In Development – Do Math, Play Better

I had trouble with counterspells for a long time.

It just seemed like the opponent always had a counterspell—or I just couldn’t tell when they did or didn’t. After a while, I started to fall into that trap of just holding back my plays, convinced that they’d be countered. It wasn’t a successful approach.

I remember the first time I actually stopped myself and ran the math instead. It was in an Extended PTQ several years ago. In a long game against some kind of blue control, I had to decide if I was going to make my big play, or wait another turn. If he had the counterspell now, that was pretty bad for me. But if I let him have another turn—well, that’s another turn to draw a counterspell, right?

But then I counted the counterspells he’d used so far, made a best guess at how many he’d have left, and did some quick math in my head. The odds were he didn’t have a counterspell in hand. Good enough. I moved to make the play, he bluffed countermagic, and I said, “I don’t believe you.”

He didn’t have it.

Today I’m going to talk about how doing the math will unburden you to make better plays, and then give you two quick cheats to let you know the odds in the most important game situations.

Why We Want to Do the Math

We get a lot of advantages for our game play out of running some math during the match. It lets us pick the numerically best play, reduces our cognitive burden, keeps us from tilting, and helps us get the read on our opponent.

Picking the Most Logical Play

If you imagine an opponent who just played their cards and never bluffed, you would find that there are often statistically “best” plays for different situations.

For example, imagine an opponent who never bluffs countermagic, but always counters the next spell you cast. If you could calculate the likelihood that they actually had a counterspell in hand, you’d know whether to cast your big spell or not.

As it happens, this is a pretty good way to make that decision anyway. Most of the time, your opponent will make the logical play, so what matters is whether they have the card in hand.

Reduce Cognitive Burden

Humans have limited cognitive capacity.

If you’re thinking about whether you locked your car, that’s an actual part of your brain supporting that thought instead of figuring out whether or not to attack with all your creatures.

Knowing the likelihood that they have a card in hand—or might topdeck it—means that you don’t have to spend time trying to randomly decide whether or not that’s going to happen. More to the point, it lets you actually focus on how your opponent plays the game.

How often do they bluff, for example? It’s easier to figure that out once you know how likely they were to have the card in hand.

Basically, knowing the likelihoods gives you a base for all of your understanding of what your opponent is doing, so you can focus on their plays rather than trying to deconstruct the basic math of their position.

Reducing Tilt

Knowing the numbers really helps reduce tilt.

I’ve written about this at length in an earlier article, but the basic idea is this:

Knowing how likely an event is helps you understand that you (usually) shouldn’t be upset by it.

When Gab Nassif topdecked [card]Cruel Ultimatum[/card] to turn the game around and crush Matteo Orsini-Jones at PT Kyoto, that felt like a disastrously bad beat on very long odds. In reality, Nassif had about a 4.3% chance of topdecking that Cruel Ultimatum—not odds to plan on, by any means, but also just about a 1 in 20 chance, rather than some more extravagant piece of luck.

In my experience, players tend to tilt when their opponent “lucks out” with odds much better than this—say in the 20-25% range. Once you can do the basic math and know that your opponent was 25% to hit their out, you can stop feeling like reality hates you and get on with the game.

Do They Have It?

The first of today’s two situations that benefit from some basic math is the question of whether your opponent “has it” or not.

That is, are they holding that one card, be it counterspell, removal, or something else, that will totally destroy the play you’re about to make?

The Core Question—Take a Risk, or Not?

The whole reason we want to know whether our opponent “has it” or not, is that we’re trying to decide whether to place our ability to win in harm’s way.

This may mean that we’re playing a combo deck and need to decide whether we can go off this turn, or need to wait for a turn in which we can lead with disruption. It could mean that we want to tap out for a [card]Baneslayer Angel[/card] to hold off that aggro opponent, but if they have removal for it we’re dead. Or it could be that we’re deciding whether to lead with a test spell and go for a slower win, or lead with our power card and go for a win right now.

Part of what we need to know to answer this question is how likely it is that the opponent has the counter to our play in hand (or will next turn). This might be a literal counterspell, or it could be removal, disruption, or pretty much anything else that takes apart our game plan.

The Real Math Behind this Question

If you’ve played around with variations on this question before, you know that the math we use to handle it is the hypergeometric distribution. This is a statistical approach to determining how likely it is that you’ll have some item or items you care about when you’re pulling them from a finite pull.

Like cards in hand drawn from a deck.

Although we want to ask, “Do they have it?” our lack of information about the opponent’s hand means that we’re really asking, “How likely is it that they have it?” To do this, we need to know how many relevant cards they have left in their deck, how many cards are left in the deck overall, and how many cards they have in hand. Equipped with all of this information, you get probabilities for the early-to-mid game that look like this:

For the sake of convenience, this chart looks at cards in hand and in the deck in increments of 2.

The numbers across the top are how many “relevant” cards are left in the deck. A relevant card is one that crushes your game plan. The classic case here is countermagic. This is the number left in their deck, which means taking what you know about their list (most likely your best guess based on their archetype, in a typical tournament match) to give you a basic count of relevant cards. From that, subtract out the number you know they’ve used to get your total.

The numbers on the left are the number of cards they have in hand at the point that matters. This could be right now, for a counterspell on your turn, or the number they’ll have in their following turn, for something like removal.

So if you’re playing against an opponent who has four cards in hand, and who you estimate had six counterspells in her deck and who has used two of them, you can cross-reference “4” (top row) with “4” (side column) to find out that there’s a 31% chance she has a counterspell in hand right now.

This chart doesn’t ask you how many cards are left in the deck, as it’s based on an assumed amount that fits the early to mid-game.

The Quick Way to Get a Best Guess

You’re not going to run a hypergeometric distribution in your head, at least not unless you can actually calculate factorials that way.

Conveniently, there’s a quick hack that is typically more than enough to answer the basic question of, “Should I make this play or not?”

That hack is:

In other words, the percent likelihood that they have it is approximately equal to the number of relevant cards left in their deck, multiplied by the number of cards in their hand, multiplied by two.

Since this little hack assumes a certain deck size—and the odds don’t vary much for decks within the normal expected range for Constructed decks in the early to mid-game—you don’t need to know how many cards are left in their deck.

All you need is the number of cards in their hand and your best guess about how many relevant cards they have left in their deck.

If you’re thinking of dropping that key spell and your opponent has 4 cards in hand, and you think they probably have 6 counterspells left in their deck, then the chance that they have a counterspell right now can be estimated at:

4 x 6 x 2 = 48%

Limitations for this Hack

You may have noticed that 48% doesn’t quite match the 45% shown for the same cards in hand and in deck in the hypergeometric distribution above. Like most hacks, it drifts a little bit:

For higher numbers of cards in hand and relevant cards left in the deck, this approximation overestimates the chance they’ll “have it.” Here’s how bad the overestimation is:

Some of those overestimates can seem pretty bad, but they really aren’t a problem because they only start happening when it’s 50% or better that the opponent has it.

In other words, all those cases where you’re better than a coin flip to run into their countermagic or removal anyway.

For that reason, I think this is a great little trick to help you decide how likely it is that you’re about to run into countermagic. If your quick estimate says 16%, then it’s likely right, and you’re 5 chances in 6 to not run into trouble.

Will I Draw It?

The other big likelihood-based question we find ourselves caring about from time to time is, “Will I draw it?”

The question generalizes to, “Will they draw it?” since we care about topdecks both from us and against us. Should we bank on the opponent drawing removal next turn, or can we plan without that possibility in mind? Should we play to topdeck burn for the win, or no?

The basic math here is easy, but probably a bit much for most of us to do on the fly—so it’s handy that there’s another trick we can use.

The Real Math Behind Topdecks

The math for topdecks is much easier than for cards in hand. It looks like this:

Yup. That’s just the number of relevant cards left in the deck, divided by the total number of cards left in the deck.

It’s basic, but not always speedy for us to do while trying to focus on playing the game.

The Quick Way to Estimate Topdecks

For this super simple hack, you just need to know the number of cards that serve your topdeck needs.

That’s it.

Here it is expressed another way:

Yup. That easy.

For example, say you’re playing a burn deck and literally any 3-damage spell will win the game for you. You know you have seven 3-damage spells left in your deck.

7 x 2 = 14%

For this hack, we don’t even care how many cards are left in your deck because in most cases, the number of cards left that work for you is much smaller than the number of cards in the deck. As a consequence, pushing the deck size a little bit in one or the other direction—say, whether it’s turn 4, 5, 7, or 9—has little impact on the odds.

Good Enough Is Good Enough

The whole point of these quick hacks is to give us enough information to meet our decision-making needs while relieving us of the cognitive burdens of not knowing and of trying to arithmetic (or factorials!) on the fly. In these situations, a reasonable estimate goes a long way toward making our decisions way better, both by giving us information we need and by freeing up brain space to think about the game instead of the game state.

Do you have favorite tricks, estimates, or tools you use to inform yourself and free up cognitive capacity while you play?

magic (at) alexandershearer.com
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