## Sleeping Beauty and Mondays

### by Tanya Khovanova and Alexey Radul

Sleeping Beauty participates in the following experiment. On Sunday she is put to sleep, and a fair coin is flipped. Regardless of the result of the coin flip, she is awakened on Monday and is offered a bet. She may pay \$550 in which case she will get \$1000 if the coin was tails. If the coin was tails, she is put back to sleep with her memory erased, and awakened on Tuesday and given the same bet again. She knows the protocol. Should she take the bet?

As we discussed in our first essay about Sleeping Beauty, she should take the bet. Indeed, if the coin was heads her loss is \$550. But if the coin was tails her gain is \$900.

To tell you the truth, when Beauty is offered the bet, she dreams: “It would be nice to know the day of the week. If it were Tuesday, then the coin must have been tails and I would gladly take the winning bet.”

In our next variation of the riddle her dream comes true.

Every time she is awakened she is offered to buy the knowledge of the day of the week. How much should she be willing to pay to know the day of the week?

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1. #### Bill:

Great question!

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.

2. #### Peter Gerdes:

So if sleeping beauty pays \$x every time she is woken up to find out the day of the weak and chooses not to bet on mondays her expectation becomes

.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.

3. #### misael:

i think that https://youtu.be/fQH6L4atLuY

4. #### Jeff:

I recall reading that Erdos had trouble with conditional probabilities. An off topic thought I have concerns his concept of the book, the one containing the best proofs of all mathematical results. I wonder, is the number of pages of such a book finite? I know the P vs NP question is related if we think of the book as a proof finding program of finite code length. Any thoughts anyone?

5. #### Francisco:

Nice puzzles! Nice comments about math books! How in the World didn’t I find this place earlier? Congratulations on a very sharp and entertaining blog.