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Rule of Law – OBP and The Mulligan Decision

 

What is Optimization By Proxy (“OBP”)?

Optimization by proxy sounds like a playtesting method by which you write on cards with a Sharpie. It isn’t.

I was just recently introduced to this concept by an article on the rationality blog “Less Wrong.” That blog post can be found here.

The Less Wrong post discusses a fascinating study about what makes herring gull hatchlings peck their mother’s beak to obtain food:

The first thing a newly-hatched herring gull does after breaking out of its shell is to peck on its mother’s beak, which causes her to give it its first feeding. Puzzled by this apparent automatic recognition of its mother, Dutch ethologist and ornithologist Nikolaas Tinbergen conducted a sequence of experiments designed to determine what precisely it was that the newborn herring gull was attracted to. After experimenting with facsimiles of adult female herring gulls, he realized that the beak alone, without the bird, would elicit the response. Through multiple further iterations he found that the characteristics that the newborns were attracted to were thinness, elongation, redness and an area with high contrast. Thus, the birds would react much more intensely to a long red stick-like beak with painted stripes on the tip than they would to a real female herring gull. It turns out that the chicks don’t have an ingrained definition of ‘motherness’ but rather determine their initial actions by obeying very simple rules, and are liable to radically miss the mark in the presence of objects that are explicitly designed to the specification of these rules.

“¦

Generalising the above example, we can say that Optimization by Proxy occurs when an algorithm substitutes the problem of measuring a hard to quantify attribute, with a usually co-occurring a proxy that is computationally efficient to measure.

I started thinking about the many ways this applies to Magic: The Gathering strategy, and the most salient one for me was the mulligan decision. Conley Woods and PV recently wrote about the mulligan decision (with the articles here and here), and their examples and analysis were a great read. I do, however, feel that those articles lacked an ability to generalize across formats, situations, and even other decisions besides mulliganing. Walking through examples is helpful, but what do people use when they encounter a novel situation (besides ask it where Snookie is at)? It turns out people encounter novel situations all the time, and each one is kind of like hatching out of a herring gull egg in a way. In each situation, we have heuristics, or shortcuts, with which we try to optimize our decision. We can’t do a complete analysis of a Magic game decision any more than a baby gull can do a maternity test. What we can do is understand the shortcomings of our heuristics, which then helps us identify and avoid traps.

THE MULLIGAN DECISION

Whether or not to mulligan is such a complex problem that it is no surprise we all use heuristics to help us solve it. To correctly determine whether or not to mulligan a given hand, we would need to determine the likelihood of winning if we mulligan, and compare that to the likelihood of winning if we don’t mulligan (for the statisticians among you, p(win|mulligan) vs. p(win|~mulligan)). Breaking that down a bit, we’ll need to know the probability of every possible game that could occur given our present hand, as well as the win/loss outcome for each of those games. We’ll also need the probability of every possible post-mulligan game and corresponding outcome, and then we can compare the probability of winning given a mulligan to the probability of winning given a keep. It should be obvious that we cannot complete this task, and we certainly can’t even come close to figuring all this out in the minute or so we have to make a decision. Remember though, that the baby herring gull isn’t equipped to definitively identify its mother moments after hatching. To do that it would need to perform an analysis far too complicated given its ability and the time within which it must make a decision and act. The gull instead uses an algorithm that identifies thinness, elongation, redness and an area with high contrast as a proxy for a process which would identify its mother. What are the algorithms we use to determine the mulligan decision in place of the complex analysis given above? What are our “long red sticks” that we look for since we can’t look for “what really matters”?

Below is a list of things we often look for in our hand, and a discussion of when optimization by proxy with regard to this particular stimulus can lead us astray. If you haven’t yet learned to look for these things, this list is a good primer on mulligan strategy. If you’re just learning how to mulligan more effectively, be particularly wary of the “all my colors” and “particular important cards” methods, for now, since those are common traps for beginners. Everyone should be looking to avoid the situations where our OBP can do more harm than good, discussed below each method. I proceed (roughly) starting with simple heuristics and working my way up to the more nuanced, as a beginning player might in his Magic career.

Land-to-spell ratio

We’ll start with perhaps the most common heuristic, “what is my land-to-spell ratio?” A beginner using this method might look at whether he/she has a “good” or “bad” ratio of lands-to-spells, with 1:6 being bad, for example, and 3:4 being “good.” A slightly more experienced player might view all the ratios on a continuum from 0:7 to 7:0, rather than binary categories, but this player is still using the land-to-spell ratio as a proxy for the strength of the hand.

OBP based on land-to-spell ratio can be problematic in a number of ways. The most obvious is that it disregards which lands and which spells have been drawn. Taking the Standard Naya deck as an example:

 

We wouldn’t want to keep this, but the land-to-spell ratio doesn’t tell us that. That’s the obvious case, so without much analysis let’s look next at a hand like so:

 

I imagine this is an easy mulligan for top Naya players like The Boss or LSV, yet I suspect many beginning Naya players will keep this hand. I urge them not to. Card quality matters. Even if you have enough land to cast all of your spells, and you’ve got three spells to cast, this is too little information to reliably arrive at a correct decision by proxy.

“All my colors”

In a multicolor deck, such as a 3- or even 2-color Limited deck, there is a temptation to use the following algorithm: If the hand can produce all my colors of mana, then do not mulligan the hand. Players might add a level of sophistication by running the hand through the land-to-spell algorithm as well before deciding to keep, resulting in: If all my colors of mana can be produced AND the ratio of lands to spells isn’t extreme, then do not mulligan the hand. The above example of Mountain, Forest, Plains, Sejiri Steppe, Stoneforge Mystic, Basilisk Collar, Behemoth Sledge again is illuminating here as well. I mentioned above that I suspect many beginners would keep this hand, but I didn’t say why. It is the commonly used algorithm I just described that explains it. The “If all my colors of mana can be produced AND the ratio of lands to spells isn’t extreme, then do not mulligan the hand” algorithm has all the makings of a great OBP trap. It very often accurately approximates the optimal result; lands that can produce all your colors and have 3-4 spells in them tend to be keepable hands. Furthermore, in the situations in which the algorithm leads us astray, it is often non-obvious that our mulligan decision was flawed. Since the hand does allow us to play lands and some spells, by definition, we won’t lose due to mana screw very often. How can we tell which of our decisions was flawed when we look back on the game? Many of us use whether we got mana screwed as a proxy for whether our decision to keep was good or bad! Thus, in another example of OBP, we don’t even examine our poor mulligan decision because this “wasn’t one of those games where I kept a 2 lander and never drew a land” or the like. If you’re one of the players who uses this kind of mulligan strategy or a similar variant, take a close look at whether you’re considering more than just colors of mana and land-to-spell ratio.

Productive early turns

Some people realize that “how the game is likely to play out” is an important element in predicting the outcome of the game. Since no one can fully predict how the game is going to play out (and if you could, that would be your answer), one proxy for this is to answer the much simpler version, “How are the early turns of this game likely to play out.” This is a productive exercise, but as with any OBP, there is a downside. The downside here should be fairly obvious: the early game isn’t the entire game. This error shows up as a sort of “overcompensation.” I say overcompensation because the natural tendency for Magic players seems to be to mulligan too infrequently. Well, some players have figured this out, and they use methods like looking for “productive early turns” to make sure they aren’t keeping poor hands. In doing this they can develop algorithms that instruct them to mulligan too much. Fortunately for me, Rise of the Eldrazi is the perfect context to illustrate such an error. In ROE sealed deck it could be perfectly acceptable to keep a hand with 5 land, a 4 drop such as Kozilek’s Predator or Ondu Giant, and an 8 mana spell, such as Ulamog’s Crusher. In Zendikar x6 sealed deck, you would be crazy to keep such a hand, but that’s the nature of OBP: contextual differences are what make your heuristic fail.

Particular important cards

This example of OBP can trick the beginner and the expert alike. Sometimes a card is perceived to be so important that its presence in the hand is used as a proxy for whether the hand should be kept or mulliganed. Of course, you wouldn’t keep a 7-spell hand, so we can assume you’re running the hand through one or more other algorithms, like “all my colors” or “land-to-spell ratio” or both. Still, some draws won’t be good enough to allow you to survive to cast the bomb you’ve drawn. Even more subtly, sometimes the card you think is of supreme importance isn’t as important as you think! The opponent may have sideboarded 5 cards that kill your Baneslayer or counter your Ajani, so what was critical game 1 might be a trap game 2. Looking for important cards is something to be aware of, but we need to be careful to only let it be a proxy for strength of hand in very limited circumstances, such as when your only way to have a reasonable chance at victory is to begin the game with a particular card (think Leyline of the Void in constructed or a bomb rare in limited in a very, very lop-sided matchup).

“Is a 6 card hand likely better?”

This heuristic is so broad that is almost no help at all. It is perhaps more question than answer (it even contains punctuation to that effect). However, this can still be OBP since it isn’t the same thing as probability of winning given mulligan versus probability of winning given no mulligan. There are several ways “If a 6-card hand is likely better than these 7 cards, then mulligan” is a flawed operation. First, it doesn’t account for how much better the typical 6-card hand is. Imagine hand A has a 40% chance of winning. 51% of my 6-card hands will have a 45% chance of winning, while the other 49% will have a 10% chance of winning. This example is too simplified to be a real scenario obviously, but we can see the problem nonetheless. (.51 x .45) + (.49 x .1) = .3145, so in other words our chances of winning when we mulligan are only around 31%, and we gave up 40% to get there. This is true even though it is likely that a 6-card hand is better than our current 7-card hand. You can adjust the heuristic to account for this deficiency, but you’ll find that either a) there is still something you aren’t accounting for, or b) your heuristic isn’t a shortcut at all, it is a restatement of the original problem “should I mulligan this hand?” Perhaps surprisingly, I still find this kind of heuristic to be helpful. Framing the problem in terms of your “likely” or “average” hand of 1 less card that you presently hold at least means you aren’t using only the more seriously flawed heuristics listed above, and it means that you are taking a big-picture approach rather than focusing on certain details by proxy.

“If I draw a land, will I win? If yes, the odds of drawing land determines whether I keep.”

This is a popular shortcut for the land-light hand. The probability of drawing a land (or two straight lands, or whatever the case may be) becomes a proxy for the odds of winning. This has at least two problems. 1) the odds of winning if you keep are only half the equation, it doesn’t say anything about the odds of winning if you mulligan, and you need both sides of the coin to do a comparison, and 2) it is almost never the case that if you draw your land or two land, your chances of winning rise to 100%, which is what is implied in the stated shortcut. As with all the other heuristics, knowing the exceptions allows you to better utilize the rule. There are plenty of times you’ll want to go ahead and use the odds of drawing a third land as a proxy for the odds of winning, but remember to view the 6-card hand’s chances of winning as the other side of the coin, and that you need to back off the chance of winning by some amount that reflects the chances of drawing a land and still losing the game.

CLOSING THOUGHTS

So how do I advocate approaching the mulligan decision? What we need to keep in mind is that all we can hope to do is find the correct OBP, we can’t hope to avoid OBP altogether. What works for me is a combination of intuition, “is the average 6-card hand better than these 7,” and trying to figure out how the game will play out, as far out as I can reliably see. I’ve written previously about the value of intuition and of listening to and refining your “gut” instinct. I stand behind that sentiment, and once again add that intuition isn’t the only thing to consider; you need to check your intuition against rational thinking, and check your rational thinking against your intuition. If they agree, great, act. If they disagree, you must be prepared to decide which tool seems most likely to fail in the given context, and act accordingly.

I usually start with a “feeling” or gut reaction about the hand, but I then make sure to be cognizant of how my deck desires to have the game go. Do I want early plays? Do I need early plays? Is finding my one [card]Sphinx of Jwar Isle[/card] more important than having a turn 2 play? Does my opponent’s deck put me on a fast clock? Over time you learn which questions to be asking, and I recommend these 2 recent articles from Paulo and Conley to begin to think about what questions to ask. I’m more concerned here with the process, and those articles largely only talk about one step, the rational explanations for keeping a hand. I wouldn’t even use that as a starting point, rather, I get a feel for the hand first, then do some analysis about whether a 6 card (or 5 card if I’m already at 6, etc.) presents a better chance of winning, given all the heuristics I have available, while being cognizant of their shortcomings. If I feel that the “all of my colors” aspect of my hand is one of the reasons it “feels” like a keep, I’ll bring to my conscious mind the ways I can lose a game even when I have all my colors of mana. You could say I consider these things: intuition, applicable heuristics, and contexts where those heuristics may fail.

-Matt Sperling
@mtg_law_etc on twitter

41 thoughts on “Rule of Law – OBP and The Mulligan Decision”

  1. good read, good article.

    mulliganing is hard.

    just thought i throw that out there.

  2. What about:
    I’m on the play/draw
    And
    My opponent mulled as well.

    Do you consider these legit factors without a trap, or are they covered within another point?

    Nice stuff btw, but i dont think conleys article was to example based. It just kopjes that way by al the comments.

  3. Interesting read and pretty helpful. Unfortunetely these things tend to fall into category of “X is bad, but we don’t know what is good” so they can only be so helpful. However, recognizing these traps is the first step to solving the problem.

  4. Good Read… Not great… GOOD. Why? Because first off do or do not there is no try. AND BUT negates everything before it. Which I see a lot of on this site… If you are going to go off with your amazing vocabulary at least master the simple words. 🙂

    Good Read. Slightly pompous… Peace!

  5. @morkje,

    Yes these are things to consider. Whether your opponent mulliganned, whether you are on the play or draw, etc etc. There are too many things to list, but I recommend the Conley and PV articles for tips on a few things to keep in mind. These factors don’t tend to lead to “rules” that trap players. They are factors but not really anything that people use as a “proxy” for the decision itself.

  6. Interesting. Very cool article. Today my opponent mulled and I was staring at a somewhat crappy hand (I don’t remember it, sorry), that I should probably mull as well. I decided to, and said, “Well, if you’re going to, then I guess I will too.” He stopped and said, “That’s a terrible reason to mulligan.” As soon as he said that I realized how true it was.

    While operating on that obp alone is a bad idea, I will add that I think your opponent’s choice ca

  7. Interesting. Very cool article. Today my opponent mulled and I was staring at a somewhat crappy hand (I don’t remember it, sorry), that I should probably mull as well. I decided to, and said, “Well, if you’re going to, then I guess I will too.” He stopped and said, “That’s a terrible reason to mulligan.” As soon as he said that I realized how true it was.

    While operating on that obp alone is a bad idea, I will add that I think your opponent’s choice to keep or mull can be one of a few factors. However, I only think that intuitively, I can’t explain why it’s a factor. Maybe someone else can.

  8. It’s a factor because it tells you something about thier hand. While a mulligan from a bad hand could lead into a 6-card “god draw,” an average keepable 6-card hand is inherently worse than an average keepable 7-card hand for reasons of card advantage.

    It should never be used as a major reason for your decision to mulligan or not, but if your hand is somewhat on the fence, it should definitely be a factor. There are far to many variables to outline, but it will often come down, basically, to how strong is the need to mulligan. If the hand is closer to a solid keeper, then your opponent’s card disadvantage will often solidify it’s strength. If it was a strong candidate for a mull, but you were hesitating to take the card disadvantage, then your opponent doing so might give you that breathing space to dig for a better, though smaller, hand.

    Not hard and fast rules, but things to keep in mind if your hand is on the fence otherwise.

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  10. I almost reason the other way:

    If I have a shaky hand, and my opponent mulls I am more inqlined to keep since there is more of a chance I can get away with it.
    This is mostly true when I keep a slow hand, or need another colour land ( talking limited here, dont play constrcuuted that much.

    With the old mull rules it was even better. With an opponent going to 5 on the play, many 7 cards hands become a keep for me, but I can see why you would want to mull as well.
    It obviously depends on the decks and matchups as well.

    I would love to read more on this topic:
    Are you more inqlined to mull if your opponent mulled or would you rather keep?

  11. When my opponent mulligans I am more likey to keep a land light but otherwise good hand, especially on the draw, hoping that ill have time to find it.

    I’m more likely to mulligan a below average hand with land and spells that doesnt really do anything becuase ill still be on equal footing with my opponent in cards and hopefully be able to get a better hand.

    I don’t know if that is correct but its part of my OBP.

  12. Another important thing is looking at the top card(s).
    Online, this isnt possible, but IRL I see people do this all the time.

    They have 5 spell/2 land hand, en decide to mull but look at the top 2 cards. They see 2 land and say “I should have kept”.
    Something is either the right or wrong decision but that’s never based on the top card(s) of your deck.
    Doing this will influence any future mulls by “giving” you information that’s not true.
    If you look 3 times and see the lands you needed, the 4th time you get a similar hand you are more inclined to keep cause “it would have worked the last time”

    So never look, or if you do look, stand by your decission.

  13. @morkje: the whole peeking at cards thing, it’s called results oriented thinking, and countless articles have been written on the subject pertaining to magic.

    I thought this was a great article, the connection between the bird’s recognition of their mothers and mulligan decisions was excellent.

  14. “Generalising the above example, we can say that Optimization by Proxy occurs when an algorithm substitutes the problem of measuring a hard to quantify attribute, with a usually co-occurring a proxy that is computationally efficient to measure.”

    im an idiot.

  15. I think articles on mulliganing are awkward because so many pro’s have different idea’s on what to mulligan.

  16. “Generalising the above example, we can say that Optimization by Proxy occurs when an algorithm substitutes the problem of measuring a hard to quantify attribute, with a usually co-occurring a proxy that is computationally efficient to measure.”

    Hilarious that you copied this word-for-word from the linked article. If you actually understand this concept, I’m sure you can come up with a paraphrase that makes more sense to the casual reader. As it is — FAIL.

  17. I prefer to use OBV for my mulligan decisions: Optimization by Visualization. I visualize the possible results of each choice and generate a decision based on my visions. So when I’m about to make the choice I usually think, “Stop! Should I mulligan? OBV.” to trigger this mode of decision making.

  18. I agree with you that my mulligans article was kind of specific, and that you will often encounter very different situations, which was why I linked to a previous article (this one http://www.starcitygames.com/magic/fundamentals/17391_PVs_Playhouse_Mulligans.html ) and said “this article is just applications from that other article – theory in practice”. Apparently no one bothered to read it and I should have just C/Ped the entire article again as it would have prevented many of the comments I got.

    I think that if you read it, you will agree that it is very broad and includes basically everything you’ve written in here 😉

  19. great article. gonna read it again later because i feel like there’s alot here. good dense text.

    also, your slip is showing. computer scientist?

  20. One of the best players in Brazil once enlightened me on another criteria envolving mulligan decisions.

    I was testing a Project X deck, in Standard with RAV block + TSP block, and had the following opening hand:
    Birds, Teysa, Nekrataal, Mortify, GW shockland, BW shockland, Forest.

    I snap kept and he immediately berated me for doing it… I had all my colors, a good land:spell ratio, a good curve… I couldn’t understand what was wrong.

    After losing the game, he told me:
    If you can play all the cards in your opening hand, it doesn’t automatically make it a keep. Can you sculpt a game plan with your hand? Most importantly, can you WIN with it?

    Of all the sick things that my deck could do, I stuck to a Teysa + Nekrataal beatdown plan. Not the brightest decision…

  21. This paragraph is a quote from the article I linked to. I submitted a draft in Microsoft Word with the entire thing block quoted, but in the conversion to HTML there appears to have been a typo which prematurely turned off the italics.

    “Generalising the above example, we can say that Optimization by Proxy occurs when an algorithm substitutes the problem of measuring a hard to quantify attribute, with a usually co-occurring a proxy that is computationally efficient to measure.”

    Sorry for any confusion this typo caused.

  22. Hey Matt,

    Good article, definetly helped my mull decisions. And I took your advice from your other article about mythic, played it and won my National qualifier.

  23. Hey Matt,

    Good article, definetly helped my mull decisions. And I took your advice from your other article about mythic, played it and won my National qualifier.

  24. Very good read! There was an article posted on the Mothership by Steve Sadin a couple months back that I personally use a lot as my guide (then again I play a lot more Limited than Constructed). I don’t remember which article it was exactly, but the point was something like this (keep in mind the context is a Limited deck, but it sometimes works in Constructed too):

    If you have a fast deck and draw a slow hand, you should probably mulligan. If you have a slow deck and draw a slow hand, use some other algorithm.

    I believe the exact example was some kind of Green/Red concoction in ZZZ with Goblin Shortcutters, River Boas, Oran-Rief Survivalists, etc, and the hand was 4 land, a Baloth Woodcrasher, a Territorial Baloth, and a Shatterskull Giant, or something like that. Forgetting for a moment the speed of ZZZ, the chances were (if you had the entire decklist) that your 6 would have roughly 2 2-drops in it with which you could actually do stuff.

    On the other hand, in my RRR draft last night, I drafted a deck that was very low on the 1s and 2s but had lots of quality cards at 3 and 4, and I often found myself keeping 4 land hands with nothing to do until turn 4 because I knew my deck wasn’t getting there if I mulled.

    Additionally, another algorithm I use is how many “mulligans” are already in my hand. By “mulligan” I mean a card that is very difficult or impossible to cast with the resources in my hand. A hand with Drana, Kalastria Bloodchief, 3 Mountains, and 3 red spells, for example, has 1 mulligan in it, since Drana is really hard to cast off 3 Mountains, even if you do draw an additional 2 land. In this case, you would evaluate your other 6 cards vs. what you think of any random 6 in your deck, and if you think you have a better average 6 than the 6 you have now, you pitch it back.

    Yesterday in RRR I made the mistake of keeping a 3-mulligan 4-land hand in game 2 after winning game 1 and got smashed. It was an easy mulligan but it had a lot of potential so I fell into the trap and lost.

  25. I think the biggest mistake people make when making mulligan decisions is that they try to treat every deck they play the same. Pointing out which shortcuts are potential traps only gets you so far, and really doesn’t give you the whole picure of shat you SHOULD be doing (verus what I should NOT be doing). Each deck mulligans under different rules, and only through playtesting can you truly know which hands to keep and which to mull.

    I think that rather than talking about mulligans in a general sense that it makes much more sense to discuss mulligan strategy within the context of a particular deck, just as you would talk about sideboard strategy or matchup analysis. This would give the most insight to players on the subject, and to be honest it is the one thing that I have paid more attention to in my game that has made me improve.

  26. @Jon Lewis,

    You’re right that mulligan strategy can be very deck-dependent. If you start to learn or create deck-specific algorithms, it is then important to recognize the ways those algorithms could lead to an undesired result. The more you lean on a rule, the greater your exposure to the rules’ exceptions. That’s what my article is all about really. I don’t present a definitive set of rules for all decks and all situations (for limited especially this is impossible), I merely present some common rules in an attempt to illustrate that ANY optimization by proxy is bound to be inaccurate in some contexts. We should all work on recognizing these contexts whenever we find ourselves using a pattern or rule.

    The "what not to do" theme of the article is important. Other people will tell you what TO do, i.e. what rules to follow or shortcuts. You yourself will pick up on patterns and will discover you have tendencies in response to certain characteristics of an opening hand. I'm trying to provide the proper balance to those heuristics or shortcuts or patterns. Only you know what deck you'll be playing and what algorithms you use to mulligan. I'm trying to demonstrate the methodology for becoming aware of the shortcomings of those algorithms. For example, say you find yourself keeping too many 2-land hands. It just feels to you as if you lose too often after one of these keeps. Hopefully my article can help you explain your error. Perhaps you are too focused on the odds of drawing a land in the next 2 or 3 turns, using that as a proxy for the chances of winning, when instead you should be thinking about how well your deck mulligans and how good this matchup is when you get a normal mana draw.

    As for whether discussion should be deck-specific, some people intend to improve in limited as well as constructed. For these people especially, discussion of a Jund deck or UW Control deck’s mulligan algorithms isn’t going to leave them as much as a general lesson.

  27. Great article Matt. I think some people confuse smart with pompous. Also, don’t worry about the quotation editing error…I went to lesswrong to read the original article, noticed the continued quotation, and it was pretty obvious that you were not trying to plagarize and also include a link to the original source all at once.

    To the haters: Just because Sperling uses words you may not understand doesn’t make him a bad person. Dictionary.com is your friend.

    Keep up the good work.

  28. Matt,
    Really starting to enjoy your articles, though I don’t always agree,
    Here is a big trap someone should mention about mulligans: “would I keep this hand if that’s what I mulliganed into?”
    I know ppl who practise this and for all the apparent logic it makes no sense whatsoever

  29. Good read… I really enjoy it when people make it easier for me to step back and remember to just do the math as best I can. There were a number of points you raised about the things against which you must weigh your calculated percentages that made me realize that often I am not completing a comprehensive enough set of calculations to really make a sound decision.

    Also… very well written introductory segment.

    As brought up by another comment, I often get told “oh, you shouldn’t look” when I mulligan but peel off the top 3-5 cards to see how it was likely that the game would have played out from my end. “Oh, that’s being awful… never look” they’ll chastise. But I contend that, as long as I’m making sound mulligan decisions, I can use the positive reinforcement from looking as a reinforcement of my correct decision, and in theory it outweighs the times I receive a negative result. But I’ve realized now that this isn’t fair, and that, whether looking at the top of my library or not, poorly evaluated mulligan decisions, while still positive EV in my experiences, are likely a few percent behind what they should be.

    Lately, I’ve been working on really working out the numbers on a deck with fairly easy mulligan decisions (for example: Vintage Dredge), and it has made me realize some interesting things about the character of 4, 5, 6, and 7 card hands (I seldom go to 3 or fewer cards in constructed magic when not playing dredge, but maybe that’s me being bad). I’d advocate toying around in Excel or some other software package to anyone who really is interested in learning how to maximize percentages, as it has been illuminating.

  30. Something I’ve found helpful is creating guidelines for any deck I create that I play heavily. The important thing is to use them as a starting point for your thinking to save some time, not let them become OBP traps.

    In my extended deck with a 3 piece combo, for example, I came up with a rough starting guideline of “I want at least 2 of the 3 pieces in my opener, plus 2-3 mana”. I would count a tutor as equivalent to the piece I would get with it, but prefer to avoid hands that have two tutors. Once I had my baseline opinion I would modify it based on what I knew of the matchup. Are my counterspells important or not in this matchup, how important is turn 1 birds, am I on the play or draw, have I sided in a silver bullet for this matchup, do I have it or a tutor for it and how important is the silver bullet for this matchup, how fast is the opponent’s deck, how much should I value thoughtseize/duress in this particular matchup, etc.

    A borderline hand would get modified up/down based on those factors. But I had a baseline set of principles that would let me make an initial estimation of how likely the deck was to hit its combo quickly, or slowly. Ponder in the opener would modify my odds of keeping up slightly, too, though not as much as just having the cards I would want to ponder into.

  31. I want to comment before I read the remainder of this article…

    It makes me so excited to see the word/concept of heuristics in a magic strategy article. Matt Sperling and also Alexander Shearer are the brains of this operation and really class up the joint.

    However, I would wager a guess that some commenters will bash this article for being laden with ‘mumbo-jumbo’.

  32. Now that I have finished reading the entire article, I definitely enjoyed it. Also, I’m surprised at the lack of trolls present in the comments…however they might have been scared away by Sperling’s last article.

    “but that's the nature of OBP: contextual differences are what make your heuristic fail.”

    this.

  33. @Matt

    What it breaks down to is “The only hard and fast rule about mulligans is that there are no hard and fast rules.” I am in agreement with this.

    What I was suggesting is that for future writers to talk about mulligan strategy when discussing decklists and/or deck archetypes, and that future writers who talk about mulligan strategy take the concept further by explaining how specific decks should be mulliganed in order to demonstrate where normal mulligan conventions may not work and to demonstrate what type of mulligan conventions will work better. This will help players develop their own deck or archetype specific mulligan rules that are much more contextually relevant.

    And as for Limited, the rules for mulligans are also drastically different than for constructed and they differ from environment to environment. Just as people are talking more and more about draft strategy related to certain color combinations and drft archetypes, mulligan strategy would still be deck dependant to a certain extent.

  34. Just in case anyone was wondering what the quote

    "Generalising the above example, we can say that Optimization by Proxy occurs when an algorithm substitutes the problem of measuring a hard to quantify attribute, with a usually co-occurring a proxy that is computationally efficient to measure."

    actually means…

    It simply means that you’re using a basic set of equations (in your head) that are easy to figure out (such as land/spell ratio, color fixing, power ratio etc.) as a proxy for the answer the more complex issue of mulliganing.

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