Well, obviously there is no official solution. If you're asking what the result of Elga's method would be, you're right. But I would write "P(H and day 1)" not "P(H on day 1)", that can be confused with the conditional probability "P(H | day 1)".

I think Elga's method makes sense if one considers what Sleeping Beauty should do if she could bet on heads or tails (so that she wins 1$ if she is right and loses 1$ if she is wrong).

She bets on heads:
If the coin showed heads, she is right once (on day 1) and wins 1$. If the coin showed tails she is wrong on day 1 to 7, and loses 7$.

She bets on tails:
If the coin showed heads, she is wrong once (on day 1) and loses 1$. If the coin showed tails she is right on day 1 to 7, and wins 7$.

So it makes more sense for Sleeping Beauty to bet on tails and this would support Elga's method.

As you probably know, there is no consensus which is the right solution to the original problem. Some people, including Adam Elga say it is 1/3, other people including David Lewis say it is 1/2. 

If you're asking what the result of Elga's method would be, you're right, it would be 1/8. But I would write "P(H and day 1)" not "P(H on day 1)", that can be confused with the conditional probability "P(H | day 1)".

Well, obviously there is no official solution. If you're asking what the result of Elga's method would be, you're right. But I would write "P(H and day 1)" not "P(H on day 1)", that can be confused with the conditional probability "P(H | day 1)".

I think Elga's method makes sense if one asks considers what Sleeping Beauty should do if she could bet on heads or tails (so that she wins 1$ if she is right and loses 1$ if she is wrong).

She bets on heads:
If the coin showed heads, she is right once (on day 1) and wins 1$. If the coin showed tails she is wrong on day 1 to 7, and loses 7$.

She bets on tails:
If the coin showed heads, she is wrong once (on day 1) and loses 1$. If the coin showed tails she is right on day 1 to 7, and wins 7$.

So it makes more sense for Sleeping Beauty to bet on tails and this would support Elga's method.

Well, obviously there is no official solution. If you're asking what the result of Elga's method would be, you're right. But I would write "P(H and day 1)" not "P(H on day 1)", that can be confused with the conditional probability "P(H | day 1)".

I think Elga's method makes sense if one considers what Sleeping Beauty should do if she could bet on heads or tails (so that she wins 1$ if she is right and loses 1$ if she is wrong).

She bets on heads:
If the coin showed heads, she is right once (on day 1) and wins 1$. If the coin showed tails she is wrong on day 1 to 7, and loses 7$.

She bets on tails:
If the coin showed heads, she is wrong once (on day 1) and loses 1$. If the coin showed tails she is right on day 1 to 7, and wins 7$.

So it makes more sense for Sleeping Beauty to bet on tails and this would support Elga's method.