The best way to get an edge in Magic is not in deck selection, but in playing a deck to its fullest capacity. There is a lot more to a tournament than picking a deck, signing up for the event, and filling out the decklist sheet. Every game is filled with decisions on each and every turn, and making the correct decisions is key for maximizing the potential of a deck. Making the correct decisions leads to a deck operating at its highest level, actualizing its theoretical potential, and performing at its peak in a given game. Decisions go beyond specific individual in-game plays, including one underlooked factor that's among the most important decisions of all: when to mulligan. Mulliganing is extremely contextual.
The following article is a mixture of theory, math, and practical advice regarding mulligans.Mulligan Theory
Mulliganing is complex on many levels. Boiled down to its simplest level, the question to ask before each mulligan decision is "am I more likely to win if I mulligan?" as opposed to "is this hand a favorite to win?" It's a fallacy to throw back a hand just because it doesn't seem likely to win a particular game of a particular matchup, because it's not necessarily true that the next hand will be any more likely to win.
Assume a theoretical matchup is 50/50, with both sides equally like to win. This number applies before the die is rolled or coin is flipped, and before hands are dealt. The die is rolled, and theoretically the winner has earned some number of percentage points, meaning the opponent has lost some percentage points. Things are never actually equal.
Once the cards are dealt out, the odds again change significantly. Imagine watching poker on television, where a camera shows both players' cards and the screen typically show a statistics box with the likelihood of each player to win the hand. This stats box exists for each game of Magic, it's just invisible and much harder to calculate with accuracy.
Just like the poker player must figure out their own odds, and then use these odds to make informed decisions throughout the game, so must the Magic player. The biggest difference here is that the Magic player has control over their own cards, and in a specific game can throw back their hand for a new one.
In actual games the opposing hand is unknown, but a player can figure out their own odds within the context of their own deck with accuracy. Combined with knowledge of the opponent and the matchup, this information can be used to make an educated mulligan decision.
To mulligan with precision, one needs to calculate their odds of winning the best they can, then compare it with their new odds of winning after a mulligan. If the starting hand is more likely to win than a mulliganed hand, the hand is a keep. If the mulliganed hand is more likely to win than the original starting hand, the hand is a mulligan.
The usual first step when deciding whether or not to mulligan is the land/spell ratio of the hand. Some number of lands (or other mana sources) are necessary to cast spells or creatures and execute a game plan. An insufficient number of lands is going to lead to a low chance of winning. Conversely, a hand filled with lands but lacking sufficient action will similarly lead to a low chance of winning. Hands of either extreme are typically mulligans.
Interesting mulligan decisions come somewhere in between the extremes, when a hand has a near-borderline number of lands or spells. Assuming knowledge of the deck and matchup, the pilot knows what is necessary to "make" the hand, whether it be another land or two, a specific spell, a specific subset or combination of spells, etc. These situations highlight the simplest form of mulligan decisions: calculating the odds.Calculating the Odds
A typical common scenario is a one or two-land hand, where one needs to hit a land to make their hand and execute their game plan. First figure out the number of live draws compared to the number of dead draws, then compute the odds based on the number of draw steps before the land must drawn, and come up with a percentage.
As an example, assume a player needs to hit their first few consecutive land drops: needing to draw one land on the play yields one draw step and on the draw, two draw steps. A two land hand on the play will yield two draws steps and on the draw, three draw steps. Another scenario is a flooded hand that needs to draw a specific class of card, say a two or three-drop creature by turn three, or a two-drop removal spell by turn two, a combo piece by turn four, a specific hate card in the opening hand, etc. It could be more complex, such as needing to draw two lands or two spells. The possibilities are endless.
Determining these odds precisely at the table can be difficult, but coming up with a close approximation is basic Magic mathematics and will provide a reasonably accurate number. Here's an equation, taken from the work of Patrick Chapin and widely shared online, including Craig Wescoe's piece on Mulligans two years ago that can be used to figure out the approximate odds in-game, at the table, of drawing a specific card or class of card, such as a land, within a given timeframe.
100%-[(Z/X)^Y] = odds of drawing a live card Z = dead drawsX = cards in libraryY = number of draw steps
This equation is not exact, but an estimate, and it starts to fall apart as the number of draw steps increases because it does not account for the continually shrinking deck.
This is not an in-depth math article, but the scientific way to determine the odds exactly is called hypergeometric distribution, and I recommend checking out the link for more in-depth detail.
Here's a link to a handy online calculator for doing said math.
This is useful for playing around, or for figuring out specific numbers for a specific deck before a tournament. It's a great idea familiarize oneself with the particulars of that deck, because each and every deck has its own particular percentages. It's also a very useful tool for anyone playing Magic Online!
Once the odds of making a hand are figured out, this must be compared to the odds that a mulligan will yield a better hand. Mulliganing for Specific Cards
In practice, figuring out the odds of a hand to win may come down to something more specific. An extreme example is the case of hate cards. An extreme and simple case of hate cards would be Leyline of the Void, which requires no mana and ignores the play/draw, against a manaless Dredge deck, which has no out to the hate card. Assuming the game is a sure loss without the card, and a sure win with the card, the mulligan decision would already be made, mulligan until the hate card is in the opening hand, regardless of the odds of hitting it. A more practical example is something like Kataki, War's Wage from Modern GW Hate Bears against Modern Affinity, where the odds of winning may be very unfavorable without the card but highly favorable with it.
For this example, assume 30% to win without the hate card, and 70% to win with the card. Determining the actual number in practice would require in-depth knowledge of the decklist with significant testing, but this is a realistic approximation for this example. With four copies of Kataki, War's Wage in the deck, the odds of it being in the opening hand are approximately 40%. The odds of it being in the six card hand are 35%. Assuming the hate card is not in the opening hand, this would be a case of a clear mulligan, because the average six card hand is going to provide a 35% chance of winning, much greater than the 30% chance the 7-card hand provides. This becomes more slightly more complicated when considering that lands must also be in hand to cast said hate card, which slightly lowers the theoretical 35% win rate some number for each required land, but it's not significant for practical purposes. More difficult is knowing the actual odds of a matchup and how much a specific card helps, which is again a place where experience and intuition come into play.The Average Hand
Every deck has an approximate average hand based upon its specific composition, and the average six card hand is going to have 1/7th less of each component than a seven card hand, the average five card hand 1/6th less than the six card hand, and so on down to a one-card hand. It's nearly impossible to compute this mulligan math precisely in a timely fashion at the table, so it requires a wealth of experience that has honed an intuition, or some calculations beforehand. It's generally assumed that if the hand has land and a mixture spells, the hand is a keeper. Decks are designed with this philosophy in mind, so given a sound, well-constructed deck, much of this work is really done before a tournament ever begins. Much of playtesting and familiarizing oneself with a deck is learning the average hand, how it compares to the average hand of a specific opposing archetype, and being able to identify the hands that fall far enough below the average to warrant a mulligan.
As a quick and rough guide, the average hand from a deck with 25 land will provide approximately three land in seven, two and a half land in six, and two in five. Assuming a deck has 25 creatures, or 25 Counterspells, then the ratios will be the same for those classes. These numbers can be broken down into smaller categories such as colored sources, mana-curve of creatures, types of spells, Goblins, and so on.The Extreme Hand
It's also very important to be able to identify and properly evaluate hands beyond the average. Not all cards are created equal, especially not in the context of a game. One important aspect to consider is the particular matchup, while another thing to consider is who is on the play or draw. Again context is everything.
Cards work together in what we call synergy, and having a specific set of cards in the opening hand may lead to above-average results.
In Standard, the curve of Thoughtseize into Pack Rat on the play may be good enough to beat anyone, even if the other five cards are lands. On the draw, it might be a mulligan. If that hand was two Gray Merchant of Asphodel and five lands, it wouldn't ever be close to a keep against anyone, play or draw.
Nearly every card in Faeries works better with Bitterblossom in play. Some, like Spellstutter Sprite, become much more efficient while a card like Mistbind Clique all but requires it. A Faeries player will keep nearly every hand with a Bitterblossom and realistic expectation that that they will be able to cast it.
There are also decks that function completely differently based on a specific card in the opening hand. An extreme example is Vintage Dredge, which simply mulligans every hand until it finds Bazaar of Baghdad.
In Legacy, having access to Sensei's Divining Top makes decks with Counterbalance, such as Miracles, operate much more efficiently, to the point that nearly every hand with a land and an Sensei's Divining Top will be a keep, even if it would otherwise be far too extreme below the average.
There is also the negative case, when a card in the opening hand creates significant negative synergy. Consider Legacy Elves that sideboards in Progenitus to combine with their Natural Order to present a difficult to deal with threat. The deck has no card manipulation like Brainstorm, so assuming the Elf player draws the Progenitus, it has no way to cheat it into play with Natural Order. This is a risk the deck takes, but assuming the Progenitus is in the opening hand, and assuming the Natural Order plan is an important part of their post-sideboard plan for the matchup, this hand would have to be a mulligan.To Mulligan, or Not to Mulligan
Compared to other decisions, the decision of whether or not to mulligan occurs each and every game, and often more than once. It represents the single greatest control a player has over the cards they will work with in a given game, and it's a tremendously important aspect of Magic: The Gathering. Magic is a game with a considerable amount of chance, but the mulligan decision does a great deal in mitigating variance and creating a stable foundation for the game. Often times players are seen at the top again, and again, and again, and it's certain that these players have a strong grasp over the mulligan.
Please turn to the comments with thoughts on this article; it's a departure from the usual pieces I've been sharing lately but I hope everyone can take some wisdom from it. If you have additional questions, or even ideas for a future article, please share. I'm also interested in hearing your own thoughts on mulligans, tips and tricks you use, theory, math, practical advice, and everything in between.