I spent the day at USC today – more accurately at the USC Health Sciences Campus, where I gave a talk and spoke with other researchers who were interested in my work and how it might impact theirs. All in all, it’s a lot like meeting up with other Magic players at a big event and discussing tech and strategy.

The core point of today’s talk was that “it’s about asking smarter questions, not finding exact answers.” In the context of the talk, it was about using computational methods to go from having to test ten million things (which is basically impossible) to having to test twenty (which you could knock off in a couple days). But even with that massive change in scale, we still expect that some of our predictions are wrong.

It’s just that being wrong five times out of twenty wastes way less time than being wrong a few million times out of ten million.

So that brings us to a question that comes up again and again in a game that involves a shuffled deck of cards.

When do we need to suck it up and reshuffle? When can we realistically expect to hold on and get there?

When, in a word, do we mulligan?

The mulligan basics

If you’ve read any strategy articles on the topic of mulligans, there are some core things that you should already have encountered. By now, they’re probably engraved in your brain – at least in theory. Even so, it’s worth checking in with our received wisdom before we wander off into new territory and get to the point of today’s piece. So bear with me as I rehash a few points for the next little bit.

Mulligan for the game you’re playing

The decision point for whether or not to mulligan a hand of cards does not simply come down to asking “Do I have both lands and spells?” We know this. We also know that it shouldn’t be simply about “Do I have 2-4 lands and spells I can cast with those lands?”

But a lot of us do stop there.

Instead, it’s about using our ability to mulligan to position ourselves to win the actual game we’re playing right now. It’s about having a plan for how we’re going to win that specific matchup, and making sure we engineer a hand that has a chance to do so.

So mulligan to fit the actual game we’re playing right now. Check. It’s covered, and we move on.

Will six cards be better?

The second idea, that we hopefully also all know already, is that you need to seriously consider if you’re likely to have a legitimately better chance of winning a game with a six-card hand. This is a more philosophical question, but it boils down to asking two things:

First, just how bad are these seven cards?

Second, how much better could any other hand be?

Sometimes we have a snap mulligan, of course. All spells, all lands, that kind of thing. These hands are clear losers. But what about the CawBlade hand that has [card]Dismember[/card] and [card]Mana Leak[/card], but misses out on [card]Stoneforge Mystic[/card], [card]Squadron Hawk[/card], and [card]Preordain[/card]? Is that keepable? Will six cards be better?

As I said, now we’ve ventured into philosophical territory. Although I won’t dwell on this aspect of theory all that much right now, you basically need to ask:

Are there any dealbreaker cards I’m missing?

In other words, is this a matchup where you must have a specific card in hand or within the first few turns to have a chance of winning? Or can you more generally look for a type of hand (e.g. “with some removal” versus certain aggro decks) and that’s good enough?

Does this hand have the right tempo for this matchup?

Some hands are “fine” but too slow. It happens, and it just might be worth trading down to six cards for a chance at being faster.

Special cases (always occur)

This is a sort of extension of the first point – the one about playing to a specific matchup. Sometimes you need to consider the pressures that push toward not mulliganing an otherwise lame hand.

For example, a Dredge player who knows they’re facing down facilitator hate may want to avoid a mulligan so they’ll have a shorter lag in reloading their hand and simply discarding to start the dredging.

Similarly, if you’re facing down a heavy discard deck, the smart move could easily be to have a full hand, even if the quality of the cards or even the tempo of the game plan they represent are deficient. Sometimes, it’s just good to have more hit points instead of better defenses. It happens.

Hang a number on that thought

If my writing has themes, one of them is assuredly quantifying – that is, figuring out the numbers behind things. This reflects my day job, but it also reflects that these numbers exist anyway, so we might as well think about them from time to time and let them guide our decisions.

But there are reasons behind this, and they drive how and when we tinker with those numbers.

You aren’t a computer, and that’s fine

I don’t run stats in my head.

I certainly don’t run stats – not real ones – in my head during a game. No way. First of all, I can’t process a hypergeometric distribution in my head even if I thought that would be a good idea. Second, I don’t think that’s a good idea, really.

But you do want a ballpark understanding of stats.

If you have that ballpark understanding, you’ll know that your opponent’s topdeck was reasonable, and maybe you won’t go on tilt as a result. You’ll also have a good understanding of which of your outs to play for, based on a combination of how likely you are to draw into each one plus how effective it’s going to be.

A little prep, then don’t worry about it

The upshot of this idea is that you probably just want to think about stats during a time when you’re not actively engaged in trying to outmaneuver an opponent.

Now, for example, when you’re just taking some time to think about the game.

This is why you see articles from me about evaluating mana base depth or figuring out whether Chancellor of the Tangle is worth playing. I’d hate to be stuck figuring out the odds on these things on the fly, but if there’s some time to burn while I’m not actively slinging spells, it’s good to get a handle on how plays are likely to go.

Running the numbers is not meant to be a replacement for intuitive play. Rather, they bolster our intuition, and let us shortcut a lot of practice time by simply letting us know the likelihoods for certain things, instead of requiring our brain to slowly run its own statistical modeling and figure them out.

It’s basically like being told what your car’s estimated mileage is when you buy it, instead of forcing you to figure it out by experience over time. And, just like that mileage, the numbers we work out ahead of time may end up being a little bit off from reality – but they still provide a sounder starting for us to learn from than having no numbers at all.

…and then once we’ve run those numbers, we can file that knowledge away and just play the game.

Mull or get there?

So that brings us to this week’s exercise in running the numbers. Let’s return to that question I mentioned above.

Do we mulligan or keep?

Let’s set aside the snap keeps and snap mulligans and ask about those tricky situations that live in the mdidle. I’ll split the tricky stuff into a few types and then take a stab at figuring out how to address them.

Searching for (possibly) greener pastures

Let’s say you’re playing Caw-Blade, running Paulo’s deck from GP Singapore. You’re on the play, and you’ve just drawn this hand of seven cards:

[draft]Island
Seachrome Coast
Tectonic Edge
Dismember
Dismember
Spell Pierce
Consecrated Sphinx[/draft]

Do you keep that? Do you mulligan? It’s neither a snap keep nor a snap mulligan, at least as long as you’re going up against a generic opponent. It has some early disruption and point removal, but not much in the way of action. If we synthesize the advice we’ve received from seasoned CawBlade players, this is a passable hand, if not an exceptional one.

Would six cards be better?

The real question here is not just “Would six cards be better?” It’s actually the more elaborate “Which six-card hand would be markedly better than this seven-card hand?”

The easy answer to that question is “Any hand with a [card]Stoneforge Mystic[/card] and the land to play it.”

Conveniently, I can tell you right now that your chance of mulling into that Mystic and the lands to play it is about 25% in a typical CawBlade deck. Should you trade what you have in hand for a 25% shot at a Mystic?

I’ll go with “probably not.” However, other situations are less straightforward. Consider:

[draft]Celestial Colonnade
Seachrome Coast
Plains
Mana Leak
Dismember
Into the Roil
Glacial Fortress[/draft]

This hand is pretty annoying, seeing as the mana is solid but the hand is essentially action-free.

Mulligans versus the power of waiting

As it happens, this example isn’t just asking you if those seven cards are superior to a potential six-card hand. What it’s really asking is this:

Will I draw the card I need in time, or should I mulligan to raise my odds of seeing it?

To answer that, we’re going to want to have some, you know, numbers.

This table is super straightforward, and it asks just one thing – if there are a certain number of cards that I’m looking to draw one or more of, what’s the chance I draw into them when I dig so many cards deep into my deck.

For example, if I have four target cards – say, those Stoneforge Mystics – and I’m on the draw, then the first card I draw off the top of my deck has about a 7-8% chance of being a Mystic. Two cards in, I’m approaching 15% to have a Mystic, and so forth.

Now, let’s add in a table about mulligans:

Now, we’ve mulliganed into six cards. Once again, given a certain number of cards we’re looking for in our deck, we have some odds. This time, the first column is the chance that we mull straight into our card of choice – in a hand with otherwise reasonable mana – and the subsequent columns cover the odds of our drawing into our card of interest.

I know, it’s a bunch of numbers. As I said above, the point is certainly not to memorize this stuff – I know I’m not going to. Instead, the point is to get a feel for how our deck’s going to run.

With that in mind, let’s return to those hands I described above.

But what do I do?

Once again, here’s our opening seven:

[draft]Celestial Colonnade
Seachrome Coast
Plains
Mana Leak
Dismember
Into the Roil
Glacial Fortress[/draft]

The simple question we want to ask here is “Do we mull to six, or do we wait it out?”

If we’re content to have either a Mystic or a Hawk, then we have eight cards that we’d like to get out of our deck.

As you can see, this means that our hitting one in our first three turns (on the play) looks like this:

Turn 1 – 0% (we don’t get to draw!)
Turn 2 – 15%
Turn 3 – 28%

If we mull to six, our odds look like this:

Turn 1 – 42%
Turn 2 – 46%
Turn 3 – 50%

The intuitive reaction here is to think that I’m peddling something curious here, since I seem to be suggesting that all sorts of weird magic happens when you mull to six. So to divert that very natural reaction, consider what we’re saying here.

First, we already know what our opening hand is, so this has nothing to do with how likely that is.

Second, we don’t know what hands we’re mulliganing into, so answering that part of the question very much does have to do with how likely certain hands are.

Essentially, these numbers tell me that if I choose to mulligan, I’m trading away a 28% chance of seeing one of the cards I want by turn three for a 50% chance of seeing the thing by turn three. Double the odds seems great…until we consider that all the new hands that don’t have a Mystic or Hawk may be even more “meh” than the reactive hand we’re considering, and may even be unplayable.

Sounds like it’s time to write out the facts as they stand.

If we keep…

We get [card]Mana Leak[/card], [card]Dismember[/card], [card]Into the Roil[/card], and 28% chance of a Hawk/Mystic by turn 3

If we mulligan…

We get uncertain card quality and a 50% chance of a Hawk/Mystic by turn 3

…and this brings us to a fun, extraordinarily rough, rule of thumb. We’re about two to three times as likely to hit the card we want in the first few turns if we mull to six than if we keep our otherwise playable seven.

Yes, this is a super-rough rule of thumb. Yes, sometimes we’re 1.5 times as likely, sometimes three. But for quick-and-dirty planning purposes, “two to three times as likely” is a good rule…and it means we only really need to remember the ballpark odds of hitting the card we need a few turns after we keep our seven.

So, if we wanted to hit our Mystic or Hawk by turn two instead of three, we’d be 15% to get one by keeping our seven, and roughly 30-45% to get a Hawk or Mystic if we take the risk and go to six.

…versus…

That’s as clear as it’s going to get – the rest is up to you. As for me, I think I’d have to stick to my Taleb and choose to keep.

Get there, get there

What if the hand were this instead?

[draft]Celestial Colonnade
Seachrome Coast
Plains
Mana Leak
Dismember
Preordain
Glacial Fortress[/draft]

Now our decision is a little bit more subtle. We can’t just glibly double or triple the odds from keeping and call it a day. Instead, we have to consider the fact that the hand we’re holding onto now has the ability to dig two cards into the deck.

As it happens, that’s simple enough to deal with – we just move two cards over on our handy table.

Instead of this by turn two on the play:

We have this:

The ability to dig two cards deeper with Preordain pretty much brings us to parity in terms of finding one of our power two-drops…and that means the comparison looks like this:

[40% to hit M/H versus 46% to hit M/H and 54% unknown]

Suddenly, the decision seems pretty clear. And, fascinatingly, it looks like a mulligan gives us about the same odds of hitting the card we want as having a seven-card hand and something that digs us two cards deeper.

That is, a mulligan is worth three cards.

Or three cards and a big risk, at any rate.

The take home – rules, thumbs, and so forth

The point today is not to drag your brain through all these numbers and then force you to spreadsheet your way to victory. If you are interested, I may write up some supplementary notes on how I came up with today’s numbers – although that will have to wait for when I’m not on the road talking up science.

Instead, the point is to get you thinking about the pertinent questions and tradeoffs when the decision to mulligan is unclear – and to get you thinking about these situations while you’re pondering the deck, and not simply once you’re there in your seat at your next big tournament, eating time on the clock and earning yourself a Slow Play warning.

You’ll come up with your own rules of thumb for each deck you play, but for now, consider starting with these ideas:

A mull to six may be a good trade for a passable hand

A mull to six is probably not a good trade for a passable hand with access to card filtering

…and these questions:

How risky is a new hand for me?

What is the absolute risk of a junk hand versus the nuanced risk of keeping my seven?

In general, the best meanderings through complexity seem to end up in simple ideas like these, or a new understanding like, “Maybe a Preordain really can power up an otherwise passable hand.” Even if you don’t feel like mucking about with hypergeometric distributions, I think a little spare time spent taking even the roughest of looks at the odds will help you build your own rules of thumb – and let you get down to the business of playing instead of burning through the clock.

So what’ll it be? Keep or mull?

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