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