I think another way to frame @chubbyrain1 's point of view is as this:

The publishing of this card is approximately equivalent to adding the following card errata on Alhammaret's Archive:

• This card's mana cost is reduced by (3) and increased by (UU).
• This card can now be Pyroblast'ed.

How many cards in Magic, if given the above errata modification, would move the card from vintage-unplayable territory to vintage-playable territory? Is there reason to believe that Alhammaret's Archive is one of those cards?

There is another consideration worth bringing up.

In the past, players have complained that vintage had become a format where there are no meaningful decisions to be made after the sideboarding and mulligan phases of the game.

From what I've seen, it appears that Lurrus leads to increased interactivity, and leads to the better player winning more often.

For instance, in this video, Andrea Mengucci says that he achieved an overall record of 33-3 with a Lurrus deck, facing a lot of other Lurrus, and that the games are very "grindy", with many games finishing due to a player running out of time due to the sheer weight of so many difficult decision points. Lsv tweets here that he went 8-1 in a challenge with the same deck, mostly facing Lurrus.

One could reasonably hold the opinion that the pro of increased interactivity/gameplay-skill outweighs the con of decreased deck diversity. Perhaps future companion printings can increase diversity and we can get the best of both worlds.

@ten-ten I don’t think this is necessarily so. Imagine hatebears that strongly counter Lurrus are printed, but come with deck building restrictions like no blue cards, or no artifacts, or no spells with odd CMC, or color with maximum devotion is red. You could have a wide palette of available hatebears, perhaps one against every major archetype of the format in your sideboard, but doing so might require thinking outside the box of existing archetypes. This could lead to format diversity.

New companions can be printed to mitigate Lurrus domination without introducing power creep. To see this, imagine they print a companion that is a Relic of Progenitus or Nihil Spellbomb on a beating stick. Now running Lurrus and all those baubles (instead of Mentor or Tinker) might become a bit of a glass cannon gambit.

Or the companion could be a bear with an ETB “gain control of target legendary permanent” effect, or a “moon” bear with a static effect like “all legendary permanents are Mountains”.

With enough hatebear companions, I could envision an interesting meta where a highly defensive deck with multiple companions in the sideboard (anti-blue companion, anti-graveyard companion, anti-workshops companion) becomes viable. This could open up the deck building design space, and increase the pool of vintage viable cards (just like Lurrus brought back previously unused cards like baubles and Dead Weight).

It sounds like in your simulation you changed the card Force of Vigor to only have 1 target instead of 2. It is no wonder then that it performed worse than alternatives.

What if two people sit down to play Vintage, and rather than bringing an already constructed set of 60+15, they take turns constructing their decks on the spot, one card at a time? This would look like, for example:

Alice: "I pick Mishra's Workshop"
Bob: "I pick Black Lotus"
Alice: "I pick Black Lotus"
Bob: "I pick Ancestral Recall"
Alice: "I pick a second Mishra's Workshop"
...

After this construction they would engage in a standard best-of-3 match. Would this make for an interesting format?

In the above example, Bob gets to optimize for his opponent by for example not including Mental Misstep.

This can be extended to, say, 4 players, followed by a round-robin set of matches between them, although you'd probably want simultaneous card selections rather than round-robin to save time. Maybe card selections can be made in groups of 4 cards at a time or something.

What would the "meta" be in this format?

To be clear, just because the game enters an infinite-loop within the rules of Magic doesn't mean that a software program implementing those rules must also enter an infinite-loop. A program like MTGO can, in principle, perform logical deduction on the game state to determine that the game rules yield an infinite-loop, and then output the correct game result (the rules of Magic specify that infinite-loops result in draws). The paper shows that a program like MTGO cannot do this deduction perfectly. Any such attempt will be "buggy" - not due to incompetence of the programmers, but due to the laws of mathematics. But it is certainly plausible that, if well programmed, such a program can do it well-enough to cover all "real-world" situations.

@inkfathombiomage That’s like asking, why doesn’t an integer that is theoretically even and odd count? No such thing can exist.

The current MODO will either output an incorrect value on some inputs, or infinite loop on some inputs. Any theoretical modified version of MODO will have the same property.

@cuikui I'm repeating myself a bit, but the result of the paper is not concerned with the number of possible moves or number of possible game states. You are right to note that these numbers are large, but the authors don't care about these numbers at all. They are not considering algorithms that try to compute these numbers.

Rather, they only consider game states where neither player has any more decisions to make. In MTGO terms, both players are essentially in "F6" mode. The question is, if we restrict ourselves to only those types of game states, is it possible to write an algorithm that determines who will win the game after the remaining forced actions and triggered effects are processed? The paper proves that the answer to this question is no.

To be very precise on what this means, consider the problem of adding two numbers, x and y. Here are 3 candidate algorithms (written in python) to solve this problem:

• algorithm1 will always return something. The problem is, that something is not always correct - it will output 4 as the sum of 1 and 2, which is incorrect.
• algorithm2 is better in some sense - it never returns an incorrect output. However, it has another problem - it sometimes runs forever, never returning anything. For instance, if you ask for the sum of 0 and 1, it recursively calls itself, which calls itself, etc., going into an infinite loop.
• algorithm3 gets the best of both worlds - it always returns something, and it never returns an incorrect value

The authors proved that if you consider the universe of all possible algorithms that accept a game state of Magic as input (even if we restrict ourselves to those "F6-mode" states where there are no more decisions left to be made by either player), and outputs a winner as output, then every single algorithm in that universe will be like algorithm1 or algorithm2 in my example. In other words, any such algorithm will either return an incorrect output on some inputs, or go into an infinite loop on some inputs. If someone tells you, "hey, I wrote a computer program that correctly outputs the winner given an F6-mode game state as input, without infinite-looping", that'd be like someone saying, "hey, I discovered an integer that is neither even nor odd!" It's simply a mathematical impossibility, a claim that can be rejected on its face, without even looking at the code.

@protoaddict Yes, you are absolutely right, I forgot about basic lands when I wrote that.