Perspective Importance and Solution to the Sleeping Beauty Problem

This is a simplified argument for perspective disagreement in the sleeping beauty problem. The purpose is to present the idea in a shorter, more structured way than my first post.

Perspective Argument:

Considering the following experiment equivalent to the sleeping beauty problem:

(If you are skeptical about this equivalency please check my first post. It has more detailed explanation why my argument does not dependent on the equivalency)

Duplicating Beauty:

Beauty falls asleep as usual. The experimenter tosses a fair coin before she wakes up. If the coin landed on T then a perfect copy of beauty will be produced. The copy is precise enough that she cannot tell if herself is the clone or the original. If the coin landed on H then no copy will be made . The beauty(ies) will then be randomly put into two identical rooms. At this point another person, let’s call him the selector, randomly chooses one of the two rooms and enters. Suppose he saw a beauty in the chosen room. What should the credence for H be for the two of them? 

For the Selector his probability is easy to calculate. Because he is twice more likely to see a beauty in the room if T, simple bayesian updating gives us his probability for H as 1/3.

For Beauty, her room has the same chance of being chosen regardless if the coin landed on H or T. Therefore seeing the Selector gives her no new information about the coin toss. So her answer should be the same as in the original sleeping beauty problem: if she is a halfer 1/2, if she is a thirder 1/3. 

This means the two of them would give different answers according to halfers and would give the same answer according to thirders. Notice here the Selector and Beauty can freely communicate however they want and it won’t change their answer since they have the same information regarding the coin toss. So halving would give rise to a perspective disagreement.

 

This perspective disagreement is quite unusual (and against Aumann’s Agreement Theorem), so it could be used as an evidence against halving thus supporting Thirdrism and SIA. I would argue why SIA has its own form of perspective disagreement in a separate post. For now I want to argue that this disagreement is logically sound. 

First of all, this perspective disagreement from bayesian reasoning is also mirrored by frequentist interpretation of probability. Let’s take a frequentist’s approach and see what happens if the experiment is repeated, say 1000 times. For the Selector, this simply means someone else go through the coin toss 1000 times and let him chooses a random room after each time. On average there would be 500 H and T. He would see a beauty for all 500 times after T and see a beauty only 250 times after H. Meaning out of all the instances where he sees a beauty in the chosen room, 1/3 of which would be after H. Therefore he is correct in giving 1/3 as his answer.

From beauty’s perspective repeating the experiment simply means she goes through the experiment and wake up in a random room awaiting the Selector’s choice again.  So by her count, taking part in 1000 repetitions means she would recall 1000 coin tosses after waking up.  In those 1000 coin tosses there should be about 500 of H and T each. She would see the Selector about 500 times with equal numbers after T or H. Therefore her answer of 1/2 is also supported by long run frequency.

 

I think this is the easiest way to see the relative frequency from beauty’s perspective: Ignore the clones first, if someone experienced 1000 coin toss she would expect to see 500 Heads. Now consider how the cloning would affect her conclusion. If there is a T somewhere along the tosses thus a clone is created, both the original and the clone would have remembered exactly the same tosses happened earlier. They would also have the same expectation about coin tosses after they split. The relative frequency would be the same for both the original and the clone. Therefore  she does not have to consider whether she is physically a clone or the original. After beauty wakes up remembering a large number of tosses she shall expect about half of those landed in Heads.

Alternatively, a more complicated way would be to think about all the coin tosses happened and her own position. For example a beauty experiencing two coin tosses there are several ways this can happen from a third party’s perspective:

1. H-H: Both coin tossing landed on H. There is a 1/4 chance for this happening. Beauty experience 2 Hs.

2. H-T: First coin landed on H and second coin landed on T. There is a 1/4 chance for this happening and there would be 2 beauties at the end of the experiment. It doesn’t matter if she is any one of the two she experienced 1H and 1T.

3. T-(H,H): First toss landed on T and both resulting beauties experience H at the second toss. There is 1/8 chance for this to happen, doesn’t matter if she is any one of the two beauties at the end, she experienced 1 H and 1T.

4. T-(H,T): First coin landed on T and a H and T happened at the second round of tosses respectively. There is 1/4 chance for this to happen. Depending which coin toss she experienced at the second round, half the time (1/8) she experienced 1H and 1T, while the other half (1/8) she experienced 2 Ts. 

5. T-(T,T): All tosses landed on T. There is 1/8 chance, doesn’t matter which second toss is hers she experienced 2Ts. 

The expected number of H from beauty’s perspective: (1/4)x2+(1/4)x1+(1/8)x1+(1/8)x1=1: half the number of tosses. Obviously this calculation can be generalized to other number of tosses. 

Both the above methods correctly gives the relative frequency from beauty’s perspective. The most common mistake is for a beauty to think she is equally likely to be any one of the beauties at the end of multiple coin tosses. This seemingly intuitive conclusion is valid from a third party’s (the selector’s) perspective, aka if he randomly chooses one from all the ending beauties then all beauties are in symmetrical position. However from beauty’s first person perspective the ending beauties are not in symmetrical position especially since they have experienced different coin toss results.

If we call the creation of a new beauty a “branch off”, here we see that from Selector’s perspective experiments from all branches are considered a repetition. Where as from Beauty’s perspective only experiment from her own branch is counted as a repetition. This difference leads to the disagreement.

This disagreement can also be demonstrated by the difference in betting odds. Imagine the selector and the beauty(ies) enters a bet about the coin toss result whenever they meet. The payoff enabling them to break even in the long run would be different. For the selector, choosing any of the two rooms after T leads to the same observation: he always sees a beauty and enters another bet. However, for the two beauties the selector’s choice leads to different observations: either she can see him and enters another bet or not. Effectively the selector is twice more likely to enter a bet than a beauty if the coin landed on tails. If they bet 1 dollar on H, the longterm breakeven payoff would be 3 dollars for the selector and 2 dollars for beauty respectively. (I will discuss the use of bets as arguments for different positions in the sleeping beauty problem in a separate post in more detail.)

It should be noted by choosing a random room the selector is essentially choosing among all possible observers as stated by SIA. We can also construct the experiment so that the selector is choosing among all actual observers as stated by SSA. For example a third party who knows the content of the rooms can randomly label the two rooms 1 & 2 in his mind. The selector can pick a number as he wish. The third party would then place him into the corresponding room if it is occupied, and place him into the other room if it is not. This way the selector would always see a beauty in the room. Therefore when he sees a beauty his probability would remain at 1/2 as there is no new information. However from beauty’s perspective she is guaranteed to meet the selector in H and only has half the chance in case of T. Therefore seeing the selector is evidence confirming H from her perspective. Assuming she is a halfer, her probability of H would become 2/3. The two of them would still be in disagreement as in the previous case. In essence the two parties interpret the meeting differently which leads to the disagreement. The selector interpret it as a meeting between him and A beauty, any beauty. Whereas beauty interpret it as the meeting between the selector and THE beauty, herself. 

Another important question would be what should beauty’s probability be once learning she is actually the original. If we continue to apply the perspective reasoning then her probability would remain unchanged at 1/2. This can be shown in two ways.

One way is to put the new information into the frequentist approach mentioned above. In Duplicating Beauties, when a beauty wakes up and remembering 1000 repetitions she shall reason there are about 500 of H and T each among those 1000 tosses. The same conclusion would be reached by all beauties without knowing if she is physically the original or created somewhere along the way. Now suppose a beauty learns she is the original. She could ignore the cloning part of the experiment entirely and still conclude those 1000 tosses contain about 500 of H and T each. This means her probability of H would remains unchanged at 1/2.

Another way to see why beauty should not change her probability is to see the agreement/disagreement pattern between her and the selector. It is worth noting that beauty and the selector will be in agreement once knowing she is the original. As discussed above one way to understand the disagreement is they interpret the meeting differently. The selector interpret it as a meeting with ANY beauty whereas beauty interpret it as a meeting with a specific beauty aka herself. However once we distinguish the beauty(ies) by stating which one is the original (and possibly which one is the clone) the selector and the beauty would have to have the same interpretation, that the meeting is between the selector and a specific beauty. We can also use bets mentioned above to demonstrate this. After T, seeing either beauty is the same observation for the selector while it  is different observations for beauties. This in turn causes the selector to enter twice more bets than beauty, leading to the difference in their betting odds. However if a bet is only set between the selector and the original beauty then the selector would no longer be more likely to enter a bet. He and the Beauty would enter the bet with equal chances. Meaning their betting odds ought to be the same, aka they must be in agreement regarding the probability of H. 

Now consider the duplicating beauty problem with the selector choosing a random room. Once the selector saw a beauty in the room his probability of H changes to 1/3 while beauty’s probably remains at 1/2. At this time they are in disagreement. However, if the experimenter tells them the beauty in the room is the original then selector’s probability of H would increase to 1/2 again since he has equal chance of choosing the room with original beauty regardless of coin toss result. Because they ought to be in agreement beauty’s probability should remain unchanged at 1/2.

The same result can be shown if the selector is choosing among existing beauties as SSA suggested using the procedure mentioned previously. As shown above once they meet the beauty’s probability of H rises to 2/3 while the selector’s probability remains at 1/2. If the experimenter tells them the beauty in the room is the original then selector’s would adjust his answer. Since he would always meet with the original beauty if H and meet her half the time if T his probability of H would rises to 2/3. Because they ought to be in agreement beauty’s probability must remain unchanged at 2/3.

The above methods shows beauty should not update her answer after learning she is the original. Beauty’s antibayesianism, just like the perspective disagreement, is quite unusual. This is again due to perspective reasoning. The concept of original or clone is only relevant from a third person’s perspective. From beauty’s perspective by introspection and recollection she is every way the same person who went to sleep last night. Therefore the bayesian update can only be performed from a third person perspective, aka by the selector. 

Traditionally the answer to this problem is deemed to be between self-sampling assumption and self-indication assumption. Self-sampling assumption states beauty should reason as if she is randomly selected among all actual existent (past, present and future) beauties while self-indication assumption states beauty should reasonas if she is randomly selected among all potentially exist beauties. Both methods attempt to answer the question from a third party’s (the selector’s) perspective. As shown above beauty and the selector could disagree with each other therefore neither methods give correct answers. Applying the same logic could help us debunking other problems such as the doomsday argument, the presumptuous philosopher and  the simulation argument. 

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