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Parrondo’s paradox

Parrondo’s paradox is a double surprise. Contrary to common intuition, it is possible to mix two losing games into a winning combination. This is good news. But don’t rub your hands yet. The theory does not apply to casino games. Learning it all should be its own reward. On the bright side, but on shaky ground, Sandra Blakeslee reported last year in the NY Times that Brookhaven National Laboratory’s Dr. Sergei Maslov had shown that if an investor simultaneously shared equity between two losing portfolios, equity would rise. instead of decreasing. (On the downside, at the time of writing, it was too early to apply his model to the real stock market due to its complexity.)

Since the paradox was reported a couple of years ago, many abstract and real-world examples have been devised that make it more palatable. In fact, aside from monetary rewards, a combination of negative trends can lead to a positive outcome.

Brooke Buckley, an undergraduate student at Eastern Kentucky University, in her honors thesis that it is a well-known fact in agriculture, “that both sparrows and insects can eat all crops. However, having a combination of sparrows and insects, a healthy crop is harvested. “

In the aforementioned article, Sandra Blakeslee quotes Dr. Derek Abbott of the University of Adelaide, who saw in the public attitude towards the Monica Lewinsky affair a manifestation of a similar phenomenon. “President Clinton, who at first denied having a sexual affair with Monica S. Lewinsky, saw his popularity rise when he admitted that he had lied. The additional scandal created more profits for Mr. Clinton.”

As everyone knows, Clinton was less fortunate in her quest for the Nobel Prize, although in 1993 she helped sponsor the Nobel Peace Prize for Yassar Arafat, an arch-terrorist and pathological liar. The paradox worked for the latter.

In the insightful article by Shalosh B. Ekhad and Doron Zeilberger (then at Temple University), the authors note that the order of intermingled activities can be of great importance. Although they mostly apply their theory to mundane situations, such as walking, driving, and flying, we can use their observation with the aforementioned cases. For example, lying in public first (eg, during election campaigns) and then having an affair did not earn Mr. Clinton any points with the public.

But what is Parrondo’s paradox? Several articles are available on the Web, including the original article by Derek Abbott and Greg Harmer in the journal Nature (vol. 402, 23/30 December 1999, p 864). The magazine charges an outrageous $ 7 for a short 1-page communication that is available anyway (along with many other newspapers) on Greg’s site.

Of the two losing games, A and B, the first is simple, the other is complicated. In simple game A, one wins or loses $ 1 with odds P and 1 p, respectively. Game B is itself a combination of two games, say B1 and B2, both being as simple as Game A. In Game B1 the probability of winning $ 1 is p1, in B2 is p2. In B, game B1 is played if the current capital is a multiple of an integer M> 1, B2 is played otherwise.

The problem here is that, for the paradox to occur, the three games A, B1 and B2 cannot be losing. A typical assignment of probabilities would be P = .495, p1 = .095, and p2 = .745, making B2 a winning game. For M = 2 or 3, B is still a losing game, although it wins for M> 3.

(As in the original article by D. Abbott and G. Harmer, in the applet below, games A, B1, B2 are won with odds PEpsilon, p1Epsilon, and p2Epsilon, where Epsilon it is a small number around .005, but in fact it can also be zero).

Sets A and B can be combined in many different ways. They can be randomly combined with a prescribed probability of selecting, say A. Or, your selection may follow a periodic pattern, such as AABB, which means playing two games A deterministically, followed by two games B, followed by two A’s, and so on. . The applet allows you to define up to 7 combinations (9 is the number of different colors that I clearly recognize as different in my browser. Sets A and B occupy two of the colors). Just type the strings of A and B or real numbers (for probabilities) separated by a space in the edit control at the bottom of the applet. Each test consists of a specific number of games (100, originally), and you can also specify the number of tests (500, originally).

The ABBAB period was found to be by far the best strategy for M = 3, while AB is insurmountable for M = 2 and M = 4. This is in agreement with the results of Shalosh B. Ekhad and D. Zeilberger. The first one carries a Maple PARRONDO package, which, among other things, helps to establish these results quite precisely. For example, for M = 3 and the probabilities defined above, the random strategy is optimized when A is selected with a probability of .4145. However, even then, the random selection follows the ABBAB periodic strategy by a factor of approximately 3.

(The article is adapted from a June 2001 MAA Online column available at http://www.cut-the-knot.org/ctk/Parrondo.shtml and http://www.maa.org/editorial/knot / parrondo .html.)

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