Knowing when to mulligan and when not to mulligan is probably the hardest and least understood decision in Magic. When the modern mulligan rule was introduced, players instinctively kept all but the worst hands. Over time it became clear that the proper mulligan strategy was far more aggressive. Aggressive mulligans are often counterintuitive but they all work on the same principle: Your primary goal is not making your first few land drops, it's winning the game. It also doesn't matter how badly you lose or convincingly you win. All that matters is whether a random hand with one less card would have a better chance of winning you the game then your current hand. I'll start with the basics, then move on to the logic I use to deal with most mulligan decisions.
The Automatic Mulligans
Most people's thinking about mulligans starts with this rule: If you have no lands, throw your hand back and draw six cards. If your six card hand has no lands, throw your hand back and draw five cards. If that hand has no lands, go to four. That's a good rule. It's hard to play a game of Magic without lands. Cards like Chrome Mox that provide mana and cost zero mana can count as lands for this purpose. It's easy to understand why you automatically throw these hands back. If you don't have any mana, it will be hard to cast your spells. There are exceptions to even this rule, but these exceptions require highly unusual decks and even then they are extremely rare – I can think of only one example of a real game where it was correct for a player to keep even a five card hand that did not have any mana sources in it.
When you have all lands, six lands out of seven cards, or only one land, you should mulligan most but not all of the time. These hands are going to cast far fewer spells on average than a random six card hand and there's a very good chance that they will have a very low chance to win the game. Many people who have one-land hands think to themselves that they have a good chance to play a second land on turn two but they forget that this is unlikely to be enough. Yes, there is a reasonable chance that they will get to play a second land, but if they go on to miss their third land drop that is fatal for a lot of decks and a lot of hands, especially if it takes several turns to find that third land (which it often will). Keeping a hand with an overabundance of land is a lot like keeping a five or six card hand because those extra lands are probably not going to be much help winning the game. Once you have enough mana to cast your spells extra lands are generally very marginal cards and sometimes they are outright meaningless.
The Loss of a Card: How Bad is Going to Six?
Screams of the Damned When you mulligan you're starting with one less card. A six card hand needs to come with lowered expectations but that doesn't mean you should think of the game as lost. Being down a card is not necessarily as big a deal as you might think. Decks need to make sure that most of their seven card hands have everything they need, which means that on average you have more than enough of everything that is vital to the early game. If you fear a typical six card hand is unlikely to give you a chance to play the game then a seven card hand had far too high a failure rate.
Some decks mulligan better than others, but all good decks mulligan reasonably well because they have lots of redundancy (and/or the ability to search) and either can win without card advantage or have strong enough engines to overcome the loss of a card. That doesn't mean you should get greedy. One of the most underappreciated problems with a six card hand is that a five card hand is where you start crossing over to disaster. You no longer have enough cards to fill out your hand and mana curve properly and even if everything goes right you'll still be down two cards.
When you go to six cards you are hoping for a playable hand, nothing more. You can get a six card hand that is better than the average seven card hand, but you shouldn't expect it and you are essentially stuck with anything remotely playable. A lot of your five card hands will be flat out unplayable so if you can fix your six card hand by drawing the right card it's an all but automatic keeper.
Only experience can give you a good read on how well a deck mulligans, but you can use some things as a rough guide. A good rule of thumb is that the fewer lands a deck needs to operate properly the better it will mulligan. A deck full of creatures that cost one and two mana can often mulligan without missing a beat. It needs fewer cards to play its game and all of its cards have a lot of redundancy. That redundancy is key as well. If you have backup copies of the key components of your deck then going to six cards becomes far less dangerous, especially if your plan doesn't require you to gain card advantage. On the other hand, if you have quick sources of card advantage you can use them to climb out of the hole your mulligan has put you in. For example, decks that are built around Survival of the Fittest get so much advantage out of Survival of the Fittest that a six card hand with it is far more resilient than a seven card hand without it.
The more you play with a deck the more you get the sense of how well it mulligans. It's rare that a strong deck will fail to have the ability to mulligan well. You need to know the answer so you have a good baseline for how good your six card hands will be. This allows you to compare it to a given seven card hand and see which one is more likely to win the game.
If you're afraid to go to six cards, chances are you're playing the wrong deck. Don't ignore the warning signs.
The Rule of One Missing Piece
The core of my mulligan method is the Rule of One Missing Piece. If your hand lets you play your game no matter what you draw, you've got an automatic keeper. If there is something you'll need to draw, look at your hand and ask yourself what you need to draw in the next few turns in order to give you a good hand for now. If the answer involves two or more cards soon then you throw your hand back almost automatically. If I need to draw multiple lands, I will almost always throw the hand back. ‘All this hand needs is a land' is worth thinking about but ‘I need to draw two lands off the top' or ‘two out of my next three or four draws' (especially if some lands don't work because of color requirements) generally is not. If I'm about to miss multiple spots on my mana curve and I can't afford to miss any of them then I'll throw the hand back.
If you have exactly one missing piece then it's time to ask how good your hand is if the missing piece shows up and how bad your hand is if it does not show up (or shows up late). That then has to be balanced against how many copies of the missing piece are in your deck. When you keep a hand that could fizzle you need compensation for the risk you're assuming. The harder it will be to get what you need, the better your draw needs to be if you do get what you seek. The easiest thing to draw is any spell, but drawing any land is also reasonably easy if you don't need it right away. The key mistake people make with land-heavy draws is that they assume that all they need to do is draw spells – that's part of the problem, but frequently a lot of your spells won't do much to help the problem, and that has to be weighed in. Another thing that commonly gets missed is that you often can also have alternatives you may not have thought about. For example, drawing an additional two drop might be enough to allow you to miss your third land drop for a turn.
Deceptively Bad Hands
It is not enough to have land and spells. You must have good action and the ability to put up a fight. Look at your hand and ask the question: Will this hand win me the game against a normal enemy draw? Surprisingly often the answer is ‘probably not' and when you think about it you'll realize that you're counting on drawing a solution to your problem sooner rather than later.
Comparing the Two
Noetic Scales The key is to compare your chance of winning the game in each scenario rather than comparing the average strength of your hands against some unknown. Hands that become very strong if you draw the right card are often deceptively strong, while those that don't have good potential are often safe but far weaker than they look. The most dangerous hands of all are those that look like they aren't that bad but won't win many games because they simply don't do enough. Don't be fooled into thinking that in a given scenario you'll ‘get to play your game' if you're still probably going to lose it. Nearly everyone knows to throw away a hand with no land, so those aren't so dangerous. It's the deceptively weak hands that cost you over time if you aren't vigilant.
I don't know anyone who maths out the expected chance of winning explicitly, although some will do the calculation of how often they'll get what they need. But an approximation will do fine. With that in mind, an important final note: these decisions are far too complicated to know everything, so when it is close it will have to come down to your gut. The problem that needs to be fixed is that most peoples' guts are far too conservative. When the voice inside your head thinks that your hand is doomed, don't listen to the one that hates to mulligan. It's probably time to go to six.