I prefer to just think about utility, rather than probabilities. Then you can have 2 different “incentivized sleeping beauty problems”
Each time you are awakened, you bet on the coin toss, with $ payout. You get to spend this money on that day or save it for later or whatever
At the end of the experiment, you are paid money equal to what you would have made betting on your average probability you said when awoken.
In the first case, 1⁄3 maximizes your money, in the second case 1⁄2 maximizes it.
To me this implies that in real world analogues to the Sleeping Beauty problem, you need to ask whether your reward is per-awakening or per-world, and answer accordingly
That argument just shows that, in the second betting scenario, Beauty should say that her probability of Heads is 1⁄2. It doesn’t show that Beauty’s actual internal probability of Heads should be 1⁄2. She’s incentivized to lie.
EDIT: Actually, on considering further, Beauty probably should not say that her probability of Heads is 1⁄2. She should probably use a randomized strategy, picking what she says from some distribution (independently for each wakening). The distribution to use would depend on the details of what the bet/bets is/are.
I prefer to just think about utility, rather than probabilities. Then you can have 2 different “incentivized sleeping beauty problems”
Each time you are awakened, you bet on the coin toss, with $ payout. You get to spend this money on that day or save it for later or whatever
At the end of the experiment, you are paid money equal to what you would have made betting on your average probability you said when awoken.
In the first case, 1⁄3 maximizes your money, in the second case 1⁄2 maximizes it.
To me this implies that in real world analogues to the Sleeping Beauty problem, you need to ask whether your reward is per-awakening or per-world, and answer accordingly
That argument just shows that, in the second betting scenario, Beauty should say that her probability of Heads is 1⁄2. It doesn’t show that Beauty’s actual internal probability of Heads should be 1⁄2. She’s incentivized to lie.
EDIT: Actually, on considering further, Beauty probably should not say that her probability of Heads is 1⁄2. She should probably use a randomized strategy, picking what she says from some distribution (independently for each wakening). The distribution to use would depend on the details of what the bet/bets is/are.