Thursday, August 7, 2014

A confused sleeping beauty 2

This post contains a small twist of the original experiment discussed in the previous post, A confused sleeping beauty. The new version doesn't require putting anyone to sleep and removing her memories, because we replace memory removal with lack of information.

Confusing Sleeping Beauty without erasing her memory

Sleeping Beauty is no longer required to sleep, but she may still need to sleep, to remain beautiful.

Consider the following settings:
- We toss a fair coin.
- If it lands heads, we will ask once Sleeping Beauty her belief for the proposition that the question landed heads.
- If the coin lands tail, we ask her twice.
This is similar to the original experiment, but instead of erasing her memory, we just do the following:
- Before asking her any question, we toss the coin a large number of times.
- Then we ask Beauty, but not in the same order in which we tossed. For example, when we toss a coin, if it landed heads, we write down a question and don't ask it yet. If it landed tails, we write down two questions, and don't ask them yet. Then we shuffle the questions and we ask Beauty one at a time. We take care to keep track for each question to which toss is connected.
To prevent the possibility that she adjust her estimates by counting counting the number of heads and tails about which she was already asked, we don't tell her whether she guessed or not, until the end of the experiment.

We see that the most rational answer she can give is 1/3. On the other hand, of course she knows that the probability that when the coin was tossed it landed heads is 1/2.

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