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# New Switching Problem

## Question

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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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

The strategy with the highest expected value for money earned is to always switch, except when the envelope amount is the highest possible- 330.

If Charles is interested in beating Bob's earning most of the time, then he should switch envelope *only if* his envelope amount is 1. Most of the time Charles will earn a few more bucks than Bob. However, Bob's expected winning value is still higher because in the rare case where Bob beats Charles, his earnings will be astronomically larger than Charles'.

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

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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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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

The strategy with the highest expected value for money earned is to always switch, except when the envelope amount is the highest possible- 330.

If Charles is interested in beating Bob's earning most of the time, then he should switch envelope *only if* his envelope amount is 1. Most of the time Charles will earn a few more bucks than Bob. However, Bob's expected winning value is still higher because in the rare case where Bob beats Charles, his earnings will be astronomically larger than Charles'.

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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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

Let strategy A be the case where Charles switches only if his envelope has 1. Let's call the strategy where Charles switches whenever he gets 1 or 329 strategy B. There's a tradeoff in expected winning and the number of times that Charles beat Bob's earnings. Strategy B has higher expected winning than strategy A's, at the cost that Charles will beat Bob's earning less often.

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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

Let strategy A be the case where Charles switches only if his envelope has 1. Let's call the strategy where Charles switches whenever he gets 1 or 329 strategy B. There's a tradeoff in expected winning and the number of times that Charles beat Bob's earnings. Strategy B has higher expected winning than strategy A's, at the cost that Charles will beat Bob's earning less often.

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

By switching on 1 and 3^29, Charles will still usually beat Bob, but instead of running into trouble when the envelopes are 3^29-3^30, he lose to Bob when the envelopes are 3^28-3^29. Charles will sometimes trade down to 3^28 from 3^29, but he'll never trade up like Bob would sometimes.

So, Charles can fix his new problem by switching on 1, 3^28 and 2^29. But now he'll do poorly when the envelopes are 2^27 and 3^28. The logical fixes can be brought all the way down to switching all envelopes except 3^30 and 3. But, of course, Charles now does poorly when the envelopes are 3 and 9, so the final "fix" will be switching all envelopes except 3^30, which is exactly Bob's strategy, and turns out to be the optimal strategy.

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