show me the specific mistake in my math, rather than appeal to a verbal presentation of a non-formal, intuitive explanation.

My point was that I didn't think anything was wrong with your math. If you count tokens the answer you get is 1/3. If you count types the answer you get is 1/2 (did you need more math for that?). Similarly, you can design payouts where the right choice is 1/3 and payouts where the right choice is 1/2.

You can compute things like the expected number of green marbles left in the bag. In the bag problem, IOW, we are quantifying our uncertainty over tokens, while taking types to be a fixed feature of the situation.

b) what justifies "worrying about types rather than tokens" in this situation, where every other discussion of probability "worries about tokens" in the sense I've outlined above in reference to the bag of marbles?

This was a helpful comment for me. What we're dealing with is actually a special case of the type-token ambiguity: the tokens are actually indistinguishable. Say I flip a coin. I, If tails I put six red marbles in a bag which already contains three red marbles bag, if heads do nothing to the bag with three red marbles. I draw a marble and tell Beauty "red". And then I ask Beauty her credence for the coin landing heads. I think that is basically isomorphic to the Sleep Beauty problem. In the original she is woken up twice if heads, but thats just like having more red marbles to choose from, the experiences are indistinguishable just like the marbles.

Statements like "information is gained" or "information is lost" are vague and imprecise,

I don't really think they are. That's my major problem with the 1/3 answer. No one has ever shown me the unexpected experience Beauty must have to update from 0.5. But if you feel that way I'll try other methods.

c) how do you apply the type-token distinction in other problems, say, in the case of the Tuesday Boy?

Off hand there is no reason to worry about types, as the possible answers to the questions "Do you have exactly two children?" and "Is one of them a boy born on a Tuesday?" are all distinguishable. But I haven't thought really hard about that problem, maybe there is something I'm missing. My approach does suggest a reason for why the Self-Indication Assumption is wrong: the necessary features of an observer are indistinguishable. So it returns 0.5 for the Presumptuous Philosopher problem.

I'll come back with an answer to (a). Bug me about it if I don't. There is admittedly a problem which I haven't worked out: I'm not sure how to relate the experience-type to the day of the week (time is a property of tokens). Basically, the type by itself doesn't seem to tell us anything about the day (just like picking the red marble doesn't tell us whether or not it was added after the coin flip. And maybe that's a reason to reject my approach. I don't know.