Background: the Snake Eyes Paradox market.

The anthropic version of the Snake Eyes Paradox is like so:

Initially the universe is empty. 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. Once again, those people die on snake eyes and the game ends. This repeats as long as non-snake-eyes are rolled, with the group size doubling each time. As soon as snake eyes is rolled the latest group dies, and the game ends.

The question: 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?

Argument for YES: No one ever dies except by fair dice roll. So your chances of dying are the chances of rolling snake eyes on your round: 1/36.

Argument for NO: Due to the doubling, the final group that dies is slightly bigger than all the surviving groups put together. So if you exist, you could be anyone and so you have about a 50% chance of dying.

Resolution criteria: I'll defer to Martin Randall's opinion unless I can articulate why I'm certain he's wrong. Since this could be subjective, I won't trade in this market.

## FAQ

### 1. What if snake eyes is never rolled?

Eventually, with probability 1, snake eyes will be rolled but if hypothetically we rolled non-snake-eyes forever then an infinite number of people would be created and none would die.

### 2. What if the probability is undefined?

That's a no in this market. (I tragically failed to anticipate that possibility in the original snake eyes market, which caused lots of consternation!)

### 3. Should we use the self-sampling assumption (SSA) or the self-indication assumption (SIA)?

TBD -- please discuss in the comments if it's necessary to clarify this. So far SSA seems to me to miss the whole point of anthropic reasoning but let's discuss.

### 4. Is this different from the regular Snake Eyes Paradox?

I think so but the point of this market is to clarify that. In my interpretation of the original, there's a fixed pool (maybe an infinite one) of people, a subset of which is chosen to play. In this version you don't exist at all unless you're chosen to play. I meant the original to be as realistic as possible. This version is explicitly in philosophy-thought-experiment-fantasy-land. But, again, I'm hoping that understanding the differences between this version and the original will help pull us out of the rabbit hole (or push through it to come out the other end?).

### 5. Priors?

Uniform. And the dice are fair and i.i.d.

--

Please ask other clarifying questions before trading! And huge thanks to everyone for all the work on resolving this paradox.

## Related questions

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 "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.

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.

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?

@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

@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.

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?