Beating Four Dead Horsemen
There used to be a deck in Legacy called Four Horsemen. The deck set up a way to mill itself infinitely (Basalt Monolth + Mesmeric Orb), and then needed to mill cards in a specific order; Sharuum, Blasting Station, and Dread Return need to go into the graveyard before Emrakul, the Aeons Torn.
This combo is guaranteed to kill the opponent eventually, but not after any particular number of iterations; it's always possible for the Emrakul to get milled before the Dread Return, Sharuum, and Blasting Station, forcing the player to reshuffle and try again.
This runs afoul of Magic's tournament policy on nondeterministic loops; in short, they're not allowed to continue past the first iteration.
I would like to be the 5000th person to argue that this is a mistake.
First, let's think about what the goal is here. Magic is a complicated game, and this is an intentional part of its design; cards can do almost anything imaginable, and the number of potential combinations between these cards leads to a virtually inexhaustible design space for people brewing decks. The goal of tournament policy is not to tell people what decks they're allowed to play, but to facilitate the players making their own choices.
Logistical constraints are imposed not because they are inherently good, but because they are necessary to accomplish other goals. Ideally we'd let players think for as long as they need to make a decision, since that maximizes the relevance of skill, which is the point of tournaments. But one person taking too long would delay other people, so we limit rounds to 50 minutes.
Constraints like these are not fundamental parts of the game of Magic, but are unfortunate compromises we make, imposed only due to the substrate on which the game is being played; if you bring a 300-card Battle of Wits deck to a tabletop tournament, you likely won't be able to shuffle it in time. But play it on MTGO or Arena and it's perfectly fine.
What this philosophy tells us is that by default, any deck strategy should be permitted, and the only good reason to disallow one is if there's no way to allow it without compromising on some other goal. Having an entire archetype that’s legal in theory but banned in practice is a pretty wide deviation from that goal.
And Four Horsemen isn't the worst example of problems caused by disallowing nondeterministic loops. If you control Frenetic Efreet and activate its ability a million times, you have to actually flip a million coins by hand. If you control Krark, the Thumbless and cast a free spell, losing the flip and returning it to your hand, you can't cast it again. If you loop casting Lotus Petals from your graveyard and then announce you're Brain Freezing your opponent for 1000, not realizing that they have an Emrakul in their library, you have to resolve those Brain Freezes one-at-a-time.
Is there any benefit of the current policy that justifies such terrible outcomes? I don't think so. There are several common reasons given for why the policy needs to be this way, but upon closer inspection I don't think any of them hold up.
1. "We don’t want players/judges to have to understand complex mathematics in order to play the game"
It is true that Four Horsemen, and most other nondeterministic loops, touch on a technical mathematical concept. But this isn't unique to them.
Much ado has been made about how Magic is Turing-Complete. This means it can run any possible computer program, and have arbitrarily complicated behavior. It's entirely possible to set up a game state that has unknown outcomes and cannot be shortcut, despite using only deterministic mechanics. The tournament policy doesn't even attempt to cover this situation.
From this fact, we can derive that Wizards is not concerned with covering every possible corner case, and only cares about handling practical situations. But in practical situations, nondeterministic loops are far simpler to explain than many other combos that Wizards has chosen to allow!
Consider the Duskwatch Recruiter + infinite mana loop, or Omniscience + Petals of Insight. Wizards has ruled that as long as the number of cards in the library isn't a multiple of 3, it's acceptable for a player to shortcut this to "I get to put my library in the order of my choice". Why the restriction on not being a multiple of 3? Well I could explain it to you, but you'd have to understand modular arithmetic, and as we know, advanced math has no place in Magic: The Gathering.
It gets worse when we imagine actually executing the combo. There is indeed a specific algorithm that you can follow to get your library into an order of your choice, but can you derive it? If you were told to execute the combo manually, and get even just, say, 4 cards in the order you want on top your library, could you do that? Do you even know how much times you'll need to cast Petals of Insight (a requirement in order to be able to shortcut it)? Not without using any math!
In fact Wizards has helpfully done part of the math for us, stating that players can say "I cast Petals 410,758 times, and the final order is [whatever I want it to be]". Note how handwavy this is; officially, players who are executing a combo are required to state exactly how it works, how many times they're doing it, and what all the intermediate states will be.
These latter two facts are exactly the justifications given for why Four Horsemen is banned, since the number of iterations is indeterminate, and the intermediate states unknown. But players doing Petals combo aren't required to describe or even understand the process themselves! Nor are they required to state what order they're making the changes in, meaning that if an opponent wants to interact at some point with, say, Predict, it will be undefined what card is currently on top of the opponent's library. And Wizards is clearly ok with some math being done to prove it's possible, since they've supplied the work themselves.
Compare this to what's necessary to show that Four Horsemen combo can eventually mill cards in the right order. That's right: nothing. It's intuitively obvious to anyone that if you keep rolling a die over and over, you will eventually get every possible result. The chance that a Magic player would get confused at the idea of "I'm gonna keep shuffling until I get the cards I want" is negligible - certainly much lower than the chance that someone misunderstands Petals combo and e.g. thinks it works even with a multiple of 3 number of cards.
In fact, by using the current distinction of whether the probability goes to 1 in finite vs. infinite time to determine whether a loop can be shortcut, Wizards is requiring players to understand exactly the complex mathematics that they were purportedly trying to avoid! A world where Four Horsemen were legal would require fewer math explanations, not more.
2. "It violates Magic's rules against infinity"
In order for Four Horsemen to be guaranteed to win, one must be able to execute it an unbounded number of times. This is different from executing it an infinite number of times, but the standard rules for loops are "you have to pick a number", and picking a single number doesn't work for Four Horsemen, since it's always possible that all shuffles up to that point happen to not get you the order you want.
Of course the whole point here is that Wizards can change the rules, so simply pointing to the current rules is unhelpful. But perhaps Wizards of the Coast just has a sincere ideological commitment to strict finitism, and won't accept a statement like "the probability of success approaches 1 as the number of shuffles approaches infinity" as meaningful.
To this I say: balderdash! We need only consider rule 107.1b:
Most of the time, the Magic game uses only positive numbers and zero. You can't choose a negative number, deal negative damage, gain negative life, and so on. However, it's possible for a game value, such as a creature's power, to be less than zero. If a calculation or comparison needs to use a negative value, it does so. If a calculation that would determine the result of an effect yields a negative number, zero is used instead, unless that effect doubles or sets to a specific value a player's life total or the power and/or toughness of a creature or creature card.
Firstly, by talking about the "positive numbers", the Magic Comprehensive Rules admits the existence of infinite sets, thus disproving any commitment to strict finitism.
Secondly, consider that last sentence; the one that means that Scourge of the Skyclaves can become bigger than a 20/20, but Death's Shadow's is capped at 13/13. No lover of mathematical elegance or consistency would ever write such a sentence into their rulebook.
But even if we grant that the rule needs to remain
3. "Probability 1 isn't the same as guaranteed"
Some say that, since there's a zero-probability-but-technically-within-the-set-of-legal-outcomes event that results in the player winning (shuffling an infinite number of times and never getting the order they need), it would be unfair to let them shortcut to a win.
I would be happy to let any opponent with this concern simulate it properly for themselves, so they have their full chance at the combo fizzling. Just use the die-rolling program in many Magic phone apps, and keep hitting the "d4" button until the player gets a 4. I'd bet that after a few times of this simulation always succeeding eventually, these opponents will get tired of asking.
But again, none of this matters. Even if you reject the premise that 0 means 0 and don't want to let it be simulated, the four horsemen player can just lower their expectations and accept the ~1/10124,938,736 chance of failure that comes from limiting themselves to a billion iterations.
4. "What if the opponent wants to interact?"
This one is actually a problem, sort of.
What if the opponent has Nurgle's Conscription, and they want to reanimate the Four Horsemen player's Dauthi Voidwalker in response to Emrakul's shuffle trigger? Do they get to? There's no guarantee that the Voidwalker would ever be milled before the target graveyard order is reached.
In general, if we let the player shortcut the combo and then the opponent wants to interact at some point in between, how do we figure out the exact game state for the interaction?
Luckily there's a simple answer: just let the opponent pick. That is, if a player wishes to shortcut a nondeterministic sequence of actions, they are giving up the right to know exactly what goes on during the time they're skipping over, so the opponent gets to assume it's whatever benefits themselves the most. In the example above, the opponent says "well, turns out you get unlucky and have to show every card from your deck before you get the order you want, and I will interrupt when you mill Dauthi Voidwalker".
This is not a perfect solution; it still results in the four horsemen player being at an unfair disadvantage. But it's obviously much less disadvantageous than "you can't play the deck at all", so it's strictly an improvement over the status quo.
(And in practice, these sorts of situations where the opponent would need to know the result of random events are actually quite rare. Notice how I had to use famous legacy staple Nurgle's Conscription in order to come up with an example. So under the aforementioned philosophy of "let's write policy primarily to cover the things that actually happen", this is even more clearly an improvement.
In a game of infinite complexity, there will likely always be corner cases that a reasonably-sized rulebook can't handle. But this is not an excuse to throw up one's hands and say "well, guess we can just leave it arbitrarily broken then".
Quite the contrary in fact; it's an argument to use our limited rulebook length as efficiently as possible, focusing on the cases that are likely to actually come up.
The fact is, while no nondeterministic loop deck would currently be a top tier deck if allowed in an officially-supported format
I think it's high time we stopped having de-facto bans on entire archetypes that are otherwise entirely non-problematic. Let's let Magic players play what they want to play.
With thanks to Pi Fisher for suggesting the elegant solution to problem #4.
