This can resolve NO.

Any percentage is a solution to the snipgiblets problem: Any nonnegative c is a valid solution to

c * 0 = 36 * 0, so
0/0 if it exists equals c/36, which can again be any real number.

So 0%, 100%, and 1/36 are all valid percentages, if such percentage exists.

You haven't given an argument that there is a unique percentage, beyond "it would be nice if this limit existed and we could define a constant".

Also, the same thing but in words: "If you imagine a mirror problem where the pope has 35x as many snibgiblets as you, then the 1/36 argument goes through just the same to say that the percentage is 1/35. But you and the pope have the same number of snibgiblets as in the 1/36 problem, so clearly the percentage isn't a function of (your snibgiblet count, the pope's snibgiblet count) but has some extra context-dependent parameter. So there is no solution, if you are just comparing snibgiblet counts."

You gain continuity within a problem instance by saying it's 1/36, but lose continuity across problem instances. Saying it's 0 or 1 keeps continuity across problem instances but loses it within. So it's not clearly a cleaner solution to say 1/36.

@Mira I like your point about continuity between instances vs continuity within a problem instance. But I feel the clarification in parentheses pins us down. As posed, we're going with continuity within the problem instance.

@dreev I usually set 0/0 = 1 when I'm writing code, for example if I'm reporting progress on 0 tasks out of 0. So if I needed 36 tasks to be done per CPU and "scheduling a task" is a task, maybe I would have this family of situations:

0 CPUs: 0/0 = 100%

1 CPU: 1/36 = 2.7%

2 CPUs: 2/72 = 2.7%

where the 0/0 instantly finishes because there is no work to be assigned out. (this uses integers but I could e.g. take a rational sequence of "this CPU has N cores so there's a 1/N multiplier".)

There actually is a binary discontinuity in having work to be assigned out, no matter how small the work is.

@mira I can't tell if that's an elaborate joke or not.

@dreev If you're just asking "what is lim_x->0 of x/(36*x)?", then the answer is 1/36. But the association of your word problem to that expression isn't obvious. Why bind the variables together during the limit, for example?

"lim_x->0 of 0/(36*x)", so taking the limit as the pope exits his snibgiblet position to restore the invariant we see that the answer should be 0.

"But the invariant isn't held during that time"

In some contexts, one variable moves first and the second follows to restore it.

The problem here says that the pope currently has 36x as many snibgiblets as me. Doesn't say anything about the moments prior to ending up in that position. Why should the invariant be assumed to hold at all times for eternity?

You can make the limit 0, 1/36, or infinity depending on how you end up at (0,0).

This question is poorly defined. It's true that the limit of x/(36x) as x -> 0 is 1/36, and so the question is just "is taking that limit the correct way to answer the question". But the question is so vague in trying to speak in both abstract mathematical terms ("real number") and concrete/real-life terms ("the pope") that there is no single "correct" answer. The answer is there is no answer.

@Shelvacu I would agree with this without the clarification in parentheses. With that clarification, "what is the limit?" is pretty explicitly the question.

But of course the question this market is asking is whether commenters will agree with that. Seeing that they do not sheds light on at least some of the disagreement in the original Snake Eyes market, so this has been useful.

it sheds light by demonstrating people will disagree with basically anything if you get enough of them in one place?

@dreev So this is a word-lawyering question, right?

For the record: If the whole question was

Say your snibgiblet number is x. By definition the pope's number is 36x. Is the limit as x goes to 0 of x/(36x) 1/36?

my answer would be Yes.

But that's not the question at all.

@Primer Not a word-lawyering question!

@dreev Yeah, ok, technically there are a few layers of "how will other bettors anticipate you will judge comments", "will there be comments", "will it be handled in good faith", "when will it be resolved, if at all", "how will it be used in connection with the original market", etc., which I'll leave to others whether those can be subsumed into word-lawyering or into math or into something else.

Background: My market on the snake eyes paradox has become a bit of a rabbit hole. It seemed like a lot of the confusion was about my suboptimal statement of the problem so I've tried to mirror that here but where the question itself is non-controversial. As I said in the snake eyes comments, I think if you don't agree that the answer here is 1/36 then we're kind of on different planets.

But then an actual mathematician said they didn't agree that the answer here is 1/36 (but does agree that the answer to snake eyes is 1/36?!) so now I don't know what to think. I think they might be saying that snake eyes is an interesting paradox and 1/36 is ultimately the correct answer and they're not fussed about technicalities of the problem statement. But with this question it's just inviting language-lawyering and if you language-lawyer it, you don't get an answer of 1/36? Something like that?

Anyway, I'm frustrated by the NO bettors from the snake eyes market who seem to be refusing to establish any common ground here. Commendable exception: @MartinRandall. I'd also love to hear from @FintanCostello about whether we're on the same page about this snibgiblets problem. Again, I think this is a helpful sanity check on where we're diverging on the snake eyes problem.

See also my market on pinning down the answer for the two-round version of the snake eyes problem.

• I think this can resolve No, as there are unhidden comments which disagree with Yes

• I won't bet here as your comment below says we can "discount" this market's No-bettor's views in the original Snake Eyes

• Do you notice that the attempt at creating a very clear-cut market intended to get rid of disagreements ended up with a contradictory problem statement?

@Primer I agree it's looking like no. I'm eager to hear your own view of this question. I don't think the problem statement is very contradictory. "What's such-and-such ratio, taking a limit as follows to make the answer not just be 'undefined'?".

Like, sure, technically it asks a question to which the most technical answer is 'undefined' and then proceeds to contradict itself to make the answer not be undefined. But, I mean, c'mon. Like @jacksonpolack said, anyone outside Manifold would just give the natural answer, wouldn't they?

(This reminds me of the time I made a market to experiment with auto-resolving and picked "is the sky blue?" as the question so there'd be a clear ground-truth answer to compare with how the market auto-resolved. People had a heyday language-lawyering the question of what color the sky really is.)

Point being, this is not about language-lawyering, it's about seeking mathematical truth and figuring out how to actually resolve the snake eyes paradox. I'm anxious to hear your true answer to the spirit of the question! And if you think I've hopelessly mangled the problem statement, let's fix it together or make a new market that gets it right. Truth over mana! Or mana in service of Truth. Or something.

@dreev To get this out of the way: I remember whole articles about the sky not being blue, but maybe cyan or green, all this being very culturally diverse.

(1) You have experience with those kind of problems. How did you adjust?

anyone outside Manifold would just give the natural answer, wouldn't they?

How would I know? I haven't talked to anyone outside Manifold since Snake Eyes took off 😂

But seriously: Most people would respond with "What is a snipgiblet?", "Why should I care?", "How do I 'convert to a percentage'?" or "What is a limit?" and I do think we only care about those with sufficient math knowledge. Many of those will probably just state that the question is poorly defined, or rightout contradictory, as they are used to clear, math-y language.

Point being, this is not about language-lawyering, it's about seeking mathematical truth

(2) If you seek for mathematical truth, why don't you define a function or a limiting process or whatever and ask for the solution, without any mentions of paradoxes?

(3) This is as much about language-lawyering and intended meaning and premises and wordings and philosophy as it gets!

(4) Has there been any disagreement about actual maths so far?

Not about which maths to use, but like from one line of mathy symbols and numbers to the next, or about a coding error, which is not about the mapping of words to maths/code?

I'm anxious to hear your true answer to the spirit of the question

My credence is: The spirit of the question is to provide a sufficiently ambiguous question which partly covers solely the Yes-holders' interpretation of Snake Eyes for the purpose of being able to point to it later. "Look at this question, it may also be ambiguously phrased, but without the context of Snake Eyes, everyone agrees the Yes position is the correct one!" That's ok, I guess, for a debate on intent and words, which I'd hope you'll agree sometime this is.

All in all, this feels unfair and I'm genuinely not sure how much of it is intentional versus just not at all "getting" the opposing point of view versus not wanting to change anything seeming relevant.

Meta: Please tell me if you feel my opposition starts to cross towards hostility. I've seen so many ambiguously phrased and poorly operationalized markets, which gain a lot of traction, supposedly not in spite but rather because of the ambiguous phrasing and poor operationalization. Markets with potential for drama flourish. Markets which might resolve to either the creator's intent or the description's wording, no way to know in advance, bring in the trader bonuses. I find this annoying and this might influence my attitude here as well: I feel some pushback is justified.

@Primer laughing at your description of normal people. There's also "But I already don't have any snipgiblets".

@MartinRandall

"You're a snipgiblet!"

@Primer It seems like you're mainly defending the max-language-lawyering position here. Localized replies:

(1) You have experience with those kind of problems. How did you adjust?

I guess I've failed to; fair point.

[many of] those with sufficient math knowledge [...] will probably just state that the question is poorly defined, or rightout contradictory

Excellent, yes, an empirical question. I'm predicting they'll mostly say 1/36 even though the one mathematician so far who's answered has said the opposite.

If you seek for mathematical truth, why don't you define a function or a limiting process or whatever and ask for the solution, without any mentions of paradoxes?

The snibgiblets problem does so, just with the suboptimal phrasing, I think.

Has there been any disagreement about actual maths so far?

Oh yeah, tons.

not wanting to change anything seeming relevant

I can't tell what this bit means.

As for the seeming unfairness, I've added more background as a new top-level comment here. I'm trying to be totally transparent about my thinking at every step. I did expect people would agree on 1/36 in this context and that that could invalidate some of the more seemingly (to me) disingenuous NO arguments in snake eyes. Maybe I should've been clearer from the start that answering 1/36 here might be something of a concession in the snake eyes market? (Again, hat-tip to @MartinRandall for being perfectly happy to make that concession and focusing on the things he and I actually do disagree about with snake eyes.) Anyway, I thought all that was already clear, especially given my comments in snake eyes, but I shouldn't assume people see the comments buried in the snake eyes market!

Markets which might resolve to either the creator's intent or the description's wording

Yeah, that's rough. I'm doing my best and care a lot about maximizing fairness! That's been what's motivated my edits and clarifications. In retrospect I shouldn't have traded in the snake eyes market myself, but of course at the time I failed to anticipate the ambiguities and thought that "we'll get a mathematician to judge" made for an objective resolution. Which it still does. But then my attempts to clarify the unanticipated ambiguities can be viewed as attempts to bias the resolution. I guess it's especially easy for them to seem that way when I'm trying to clarify to align the wording with "the creator's intent". That's why I've undone those edits in the meantime. Thanks for bearing with me on all this!

And please do keep helping me understand other potential unfairnesses. I will keep doing my best to make things fair.

Well, it's looking like Kongo and Floris sincerely disagree that 1/36 is the right answer here. I don't think their arguments are coherent, but that's fine, that's not what this market is asking about.

(To summarize, they seem to be hung up on the self-contradiction in the question: the answer before the clarification is "undefined" and then the clarification says "but wait, let's change the question so we can get a real number answer". Which is maybe an annoying presentation and the question should be framed in terms of limits from the start to avoid that. It all seems reasonable to me the way it is but maybe my brain is hopelessly corrupted at this point.)

I'm encouraged by this because I'm pretty sure mathematicians will view this the way I do. Which means we can discount Kongo and Floris's views in the main snake eyes market, since they're disagreeing with what's standard in math and a mathematician will be determining the resolution of that market.

In other words, this market has shown that we're disagreeing about fundamental math stuff, not the snake eyes paradox specifically.

But this is just one source of disagreement about snake eyes. Others, like Martin Randall, agree that 1/36 is the answer here but still disagree that it's the answer to the snake eyes paradox. So that's still very interesting!

@dreev I don't know if it's relevant, but I am a professional mathematician, and I do believe that the answer to the snake eyes market is 1/36 (but I didn't read much comments and some parts of the market might be contradictory). I just think that this market is not a good translation of a problem in that market.

Also, I think that most of the difficulty of the snake eyes market lies in translating the English words into a precise mathematical meaning. Anything after that is easy (from a mathematician's perspective). I'm not sure if the first problem is "better" solved by mathematicians.

@FlorisvanDoorn Dang, now my head is spinning. You're (a) a mathematician yourself, (b) are proving me wrong that mathematicians would all agree on the snibgiblets answer being 1/36 as stated, and (c) do agree that the answer to the snake eyes paradox is 1/36.

Did I get all that right? Really appreciate your engagement here! Super valuable for figuring out the disagreement in the snake eyes market, which has become quite the rabbit hole.

@dreev Yes, that's about right. I think if you showed this question to mathematicians at least some of them would say "that's undefined" or at the very least "that clarification asks a different question / is inconsistent with what you wrote before".

The snake eyes paradox is a bit more tricky. If you focus on FAQ 3, that gives you a precise mathematical problem to which the answer is 1/36. If you encode the infinite problem directly into probability theory [is that was FAQ 4 is describing?] then I think one way to encode this indeed gives you the answer roughly 1/2, but in my opinion that encoding doesn't capture the problem fairly: you assign infinite mass to some a set of probability 0, and in probability theory sets of probability 0 don't matter for the outcome, but in this problem that set of 0 probability does (IMO) matter (it definitely matters for all finite truncations of the problem).

I maintain that if Kongo and Floris came across the exact text of this question outside of manifold, they would agree that the answer was 1/36, and it's only the context of this market where it basically openly asks to be lawyered over and gives you fake money for it that leads to the disagreements. It's just a language game, you can play the same language game with any common english sentence, including casual speech between professional mathematicians. (I don't think this is bad faith in an ethical sense, just bad faith in the narrow sense of 'is it within the spirit of the question')

@jacksonpolack Beautifully said. The snibgiblets question could be articulated better but what I think reasonable would agree it's trying to say is, like, "here's a question with answer 0/0 but we want a real number so we'll take a limit; what's that limit?" Do you buy that, @FlorisvanDoorn? I guess you've said you don't, quite, but maybe framed this way? We could call it an empirical question, like @jacksonpolack says, about what fraction of working mathematicians would read it that way. I'm predicting very few, which I suppose, if I wanted to language-lawyer, is not technically incompatible with Floris's "at least some"!