.5*-x + .5*(-x-x + 450) = 225-1.5x

But her expectation from following the always bet strategy is already 225 so if she is indifferent to risk she shouldn’t pay anything.

Though it should be noted that the outcomes where she pays $x are -x and 450-2x respectively so if we want to take risk aversion/non-linear utility in money into account it may very well be reasonable to pay some nonzero amount.

]]>Without knowledge of the day of the week, Sleeping Beauty’s expected gain is $175.

If she knows the day, she won’t bet on Monday, and will bet on Tuesday. If it’s heads, she’ll buy the day once and not bet. So her total loss would be the fee. If it’s tails, she’ll pay the fee on Monday and not bet, and then pay the fee on Tuesday and make the bet. She’ll gain $450, minus the two fees. So her expected gain is $450 minus three fees, all divided by two. For that to equal $175, the fee would have to be $100/3.

Therefore, she should be willing to pay the fee if it is anything up to $33.33, but not a penny more.

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