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 just yet. The theory does not apply to casino games. Learning about all this should be its own reward. On the bright but shaky side, Sandra Blakeslee reported last year in the NY Times that Dr. Sergei Maslov of the Brookhaven National Laboratory had shown that if an investor simultaneously shared capital between two losing stock portfolios, capital would increase rather than decrease. diminish. (On the downside, at the time of this 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, monetary rewards aside, 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 article mentioned above, Sandra Blakeslee cites 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 initially denied having a sexual affair with Monica S. Lewinsky, saw his popularity increase when he admitted that he had lied. The additional scandal created more profit for Mr. Clinton.”
As everyone knows, Clinton had less luck in her pursuit of 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 point out that the order of intermingled activities may be of real 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 publicly first (eg, during election campaigns) and then having an extramarital 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 Nature (vol. 402, Dec 23/30, 1999, p. 864). The magazine charges an outrageous $7 for a short 1-page communication that is readily available (along with many other documents) on Greg’s site.
Of the two lost games, A and B, the first is simple, the other is complicated. In the simple game A, one wins or loses $1 with odds p Y 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 p1in 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, all three games A, B1, and B2 cannot be losing. A typical assignment of probabilities would be p = .495, p1 = .095, and p2 = .745, which makes B2 a winning game. For M = 2 or 3, B still loses, although he 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 p – epsilon, p1 – epsilonY p2 – epsilonwhere epsilon is a small number around .005, but in fact it might as well be zero).
Sets A and B can be combined in many different ways. They can be randomly combined with a prescribed probability of selection, say A. Or, their selection can follow a periodic pattern, such as AABB, which means deterministically playing two A games, followed by two B games, followed by two A, 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. Games A and B occupy two of the colors). Simply type the strings of A and B or real numbers (for probabilities) separated by spaces in the edit control at the bottom of the applet. Each trial consists of a specific number of games (100, originally), and you can also specify the number of trials (500, originally).
It was found that the ABBAB period is by far the best strategy for M = 3, while AB is insurmountable for M = 2 and M = 4. This agrees with the results of Shalosh B. Ekhad and D. Zeilberger. The former carries a pack of PARRONDO Maple, which, among other things, helps 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 0.4145. However, even in that case, the random selection lags behind the ABBAB periodic strategy by a factor of about 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.)
