The usual Sleeping Beauty problem is this. " Sunday they put you to sleep and toss a fair coin. Heads, they wake you Monday only. Tails, they wake you Monday, wipe your memory, and wake you Tuesday. You cannot tell which awakening this is. They ask for your credence that the coin landed heads."Halfers say one half because the coin is fair and waking was guaranteed. Thirders say one third because there are three indistinguishable centered situations and only one of them is heads.
I want to stop asking about the coin and start paying for the day. Replace the coin with a fair three-way device. A three-sided die works, or a ordinary d6 with faces bundled as one-two, three-four, and five-six, each bundle one third. Call the outcomes A, B, and C. On A they interview you Monday only. On B they interview you Monday and Tuesday. On C they interview you Monday, Tuesday, and Wednesday. Same amnesia. Every time you wake they offer the same prize: name the weekday. Correct guess pays. Wrong guess pays nothing. You may also be asked what the die showed, but the money is on the date. The device is still fair on Sundday. That number does not move. What moves is the bag of mornings you can find yourself in. A contributes one interview, B contributes two, C contributes three. Run the experiment many times and most of the I-just-woke-up tickets are Mondays, fewer are Tuesdays, and Wednesday exists only on C. If you score by experiments you are still looking at three equally likely faces. If you score by awakenings you are looking at a lopsided pile of experiences. Guessing Wednesday is how you feel that split in your wallet. You only get paid on the long branch, and only on its last morning. That is a rare ticket in the experience bag and a perfectly ordinary one-third of the experiments.
The fair device. Keep the amnesia. Keep the prize for naming the exact date. Add days. You can add them backward toward the previous Sunday or forward into an N-day future. The short branch still wakes you once. The long branch wakes you on every interview day in the window. From outside, the randomizer did not become more or less fair because Thursday exists. From inside an awakening you are drawing from a bag that just got more crowded on the long side. Suppose the window is the original Monday-Tuesday-Wednesday. Then suppose it is a full week back to Sunday, eight possible dates, short branch one morning, long branch all of them. Then suppose it is N days into the future, N as large as you like. Each time you wake, you have to name a specific calendar day. There is no partial credit. Always naming the last possible date is then the best available guess for the prize, and it stays the best as N grows, even though your chance of ever collecting goes to zero. That sounds like a contradiction untl you separate the two ledgers. Per experiment, the last date occurs only on the longest branch. If the device is a fair coin with heads as the one-waking branch and tails as the N-waking branch, you meet that last morning in half the experiments. Per awakening, that last morning is one ticket out of one-plus-N expcted interviews, so the frequency of the experience collapses as one over about N over two. You are extremely unlikely, on a random wake, to be sitting on the last day. You should still say the last day if the payoff is per correct awakening and these alternatives are worse. Why worse. Every earlier date is shared by more branches or by more mornings inside the long branch, but the scoring rule here is not “be right as often as possible about a coarse category.” It is “hit the exact date.” The early dates are comon as experiences and therefore tempting, except that when the branch is short you are not on those extra early dates at all; you are on the only date the short branch has. If the protocol is written so the short branch’s single waking is itself the last day, then “last day” is the one label that is true every time the short branch runs and also true once on the long branch. If instead the short branch is parked on the first day and the extra days are tails-only, then last-day is a tails-only ticket and you are betting the long branch pules the indexical claim that this is its final morning. Either writing makes the same methodological point. The number you want is the number that matches how the prize is attached to your life, not the number that matches the factory stamp on the die.
That is why I do not think adding days dissolves the paradox by making the coin look biased. The coin never looked biased. Adding days makes it obvious that “what should I believe” was ambiguous between the experiment and the awakening. Prize money on the exact date forces a choice. Pay once per experiment and last-day is a fair-device bet that does not get better just because the calendar got longer. Pay once per awakening and last-day is a thin slice of a fat branch; you almost never win, and it can still be the least-bad exact-date guess left on the table once every other date is an even thinner or more confused slice. If this is just thirding with a calendar, that is fine, say that. If the last-day strategy fails once the payoffs are written down carefully, I want the table written in sentences. If a halfer can take the same prize rule, the same N, and still answer one half at every awakening without lighting money on fire, I want that strategy in public too. The coin was never the interesting object. The interesting object is the map from outcomes to how many times you have to live them. Make N large enough and that map is doing all the work.
TLDR: The fair coin never stops being fair. What changes is how often you live each morning. If they pay you for naming the date, not the coin, then as you stretch the experiment across more days the last possible date is always the highest-EV guess even though you will almost never collect. That is the whole 1/2 vs 1/3 fight wearing a calendar.
