Hello everyone. I, Pooja Srihari (1933466) with my co-authors - S Dhanush (1933434), Uzair Muzaffar (1933442) and Khushi Pareek (1933463) in this blog are going to talk about a very interesting and a challenging probability paradox - The Sleeping Beauty. So get up and look for a pen and a piece of paper because things are about to get real (probably).
The Sleeping Beauty Paradox
History of the problem:
The Sleeping Beauty Paradox was originally formulated in the mid-1980s by an American Philosopher Arnold Stuart Zuboff in an unpublished paper. The name “Sleeping Beauty” was given to the paradox by an American Philosopher Robert C. Stalnaker. The paradox is a thought experiment in decision theory and is very similar to the infamous Bertrand's Paradox. Both the paradoxes are counter-intuitive and have several solutions depending on the way we approach the paradox.
Introduction to the paradox:
Who here has not promised themselves to sleep before 10 PM, but ended up watching “10 Mysterious Things Science Cannot Explain” on YouTube at 2 AM. It has always been fascinating to watch things that challenge our intuition. Today we will explain one such counter-intuitive probability paradox that will make you read this blog at least twice. So here goes the infamous, “Sleeping Beauty Paradox”.
Imagine an experiment where the sleeping beauty sleeps (as usual) on a Sunday. To decide when to wake her up, we toss a fair coin. I know this is a weird way of making decisions and real decisions are made using stone-paper-scissors, but we want you to cooperate with us to construct the paradox. Every time we wake her up, she will not know about the previous awakenings and what day it is. So we toss a coin and based on the results we follow the following given instructions:
If we get Heads:
We will wake her up on Monday. We’ll then ask her what she thinks showed up on the coin we tossed. Then she’ll sleep again.
Then we’ll NOT wake her up on Tuesday and she’ll wake up on Wednesday when the experiment ends. She can leave and probably sleep peacefully without anyone waking her up for no reason.
If we get Tails:
We will wake her up on Monday. We’ll then ask her what she thinks showed up on the coin we tossed. Then she’ll sleep again.
We will also wake her up on Tuesday and ask her what she thinks showed up on the coin. Make sure you remember that she will not know of the previous awakening. Now she’ll sleep again and wake up on Wednesday when the experiment ends.
Now a question that won’t let you sleep is:
At any given awakening, if the sleeping beauty were to use probability to answer what showed up on the coin, what is the credence (probability) that a Head showed up. Please don’t think of this as an easy question because who writes a blog on a problem everyone can do?
Most people will say, if we ignore all the seemingly unimportant extra information you are giving us, the question boils down to what is the probability that heads showed up on a fair coin tossed once. And the answer is ½. That is what our intuition says, but as you may have guessed that this is wrong because why would I add all this extra information if it was unimportant. In probability, every piece of information is important.
Why is the paradox counter-intuitive?
Before jumping into the two debatable solutions to the paradox, let us see where our intuition goes wrong in answering this question. When we say a fair coin is tossed, we know the chances of heads or tails showing up are equal i.e., 50%. How does sleeping beauty’s sleep or awakenings make any difference?
Also, you may notice that the Monday awakenings do not depend on the result of the coin. Whether the coin shows a Head or a Tail, sleeping beauty will anyway be awakened on Monday. So some of you may also think that sleeping beauty can be awakened on Monday even before tossing the coin. We can toss the coin on Monday evening when the sleeping beauty has already slept again after the Monday awakening. And for the Tuesday awakening, we rely on a fair coin. This makes us think that the probability of heads showing up and therefore the probability of Tuesday awakening is unchanged at 50% (both Heads and Tails have an equal chance of showing up on a fair coin). Well, I appreciate the thought process you all went through, but this way of thinking about the paradox is not complete. Surprised right? I was also surprised at first, but things will get clear (I hope), once you look at the two major solutions that are proposed for this paradox. So pay attention because from now on, it is only going to get more and more challenging. All the best!
Solution:
Now we look at the Frazier vs. Ali, the Yankees vs. Rox of the Sleeping Beauty Paradox - Halfers vs. Thirders. Anyone who’s looked at this paradox has invariably been divided among these two groups. While both present logical arguments, they are certain that the other one is wrong. Yes, very much like our neighbours. But we’ll let you decide which one is correct for yourself.
Halfers:
This approach was first given by the famous American Philosopher David Lewis in 2001. He asserts that the answer is ½.
So, according to him when the sleeping beauty is woken up on Monday what credence ( probability derived from personal judgment or own experience) does beauty assign to heads being the outcome is
P( Heads)= P (Tails)= ½.
The coin here is fair thus the probability of getting heads when you toss the coin is ½.
Further Beauty receives no information about the outcome of the coin toss when she is awakened. Making her credence that the coin came up heads should continue to be ½.
Confused?
We can better understand this through a small graph
Here on the row side, H is for tossing the coin and the outcome being head. T is for tossing the coin and the outcome being tails.
On the column side, M stands for Monday and T stands for Tuesday.
In the boxes, A stands for Sleeping Beauty woken up and S stands for Sleeping Beauty sleeping.
So, now if the Sleeping Beauty is woken up on Monday and asked her credence of the coin landed heads, then she would blanky reply ½ since the coin is fair and when you toss the coin your chance of getting a head is equal to the chance of getting a tail i.e there is a 50-50 chance-
P ( H) + P (T) = 1
Sum of the probability of heads and tails should be equal to 1.
P (H)=P (T)= ½
Now splitting the probability of getting tails evenly between Monday and Tuesday we get the following plot
The key phrase here is that the coin “landed head” when we refer to the past event, how much background do we “import”.
Halfers follow action interpretation as they import the past.
What is this “action interpretation” now?
Let me ask you a question, what is your belief that Cristiano Ronaldo spent a full year’s earning in his first house, here we are not talking about his earnings today because he is legendary footballer but his earnings when he was starting his football career- we are importing the past thus, this is called action interpretation.
Since it is assumed here that Sleeping Beauty knows all the rules of the experiment, she is aware of them before she is put off to her “beauty” sleep and her credence before drinking the potion that is
P (Head)= ½.
In the perspectives of Halfesr, the Sleeping Beauty doesn’t get any new information as to whether the coin tossed and came up heads or tails. Leading her credence that the coin landed in heads should remain ½ and not change.
Halfers here go on to take up the big shot view of being experiments and provide justification by taking the experimenters position. From the experimenters’ view, this whole amnesia drama and the number of awakening is irrelevant, to them, this doesn’t matter. One of the halters- Peter Winker states that this whole amnesia and number of awakening is “a snare and delusional”.
If this experiment was repeated multiple times the halfers would only count how many times the coin landed up in heads relative to the total number of trials. Ultimately since this coin was fair, it is obvious that this will happen only in one-half of the trials.
Thirders:
While the halfers claim that Sleeping Beauty hasn’t received any new information since the beginning of the experiment, there is obviously someone who disagrees. These are the thirders, they believe that being within the experiment is new.
The original advocate of the thirder position was Adam Elga who said the probability of heads landing is 1/3. Yes, he’s partly to blame for our tangled neurons. Let’s look at the argument he presented.
If Sleeping Beauty somehow knows that the coin had landed on tails. Don’t ask us how. She was probably omniscient. Now by the principle of indifference, there is no reason for her to believe that there is a greater probability that it is a Monday or a Tuesday. For the sake of this experiment, her omniscience is limited. This would mean that P(Monday | Tails) = P(Tuesday | Tails) and by extension P(Monday and Tails) = P(Tuesday and Tails).
Now let’s turn the tables and look at a situation where sleeping beauty is absolutely convinced that it is a Monday and given that there is an equal probability of a head or a tail landing when a fair coin is tossed, P(Tails | Monday) = P(Heads | Monday) and by extension P(Tails and Monday) = P(Heads and Monday).
By this logic, we can conclude that P(Tails and Monday) = P(Tails and Tuesday) = P(Heads and Monday). Considering that these events are mutually exclusive and exhaustive, the probability of Heads landing is ⅓.
In other words, there are four possibilities, the coin lands on heads and it’s either Monday and she is awakened or it’s Tuesday and she’s asleep or the coin lands on tails and it’s either Monday and she’s awakened or it’s Tuesday and she’s awakened. Now given that she’s awakened, it rules out the event of a heads and Tuesday. Now among the three remaining events, the probability of heads landing was ⅓. We know that’s a lot of ‘ands’ and ‘ors’ and we don’t want to confuse you, at least not more than you already are. So here’s a picture to make it a little easier.
Conclusion
The Sleeping Beauty paradox is a seemingly simple problem, whose solutions have required philosophers to revise their concepts and principles of rationality in order to better how and when our credences ought to change. The problem has created two clear-cut wings known as the halfers and the thirders.
It is shocking and a little offsetting to some that such a situation can arise in a branch of mathematics, in this case probability. One reason for the difference between the two wings is that both the wings interpret the problem in two different, though equally logical ways and translate it into different mathematical problems. Thus credence is defined by the two wings.
The fact that both the wings provide a logical argument makes this puzzle an interesting paradox. Though most of the philosophers support the thirder’s position, there has been no agreement on what is the best argument for that position.
While fascinating in its own situation, the problem challenges us to answer some fundamental questions in epistemology: Under what conditions are we rationally required to change our credences? And what can be qualified as ‘new evidence’ that should lead us to alter our credences? Answers to these questions matter not just to the Sleeping Beauty problem but also for other branches of philosophy and rationality.
References
Elga, A. (2000). Self-Locating Belief and the Sleeping Beauty Problem. Analysis, 60(2), 143-147. Retrieved August 5, 2020, from www.jstor.org/stable/3329167
G4G Celebration. (2019, April 10). The Sleeping Beauty Paradox Resolved – Pradeep Mutalik [Video File]. Retrieved from https://www.youtube.com/watch?v=2hxgR54X76s&feature=youtu.be
Lewis, D. (2001). Sleeping Beauty: Reply to Elga. Analysis, 61(3), 171-176. Retrieved August 5, 2020, from www.jstor.org/stable/3329230
Bostrom, N. (2007). Sleeping Beauty and Self-Location: A Hybrid Model. Synthese, 157(1), 59-78. Retrieved August 5, 2020, from www.jstor.org/stable/27653543
Mutalik, P. (2016, January 29). Solution: ‘Sleeping Beauty’s Dilemma’. Quanta magazine. Retrieved from https://www.quantamagazine.org/solution-sleeping-beautys-dilemma-20160129/
Peterson, D. (2019, June 15). The Sleeping Beauty Problem. Retrieved from 1000 Word Philosophy: https://1000wordphilosophy.com/2019/06/15/the-sleeping-beauty-problem/#_ftn10
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