I was going to bet YES because the probability given by SIA is 1/36, and you seem to favor SIA over SSA (as do I). But then I realized that this market is actually asking about Anthropic Snake Eyes Variant N, since it explicitly says that God is about to kill the group that got snake eyes. SIA gives a probability of 1/2 for dying in this case.

(If you're wondering where those numbers come from, I am currently writing a blog post that explains them. Anthropic Snake Eyes with SSA is exactly equivalent to the ball-and-urn variant, so it gives an answer of 52%, regardless of whether it's the regular variant or Variant N. This answer is unambiguously the correct SSA answer. SIA breaks down for Anthropic Snake Eyes in classical probability theory, but it is equivalent to the answer we get in regular snake eyes when we use a non-standard probability theory that allows for a uniform distribution over the naturals. So SIA gives a 17/18 chance that God never rolls snake eyes, conditional on you existing, and a 1/36 chance that you will die if you don't know whether God ever rolled snake eyes. But if you do know that he rolled Snake Eyes, p(death) is 1/2).

Here's the post: https://plasmabloggin.substack.com/p/the-snake-eyes-paradox

I meant the original to be as realistic as possible.

I think this one is completely realistic, if we talk about robots instead of people and an Engineer instead of God.

There is "infinite amount of possible robot designs to be created" but only some of them are created and we don't care what was "the decision process".

That is equivalent to the original market, where we had infinite pool and somehow picked some people from it.

Being picked from the infinite pool is the same as being created!

Both markets should resolve to NO.

Once you accept that a rational agent with the same priors has a posterior probability q you must update your posterior to q if you are also a rational agent. In this example, we can accurately predict the credence an agent will have (the probability q that an agent will be in the last group which is chosen) despite gaining no additional information. We need only know that

1. We are a player in this game

2. Slightly over half of players are chosen

We know this before, during, and after the game, we need no other additional information. The specifics of the game are not relevant. We can only collapse this probability to a certainty if we gain information about what kind of player we are. Since we have this information before, during, and after the game our probability q must be the same at all points in the game. This doesn't feel correct, for example, I understand the objection to saying that Snake Eyes will be rolled with probability ~1/2 despite using fair dice. Nevertheless, reporting a probability of 1/36 is failing to update on all available information. Specifically, agents who fully update can accurately predict the magnitude and direction that other agents who do not fully update will update in. If you can correctly be told that you will believe that you are chosen with probability ~1/2, after all agents are created and whatever random process is carried out, you must then already believe this is true with probability ~1/2. The specifics of the random process do not matter because you already have information which obviates it, I believe this is the apparent paradox. I don't think that there is any other way to reconcile these positions, once you agree on a posterior probability and that the agents have shared priors. That individuals are rational agents with shared priors is a given and the posterior probability is easily calculated. We can reason about the outcome of a fair random process before the process has happened, because we are actually reasoning about our place in the distribution of outcomes.

@Sailfish I'm exited to hear how you'll apply this to Sleeping Beauty

@Sailfish "Once you accept that a rational agent with the same priors has a posterior probability q you must update your posterior to q if you are also a rational agent."

Sorry, but did you mean "and the rational agent has the same (and/or more) evidence as you" ?

Aumann's agreement theorem is over agents with the same prior (agree on uniqueness of ideal prior in same universe) *and* common knowledge of all relevant (entangled) probabilistic evidence.

I don't have to immediately update to q, not under those conditions. The rational agent with the same priors could have been staring at a blank wall for their entire existence and updated to a posterior that has way less entanglement with reality than my own.

I suppose you could say that I could update to their posterior and then refactor all of my other evidence. You could also say that even if they've been staring at a blank wall, their belief state is still Baysean evidence and should be updated on.

Aumman's agreement theorum is about convergence under conditions where the agents get access to the same evidence somehow - either by sharing, or via both getting a random ordering of evidence from the same set.

non-SIA agents predictably update here, > 1/2 of all agents die, you are an agent indistinguishable from any other agent. The specifics of the process aren't relevant. SIA agents can make accurate predictions about what other agents will believe given exactly the same information.

I thought that I would try making one.

The expected number of people expected to be created is the infinite sum of (2(35/36))^n which is unbounded.

The expected number of victims is the infinite sum of (2(35/36))^n * (2/36) which is also unbounded.

But you can write the second series as each term being 1/36 of the corresponding term of the first series, so in this case infinity/infinity = 1/36.

@JonathanRay Sure, but you can also write the second and first series in other ways such that infinity/infinity = literally any ratio you want, see https://en.wikipedia.org/wiki/Riemann_series_theorem

@MartinRandall the Riemann series theorem has to do with challenging the convergence of a series, and whether it converges under any/all arrangements. Taken in the arrangement of the serries of games n, None of these series converge, they are unbounded. Johnathan is is noting that they are growing unbounded at a fixed ratio to one another. Are you proposing an another arrangement that would cause these to converge rather than grow unbounded? Or an arrangement that would arrive at a different relative growth ratio?

@ShitakiIntaki Both series diverge, I agree. Riemann isn't quite right here.

The ratio between created people and victims depends on how they are placed in correspondence with each other. So there needs to be some justification for why any proposed correspondence is correct.

@MartinRandall The justification for this particular correspondence is the order in decreasing likelihood which games would end at a certain state, growing out of the rules for how the game is played. The series is not conditionally convergent so I am not sure that you can justify an assertion that there is any dependance on how they are placed in correspondence with each other, unless you are prepared to demonstrate by counter example, that there exists another arrangement that exploits some dependency. Since your assertion is based upon the existence of a theorem for conditionally convergent series which does not readily apply here, I would believe that the burden of proof would be on you to show that there is another arrangement.

I think an undefined question in this market is what our prior should be of having an experience. Crucially a uniform prior over the set of natural numbers is not well-defined. Also this market doesn't specify a prior (the original specified a uniform prior).

I also think SSA vs SIA applies to this market.

I think there's also an ambiguity about whether the deity is creating people and rolling dice as they go, or pre-rolling dice and creating everyone at once.

Given these ambiguities, I think the current answer is "undefined" or "it depends". But I also see that the description implies that clarifications can be added so bettors should not rely on that until Daniel has concluded any edits.

@MartinRandall Sounds good. Do you have recommendations for which choices to make for each of those? Uniform prior, definitely, for starters, right?

And SIA vs SSA won't end up mattering if neither yield an answer of 1/36, right? In this market it's just YES for 1/36 and NO for anything else, including "undefined".

As for pre-rolling, the description here already says God does pre-roll, so you exist as one of the set of all people created in all rounds. That seemed closest to the original snake eyes where you're just part of a pool of potential players. Are you sure it needs further clarification?

I'm also anxious to understand if/how this version does end up different, of course.

@dreev

This implies not pre-rolling:

God creates 1 person and rolls fair dice. If they come up snake eyes, God kills the person and the game ends. If they don't come up snake eyes, the person lives and God creates 2 new people and rolls the dice again.

This implies pre-rolling:

Suppose God pre-rolls all the dice, creates all the requisite people, and then kills the final group.

I don't think they can both be true. If the second sentence overrides the first it's not doing so explicitly.

@MartinRandall I see, yeah, I was thinking in terms of the original when I started and then realized we needed a still-alive group of people for the target "you" to consider yourself part of. How about this edit:

Suppose God has created all the requisite people, rolled all the dice, and is about to kill the final group that eventually got snake eyes. You're one of those people, with no idea which group you were in or how many other people were created. What's your subjective probability that you'll be killed?

@dreev That clears it up I think

@MartinRandall Edits in place now for uniform priors and when the dice are rolled.

Next question: Does your writeup fully apply to this version?

@dreev No, for example this version doesn't have the quoted text that implies SSA by making claims that a limit can be taken and a probability goes to zero. So it could be SIA.

@MartinRandall This version so far explicitly leaves the SSA vs SIA question open. And given that your writeup addresses both, I thought that you thought the original did too.

It now occurs to me that we could more definitively factor out the anthropics question from the original... Ok, here I go with another market:

https://manifold.markets/dreev/is-the-probability-of-puncture-in-t

Is the probability of “death” in the ball-and-urn version of the Snake Eyes Paradox 1/36?

50% chance. The snake eyes market has generated a lot of debate (1.1K comments and counting!). One aspect of the debate is the question of anthropics. So we have an explicitly anthropic version of snake eyes. Here’s a version that attempts to factor out the question of anthropics altogether: We ha…

@MartinRandall I'll register that I actually don't think it matters if the dice are pre-rolled or not. I may write something to this effect in the Sleeping Beauty market, but if the coin for example is flipped Monday after Beauty is put to sleep, my answer of 1/3 does not change. Similarly here, my answer does not change regardless of when the dice are rolled. ( I would have bet NO here)

@dreev I believe SSA answers 1/36 here, if I select simply from people who exist, I know only that there are or are not groups before me who did not role snake eyes. A prior string (or not) of no snake eyes does not make snake eyes any more or less likely. I'm not certain that this is the most correct framing of SSA, however.

Again with the death! Why do philosophers try so hard to be metal?

@MartinRandall

Recontextualize:

Iff snake eyes,

the cohort of people are each named Bob,

else

the cohort of people are each named named Aaron.

Where cohort references all people created in a single round and receive a name determined by one shared roll of the dice.

If you don't know how many, if any, other people exist, only knowing that you exist, what do you suppose the chances are that your name is Bob?