On Parrondo's Paradox

- Optimal Adaptive Strategies for Games of the Parrondo Type -

(c) March 2002 by Sven Rahmann

Since the publication of an article in Nature, the so-called Parrondo Paradox has generated some attention, even in non-scientific journals such as the New York Times. The paradoxon has been summarized as ``Losing strategies can win'', but this simplified statement seems to be a source of confusion.

Consider the following situation: An investor has the choice between two funds A and B. Both funds decrease in value in the long-term average, but there are some intermediate periods where each fund increases its value. Assume now that fund A is likely to make a profit in a high-interest market, while for fund B this is more likely in a low-interest market. It should not come as a surprise that one can make money by alternating between fund A and fund B, according to market conditions. Maybe a little more surprising is the fact that (depending on the conditions of the "games" involved), even by randomly choosing A or B in each step, a profit can be made.

This is the essence of the so-called ``paradox'': The right combination of two losing strategies can be a winning strategy. The key is to know the "rules" or conditions of the games exactly so one can decide which game to play in which situation. This is also the reason why one cannot gain momeny by applying Parrondo's paradox to the stock market. It is well known that by only buying or only selling one cannot make money. In fact, the strategy ``buy low, sell high'' has been known for centuries, but obviously is hard to follow in the real world.

Nevertheless, when we do know the precise rules of the games involved, we are faced with the problem to find an optimal strategy when a sequence of N games is to be played. In the technical report that is available for download here, we introduce a formalism for Parrondo games, state a number of variations of the optimization problem, and present an efficient solution for so-called adaptive strategies. The report is accompanied by a collection of MATLAB functions that can be used for experiments.

I was motivated to investigate optimal strategies for Parrondo games by a talk by Prof. Dr. Behrends from the Free University of Berlin at the German Open Conference on Probability and Statistics 2002 in Mageburg.