To Mulligan, or Not To Mulligan

October 7, 2016

To mulligan, or not to mulligan, that is the question. Tournaments are won and lost based on this decision. My analysis will focus strictly on the role of the mulligan in limited.
My good friend, teammate and (two-time) GP champion Seth Manfield got the following hand on the draw when going to game 3 in a limited PTQ:
4 x Swamp
1 x Disciple of Phenax
1 x Gainsay
1 x Daxos of Meletis
His deck contains 10 swamps, 6 island, 2 plains and an Opaline Unicorn. Furthermore, you can assume that his opponent is Blue because, conceivably, Seth would not have boarded in Gainsay otherwise.
Casting Gainsay
The odds of drawing a non-Island card on turn 1 are 27/33 or 81.81%. The chances of drawing a non-Island card on turn 2, assuming that you did not draw one on turn 1, are 26/32 or 81.25%. Therefore, the combined odds are 66.48%, or approximately 2/3 of the time. In conclusion, you will only be able to cast the Gainsay on turn 2 33.52% of the time (I concede that you will be able to use Gainsay later on in the game but it does not advance your board position when you are likely to already be behind).
Casting Daxos of Meletis
Extrapolating this math to turn 5 and including the Opaline Unicorn as a source of both Blue and White, the odds of drawing a non-Unicorn, non-Island card on turn 1 are 26/33 or 78.79%, turn 2 is 25/32 or 78.13%, turn 3 is 24/31 or 77.42%, turn 4 is 23/30 or 76.67%, and finally turn 5 is 23/29 or 75.86% (Unicorn excluded). Therefore, the odds of still not having a source of Blue by turn 5 are 28.98%. Performing the same calculation for White, the odds of not having a source of White by turn 5 are 62.35%. Because each of the conditions above prevent you from casting Daxos, there is approximately a 90% chance that you won’t be able to cast the Daxos early enough in the game to turn this draw into a winner.
Mulliganing in limited is a dynamic exercise. Typically in constructed you are able to look at your opening hand and establish whether or not it is capable of doing what your deck has been designed to do. Limited is not so easy. Decks are not designed perfectly and don’t always present you with a clear path to victory instantaneously. For example, think about a UW flyer deck consisting of slightly inefficiently costed creatures with flying and low-power/high-toughness ground creatures for blocking. These decks have existed in virtually every format in the last 15 years. On many occasions you will draw hands that include either all of the Horned Turtles or all of the Wind Drakes. These draws are vulnerable to different strategies. The hand with all the Wind Drakes is going to have a hard time beating an aggressive early game deck that has a couple of removals like a BR aggressive deck. It may also struggle with a Green deck that draws efficiently costed creatures and a single Giant Spider. Alternatively, it can be very effective at combating defensive decks with high toughness ground creatures and a powerful late game. Conversely, the Horned Turtle hand will be quite good against the early onslaught of BR but has no chance of beating a deck that has a more powerful late game because it is unable to provide sufficient early pressure. So how do we know whether or not to keep the hand?
This is where I think a fundamental misconception of Magic: the Gathering first comes into play. Magic: the Gathering is a resource allotment game. You have spells, mana, life, cards in library and finite turns to vanquish your opponent. When deciding whether or not to mulligan you need to analyze if the resources in your hand will be able to effectively place you on the path to victory. I often see mulligan decisions analyzed incorrectly when the primary deterring factor is fear.
Now if we apply this principle of resource allotment to Seth’s hand (listed above), should we mulligan?
First, we need to see whether or not we have a productive ratio of resources. This hand does not. We have 4 lands in our hand already and in order to cast our spells effectively we must draw a U and a W mana. Therefore, we are looking for a best case scenario of 3 spells and 6 lands so that we can cast everything on time. This is unlikely to be enough action to win.
Second, is our hand all Horned Turtles or all Wind Drakes?
Neither: This hand is essentially 1 defensive creature, 1 offensive creature and 1 removal spell. That would usually be a strong keep. You have a creature which can apply pressure, you have a creature which can avert your opponent’s pressure and you have a spell that can be protective or aggressive as needed. The problem is that the offensive creature will not be played in a timely manner the vast majority of the time, so there will be no pressure on your opponent. Furthermore, Disciple of Phenax‘s power and toughness are minor in comparison to its cost in order to provide any real combat deterrent in early action. Essentially this draw doesn’t have game against a marginal draw from an aggressive deck or a powerful late game deck.
If I mulliganed and got the same hand with one less Swamp, I would keep the hand. Why wouldn’t I keep this one then?
This line of thinking involves a simple logic error: while the hand that you are considering is essentially the same as the 6 card hand, it is still an inadequate comparison because of the drastically different alternatives. The consideration should be whether or not you should take a free mulligan if you have a 6 card hand of 3 Swamp, Disciple, Daxos and Gainsay (in this case I think you should). The flaw is in the fact that if you mulligan a 6 card hand you have to go to 5, while if you mulligan a 7 card hand you go to 6. I have never understood this logic and I laugh to myself every time someone uses it when considering whether or not to mulligan.
While this may not have as great a following in Europe, the American football coach Dennis Green said it best: “We play to win the game!”
When looking at a hand it is essentially a simple expected value calculation. You want to calculate the chances you have to win, given the particular cards you have, against the chances you have to win with a random 6 card hand. The numerical analysis above can be used to aid in these decisions. For example, let’s assume that your opponent has an Ashiok, Nightmare Weaver in his deck. He is on the play and you have very little early game pressure, so you probably can’t beat it without a turn 2 Gainsay. Furthermore, let’s assume that other than the troublesome planeswalker his deck is rather weak and should not cause you many issues. This means that you are likely to win if you are able to Gainsay his Ashiok and lose if not (for the sake of simplicity, let us assume that you are 100% likely to win if you counter it and 0% if it is resolved, which is certainly not the case). The above calculation tells us that this hand was 33.52% likely to be able to cast Gainsay on turn 2. The chances of drawing the card Gainsay in a 7 card hand is 17.5% and for a 6 card hand the chances are 15%. Therefore, it is more likely that you will be able to counter the Ashiok with your current hand than after a mulligan. Normally, we would have to calculate the chance that your opponent will draw Ashiok (which is around 20% by turn 3), but it is irrelevant in this case given the premise: “If he doesn’t draw Ashiok, we win.” This is a fundamental, simplifying assumption. Using this information, it would be correct to keep the hand because the particular importance of a certain spell improves your chances of winning by keeping the hand.
No player ever has to mulligan. It is a privilege to be able to draw a new hand when your first one is incapable of winning. However, it is a daily occurrence to see how the demeanor of both the player taking the mulligan and his opponent changes drastically when the mulliganer throws his hand. The mulliganer experiences anger and disgust while his opponent perks up and exudes confidence. This psychological effect has significant impact on the outcome of matches: your body physiologically changes when you experience anger or sadness and your opponent senses weakness and becomes increasingly aggressive and confident. Most games of limited do NOT come down to the utilization of every card from both players. Therefore, my advice is to battle your opponent’s confidence with unflappable stoicism and show him that not only do you believe you can win with 6 cards but that you WILL win with 6 cards. Furthermore, your opponent will play more aggressively when he senses submission. This could cause him to kill you a turn faster and take away an opportunity for you to draw to narrow outs. When extrapolated over time this could change a GP top 32 player to someone that cracks his first top 8 and finds his way on to the pro tour.
I hope you found this article insightful. Please feel free to leave any comments or email me with topics for discussion at I graciously thank you for your time.
P.S. Seth kept the hand, drew an Island on turn 1, Plains on turn 2, hit many times with Daxos and won easily. It is good to be blessed.
Signing off,
The Gatormage
Chris Fennell

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