Winning from Two Losing Games
In the article, it discusses the newly-discovered (15 years ago) paradox discovered by Parrondo that by carefully choosing a strategy, it is able to in the long run win money with two games that, played independently, always lose money in the long run.
First, consider two games, A and B. In game A, there is a tails-biased coin where if heads, you win $1 and if tails you lose $1. In game B, there are two coins. If your money is divisible by three, you flip the first coin and if not, flip the second coin. The first coin is slightly biased at winning while the second coin is heavily biased in losing. For both games, if you play long enough, you will lose all your money.
Now, rather than play each game independently, alternate between playing the two games (such as AAB). Surprisingly, according to simulations done by Parrondo, one will actually accumulate money than lose money, even though both games are biased toward losing money. The reason why money can accumulate is because game B depends on the person’s current money. One way to think about it is although it’s more likely to lose money from game A, it makes the winning coin occur more frequently and rebound any time the amount of money goes down. Parrondo relates the effect to a ratchet, in the way that the earnings are prevented into going in only one direction like turning a ratchet.
Looking at the games, it seemed easy enough to implement, so I wrote a MATLAB program that tests whether the paradox is true or not. After a little bit of coding, I tested it out and, sure enough, the paradox stands. I found out that not any probabilities works, but it worked with the first coin as .495 at winning and .745 for the win-biased coin and .095 for the lose-biased coin. (The results for one trial were for 1 million games, game A lost 9850, game B lost 8560 and doing the sequence BAA gained 8385).
This paradox shows that games are not always intuitive. Just like the prisoner’s dilemma where the Nash equilibrium is where both people lose when they have the possibility of both having better outcomes, Parrondo’s Paradox shows the complexity in games. Just as in the games in class, they correspond to real world applications. For example, Parrondo’s is currently being analyzed to links with falling stocks and the formation of animo acids that normally break apart but may stay together due to ratchets.
Here is another good simulation link: http://www.maa.org/editorial/knot/parrondo.html