I like this question a lot. My base position is “It’s a modeling choice—either one can make appropriate betting choices, so it’s unclear what true probability even means in correlated-existence cases”. Though in private, I’ll admit that 1⁄3 is simpler most of the time, and in public I argue for 1⁄2 fairly often because die-hard thirders seem to be so common.
The difficulty with betting analyses is how to handle the duplicate days, where Beauty’s experience is doubled on tails, but you want that to be irrelevant to the wager (to put focus on the fact that it’s a coin flip, so you have to be indifferent on Wednesday). Saying “you must agree or be put to death” is one way, which leads to a static decision that’s indifferent to heads or tails, which is justified by either halfer or thirder. Saying “bets are off if they disagree” doesn’t work, since only tails can disagree. Saying “Monday takes precedence” kind of obviates the whole excercise.
Your mechanism of “days are ORed together” would normally show the weirdness of the thirder position, because a random choice by Beauty would lead to taking the bet 50% of the time for heads and 75% for tails. But painting the walls gives a coordination mechanism between Monday and Tuesday, without revealing which day it actually is.
I think “bet if the wall is red” works here. If it’s heads (50% of the time, to an outside observer, and to the bet payoff), you’ll take the bet 50% of the time and lose $200. If it’s tails, you’ll take the bet 100% of the time and win $300. 0.5*0.5*-200 + 0.5*300 = +100.
I don’t think it sheds any light on the halfer vs thirder debate—this encourages an outside view, because the payoff is outside and the coordination is outside (which color to pick in any world on any day). But I guess that means it supports my position that it’s a modeling choice rather than a true distinction.
I like this question a lot. My base position is “It’s a modeling choice—either one can make appropriate betting choices, so it’s unclear what true probability even means in correlated-existence cases”. Though in private, I’ll admit that 1⁄3 is simpler most of the time, and in public I argue for 1⁄2 fairly often because die-hard thirders seem to be so common.
The difficulty with betting analyses is how to handle the duplicate days, where Beauty’s experience is doubled on tails, but you want that to be irrelevant to the wager (to put focus on the fact that it’s a coin flip, so you have to be indifferent on Wednesday). Saying “you must agree or be put to death” is one way, which leads to a static decision that’s indifferent to heads or tails, which is justified by either halfer or thirder. Saying “bets are off if they disagree” doesn’t work, since only tails can disagree. Saying “Monday takes precedence” kind of obviates the whole excercise.
Your mechanism of “days are ORed together” would normally show the weirdness of the thirder position, because a random choice by Beauty would lead to taking the bet 50% of the time for heads and 75% for tails. But painting the walls gives a coordination mechanism between Monday and Tuesday, without revealing which day it actually is.
I think “bet if the wall is red” works here. If it’s heads (50% of the time, to an outside observer, and to the bet payoff), you’ll take the bet 50% of the time and lose $200. If it’s tails, you’ll take the bet 100% of the time and win $300. 0.5*0.5*-200 + 0.5*300 = +100.
I don’t think it sheds any light on the halfer vs thirder debate—this encourages an outside view, because the payoff is outside and the coordination is outside (which color to pick in any world on any day). But I guess that means it supports my position that it’s a modeling choice rather than a true distinction.