This is part of my quest to untangle the controversy in The Snake Eyes Paradox. It's called Variant Y because it's how YES bettors interpreted the original.
You're offered a gamble where a pair of six-sided dice are rolled and unless they come up snake eyes you get a bajillion dollars. If they do come up snake eyes, you're devoured by snakes.
So far it sounds like you have a 1/36 chance of dying, right?
Now the twist. First, I gather up N people willing to play the game. I take 1 person from that pool and let them play. Then I take 2 people and have them play together, where they share a dice roll and either get the bajillion dollars each or both get devoured. Then I do the same with 4 people, and then 8, 16, and so on.
At some point I either roll snake eyes and a group will be devoured by snakes, or I run out of people. Either way, I stop.
In the limit as N goes to infinity โ an unlimited number of people โ what is your chance of dying, given that you're chosen to play?
Argument for ~1/2
In the limit there's zero chance of running out of people which means a guarantee of a final group dying. That group is slightly bigger than all the surviving groups put together. So if you're chosen to play you have about a 50% chance of dying! ๐ฌ ๐
Argument for 1/36
The dice rolls are independent and whenever you're chosen, whatever happened in earlier rounds is irrelevant. Your chances of death are the chances of snake eyes on your round: 1/36. ๐
Clarifications and FAQ
The game is not adversarial and the dice rolls are independent and truly random.
Choosing each group also happens uniformly randomly and without replacement.
In the limit, if I somehow never rolled snake eyes, then an infinite number of people would all play and none of them of would die. That has 0% probability but, y'know, hypothetically.
Resolution Criteria
This resolves to the mathematically correct probability, Pr(death | chosen). How do we decide what that is? I'm naively hopeful that this version, spelling out the limiting process up front, won't be controversial and we can resolve to consensus in the comments. If that's not realistic, I'll hide comments from everyone who disagrees with me and resolve to the consensus of what's left. Just kidding. Here's my new idea: a council of 2 people who've argued coherently for NO in the original market (eg, Martin Randall and Primer), 2 people who've argued coherently for YES (eg, David Pennock and Wamba Ivanhoe), plus a mathematician who hasn't bet (eg, Lorxus Mathfox). I predict we'll reach consensus but we'll vote if necessary. I won't trade in this market.
See also Snake Eyes Variant N.
Related questions
Argument that this should resolve to 1/36 = 2.8%:
Try it with N=2, see that it comes out to 1/36
Math it out with generic N, still 1/36 (see YES writeups from original market)
Unless there's some crazy staircase paradox problem, the limit is 1/36
I'd forgotten when nominating @Primer here that they already believe the answer to Variant Y is 1/36. Is that true for @MartinRandall as well? If so, this will be an easy resolution. Maybe we can persuade @KongoLandwalker as well?
Here's my new idea: a council of 2 people who've argued coherently for NO in the original market (eg, Martin Randall and Primer), 2 people who've argued coherently for YES (eg, David Pennock and Wamba Ivanhoe), plus a mathematician who hasn't bet (eg, Lorxus Mathfox).
I am glad I read the resolution criteria.
I suppose I shouldn't bet either, if I am supposed to participate in a resolution council, although, there probably is little I would say that I have not already explicitly written out in my write up.
@DavidPennock well so has @Primer. The intent is that these markets are supposed to be non controversial sooo there is that.
@ShitakiIntaki That was one of the reasons I didn't create these markets myself. Also, I'm on record already for the solution to variant Y being 1/36.
My preferred resolution would have been for someone else to give (the original, without Daniel's modifications) variants Y and N to a mathematician or a philosopher, and then let a linguist decide whether one of them does not map onto the original Snake Eyes.
@ShitakiIntaki @DavidPennock Oh I think it's totally fine that you are betting! Like @Primer says, your opinions are already public and you're even identified by your opinions in the resolution criteria.
I wished you hadn't included the clarification and FAQ from the original, as those are not unambiguous.
Also, what does buying Yes or No shares here correspond to? Or does this resolve to percentage?
@Primer Oh. Well, we can still change it! No one has traded yet. Let me copy over the ones I think are needed here and you can tell me what you think. [EDIT: and done]
Yes to this resolving to PROB.