Thirder here (with acknowledgement that the real answer is to taboo ‘probability’ and figure out why we actually care)
The subjective indistinguishability of the two Tails wakeups is not a counterargument - it’s part of the basic premise of the problem. If the two wakeups were distinguishable, being a halfer would be the right answer (for the first wakeup).
Your simplified example/analogies really depend on that fact of distinguishability. Since you didn’t specify whether or not you have it in your examples, it would change the payoff structure.
I’ll also note you are being a little loose with your notion of ‘payoff’. You are calculating the payoff for the entire experiment, whereas I define the ‘payoff’ as being the odds being offered at each wakeup. (since there’s no rule saying that Beauty has to bet the same each time!)
To be concise, here’s my overall rationale:
Upon each (indistinguishable) wakeup, you are given the following offer:
If you bet H and win, you get N dollars.
If you bet T and win, you get 1+ϵ dollars.
If you believe T yields a higher EV, then you have a credence P(T)≥NN+1
You get a positive EV for all N up to 2, so P(T)=23. Thus you should be a thirder.
Here’s a clarifying example where this interpretation becomes more useful than yours:
The experimenter flips a second coin. If the second coin is Heads (H2), then N= 1.50 on Monday and 2.50 on Tuesday. If the second coin is Tails, then the order is reversed.
I’ll maximize my EV if I bet T when N=1.5, and H when N=2.5. Both of these fall cleanly out of ‘thirder’ logic.
What’s the ‘halfer’ story here? Your earlier logic doesn’t allow for separate bets on each awakening.
Thirder here (with acknowledgement that the real answer is to taboo ‘probability’ and figure out why we actually care)
The subjective indistinguishability of the two Tails wakeups is not a counterargument - it’s part of the basic premise of the problem. If the two wakeups were distinguishable, being a halfer would be the right answer (for the first wakeup).
Your simplified example/analogies really depend on that fact of distinguishability. Since you didn’t specify whether or not you have it in your examples, it would change the payoff structure.
I’ll also note you are being a little loose with your notion of ‘payoff’. You are calculating the payoff for the entire experiment, whereas I define the ‘payoff’ as being the odds being offered at each wakeup. (since there’s no rule saying that Beauty has to bet the same each time!)
To be concise, here’s my overall rationale:
Upon each (indistinguishable) wakeup, you are given the following offer:
If you bet H and win, you get N dollars.
If you bet T and win, you get 1+ϵ dollars.
If you believe T yields a higher EV, then you have a credence P(T)≥NN+1
You get a positive EV for all N up to 2, so P(T)=23. Thus you should be a thirder.
Here’s a clarifying example where this interpretation becomes more useful than yours:
The experimenter flips a second coin. If the second coin is Heads (H2), then N= 1.50 on Monday and 2.50 on Tuesday. If the second coin is Tails, then the order is reversed.
I’ll maximize my EV if I bet T when N=1.5, and H when N=2.5. Both of these fall cleanly out of ‘thirder’ logic.
What’s the ‘halfer’ story here? Your earlier logic doesn’t allow for separate bets on each awakening.