Defining probabilities in terms of bets is one way to do things, but not the only way—one can also define them in terms of limiting frequencies, or as numbers that allow you to efficiently encode the environment, or as objects that follow some axioms you think numerical confidence-objects should follow.
I have witnessed people arguing for betting with probability 1⁄2 in case 2. After all, they say, the probability is 1⁄2, so that’s how you should bet. Most people who approach this problem for the first time (whether thirders or halfers) use the same decision-making algorithm: first, compute the probability (perhaps wrongly) of winning the bet for the person inside the experiment, second, use that probability to determine the value of the bet.
When you say it’s obvious that in case 2 you should bet a certain way, I think you’re choosing how to bet in a different way: from the viewpoint of someone on the outside, what strategy should the person on the inside follow to maximize their gains? This viewpoint becomes a lot more obvious after being exposed to LessWrong for a couple of years.
And there’s one tricky thing here, which is that if you use this perspective, you as the outside person have some probabilities, but the person inside the experiment also might have probabilities, which do not have to be simply related to the optimal strategy. So you have to be pretty careful with this argument that knowing the correct strategy implies knowing the correct probabilities.
Defining probabilities in terms of bets is one way to do things, but not the only way—one can also define them in terms of limiting frequencies, or as numbers that allow you to efficiently encode the environment, or as objects that follow some axioms you think numerical confidence-objects should follow.
I have witnessed people arguing for betting with probability 1⁄2 in case 2. After all, they say, the probability is 1⁄2, so that’s how you should bet. Most people who approach this problem for the first time (whether thirders or halfers) use the same decision-making algorithm: first, compute the probability (perhaps wrongly) of winning the bet for the person inside the experiment, second, use that probability to determine the value of the bet.
When you say it’s obvious that in case 2 you should bet a certain way, I think you’re choosing how to bet in a different way: from the viewpoint of someone on the outside, what strategy should the person on the inside follow to maximize their gains? This viewpoint becomes a lot more obvious after being exposed to LessWrong for a couple of years.
And there’s one tricky thing here, which is that if you use this perspective, you as the outside person have some probabilities, but the person inside the experiment also might have probabilities, which do not have to be simply related to the optimal strategy. So you have to be pretty careful with this argument that knowing the correct strategy implies knowing the correct probabilities.