An infinite amount of pain
Jul. 18th, 2008 04:38 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
jackBack to the example of Magic:TG. People have often pondered the most appropriate infinity rules. There's a few things it would be nice to do:
* If you gain an infinite amount of life, you can't be killed by any finite amount of damage
* If you gain an infinite amount of life, you can be killed by an infinite amount of damage
* If you gain nought life an infinite number of times, it doesn't make any difference
In jargon, "gain an infinite amount of life" refers to a cycle of moves which gains life and you could repeat without stopping. (Eg. "When X happens, gain 1 life", "When you gain life, do Y" and "When Y happens, do X".)
The normal rules approximate these with finite numbers pretty well. If you have an infinite loop, you have to choose how many times to go round it, but then to do something else. So in fact, if you gain infinite life, you think of the biggest number you can, and gain that much, which can be exceeded if someone does infinite damage, but hopefully not by any finite combination[1]. If you have a loop that doesn't have any useful effect, you can't force a draw, you have to stop[2].
First infinity rule
But it would be nice to actually have rules that could properly get to grips with infinity. The obvious interpretation is that if there's an infinite loop, the result is what you get when you take the limit of any number that changes.
For instance, if I have a loop where everything stays the same, but my life goes 20, 22, 24..., then after the sequence is complete, I obviously have ω (infinity) life.
But what happens if player 1 can play "Target creature gains flying" and player 2 can play "target creature loses flying" for free? If player 1 gives his creature flying, and player 2 takes it away, and they repeat, what's the result?
The traditional rules give the second player the last word. The analogue would seem to be that the second player can choose any subsequence to take the limit of. So if player 1's life goes 20,22,24.. then whatever player 2 chooses, the limit is omega. But if flying goes 0,1,0,1.. then player two can choose the sequence 0,0,0,0...
What if player 1's life goes 20,10,30,10,50,10,90,10,170,10...? Then if player 1 could stop at any point, he could have any large amount of life. But if goes on to infinity, player 2 could choose 10. Maybe player 2 gets the choice only if they were able to stop the loop, otherwise player 1 chooses. Is there a better description of that?
I think you can always find some convergent subsequence (ie. your favourite infinities are compact. Is this right?) Surreal numbers may provide any limit, but I don't think it's a useful limit if the sequence isn't monotonic.
Second infinity rule
However, I believe there's a more fundamental problem. As soon as you introduce an infinity rule, you have to ask, "if I gain an infinite amount of life, can someone kill me by doing an infinite amount of damage". The interpretation we have says "no", since the limit of ω, ω-1, ω-2, ... is, if anything, a surreal number infinitesimally below infinity, so still larger than any finite number, and emphatically non-zero.
This is also the answer in the only official magic rules touching on the situation, "No. Infinity minus infinity is still infinity. Stupid infinity" from this article.[3]
However, I believe our interpretation is actually flawed. It's fine for a countably infinite amount of life, but if you have a countably infinite amount of life, and someone does an uncountably infinite amount of damage, you would still be alive! Your life would be infinitesimal just smaller than infinity smaller than any infinitesimal you can reach by subtracting one from infinity any countable number of times, but that's still larger than any finite number.
But the uncountable amount of damage is greater than countably infinite amount of life by any metric, and should totally kill you. (Uncountable infinities require some slight reinterpreting of the rules that I'll come back to later, but at least conceptually we have a problem.)
My answer is that instead of being ordinals measuring "how many times", life totals should be cardinals measuring "how many things". When you play magic, you already normally represent your life total with a number of counter, the only difference here is to make that official. Thus when you deal damage, you have to specify which of the counters you want to take away. This normally makes no difference at all, except that if you're taking away infinitely many counters from an infinite number of them, you can either choose to take away each of them in turn, or to take away each other one, thus leaving an infinite number.
This also has the advantage that it matches the finite rules. If I gain an "infinite" amount of life with the finite rules, my opponent can still kill my by doing an "infinite" amount of damage, as we both chose large finite numbers, and he can choose a bigger one. I'm not allowed to retrospectively have chosen a bigger number when I went "infinite". However, if I gain an infinite amount of life, and then spend an infinite amount of life to do something, I can easily choose the first number much bigger than the second, and thus both spend and be left with an effectively "infinite" amount of life.
Turn order
The same "take a limit" logic ought to work fine for turn order without any changes to the rules at all -- I don't think the rules ever specified only a finite number of turns. If my opponent has an infinite amount of life, but can't do anything, and I deal her one damage a turn, and neither can draw any cards or anything, what happens? Obviously, after an infinite number of turns I kill her.
Conclusions
There's definitely more to do. How to deal with infinite decks. How to deal with uncountable numbers. Do we deal with "choose a number" ok? And we know there are edge cases unspecified.
However, I get the feeling the infinity rules ought to work as they are, having made only very very minor changes to the existing rule set, and let the players play on to a defined result if one of them manufactures or is trapped in an infinite loop.
Can anyone think of any broken edge cases?
Footnotes
[1] I recall a discussion somewhere about a magic deck which could generate the largest number without being able to go infinite. Does anyone remember that?
[2] Indeed, obviously, there have been many abilities you can use at will where it doesn't make any difference if you used them a second time, you can't just say "I activate this ability" forever. There are some subtleties, like if both players are counteracting each other's moves, the second player gets to play last. If the loop is mandatory, then the game is a draw, though I don't know if that's necessary.
[3] The card in question comes from the "Unhinged" set, which is a parody of normal magic, and does all sorts of crazy shit just to see if it's possible, such as a card that gains infinite mana. The official rules team, designed to stop loop-holes in sane sets, threw up their hand at a set that included this sort of crazy zany shit, and handed the official rules off to someone who used to script-write for Roseanne[4]. Thus the "unhinged" ruleset allows for infinity, though even he drew the line at "choosing a number" including ω
[4] Mark Rosewater, working at Wizards of the Coast, and author of many humorous articles.
* If you gain an infinite amount of life, you can't be killed by any finite amount of damage
* If you gain an infinite amount of life, you can be killed by an infinite amount of damage
* If you gain nought life an infinite number of times, it doesn't make any difference
In jargon, "gain an infinite amount of life" refers to a cycle of moves which gains life and you could repeat without stopping. (Eg. "When X happens, gain 1 life", "When you gain life, do Y" and "When Y happens, do X".)
The normal rules approximate these with finite numbers pretty well. If you have an infinite loop, you have to choose how many times to go round it, but then to do something else. So in fact, if you gain infinite life, you think of the biggest number you can, and gain that much, which can be exceeded if someone does infinite damage, but hopefully not by any finite combination[1]. If you have a loop that doesn't have any useful effect, you can't force a draw, you have to stop[2].
First infinity rule
But it would be nice to actually have rules that could properly get to grips with infinity. The obvious interpretation is that if there's an infinite loop, the result is what you get when you take the limit of any number that changes.
For instance, if I have a loop where everything stays the same, but my life goes 20, 22, 24..., then after the sequence is complete, I obviously have ω (infinity) life.
But what happens if player 1 can play "Target creature gains flying" and player 2 can play "target creature loses flying" for free? If player 1 gives his creature flying, and player 2 takes it away, and they repeat, what's the result?
The traditional rules give the second player the last word. The analogue would seem to be that the second player can choose any subsequence to take the limit of. So if player 1's life goes 20,22,24.. then whatever player 2 chooses, the limit is omega. But if flying goes 0,1,0,1.. then player two can choose the sequence 0,0,0,0...
What if player 1's life goes 20,10,30,10,50,10,90,10,170,10...? Then if player 1 could stop at any point, he could have any large amount of life. But if goes on to infinity, player 2 could choose 10. Maybe player 2 gets the choice only if they were able to stop the loop, otherwise player 1 chooses. Is there a better description of that?
I think you can always find some convergent subsequence (ie. your favourite infinities are compact. Is this right?) Surreal numbers may provide any limit, but I don't think it's a useful limit if the sequence isn't monotonic.
Second infinity rule
However, I believe there's a more fundamental problem. As soon as you introduce an infinity rule, you have to ask, "if I gain an infinite amount of life, can someone kill me by doing an infinite amount of damage". The interpretation we have says "no", since the limit of ω, ω-1, ω-2, ... is, if anything, a surreal number infinitesimally below infinity, so still larger than any finite number, and emphatically non-zero.
This is also the answer in the only official magic rules touching on the situation, "No. Infinity minus infinity is still infinity. Stupid infinity" from this article.[3]
However, I believe our interpretation is actually flawed. It's fine for a countably infinite amount of life, but if you have a countably infinite amount of life, and someone does an uncountably infinite amount of damage, you would still be alive! Your life would be infinitesimal just smaller than infinity smaller than any infinitesimal you can reach by subtracting one from infinity any countable number of times, but that's still larger than any finite number.
But the uncountable amount of damage is greater than countably infinite amount of life by any metric, and should totally kill you. (Uncountable infinities require some slight reinterpreting of the rules that I'll come back to later, but at least conceptually we have a problem.)
My answer is that instead of being ordinals measuring "how many times", life totals should be cardinals measuring "how many things". When you play magic, you already normally represent your life total with a number of counter, the only difference here is to make that official. Thus when you deal damage, you have to specify which of the counters you want to take away. This normally makes no difference at all, except that if you're taking away infinitely many counters from an infinite number of them, you can either choose to take away each of them in turn, or to take away each other one, thus leaving an infinite number.
This also has the advantage that it matches the finite rules. If I gain an "infinite" amount of life with the finite rules, my opponent can still kill my by doing an "infinite" amount of damage, as we both chose large finite numbers, and he can choose a bigger one. I'm not allowed to retrospectively have chosen a bigger number when I went "infinite". However, if I gain an infinite amount of life, and then spend an infinite amount of life to do something, I can easily choose the first number much bigger than the second, and thus both spend and be left with an effectively "infinite" amount of life.
Turn order
The same "take a limit" logic ought to work fine for turn order without any changes to the rules at all -- I don't think the rules ever specified only a finite number of turns. If my opponent has an infinite amount of life, but can't do anything, and I deal her one damage a turn, and neither can draw any cards or anything, what happens? Obviously, after an infinite number of turns I kill her.
Conclusions
There's definitely more to do. How to deal with infinite decks. How to deal with uncountable numbers. Do we deal with "choose a number" ok? And we know there are edge cases unspecified.
However, I get the feeling the infinity rules ought to work as they are, having made only very very minor changes to the existing rule set, and let the players play on to a defined result if one of them manufactures or is trapped in an infinite loop.
Can anyone think of any broken edge cases?
Footnotes
[1] I recall a discussion somewhere about a magic deck which could generate the largest number without being able to go infinite. Does anyone remember that?
[2] Indeed, obviously, there have been many abilities you can use at will where it doesn't make any difference if you used them a second time, you can't just say "I activate this ability" forever. There are some subtleties, like if both players are counteracting each other's moves, the second player gets to play last. If the loop is mandatory, then the game is a draw, though I don't know if that's necessary.
[3] The card in question comes from the "Unhinged" set, which is a parody of normal magic, and does all sorts of crazy shit just to see if it's possible, such as a card that gains infinite mana. The official rules team, designed to stop loop-holes in sane sets, threw up their hand at a set that included this sort of crazy zany shit, and handed the official rules off to someone who used to script-write for Roseanne[4]. Thus the "unhinged" ruleset allows for infinity, though even he drew the line at "choosing a number" including ω
[4] Mark Rosewater, working at Wizards of the Coast, and author of many humorous articles.
no subject
Date: 2008-07-18 05:28 pm (UTC)A much, much, much bigger carrying case.
Is the "unhinged" set the one with the Chaos Confetti card in ? I think
no subject
Date: 2008-07-18 06:03 pm (UTC):)
Although, there's only a few thousand magic cards in existence, IIRC, and you're only allowed four of each, and there's no point playing any that aren't useful, so you might end up with approximately forty good cards, plus a few for special situaitons, plus an infinite number of basic lands and relentless rats you borrowed from your host :)
Is the "unhinged" set the one with the Chaos Confetti card in ?
Yep, or rather the sequel. You mean the hacks for chaos confetti, or unglued in general? I wasn't into magic at the time (I'm not now), but silly definitely sounds right :)
no subject
Date: 2008-07-19 11:21 am (UTC)no subject
Date: 2008-07-19 09:28 pm (UTC)Yep. That was me. :) It's the top entry on http://toothycat.net/wiki/wiki.pl?MagicTheGathering/ComboMeThis . You need Very Large Number notation to describe how much mana he can make on turn 6, with only 13 cards. And that's all without being able to go infinite.
no subject
Date: 2008-07-19 11:11 pm (UTC)no subject
Date: 2008-07-20 09:27 pm (UTC)no subject
Date: 2008-07-21 01:39 pm (UTC)