In Defense of Pile Shuffling

Published on April 1st, 2023. See also the follow-up article, In Defense of Not Pile Shuffling.

Pile shuffling has gotten a bad rap lately. Judges tend to look down on it as a waste of time, and it's even been restricted to occurring once per game in official tournament policy. Unfortunately, I think a lot of people are unaware of all the arguments in favor. And it turns out there are quite a few!

First, there's the meta-argument. There are a lot of people on the internet who enjoy attacking things they don't understand, and boy does this happen with pile shuffling. Statistics and randomness are legitimately complicated topics, and there's no end to people who will confidently say false things in order to win internet points. In general, we should keep a skeptical eye towards any aggressively-enforced "consensus" among the loudest voices on the internet.

Disparagement of pile shuffling and people who engage in it is ubiquitous in today's online environment. From Twitter to Reddit to Youtube, people falsely accuse pile shufflers of being cheaters, ridicule them and say they're wasting time by trying to ensure their deck is properly random. Even MTGRemy, normally a top tier content creator, has joined in on the bashing. It's become a staple of Magic Twitter to make completely uninformative posts masquerading as education, which function as a method of manufacturing social consensus without having to actually provide any justification for their arguments. I mean just look at some of this stuff:

At my very first MtG event,long ago, I saw my 1st opponent pile shuffle. I genuinely thought he had a psychological impairment.

— it's Julian (@itsJulian23) May 30, 2016

When my opponents pile shuffle, it tells me that they don't respect my time and that I need to thoroughly shuffle their deck. That's it.

— Oraymw - 🧠♿ - 💖💜💙 - Parody* (@oraymw) May 29, 2016

If you attempt to pile shuffle for 2-5 mins in between games when your opponent bricks game 1, you should be fucking ashamed. 2 differeny Chicago players I played against did this. They both got no pts. And got punished for it. Why can't people just play the game instead of 🕑

— Procircuitscrub (@procircuitscrub) March 26, 2023

At the very least, don't pile-"shuffle" more than once, esp. not after mulligans. It not only wastes time, it also makes you look stupid.

— it's Julian (@itsJulian23) February 21, 2016
A Facebook post that says 'So what do we think of the policy changes? I don't expect the pile shuffle rule to really impact much. I just hope I don't get some clown wanting to puke after every in game shuffle.'

At times, well-intentioned people have tried to defend pile shuffling. It usually doesn't go well.

This article on the Magic judges blog correctly states that pile shuffling is an optimal randomization method if we start from an unknown deck configuration, and suggests that people use it in tandem with other shuffling methods. It was then soundly criticized by a well-known L3 judge on their personal Facebook page; not for being incorrect, but for providing justification for "superstitious players" to behave in a way that that judge doesn't like.

Wizards themselves even ran an advertisement for pile shuffling, demonstrating how quickly it could be performed, how much people enjoy it, and how it's become an integral part of Magic's culture. Of course the Reddit post making fun of it got almost 1000 upvotes.

But of course all of that is tangential to the question of whether pile shuffling works. While observing the psychology of the detractors of pile shuffling can give us an idea of how they may have arrived at their conclusions, a bad reasoning process doesn't guarantee a false conclusion. So let's look at the object-level arguments here.

First, it's important to understand that when we talk about a deck being "random", that means that we don't know where any of the cards are. It has nothing to do with following a certain process of moving the cards around. Think about any type of shuffle you believe to be effective; riffle shuffling, pile shuffling, whatever. Now imagine performing that shuffle with the card faces facing towards you, so you know exactly what ends up on top of the deck. It's the exact same physical process! Yet that's clearly not random.

The purpose of a shuffling a deck is to get the player to lose the information of where the cards are. (Along with any special relationships between them, like "all my Lightning Bolts are next to each other.) When someone riffle shuffles, the reason that works is because they can't keep track of exactly where the cards are going in each shuffle, so they end up ignorant of their final positions.

Pile shuffling conceptually does the exact same thing. Most people can't remember the exact position of 60 cards as they're laid out into piles and those piles reassembled, so this process causes the player to lose information about what's in their deck just like a riffle shuffle does.

This is really not a complicated concept, yet people have a strangely hard time understanding it. The argument that pile shuffling is bad because it's deterministic makes no sense whatsoever; riffle shuffling is also deterministic, just like everything else in the universe.Except potentially quantum events. Riffle shuffling works because it seems random to the human, and so does pile shuffling.

Next let's talk about "breaking up clumps". Critics of pile shuffling and mana weaving generally point out that a statistically random deck will still have clumps, because an even distribution of lands and spells is a pattern, and a random distribution shouldn't have patterns. This is in fact true; if your deck is exactly land-spell-land-spell etc., then it's not random. But what often gets missed is that all decks will have some pattern; riffle shuffling a deck increases its randomness asymptotically towards the maximum, but it never quite reaches the maximum.

What we care about in Magic tournaments is that any residual patterns are very minor and unlikely to impact the game. Spreading out the different cards will indeed help the deck score more highly on traditional measures of randomness, since it increases the information-theoretic entropy of the deck. (There are more configurations that have spread out lands than ones with lands all in a large clump.) As a visual example, consider two "sorted" distributions:



Clearly neither of these is a properly random deck. But now add a small amount of randomness of top of that:



Which of those decks looks better shuffled? And this isn't just a visual thing; mathematical tests for randomness will find the second one to be significantly more random. (As a simple example, everyday compression algorithms approximate Kolmogorov complexity, and if you compress the first string into a zip file, you'll find that the result is shorter than the second.)

The concept of a deck "looking random" actually does get at something important. Randomness is in the mind of the observer, so if the deck looks random to a sufficient degree, it actually is.

Lastly, some people have claimed that pile shuffling makes it easy for someone to stack a specific card on top of their deck. While true, it's actually easier to do with mash and riffle shuffles. Just choose to interweave the deck halves such that the same half always remains on top, and you can actually multiple cards to stay in the same position throughout the whole shuffle. With pile shuffling you can only do it with one single card, so pile shuffling actually makes it harder to cheat in this way, not easier. As always, the solution to Cheating is not to force players to use a specific shuffling method, but rather to carefully watch for deck manipulation, and encourage players to always shuffle their opponent's deck.

Pile shuffling has several additional benefits as well:

While none of these are the primary purpose of shuffling, they're all very useful things to do, and pile shuffling's superiority here should be taken into account.

Indeed, many people with serious expertise in statistics believe that pile shuffling is a better option. For example, I remember talking to a player in my LGS several years ago and asking them to stop pile shuffling. At the time I was against pile shuffling, but they explained to me that it was a useful method of randomizing the deck, and backed this up by mentioning that they were a statistics student at the University of Florida. After that interaction, I looked into the matter further, and found out many of these arguments in favor that I had previously been unaware of.

It's important that we're able to admit our mistakes and change our minds when presented with new evidence.

I'd like to thank Pi Fisher and Jacob Cohen for proofreading and fact-checking the claims in this article. Pi is a Google employee with a master's in Mathematical Sciences, credited in multiple mathematics textbooks. Jacob is a New York Times published mathematical writer; best known for their math puzzles column and for popularizing the concept of "Nice Numbers".

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