240334755	0.79598406265071
    Edit:
    For similar assumptions, 3 + 10 cantrips is comparable to a 4 of.
    Yay, my estimation was right at least :P

  4. #24
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    Re: Cantrip Math regarding 3-vs-4 of's.

    Quote Originally Posted by rufus View Post
    Edit:
    For similar assumptions, 3 + 10 cantrips is comparable to a 4 of.
    Except that misses all of the "I have/found 2 cantrips" cases; which I think ~6 will put you at ~44%, but my brain isn't working very well with percentages at the moment.
    Quote Originally Posted by Nestalim View Post
    Wrong. Gideon Emblem protect you from losing and you can even open your binder and slam some cards on the board, not even the HJ can DQ you now.

  5. #25

    Re: Cantrip Math regarding 3-vs-4 of's.

    Quote Originally Posted by tescrin View Post
    Except that misses all of the "I have/found 2 cantrips" cases; which I think ~6 will put you at ~44%, but my brain isn't working very well with percentages at the moment.
    Your original described case was just a single Brainstorm, so that makes sense. Multiple Brainstorms aren't even that great unless you're assuming you always have a shuffle effect.

  6. #26
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    Re: Cantrip Math regarding 3-vs-4 of's.

    Quote Originally Posted by wonderPreaux View Post
    Your original described case was just a single Brainstorm, so that makes sense. Multiple Brainstorms aren't even that great unless you're assuming you always have a shuffle effect.
    But I've covered that consideration by considering that you are instead using preordain for the sake of making lives easy. Same goes for ponder really.
    Quote Originally Posted by Nestalim View Post
    Wrong. Gideon Emblem protect you from losing and you can even open your binder and slam some cards on the board, not even the HJ can DQ you now.

  7. #27

    Re: Cantrip Math regarding 3-vs-4 of's.

    Quote Originally Posted by tescrin View Post
    But I've covered that consideration by considering that you are instead using preordain for the sake of making lives easy. Same goes for ponder really.
    Except that with Preordain if card 9 or 10 is a cantrip (and you don't have the card you're looking for) you will then wait until the next cantrip to look at card 11.

  8. #28

    Re: Cantrip Math regarding 3-vs-4 of's.

    I should probably also do calculations for a+b type combos.

  9. #29
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    Re: Cantrip Math regarding 3-vs-4 of's.

    Quote Originally Posted by rufus View Post
    Edit:
    For similar assumptions, 3 + 10 cantrips is comparable to a 4 of.
    Does this lead to the assumption that playing 11 Three off's and 10 Cantrips is more consistent at findinf any particular card during a game than a deck playing 11 four off's?

    (No clue this is right but with simple algebra 3X+10 = 4X, where X is how many individual cards you are playing. X is 10 in this case, so running 10 three-off's plus 10 cantrips should be comparable to running 10 four off's. Any more than that, and the balance should go the other way.)

    I know this is useless - since realistically you are still playing four off's with the cantrips themselves.

  10. #30
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    Re: Cantrip Math regarding 3-vs-4 of's.

    Quote Originally Posted by rufus View Post
    Except that with Preordain if card 9 or 10 is a cantrip (and you don't have the card you're looking for) you will then wait until the next cantrip to look at card 11.
    Hm. That is a good point. I think I want to get the probability straight for Ponder and add one at a time to Brainstorm x4 to see the breaking point, because this is most realistic. I guess this breaks down to:

    Chance of natural +
    Brainstorm -> Chance of finding CARD +
    (Brainstorm -> Brainstorm, or Ponder) * (chance of those Cantrips finding CARD) +
    Ponder -> Chance of finding CARD +
    (Ponder -> Chance of finding Brainstorm (or ponder if it's more than 1 copy) * (chance of those cantrips finding card)

    Where the cantrip->cantrip have to be considered based on their individual chances of finding CARD and such. I imagine I'll resurrect this topic once I've induldged myself in the breaking points of each using the standard 4 Brainstorm + X Ponder schematic


    @Others
    You don't need 10/11, due to the multiples stacking; though we've not yet nailed a number (in part because each cantrip functions differently and tend to be played in different turns.)
    Quote Originally Posted by Nestalim View Post
    Wrong. Gideon Emblem protect you from losing and you can even open your binder and slam some cards on the board, not even the HJ can DQ you now.

  11. #31

    Re: Cantrip Math regarding 3-vs-4 of's.

    Quote Originally Posted by Cire View Post
    Does this lead to the assumption that playing 11 Three off's and 10 Cantrips is more consistent at finding any particular card during a game than a deck playing 11 four off's?

    (No clue this is right but with simple algebra 3X+10 = 4X, where X is how many individual cards you are playing. X is 10 in this case, so running 10 three-off's plus 10 cantrips should be comparable to running 10 four off's. Any more than that, and the balance should go the other way.)

    I know this is useless - since realistically you are still playing four off's with the cantrips themselves.
    The contribution from repeated cantrips should get significant at some point, so, in practice, so if you allow for repeated cantrips, 10 3-ofs and 10 cantrips is probably already more consistent.

    The trick is that it scales so that playing 7 'four of" cards and 7 cantrips is better than playing 7 5 ofs for finding a key card, even without repeated digging. (Assuming you can afford the hit to tempo.)

    I guess that chart is too small in the cantrip direction and should go at least to 12, and probably to 16.

  12. #32

    Re: Cantrip Math regarding 3-vs-4 of's.

    Quote Originally Posted by tescrin View Post
    Hm. That is a good point. I think I want to get the probability straight for Ponder and add one at a time to Brainstorm x4 to see the breaking point, because this is most realistic. I guess this breaks down to:

    ...

    @Others
    You don't need 10/11, due to the multiples stacking; though we've not yet nailed a number (in part because each cantrip functions differently and tend to be played in different turns.)
    I should like to point out that in the 'find a single card by turn 2' scenario, Ponder and Portent(!) are better than Brainstorm because the chance to hit off the shuffle if you miss is only marginally worse than seeing a fourth card. A back of the napkin estimate suggests that 8 3of cards + 4 Ponder + 4 Portent is probably going to be in the same neighborhood as playing 8 four-ofs for that specific scenario.

  13. #33

    Re: Cantrip Math regarding 3-vs-4 of's.

    This should be a more extended table. I added one where cantrips dig for 4 cards to approximate Ponder/Portent

    Code:
    3      0     1     2     3     4     5     6     7     8     9     10    11    12    13    14    15    16   
     1     0.133 0.140 0.146 0.151 0.156 0.160 0.163 0.166 0.169 0.171 0.173 0.175 0.176 0.177 0.178 0.179 0.180
     2     0.251 0.263 0.273 0.282 0.290 0.296 0.302 0.307 0.312 0.316 0.319 0.322 0.324 0.326 0.328 0.329 0.330
     3     0.354 0.369 0.382 0.394 0.404 0.413 0.420 0.427 0.432 0.437 0.441 0.445 0.448 0.450 0.452 0.454 0.455
     4     0.445 0.462 0.477 0.490 0.502 0.511 0.520 0.527 0.533 0.539 0.543 0.547 0.550 0.553 0.555 0.557 0.559
     5     0.524 0.543 0.559 0.573 0.585 0.595 0.604 0.612 0.618 0.624 0.628 0.632 0.636 0.639 0.641 0.643 0.644
     6     0.593 0.612 0.629 0.643 0.655 0.665 0.675 0.682 0.689 0.694 0.699 0.703 0.706 0.709 0.711 0.713 0.715
     7     0.654 0.672 0.689 0.702 0.714 0.725 0.734 0.741 0.747 0.753 0.757 0.761 0.764 0.767 0.769 0.771 0.772
     8     0.706 0.724 0.740 0.753 0.765 0.774 0.783 0.790 0.796 0.801 0.805 0.809 0.812 0.814 0.816 0.818 0.819
     9     0.751 0.768 0.783 0.796 0.807 0.816 0.824 0.830 0.836 0.841 0.845 0.848 0.850 0.853 0.854 0.856 0.857
     10    0.790 0.806 0.820 0.832 0.842 0.851 0.858 0.864 0.869 0.873 0.877 0.879 0.882 0.884 0.885 0.887 0.888
     11    0.824 0.839 0.851 0.862 0.871 0.879 0.886 0.891 0.896 0.899 0.903 0.905 0.907 0.909 0.910 0.911 0.912
    4      0     1     2     3     4     5     6     7     8     9     10    11    12    13    14    15    16   
     1     0.133 0.142 0.150 0.157 0.163 0.169 0.173 0.177 0.181 0.184 0.186 0.189 0.191 0.192 0.194 0.195 0.196
     2     0.251 0.266 0.280 0.292 0.302 0.311 0.319 0.326 0.331 0.336 0.341 0.344 0.347 0.350 0.352 0.354 0.356
     3     0.354 0.374 0.391 0.406 0.419 0.431 0.440 0.449 0.456 0.463 0.468 0.472 0.476 0.480 0.482 0.485 0.486
     4     0.445 0.467 0.487 0.504 0.518 0.531 0.542 0.551 0.559 0.566 0.572 0.577 0.581 0.585 0.588 0.590 0.593
     5     0.524 0.548 0.568 0.586 0.602 0.615 0.626 0.636 0.645 0.652 0.658 0.663 0.667 0.671 0.674 0.676 0.678
     6     0.593 0.617 0.638 0.656 0.672 0.685 0.696 0.706 0.714 0.721 0.727 0.732 0.737 0.740 0.743 0.745 0.747
     7     0.654 0.677 0.697 0.715 0.730 0.743 0.754 0.763 0.771 0.778 0.784 0.789 0.793 0.796 0.799 0.801 0.803
     8     0.706 0.728 0.748 0.765 0.779 0.791 0.802 0.811 0.818 0.824 0.830 0.834 0.838 0.841 0.843 0.845 0.847
     9     0.751 0.772 0.791 0.806 0.820 0.831 0.841 0.849 0.856 0.861 0.866 0.870 0.873 0.876 0.878 0.880 0.882
     10    0.790 0.810 0.827 0.841 0.853 0.864 0.873 0.880 0.886 0.891 0.896 0.899 0.902 0.904 0.906 0.908 0.909
     11    0.824 0.842 0.857 0.870 0.881 0.891 0.899 0.905 0.911 0.915 0.919 0.922 0.925 0.927 0.928 0.930 0.931
    So if I've done the math right, and my approximation is accurate, a 4 of with 5 'draw 4' cantrips is close to having a 5-of for the purposes of getting a single card.

    Edit: This is all 'on the draw' so assuming 8 initially drawn cards.

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