Having read this comment, I can now see an ambiguity in the language. I read it as “You’re offered a fair coin toss on which (you’ll win $2 on Heads if and only if it’s predicted that you will pay $1 on Tails)”. That is, you’re offered the coin flip regardless of the prediction, but can only win money on heads if it’s predicted that you would have paid on tails.
Why? Being predicted to not pay on tails is perfectly consistent with seeing a flip of tails (and not paying).
As I see it, the game proceeds as follows: You flip a coin. If it comes up tails, you are asked whether or not you want to pay $1. If it comes up heads, the predictor estimates whether you would have paid up on a result of tails: you get $2 if they predict that you would, otherwise you get nothing.
You know these rules, and that the predictor is essentially perfect for all practical purposes.
Hmm, I guess I misunderstood the setup, oops. I assumed that only those who are predicted to pay $1 on tails would be offered the game. Apparently… something else is going on? The game is offered first, and then the predictor makes the prediction?
Yes, that’s why I bracketed my interpretation as I did: in my reading, the only clause to which the prediction result applies is “you’ll win $2 on Heads”.
I’m probably misunderstanding you or I’ve worded things in a confusing way that I haven’t noticed—I don’t think anywhere it’s implied what you do on Tails? The “iff” here is just saying you would be paid on Heads iff you would pay on Tails—the flip will happen regardless and the predictor hasn’t made any prediction about the coin itself, just you’re conditional behaviour
Edit: Maybe the “iff you will pay $1 on Tails” makes it sound like the predictor is predicting both the coin and your response, I’ll edit to make more clear
Having read this comment, I can now see an ambiguity in the language. I read it as “You’re offered a fair coin toss on which (you’ll win $2 on Heads if and only if it’s predicted that you will pay $1 on Tails)”. That is, you’re offered the coin flip regardless of the prediction, but can only win money on heads if it’s predicted that you would have paid on tails.
that is my reading as well… Still means you will pay $1 as predicted if the outcome is tails, regardless of your internal decision theory.
Why? Being predicted to not pay on tails is perfectly consistent with seeing a flip of tails (and not paying).
As I see it, the game proceeds as follows: You flip a coin. If it comes up tails, you are asked whether or not you want to pay $1. If it comes up heads, the predictor estimates whether you would have paid up on a result of tails: you get $2 if they predict that you would, otherwise you get nothing.
You know these rules, and that the predictor is essentially perfect for all practical purposes.
Hmm, I guess I misunderstood the setup, oops. I assumed that only those who are predicted to pay $1 on tails would be offered the game. Apparently… something else is going on? The game is offered first, and then the predictor makes the prediction?
Yes, that’s why I bracketed my interpretation as I did: in my reading, the only clause to which the prediction result applies is “you’ll win $2 on Heads”.
I’m probably misunderstanding you or I’ve worded things in a confusing way that I haven’t noticed—I don’t think anywhere it’s implied what you do on Tails? The “iff” here is just saying you would be paid on Heads iff you would pay on Tails—the flip will happen regardless and the predictor hasn’t made any prediction about the coin itself, just you’re conditional behaviour
Edit: Maybe the “iff you will pay $1 on Tails” makes it sound like the predictor is predicting both the coin and your response, I’ll edit to make more clear