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# Deus Ex Machina

## Question

Five years ago, your rich uncle secretly commissioned the construction of a supercomputer! The purpose of this binary behemoth (the "Very Good Predictor") was to predict, as accurately as possible, every action you would take over the course of these five years. It has managed to do so with 100% accuracy for its entire runtime.
Now, your uncle has approached you with this information, and an offer to participate in a game involving the supercomputer's final prediction.
He has placed two envelopes in a room: Envelope A and Envelope B.
Envelope B contains exactly 1,000 dollars. Envelope A either contains 10,000, 10, or 0 dollars. If the supercomputer correctly predicted your selection of Envelope A alone, it will contain \$0. If the supercomputer correctly predicted your selection of both A and B then Envelope A contains only \$10. Otherwise the supercomputer predicted that you would not choose A and told your uncle that he could safely put \$10,000 in the envelope (which he would oblige).
Now since this is essentially a contest between you and the supercomputer, if the supercomputer correctly predicts your choice, the amount selected is the amount you owe your uncle, otherwise you keep the amount in the envelopes (both). The values in both envelopes is revealed once a selection is made.
What should you do?

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What a great puzzle!!!

First, the outcomes:

If I pick envelope A: I either A1) win nothing and lose nothing (if the computer predicted correctly) or A2) I win either \$10 or \$10,000

If I pick envelope B: I either B1) lose \$1000 (if the computer predicted correctly) or B2) I win \$1,000

If I pick both envelopes: I either AB1) lose \$1010 (if the computer predicted correctly) or AB2) I win either \$1,000 or \$11,000

From the above, the safest bet for me is to pick A as I can't lose, but have a chance to win. However, since my uncle's supercomputer has such a great record of predicting my actions it definitely predicted that and put \$0 in A.

If there is \$0 in A then I should pick either B or A+B. It doesn't matter which - either way I should win \$1000. Of course, the supercomputer predicted this next step in my reasoning too, so it didn't put \$0 in A. This cyclical reasoning gets us nowhere fast, so let's try a different approach...

Let's say I decide to roll a die and depending on the outcome choose A, B or A+B with the equal probability. The computer could predict that move but couldn't predict the outcome of the roll, so it should use the expected winnings to make its decision.

If A contains \$0, then my expected winnings would be \$666.67,

if A contains \$10, then my expected winnings would be \$0.

If A contains \$10,000, then my expected winnings would be \$6,666.67

So, based on that expected move, the computer should put \$10 in A. Since I only get to play this game once, I stand to either win \$10, win \$1000 or lose \$1010 with equal probability.

Another approach I could consider is to decide whether I want to maximize my winnings with some chance of losing or minimize my losses with some chance of winning. In the first case, I should pick A+B. In the latter case, I should pick A.

Conclusion:

Ultimately, I think, it boils down to computer knowing my behavior and predicting my thinking patterns. Whichever of above methods I pick the computer must have predicted that and will make sure that my chances of winning are the lowest. So, if the computer cannot be fooled then the only safe bet is to pick envelope A and call it even.

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select both envelopes.

if the computer is wrong, i get an additional 10000.

if the computer is right, i get 10 additional dollars which i owe my uncle.

either way i get envelope B for 1,000.

Edited by phil1882
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select both envelopes.

if the computer is wrong, i get an additional 10000.

if the computer is right, i get 10 additional dollars which i owe my uncle.

either way i get envelope B for 1,000.

I think the amount you owe your uncle if the computer predicts correctly is the value of the money in both envelopes, not just envelope A - if I understood the problem correctly. Also, even if the computer is wrong, you're not guaranteed an additional 10,000 since it might still be empty.

I would probably only choose envelope A if I assume everything in the problem to be true, since that seems to be the safest way to go... but why would the computer suggest leaving Envelope A empty if it thinks I'll pick Envelope A? Perhaps I've misunderstood the problem.

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select both envelopes.

if the computer is wrong, i get an additional 10000.

if the computer is right, i get 10 additional dollars which i owe my uncle.

either way i get envelope B for 1,000.

I think the amount you owe your uncle if the computer predicts correctly is the value of the money in both envelopes, not just envelope A - if I understood the problem correctly. Also, even if the computer is wrong, you're not guaranteed an additional 10,000 since it might still be empty.

I would probably only choose envelope A if I assume everything in the problem to be true, since that seems to be the safest way to go... but why would the computer suggest leaving Envelope A empty if it thinks I'll pick Envelope A? Perhaps I've misunderstood the problem.

The computer was designed for/by the uncle to work in his benefit not yours, so it would do what it can to ensure your uncle makes his money.

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select both envelopes.

if the computer is wrong, i get an additional 10000.

if the computer is right, i get 10 additional dollars which i owe my uncle.

either way i get envelope B for 1,000.

if the computer is right that you would pick both, you pay your uncle 1010 not 10.

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select both envelopes.

if the computer is wrong, i get an additional 10000.

if the computer is right, i get 10 additional dollars which i owe my uncle.

either way i get envelope B for 1,000.

I think the amount you owe your uncle if the computer predicts correctly is the value of the money in both envelopes, not just envelope A - if I understood the problem correctly. Also, even if the computer is wrong, you're not guaranteed an additional 10,000 since it might still be empty.

I would probably only choose envelope A if I assume everything in the problem to be true, since that seems to be the safest way to go... but why would the computer suggest leaving Envelope A empty if it thinks I'll pick Envelope A? Perhaps I've misunderstood the problem.

The computer was designed for/by the uncle to work in his benefit not yours, so it would do what it can to ensure your uncle makes his money.

If the computer is confident in its prediction, wouldn't it always put 10,000 in Envelope A (unless it thinks I won't pick Envelope A at all)?

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Computer's are not confident, they either compute an answer or they don't. It would play the probabilities of the human selection.

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well i personally see no good resolution to the puzzle. either way you have a good chance of ending with 0.

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okay how about this, instead of making a selection myself i simply let chance decide. for example i flip a coin or roll a dice.

it would be pretty hard for a computer to guess the outcome of such a result. though i suppose it would also be hard for a computer to guess accurately my decisions over a 5 year period. i don't see a better answer though.

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okay how about this, instead of making a selection myself i simply let chance decide. for example i flip a coin or roll a dice.

it would be pretty hard for a computer to guess the outcome of such a result. though i suppose it would also be hard for a computer to guess accurately my decisions over a 5 year period. i don't see a better answer though.

Hey, that's kinda clever. Props for thinking outside of the box. Still pretty risky though. Remember, the odds don't look good and you're looking to lose a lot of money to your uncle.

Since BMAD seems to be adamant that there is nothing wrong with the OP... For some reason, it seems that if the computer correctly predicts that I will pick only Envelope A, it will contain no money. Thus, the worst case for choosing only Envelope A is that I earn no money. If the computer makes an incorrect prediction, the envelope

will contain money, so there's also a chance that I will make a 10 or 10,000 bucks. Choosing either of the other options puts me at the risk of losing a great deal of cash. So I choose Envelope A. It's a no brainer.
Edited by gavinksong
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Clarification needed on owing the money:

If for example, I choose B and the computer also selects B, what happens?

a) I get \$1000 from the envelop and I return those \$1000

b) I get nothing but still owe \$1000 to my uncle?

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Clarification needed on owing the money:

If for example, I choose B and the computer also selects B, what happens?

a) I get \$1000 from the envelop and I return those \$1000

b) I get nothing but still owe \$1000 to my uncle?

if the computer correctly predicts your picking style, then whatever money is disclosed in the envelopes is payment to your uncle. If the computer incorrectly guesses your picking style then whatever is disclosed in the envelopes is your payment. So option B

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Clarification needed on owing the money:

If for example, I choose B and the computer also selects B, what happens?

a) I get \$1000 from the envelop and I return those \$1000

b) I get nothing but still owe \$1000 to my uncle?

if the computer correctly predicts your picking style, then whatever money is disclosed in the envelopes is payment to your uncle. If the computer incorrectly guesses your picking style then whatever is disclosed in the envelopes is your payment. So option B

So, the amount won or lost is determined by the contents of both envelopes in all cases? I was assuming that only the chosen envelope determines the amount.

since the amount you win or lose is the same once the computer decided on the contents of the envelope A, the best approach is to roll the die and let the luck decide the outcome. You have 2/3 chance of winning.

Edited by k-man
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Clarification needed on owing the money:

If for example, I choose B and the computer also selects B, what happens?

a) I get \$1000 from the envelop and I return those \$1000

b) I get nothing but still owe \$1000 to my uncle?

if the computer correctly predicts your picking style, then whatever money is disclosed in the envelopes is payment to your uncle. If the computer incorrectly guesses your picking style then whatever is disclosed in the envelopes is your payment. So option B

So, the amount won or lost is determined by the contents of both envelopes in all cases? I was assuming that only the chosen envelope determines the amount.

since the amount you win or lose is the same once the computer decided on the contents of the envelope A, the best approach is to roll the die and let the luck decide the outcome. You have 2/3 chance of winning.

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