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# New Switching Problem

Best Answer bushindo, 05 May 2013 - 01:40 AM

Alec proposes to Bob and Charles that they will play a game over a 25 year period. Each day while Bob and Charles are away from Alec's house, Alec will prepare two envelopes. Alec will flip a fair coin until the coin comes up tails OR until he has flipped his coin 30 times. Based on the number of flips he makes each day, he will prepare the two envelops as follows:

If the first flip is tails, he will prepare one envelope with \$1 in it and the other with \$3 in it. If he flips twice, the two envelopes will contain \$3 and \$9. Three flips will produce envelopes with \$9 and \$27 and so on. If he somehow flips 30 times, the larger envelope will contain \$2.0589113e+14 and the other envelope will contain \$6.8630377e+13.

Both Bob and Charles know Alec's procedure and the exact distribution of the possible envelope values.

Once Alec has set the two envelopes, he will randomly place one envelope on the left side of a table and the other on the right side. Both Bob and Charles come to the table, Bob is given the right envelope and Charles the left envelope. They are allowed to privately examine the contents of their envelope. They are then given the chance, privately, to switch. They get to keep the contents of the envelope they end up with each day. If one switches and other doesn't, they will end up with the same envelope that particular day.

Question #1: Bob wants to maximize his expected value over the course of the entire game. What strategy should he use?

Question #2: Charles is less interested in maximizing his EV. He's motivated to end up winning more money than Bob over the course of the 25 years. If he knows Bob's "perfect" strategy, what strategy can Charles use to maximize his chances of ending up ahead of Bob after the 25 years?

If Charles maximizes his chances of beating Bob, he will end up ahead of Bob nearly 100% of the time.

If each player will get the full amount if they happen to choose the same envelope, then

Spoiler for

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8 replies to this topic

### #1 bubbled

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Posted 03 May 2013 - 11:31 AM

Alec proposes to Bob and Charles that they will play a game over a 25 year period. Each day while Bob and Charles are away from Alec's house, Alec will prepare two envelopes. Alec will flip a fair coin until the coin comes up tails OR until he has flipped his coin 30 times. Based on the number of flips he makes each day, he will prepare the two envelops as follows:

If the first flip is tails, he will prepare one envelope with \$1 in it and the other with \$3 in it. If he flips twice, the two envelopes will contain \$3 and \$9. Three flips will produce envelopes with \$9 and \$27 and so on. If he somehow flips 30 times, the larger envelope will contain \$2.0589113e+14 and the other envelope will contain \$6.8630377e+13.

Both Bob and Charles know Alec's procedure and the exact distribution of the possible envelope values.

Once Alec has set the two envelopes, he will randomly place one envelope on the left side of a table and the other on the right side. Both Bob and Charles come to the table, Bob is given the right envelope and Charles the left envelope. They are allowed to privately examine the contents of their envelope. They are then given the chance, privately, to switch. They get to keep the contents of the envelope they end up with each day. If one switches and other doesn't, they will end up with the same envelope that particular day.

Question #1: Bob wants to maximize his expected value over the course of the entire game. What strategy should he use?

Question #2: Charles is less interested in maximizing his EV. He's motivated to end up winning more money than Bob over the course of the 25 years. If he knows Bob's "perfect" strategy, what strategy can Charles use to maximize his chances of ending up ahead of Bob after the 25 years?

If Charles maximizes his chances of beating Bob, he will end up ahead of Bob nearly 100% of the time.

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### #2 bonanova

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Posted 03 May 2013 - 06:31 PM

Interesting puzzle.

Clarify what happens when one switches and the other does not.
What doe ending up with the same envelope mean: which one? Do they split the contents?
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### #3 bubbled

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Posted 03 May 2013 - 11:50 PM

Interesting puzzle.

Clarify what happens when one switches and the other does not.
What doe ending up with the same envelope mean: which one? Do they split the contents?

Both players will win the full value of the same envelope if they end up with it.

Edited by bubbled, 03 May 2013 - 11:50 PM.

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### #4 bgm1961

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Posted 04 May 2013 - 07:32 PM

While Bob and Charles are maximizing their strategy against each other, Alec is getting screwed!

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### #5 bubbled

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Posted 05 May 2013 - 01:04 AM

While Bob and Charles are maximizing their strategy against each other, Alec is getting screwed!

Yes, indeed! But let's assume that Alec has all the money in the world and he's just interested in seeing what strategies are best. I think I know what the two "correct" strategies are given Bob's and Charles's different motivations. I've also run some large simulations, and the results are very interesting. I'll post answers tomorrow, if no one wants to take a crack at it.

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### #6 bushindo

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Posted 05 May 2013 - 01:40 AM   Best Answer

Alec proposes to Bob and Charles that they will play a game over a 25 year period. Each day while Bob and Charles are away from Alec's house, Alec will prepare two envelopes. Alec will flip a fair coin until the coin comes up tails OR until he has flipped his coin 30 times. Based on the number of flips he makes each day, he will prepare the two envelops as follows:

If the first flip is tails, he will prepare one envelope with \$1 in it and the other with \$3 in it. If he flips twice, the two envelopes will contain \$3 and \$9. Three flips will produce envelopes with \$9 and \$27 and so on. If he somehow flips 30 times, the larger envelope will contain \$2.0589113e+14 and the other envelope will contain \$6.8630377e+13.

Both Bob and Charles know Alec's procedure and the exact distribution of the possible envelope values.

Once Alec has set the two envelopes, he will randomly place one envelope on the left side of a table and the other on the right side. Both Bob and Charles come to the table, Bob is given the right envelope and Charles the left envelope. They are allowed to privately examine the contents of their envelope. They are then given the chance, privately, to switch. They get to keep the contents of the envelope they end up with each day. If one switches and other doesn't, they will end up with the same envelope that particular day.

Question #1: Bob wants to maximize his expected value over the course of the entire game. What strategy should he use?

Question #2: Charles is less interested in maximizing his EV. He's motivated to end up winning more money than Bob over the course of the 25 years. If he knows Bob's "perfect" strategy, what strategy can Charles use to maximize his chances of ending up ahead of Bob after the 25 years?

If Charles maximizes his chances of beating Bob, he will end up ahead of Bob nearly 100% of the time.

If each player will get the full amount if they happen to choose the same envelope, then

Spoiler for

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### #7 bubbled

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Posted 05 May 2013 - 02:15 AM

Alec proposes to Bob and Charles that they will play a game over a 25 year period. Each day while Bob and Charles are away from Alec's house, Alec will prepare two envelopes. Alec will flip a fair coin until the coin comes up tails OR until he has flipped his coin 30 times. Based on the number of flips he makes each day, he will prepare the two envelops as follows:

If the first flip is tails, he will prepare one envelope with \$1 in it and the other with \$3 in it. If he flips twice, the two envelopes will contain \$3 and \$9. Three flips will produce envelopes with \$9 and \$27 and so on. If he somehow flips 30 times, the larger envelope will contain \$2.0589113e+14 and the other envelope will contain \$6.8630377e+13.

Both Bob and Charles know Alec's procedure and the exact distribution of the possible envelope values.

Once Alec has set the two envelopes, he will randomly place one envelope on the left side of a table and the other on the right side. Both Bob and Charles come to the table, Bob is given the right envelope and Charles the left envelope. They are allowed to privately examine the contents of their envelope. They are then given the chance, privately, to switch. They get to keep the contents of the envelope they end up with each day. If one switches and other doesn't, they will end up with the same envelope that particular day.

Question #1: Bob wants to maximize his expected value over the course of the entire game. What strategy should he use?

Question #2: Charles is less interested in maximizing his EV. He's motivated to end up winning more money than Bob over the course of the 25 years. If he knows Bob's "perfect" strategy, what strategy can Charles use to maximize his chances of ending up ahead of Bob after the 25 years?

If Charles maximizes his chances of beating Bob, he will end up ahead of Bob nearly 100% of the time.

If each player will get the full amount if they happen to choose the same envelope, then

Spoiler for

Very well done, Bushindo. So, I ran a simulation in Python (I can post my code if anyone's interested). Here are the results that support Bushindo's conclusions:

Each game is played 9,131 days (25 years with 4 leap days) and I ran 100,000 games. I wanted to play enough games to have a good chance of having Bob win a game:

Bob won 1 game

Charles won 99,999 games

Bob won a total of \$570,809,817,520,192 for an average of \$625,133.96 per day!

Charles "only" won a total of \$433,549,519,222,236 for an average of \$474,810.56 per day.

But, Charles got his wish, he got to brag that he beat Bob 99,999 times, while they enjoyed their private island sipping margaritas.

Here's a follow-up question:

Charles is clearly crushing Bob whenever the the envelopes are 1-3, and he's losing all his equity on the very rare case where the envelopes are 3^29-3^30. So, why can't he simply make a small change to his strategy? Keep his exact same strategy, but in the rare case where he sees exactly \$3^29, he also switches. Wouldn't this fix his problem? If not, what's wrong with that strategy?

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### #8 bushindo

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Posted 05 May 2013 - 05:39 AM

Here's a follow-up question:

Charles is clearly crushing Bob whenever the the envelopes are 1-3, and he's losing all his equity on the very rare case where the envelopes are 3^29-3^30. So, why can't he simply make a small change to his strategy? Keep his exact same strategy, but in the rare case where he sees exactly \$3^29, he also switches. Wouldn't this fix his problem? If not, what's wrong with that strategy?

In that case

Spoiler for

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### #9 bubbled

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Posted 05 May 2013 - 11:59 AM

Here's a follow-up question:

Charles is clearly crushing Bob whenever the the envelopes are 1-3, and he's losing all his equity on the very rare case where the envelopes are 3^29-3^30. So, why can't he simply make a small change to his strategy? Keep his exact same strategy, but in the rare case where he sees exactly \$3^29, he also switches. Wouldn't this fix his problem? If not, what's wrong with that strategy?

In that case

Spoiler for

I'd agree with everything you say. Here's my take:

Spoiler for Here's my take

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