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This month's IEEE GlobalSpec newsletter challenge is:
When played separately, the probability of losing each of two different coin-toss games is higher than the probability of winning the games. Wily Will, however, claims to have concocted a scheme in which he’ll play each of the pair of games in an alternating sequence to beat the odds. Is it possible Wily Will can come out ahead by combining two losing games?
And the answer is:
Yes, Wily Will can come out ahead by playing each of the two losing games in an alternating sequence.
He can achieve this triumph because he has mastered Parrondo’s paradox. This apparent paradox states that two losing games can be turned into a winning game by combining the two.
Consider the following two coin toss games. In game 1, a weighted coin is tossed with a probability of winning of 49.5%. The player starts with $100. He gains $1 for every winning toss and loses $1 for every losing toss. Over many rounds, the player will lose all of his money since the chance of losing a round is higher than the probability of winning.
In game 2, one of two weighted coins is tossed, based on the player’s capital. If the player’s capital is a multiple of 3, a weighted coin with a probability of winning of less than 9.5% is tossed. Otherwise, a weighted coin with a probability of winning of 74.5% is tossed. Like game 1, the player starts with $100, gains $1 for every winning toss and loses $1 for every losing toss. And like game 1, the player will lose all of his money over many rounds.
When played independently, the player will lose money in either of the games. But Wily Will plans to apply Parrondo’s paradox to come out ahead by playing the games one after the other in a sequence. He’ll play two rounds of game 1 followed by two round of game 2 in a continuous cycle. This approach establishes a link between the two games so that the probability of winning game 2 is no longer independent of the probability of winning game 1. The compound game establishes a dependence of game 2 on the result of game 1 so that the player can enter game 2 in a favorable state that makes him more likely to win.
Further Reading
Harmer, G. P.; Abbott, D. Parrondo's paradox. Statist. Sci. 14 (1999), no. 2, 206--213. doi:10.1214/ss/1009212247. https://projecteuclid.org/euclid.ss/1009212247
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Re: Onerous Odds: Newsletter Challenge (February 2019)
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