If it’s tails and matches what you chose, then you win $1 otherwise lose $1. If it’s heads and matches what you chose, you win $2 otherwise you lose $2.
Both options have the expected value equal to zero though? (0.5⋅1−0.5⋅1 versus 0.5⋅2−0.5⋅2.)
If you choose heads, you either win $2 (ie win $1 twice) or lose $1. If you choose tails then you either win $1 or lose $2. It’s exactly the same as the Sleeping Beauty problem with betting, just you have to precommit to a choice of heads/tail ahead of time. Sorry that this situation is weird to describe and unclear.
Both options have the expected value equal to zero though? (0.5⋅1−0.5⋅1 versus 0.5⋅2−0.5⋅2.)
If you choose heads, you either win $2 (ie win $1 twice) or lose $1. If you choose tails then you either win $1 or lose $2. It’s exactly the same as the Sleeping Beauty problem with betting, just you have to precommit to a choice of heads/tail ahead of time. Sorry that this situation is weird to describe and unclear.