Reflecting Ruel – When To Mulligan

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Friday, December 11th – Knowing when to mulligan a sketchy hand is tricky. Of course, it’s an invaluable skill; correctly deducing that your opening seven is weaker than a random six can be the difference between a win and a loss. Olivier Ruel investigates the art and science of the mulligan, with math examples and sample hands all the way!

In a game of Magic, nearly half the cards you see are those you draw in your opening hand. It explains how important it is to make the right mulligan decision. Today, let’s talk about the type of hands you should keep, they type of hands you should mulligan, and why.

In order to be able to make the right choice, you need to be able to do some basic math, and estimate approximately your chance to win a game. I’m not an expert mathematician, so if I’ve made mistakes in the demonstrations which will follow, please feel free to correct them in the forums. [Math daggers drawn… – Craig, amused.]

I consider that, in most formats and with all other things equal, you should be able to win about 60% of games when on the play, and 40% when on the draw. So we have:

7-7: 60%

(That’s 7 cards for the player on the play, 7 cards for the player on the draw, and a 60% victory percentage for the player on the play.)

And if there are mulligans involved, the percentages should be something like this:

7-6: 67% (That’s 7 cards for the player on the play, 6 cards for the player on the draw, and a 67% victory percentage for the player on the play.)

And so on…

6-6: 58%
7-5: 85%
6-5: 70%
5-5: 50%
7-4: 95%
6-4: 80%
5-4: 55%
4-4: 40%
6-7: 50%
5-7: 30%
5-6: 37%
4-7: 10%
4-6: 15%
4-5: 35%

Those numbers are not exact science, and they don’t apply exactly to every deck/format/matchup, but they should still be a reliable indication to help you out when you see your opening hand. For instance, if you have, on the play, a hand such as:

Kabira Crossroads
Steppe Lynx
Hideous End
Nimana Sell-Sword
Heartstabber Mosquito
Adventuring Gear

The first thing to do is estimate your chances of success with these seven cards. This hand is not really good, as your Lynx is off tempo, and you need two lands in three draws to make the hand strong. Considering you’re running 18 lands, you have 16 left in 33 cards, meaning you have about 50% chance to draw a land on turn 2, and 75% to have it on turn 3. Your chance to draw at least two in three turns is about 50% (if you’re not so familiar with math, we’ll get further into the demonstration a bit later). If you do manage to find it, you’ll be the favorite. If you only find one, you will have a small disadvantage, and if you draw none, you’re dead.

Let’s say you automatically lose 25% of your games when keeping this hand, and that you’ll win 60% of those in which you do find lands. That means you’ll win about 60%x75%=45% of the time. If you check the table, it says you should win about 50% of the time with 6 cards against 7, meaning that keeping would not be outrageous, but mulliganing is slightly better.

The Zero-Lander

Here’s why you mulligan a zero-land hand, no matter what it is. Let’s say, for instance, that you’re on the draw and your hand is:

Llanowar Elves
Llanowar Elves
Llanowar Elves
Nissa’s Chosen
Garruk Wildspeaker
Nissa Revane
Eldrazi Monument

You’re running 20 Forest and 4 Oran-Rief, the Vastwood. If you draw a Forest immediately, (20/53=38%), you have a good chance to win, but you need another land right away (23/52= 44%) to have about 50% chance to win the match. Meaning that your chance to win is 38% x 44% x 50%=9%, when a mulligan would give you about 33% chance to win. And this is only when you consider a perfect zero-lander; usually, keeping is even worse.

The Six- or Seven-Lander

The same goes with six- and seven-landers. Many people tend to think, particularly in Limited, that because they run more spells than lands they will end up balancing if the game goes long. But that’s not the way it works, for several reasons.

At first, with a six- or seven-lander, you still have about one-third lands left in your deck (in Limited), so you’re very likely to draw only two or three spells in the first five draws. Then, even if you do draw spells, they usually won’t show up at the right time. A good curve gives tempo. You give that away by keeping a six- or seven-lander, ad you have no knowledge of what’s coming. You also lose card advantage, as you should draw many useless lands when your opponent will draw more spells than you.

However, six-landers can be kept once in a while. But they require a few conditions.

First, the format has to be quite slow, or the card has to be an early drop which destroys your opponent’s strategy (Leyline of the Void; Kataki, War’s Wage). Then, one of the best – one of the only – reasons to keep would be due to the fact that your one spell is a card that can almost win the game by itself (Broodmate Dragon, Martial Coup), so the loss of tempo is not such big deal. Another reason to keep would be if you have a very unstable manabase in your deck, but all the lands you need are in you opening hand, alongside one of your best spells as a seventh card.

For instance, on Day 1 of Grand Prix: Melbourne, I was running a GB splash U deck, and playing game 3 of the final round with the winner making Day 2. And that’s when I pull the following seven cards, on the draw:

Gatekeeper of Malakir

Is this hand good? Not really. But with a very unstable deck, I estimate my chances to win with a mulligan would be about 30%. On the other hand, that opener has all the mana I need, and I can almost consider I’ve two spells in hand: Grizzly Bear and Diabolic Edict. I’m pretty sure this hand gives me more than a 30% chance to win, probably close to 40%. So do I think that hand will win? No. But I still think it has better chance to send me into Day 2 than six random cards.

The One-Lander

Usually, one-landers require to draw a second land on turn 2, and a third land as quickly as possible. This means you will die if you don’t find the second land immediately. Therefore, you need to have either a hand which can survive without the second land (like the green zero-lander from earlier on), or a hand of which you have huge expectations in order to keep. Of course, keeping with only one land on the play is usually a heresy, as it only leaves you with one draw in which to find the land, when you will have twice as much chance to rip it when on the draw.

For the next example, we will say Fetchlands are Plateaus, even though it is Standard, otherwise the math becomes far more complex. Let’s say you’re running 24 lands, and open with the following on the draw:

Goblin Guide
Plated Geopede
Plated Geopede
Steppe Lynx
Ranger of Eos
Lightning Bolt

There are three likely outcomes if you keep:

1) You don’t draw a land in the first two draws. Consequence: you die. Probability it’ll happen: 32/53 x 32/52= 36%

2) You draw a land in the first two draws, but there’s no more than one land in the top 3 cards. Consequence: You have about 20, maybe 25% chance to win. Probability it’ll happen: 23/53 x 30/52 x 29/51 (probability to draw land spell spell) + 30/53 x 23/52 x 29/51 ( probability to draw spell land spell)= 24%

3) You draw two lands or more in the first three draws. Consequence: You should win about 60% of the time. Probability it’ll happen (this is more difficult, and feel free to correct me if I’m wrong!): 23/53 x 22/52 x 21/51 (probability to draw three lands) + 23/53 x 22/52 x 31/51 (probability to draw land land spell) + 23/53 x 30/52 x 22/51 (probability to draw land spell land) + 30/53 x 23/52 x 22/51 = 40%

This means your chance to win would be around 0 x 36% +25% x 24% + 60% x 40% = 30%, when a mulligan should give you a little more chance.

Of course, I’m not expecting you to make up such complex math in a minute, but you should still be able to simplify. Just avoid the “if I draw a land, I win” shortcut, which is almost always wrong and will push you to take the wrong decision.

In this example, as well as in any questionable hand, you have to consider the deck and the player you’re playing against (if you know them). If you’re playing versus Yuuya Watanabe, running a bad matchup for you, you’re definitely willing to take a risk and keep the possibly very strong hand against him. On the other hand, if your matchup is excellent and your opponent isn’t too good, your chance to win becomes a little higher when taking fewer risks and going down to 6.

The Wrong-Land Hand

You’re running 10 Forests and 8 Mountains in Zendikar Limited, and open with the following on the draw:

Plated Geopede
Goblin Shortcutter
Nissa’s Chosen
Primal Below

This hand won’t do much if you don’t draw a Green mana, but it’s still a close call. Even though you’re on the draw, such a hand should make you at least 50/50 if you draw a Forest in the first three turns (1 – 23/33 x 23/32 x 23/31= 64%). Even if you only draw it on turn 4, (73%), you’re still in pretty good shape. And even if you don’t draw it until turn 5 or 6, you still have chance as long as you draw some Mountains and Red spells to cast. Sometimes, the good choice is to accept the risk of losing outright one time out of five. I think this is the case here, as your odds of winning are definitely higher than 33%, so I’d keep.

Remember two things. First, you’re never sure you’ll only go down to six cards. If you believe you’ll have as much chance to win with a keep as with six random cards, you should usually stick with the seven-card hand. Second, in most formats, don’t assume you have 50% chance to win on the draw. It would be nice, but we rarely have a format in which the player on the draw wins more than 45% of games (again, all things being equal). In Zendikar in particular, the player who goes first will win about 65% of the time. Therefore, if you think your hand won’t win the game, it’s not necessarily a reason to ship it. Just try and think about how often it’d be able to win, and then, if you are over 33%, you should just keep.

Have a great weekend!