I feel like this is one of the cases where you need to be very precise about your language, and be careful not to use an “analogous” problem which actually changes the situation.
Consider the first “bajillion dollars vs dying” variant. We know that right now, there’s about 8B humans alive. What happens if the exponential increase exceed that number? We probably have to assume there’s an infinite number of humans, fair enough.
What does it mean that “you’ve chosen to play”? This implies some intentionality, but due to the structure of the game, where the number of players is random, it’s not really just up to you.
NOTE: I just realized that the original wording is “you’re chosen to play” rather than “you’ve chosen to play”. Damn you, English. I will keep the three variants below, but this means that the right interpretation clearly points towards option B), but the analysis of various interpretations can explain why we even see this as a paradox.
A) One interpretation is “what is the probability that I died given that I played the game?”, to which the answer is 0%, because if I died, I wouldn’t be around to ask this question.
B) Second interpretation is “Organizer told you there’s a slot for you tomorrow in the next (or first) batch. What is the probability that you will die given that you are going to play the game?”. Here the answer is pretty trivially 1⁄36. You don’t need anthropics, counterfactual worlds, blue skies. You will roll a dice, and your survival will entirely depend on the outcome of that roll.
C) The potentially interesting interpretation, that I heard somewhere (possibly here) is: “You heard that your friend participated in this game. Given this information, what is the probability that your friend died during the game?”. The probability here will be about 50% -- we know that if N people in total participated, about N/2 people will have died.
Consider now the second variant with snakes and colors. Before the god starts his wicked game, do snakes exist? Or is he creating the snakes as he goes? The first sentence “I am a god, creating snakes.” seems to imply that this is the process of how all snakes are created. This is important, because it messes with some interpretations. Another complication is that now, “losing” the roll no longer deletes you from existence, which similarly changes interpretations. Let’s look at the three variants again.
A) “What is the probability you have red eyes given that you were created in this process?”—here the answer will be ~50%, following the same global population argument as in variant C of the first variant. This is the interpretation you seem to be going with in your analysis, which is notably different than the interpretation that seems to be valid in the first variant.
B) If snakes are being created as you go with the batches, this no longer has a meaning. The snake can’t reflect on what will happen to him if he’s chosen to be created, because he doesn’t exist.
C) “Some time after this process, you befriended a snake who’s always wearing shades. You find out how he was created. Given this, what is the probability that he has red eyes?”—the answer, following again the same global population argument, is ~50%
In summary, we need to be careful switching to a “less violent” equivalent, because it can often entirely change the problem.
I definitely agree on the need for care in switching between variants. It can also be helpful that they can “change the situation” because this can reveal something unspecified about the original variant. Certainly I was helped by making a second variant, as this clarified for me that the probabilities are different from the deity view vs the snake view, because of anthropics.
In the original variant, it’s not specified when exactly players get devoured. Maybe it is instant. Maybe everyone is given a big box that contains either a bazillion dollars, or human-eating snakes, and it opens exactly a year later.
In my variant, I was imagining the god initially created a batch of snakes with uncolored eyes, then played dice, then gave them red or blue eyes. So the snakes, like the players, can have experiences prior to the dice being rolled. And yes, no snakes exist before I start.
(why is the god wicked? No love for snakes...)
I’ll update the text to clarify that no snakes exist until the god of snake creation gets to work.
C) “Some time after this process, you befriended a snake who’s always wearing shades. You find out how he was created. Given this, what is the probability that he has red eyes?”—the answer, following again the same global population argument, is ~50%
I think this is a great crystallization of the paradox. In this scenario, it seems like I should believe I have a 1⁄36 chance of red eyes, and my new friend has a 1⁄2 chance of red eyes. But my friend has had exactly the same experiences as me, and they reason that the probabilities are reversed.
I feel like this is one of the cases where you need to be very precise about your language, and be careful not to use an “analogous” problem which actually changes the situation.
Consider the first “bajillion dollars vs dying” variant. We know that right now, there’s about 8B humans alive. What happens if the exponential increase exceed that number? We probably have to assume there’s an infinite number of humans, fair enough.
What does it mean that “you’ve chosen to play”? This implies some intentionality, but due to the structure of the game, where the number of players is random, it’s not really just up to you.
NOTE: I just realized that the original wording is “you’re chosen to play” rather than “you’ve chosen to play”. Damn you, English. I will keep the three variants below, but this means that the right interpretation clearly points towards option B), but the analysis of various interpretations can explain why we even see this as a paradox.
A) One interpretation is “what is the probability that I died given that I played the game?”, to which the answer is 0%, because if I died, I wouldn’t be around to ask this question.
B) Second interpretation is “Organizer told you there’s a slot for you tomorrow in the next (or first) batch. What is the probability that you will die given that you are going to play the game?”. Here the answer is pretty trivially 1⁄36. You don’t need anthropics, counterfactual worlds, blue skies. You will roll a dice, and your survival will entirely depend on the outcome of that roll.
C) The potentially interesting interpretation, that I heard somewhere (possibly here) is: “You heard that your friend participated in this game. Given this information, what is the probability that your friend died during the game?”. The probability here will be about 50% -- we know that if N people in total participated, about N/2 people will have died.
Consider now the second variant with snakes and colors. Before the god starts his wicked game, do snakes exist? Or is he creating the snakes as he goes? The first sentence “I am a god, creating snakes.” seems to imply that this is the process of how all snakes are created. This is important, because it messes with some interpretations. Another complication is that now, “losing” the roll no longer deletes you from existence, which similarly changes interpretations. Let’s look at the three variants again.
A) “What is the probability you have red eyes given that you were created in this process?”—here the answer will be ~50%, following the same global population argument as in variant C of the first variant. This is the interpretation you seem to be going with in your analysis, which is notably different than the interpretation that seems to be valid in the first variant.
B) If snakes are being created as you go with the batches, this no longer has a meaning. The snake can’t reflect on what will happen to him if he’s chosen to be created, because he doesn’t exist.
C) “Some time after this process, you befriended a snake who’s always wearing shades. You find out how he was created. Given this, what is the probability that he has red eyes?”—the answer, following again the same global population argument, is ~50%
In summary, we need to be careful switching to a “less violent” equivalent, because it can often entirely change the problem.
I definitely agree on the need for care in switching between variants. It can also be helpful that they can “change the situation” because this can reveal something unspecified about the original variant. Certainly I was helped by making a second variant, as this clarified for me that the probabilities are different from the deity view vs the snake view, because of anthropics.
In the original variant, it’s not specified when exactly players get devoured. Maybe it is instant. Maybe everyone is given a big box that contains either a bazillion dollars, or human-eating snakes, and it opens exactly a year later.
In my variant, I was imagining the god initially created a batch of snakes with uncolored eyes, then played dice, then gave them red or blue eyes. So the snakes, like the players, can have experiences prior to the dice being rolled. And yes, no snakes exist before I start. (why is the god wicked? No love for snakes...) I’ll update the text to clarify that no snakes exist until the god of snake creation gets to work.
I think this is a great crystallization of the paradox. In this scenario, it seems like I should believe I have a 1⁄36 chance of red eyes, and my new friend has a 1⁄2 chance of red eyes. But my friend has had exactly the same experiences as me, and they reason that the probabilities are reversed.