Whether or not to mulligan can be among both the most challenging and most important decisions you make in a game of Magic.
A Magic deck is capable of many millions of different opening hands, and your first handful of draw steps will offer a comparably large number of possibilities. It's simply impossible to think through all of the possible six-card hands you might get if you mulligan or the possible sequence of draws you might have if you keep your hand. Add to this the fact that as a Magic player you will play with different decks, in different formats, against different opponents, and you will never ever face the same mulligan decision twice.
In other words, mulligans are not an exact science. Someone might tell you that you should mulligan your hand, but under normal circumstances it would be impossible for them to prove it to you. Instead, we have to rely on our instincts and a handful of helpful guidelines in order to make our decisions.
Even world-class players can have different opinions on a close mulligan question. Whether you mulligan slightly more or slightly less, so long as you're consistently falling in the "reasonable" part of the spectrum, you can feel like you're doing pretty well.
To mulligan means to shuffle your hand back into your library at the start of the game and draw a new hand with one less card. Some people use house rules that allow one or more "free" mulligans, but this is never the case in tournaments. If you're playing for fun and free mulligans make things more enjoyable for you and your opponent, then feel free to play that way. However, if you have aspirations of playing in tournaments, I recommend playing with tournament mulligan rules, even when you're just practicing. Mulligans are an important part of the game.
Even if both players mulligan, you should play the game with each starting with six cards. While both players drawing an extra card may seem like a wash, a mulligan impacts different decks in different ways, and has different consequences whether you're on the play or on the draw. In a tournament, if your opponent suggests a free mulligan, you should never agree to it.
Basic Mulligan Strategy
What I'm about to say may sound obvious, but it's an absolutely essential point that's at the core of all mulligan strategy.
Starting the game with fewer cards is bad.
Taking a mulligan is more than an annoyance or an inconvenience; it's a disadvantage, and a big one at that. We work so hard to earn card advantage, and by taking a mulligan your opponent is getting free card advantage!
It's not unrealistic to win a game when you start with six cards, but it's noticeably less likely than when you start with seven. The problem, of course, is that you're also less likely to win when you keep a bad opening hand.
So don't be afraid to mulligan, either! Whenever you're unhappy with your opening hand, treat it as what it is: a choice between two bad options. Just try to keep a level head and make the decision that you deem to give you the best chance of winning.
Mulligan decisions are challenging; they depend a lot on instincts and on the intricacies of the way two decks match up. Because of all this, it's important to have a baseline strategy when you don't have the level of experience (either in general or with a particular deck) necessary to decide. I recommend the "Two to Five Lands Strategy," which is exactly what it sounds like…
Keep your hand if you have between two and five lands. Mulligan if you have zero, one, six, or seven lands.
This is the strategy I give to people who are first learning Magic, but it's not terribly far off from the way I make my own mulligan decisions. I'd estimate that this strategy leads you to the right decision approximately 90% of the time. Even when it doesn't, you'll typically only be making a very small mistake.*
Mulligans in Limited
Compared to Constructed, games of Limited are closer, go longer, and run on slimmer margins. These are exactly the circumstances that make mulligans most painful. In Limited, try to mulligan relatively infrequently.
Questions to ask about your opening hand include: "Do I have all my colors of mana?" and "Do I have anything to play in the first three turns of the game?"
You should be slightly more inclined to mulligan if your deck is very fast. In this case, you really want to have a cheap creature in your opening hand in order to press your early-game advantage. On the flip side, if your opponent's deck is very fast, you can consider mulliganing a slow hand so you don't fall behind. You still probably shouldn't mulligan a solid hand with three lands and four spells in Limited, almost no matter what.
Mulligans in Constructed
Constructed is where you'll want to deviate from the "Two to Five Lands Strategy," because games are faster, decks are more powerful, and you're punished more quickly for keeping a sub-par opening hand. Your mulligan decisions will also be very different depending on what type of deck you're playing with and against.
Just like in Limited, the presence of a fast deck should make both players more willing to mulligan. For example, if you're playing with a red weenie deck that has the goal of unloading damage as quickly as possible, your chances of winning will go down dramatically if you don't have a creature to play in the first two turns of the game. Something like four Mountains and three burn spells is a hand that you should strongly consider mulliganing.
Control decks are on the opposite end of the spectrum. Since you don't have devastating plays to make in the early game, there's not much in particular that you're looking for in your opening hand. Moreover, your strategy is based on card advantage, and starting down a card seriously interferes with that. Control players should mulligan relatively infrequently unless they don't have enough lands, or are playing against a fast deck where they desperately need early defense.
Midrange decks are closer to control decks than aggro decks in terms of mulligan decisions. However, a key difference is that a midrange deck is likely to have some cards that are great in a given matchup and some cards that are duds. (Thoughtseize is great against control but weak against red aggro; Bile Blight is the other way around). If you don't know what deck your opponent is playing with, you should mulligan relatively infrequently. If you do, you have a little bit more flexibility in looking for your best cards. You can feel free to mulligan a hand with three removal spells against a control player who hardly has any creatures.
Recall that linear decks are the most extreme kind of decks, focused on a very single-minded game plan. They're often very reliant on a single card or combination of cards and can be very fast and powerful. Linear decks cause the "Two to Five Lands Strategy" to completely fall apart, both when playing with them and when playing against them. Let's take White-Blue (WU) Heroic as an example:
WU Heroic doesn't exactly play like a normal Magic deck. Instead of trying to fight fair, it's interested in winning the game quickly while sidestepping a lot of the common defensive measures its opponents might employ. It tries to stick a heroic creature and begin targeting it over and over again with effects that make it enormous and unblockable, all the while holding protection spells like Gods Willing to save the creature from removal.
White-Blue Heroic is excellent at employing this one game plan, but not great at very much else. If your opening hand is not conducive to the game plan, you should mulligan.
This hand has both colors of mana and a good mix of lands and spells, but it's not what you're looking for as a WU Heroic player. You already can't achieve your first goal, which is to put a heroic creature into play! Moreover, you don't have a protection spell, so even if you're fortunate enough to draw a creature, you can be shut down by a single removal spell. Your goal is to be fast; you're good at pressing an early advantage but not great at clawing back from behind. This means that you don't have time to wait around and draw out of an awkward hand. You have a much better chance of winning with a six-card hand than with this one, and you should mulligan.
Now let's say that you're playing Abzan Midrange against WU Heroic.
Trying to damage-race against WU Heroic is a hopeless cause, as is trying to defend yourself by blocking. Instead, your goal is to shut down the game plan by killing the heroic creature or Thoughtseizeing it away. Thoughtseize is your best card, and your removal spells are your next best cards.
Imagine you look at this opening hand. You're on the draw and you already know that your opponent is playing WU Heroic.
This hand doesn't have any of your best cards! If your opponent has a good hand, you're likely to be in big trouble by the time you're even casting Read the Bones. You could use it to find a removal spell, but it's very unlikely that you'll be able to beat a single Gods Willing with this hand, let alone more than one. You should mulligan, looking for something better with six cards.
There are some decks and matchups where a single card is abnormally important. Thoughtseize against WU Heroic fits the bill, but even better examples are Jeskai Ascendancy in the Jeskai Ascendancy combo deck and Anger of the Gods against a mono-red aggro deck.
When you have a card whose presence abnormally affects your chances to win the game, it should influence your mulligan decisions. If you're against WU Heroic and have a borderline hand without Thoughtseize, you can feel comfortable taking a mulligan because any six-card hand with Thoughtseize is likely to be better. On the flip side, you might consider risking a one-land hand with two Thoughtseizes because of the strength of the card in the matchup.
Mulliganing Below Six
The impact of losing a card becomes magnified as your number of cards gets smaller. Mulligans beyond the first one are very costly and should be avoided whenever possible.
Basically, after one or more mulligans, if you look at your opening hand and can see any realistic route to victory, you should keep.
For example, in Sealed Deck, you should keep a six-land six-card hand after a mulligan, so long as you have all of your colors. The chances that your first five draw steps help you are better than the chances that you'll mulligan into the exactly perfect five-card hand that you're looking for. (You'd probably be hoping for three lands with all of your colors and two relatively cheap spells—that's a pretty tall order!)
Of course, if your six-card hand has no lands, or otherwise has almost no realistic route to victory, you'll have to mulligan to five. Going below five cards might as well be the kiss of death, and you should keep almost any five-card hand that has one or more lands.
Just remember to loosen your standards as your number of cards gets lower. Also factor it in as a risk of mulliganing your seven-card hand. What if your six-card hand is bad and you have to go down to five, or even fewer, cards?
There's no easy answer to a tough mulligan question. There are a wide range of factors that can affect your decision, and a countless number of ways the game could play out with either choice you make.
However, despite the complexity in the background, every mulligan decision boils down to a simple question: Are your chances of winning the game higher if you mulligan, or if you keep your hand? Sometimes you'll feel frustration over previous mulligans or you might fear being embarrassed if you keep a risky hand and lose. If you can take these factors out of the equation, you'll be better off for it. Like everything in Magic, the best you can do is stay calm and approach your mulligan decisions with a level head.
*At high levels of competition, particularly in Constructed, you should be slightly more willing to mulligan than the "Two to Five Lands Strategy" would tell you. This contributes to a stigma that keeping a borderline hand is the sign of a weak player and mulliganing a borderline hand is the sign of a strong player. However, there's a huge risk to mulliganing too much as well. For further reading on the topic, read "Reasons to Keep Keeping."