Parrondo’s Paradox – how losing strategies can win

It is hard to believe but if you combine two losing games then, in some situations, you can turn them into a winning game. This is known as Parrondo’s paradox, named after Juan Parrondo (pictured), a Spanish physicist, who discovered the paradox in 1996.

If you have a coin which has a small bias to heads and you keep placing an even money bet that it will come up tails then eventually you will lose your money. You are playing a losing game, the odds are against you and you will lose in the long run. This will be our first game, a simple coin toss with a biased coin (coin 1)

The trick in the paradox is that the second game must involve a choice which depends on another factor. In the coin tossing example the second game might work as follows:

A player has a certain amount of money. At each turn they choose one of two coins to toss (coins 2 & 3), which coin they choose depends on the amount of money that they have at the start of that turn.

If the amount of money is a multiple of, say, 3 then the player chooses coin 2 to toss, if the amount of money in not a multiple of 3 then they choose coin 3.

Coins 2 & 3 have different likelihoods of coming up heads or tails.

The second game can be constructed as a losing game. As an example, if coin 3 is quite likely to come up tails, so that you win; and coin 2 is very likely to come up heads, so you lose, then the combination of the two can be made to be a losing game as the big losses on coin 2 are not offset by the smaller wins on coin 3.

By combining this with the first game, by for example playing the first game twice, then the second game twice, then the first game twice etc. It is possible to make the combined game a winning one.

This can work because the combination will mean that the winning coin 3 is used more often in the combined game than it is in the separate game, so the combined game becomes a winning one where the two separate games are losing ones.

This is the paradox, two losing games are combined into one winning game.

With the coin tossing example it is easy to see how this is possible if you consider it to be the combination of three games, two which are losing (coins 1 & 2) and one that is winning (coin 3). As long as you get to play coin 3 enough times you will win overall.

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