The following problem is a refinement of the Sleeping Beauty problem. We replace her by a computer so that its (formerly her) inner thinking is almost rigorously understood. And we make sure that its (her) reasoning and opinion about the state of the coin has some consequences.
The story is the following.
You took the job of a janitor at FIFA and you heard a conversation between the newly elected president and other officials. They want to kickstart a new era of FIFA by an unusual and marvelous soccer match, Holland vs Tennessee, on Wednesday. However, the match wouldn't be attractive enough because one of the teams seems stronger.
So they decide about the winner in advance, on Sunday, by a fair coin. (If you don't like these new policies to decide about soccer, you should have kept Sepp Blatter.) Heads means that Holland will win (H); tails means that Tennessee will win (T). The goalies are trained to make their deliberate failures look realistic.
The planning for a Tennessee victory is hard – because everyone in Tennessee would celebrate. Everyone needs to save some electricity, so in this case of "tails" (planned Tennessee win), there will be a short blackout on Tuesday at 8:00 am that will restart all computers. Except for the FIFA officials and you, no one knows that the blackout means that Tennessee is going to win a day later.
Your brother works in the same building, likes betting on soccer, and has a computer called Sleeping Beauty. It's a computer with RAM that loads its program from the cloud and doesn't know what time (or day) it is now. On Monday morning, the computer is normally turned on.
Different bookmakers are offering different odds for the victory of Holland or Tennessee. Everyone limits how much money you may bet – either there is some overall limitation for everyone or there's a limit on the amount of money that each client may bet or something like that. Or he has his own limits.
It's Sunday evening and your task is to write the program for the Sleeping Beauty, your brother's computer, that will earn the maximum amount of money by betting on the soccer match. You don't know the result of the coin flip – you only heard how it is going to be used.
To maximize your brother's profit, you need to write the program that will know or calculate – as accurately as possible – the probability that Holland or Tennessee wins (the value of the probability at the moment when the computer was started). A tie is not possible because what you have heard but the bookmakers may either allow a tie or not. Once the computer has an estimate for the probability, it will determine all bookmakers where your brother should bet the maximum amount of money on one team or another – and permanently show this list on the screen along with the message "my brother, please bet on the victory of H in these bookmaker offices, and on the victory of T in the following ones"...
The program will be run once – on Monday – if the coin lands "heads". The calculation will be run twice – on Monday and Tuesday – in the case of "tails" because, as I wrote previously, there is a brief blackout on Tuesday morning in the case of "tails" that restarts the computers.
Your brother reads the result on the screen at some moment and bets on the victory of Holland or Tennessee sometime on Monday or Tuesday, when he has enough time. The time doesn't really matter because the program has no "random generator" or "information about the current time" so it produces the same result (the odds that the bookmakers offer are constant on the two days) every time it is launched (just like the human Sleeping Beauty).
What the program should do to maximize your brother's profit?
Solution
The ideal program knows the rules of the game – because you have heard these rules (thanks to your being a janitor) and you wrote the program. It doesn't know what day it is now – and it doesn't know whether the computer was woken up and the program was running previously.
It's obvious what the program should do. It should realize that the probability of the "heads" (Holland victory) is 50%, the probability of "tails" is 50% (Tennessee victory), and simply show the bookmakers who offer you better odds on either Holland or Tennessee than 2:1, with the recommendation for your brother to bet on Holland or Tennessee in these companies, respectively. (Other people have no clue that the chances of Tennessee are this high.) Also, the program may choose to minimize the risks – and tell your brother to bet the same amount of money on H or T in different companies, so that the positive profit is guaranteed.
(You may modify the application of the probability in numerous ways. The program may only tell your brother whether some bet is a good idea or not or something else.)
The program has the "extra" information that it was just started. But just like in the case of the human Sleeping Beauty, this "extra" information is actually no information about the state of the coin (and winner of the match) at all because both possibilities predict that the computer will wake up at least once, and that's the only fact that the computer program knows when it's running.
Discussion
I think that almost everyone will agree that the right probability of a Holland win that the program should assume is \(P=1/2\). There might be some argumentation that the problem isn't equivalent to the original Sleeping Beauty Problem but I can't exactly predict what this argumentation might be.
Note that the situation is designed so that "the coin landed heads" and "Holland wins" are equivalent propositions, just like "the coin landed tails" and "Tennessee wins" are equivalent.
An essential feature of my formulation of the problem is that it is made very clear and explicit that just like in the original problem, the state of the coin remains the same on Monday and Tuesday.
Your brother who is the ultimate user of the recommendation by the Sleeping Beauty software can only bet once on the victory of Holland or Tennessee in each bookmaker's office. As I mentioned, there are limits that the bookmakers enforce. The limits also may arise thanks to the finiteness of your brother's banking account. Clearly, the limits aren't doubled just because a computer in someone's office was restarted.
The "thirders" – who love to promote the incorrect answer \(P=1/3\) to the original Sleeping Beauty Problem – often mention betting, too. But the mistake they are making is that when they incorrectly claim or assume that all "three kinds of awakening" are equally likely, they are changing the question about the coin to a question about the typical awakening of a woman (or a computer, in our case) – which is a totally different thing. The original question is a question about an external object, the coin, and an external event (the flip of the coin or the soccer match), so it can't be affected by how many times someone thinks about the question, or how many times she is awakened (or how many times she drinks wine).
My computer reformulation of the problem may be made even more extreme if we assume that the computer boots extremely quickly. If that's so, it means that on Tuesday morning, the display just blinks for a very short period of time and the same calculated recommendation quickly appears once again. So the blackout and restart on Tuesday morning doesn't really matter at all! It may be made literally unobservable.
In effect, the computer program is run twice in the case of "tails". In the human case, we could say that the Sleeping Beauty is forced to think twice about the state of the coin if the state is tails. But if you think twice about something in the outer world – with the same initial knowledge about the state of the external world and its rules – and if your thinking is correct, you will just do the same thing twice and you will get the same result as if you were just thinking once! If you think twice how much is \(0.3+0.3\), you get \(0.6\) in each case instead of \(1.2\).
Just like the restart of the computer that quickly returns it to the same state, the second awakening of the Sleeping Beauty just cannot affect the correctly determined probabilities of any statement about the external world!
I am a bit curious what kind of objections we will hear from the thirders and other wrongnumberists.
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