In a game of chess, pieces can sometimes be sacrificed in order to win the overall game. Similarly, engineers know that two unstable systems, if combined in the right way, can paradoxically become stable. But can two losing gambling games be set up such that, when they are played one after the other, they becoming winning? The answer is yes. This is a striking new result in game theory called Parrondo's paradox, after its discoverer, Juan Parrondo1,2. Here we model this behaviour as a flashing ratchet3, in which winning results if play alternates randomly between two games.
There are actually many ways to construct such gambling scenarios, the simplest of which uses three biased coins (Fig. 1a). Game A consists of tossing a biased coin (coin 1) that has a probability (p1) of winning of less than half, so it is a losing game. Let p1=1/2−ε, where ε, the bias, can be any small number, say 0.005.
Game B (Fig. 1a) consists of playing with two biased coins. The rule is that we play coin 2 if our capital is a multiple of an integer M and play coin 3 if it is not. The value of M is not important, but for simplicity let us say that M=3. This means that, on average, coin 3 would be played a little more often than coin 2. If we assign a poor probability of winning to coin 2, such as p2=1/10−ε, then this would outweigh the better coin 3 with p3=3/4−ε, making game B a losing game overall.
Thus both A and B are losing games, as can be seen in Fig. 1b, where the two lower lines indicate declining capital. If we play two games of A followed by two of B and so on, this periodic switching results in the upper line in Fig. 1b, showing a rapid increase in capital — this is Parrondo's paradox. What is even more remarkable is that when games A and B are played randomly, with no order in the sequence, this still produces a winning expectation (Fig. 1b).
This phenomenon was recently proved mathematically1 for a generalized M and analysed in terms of entropy based on Shannon's information theory3. We used the flashing brownian ratchet4 to explain the game by analogy. The flashing ratchet can be visualized as an uphill slope that switches back and forth between a linear and a sawtooth-shaped profile. Brownian particles on a flat or sawtooth slope always drift downwards, as expected. However, if we flash between the flat and saw-tooth slope, the particles are ‘massaged’ uphill. This is only possible if the sawtooth shape is asymmetrical in a way that favours particles spilling over a higher tooth.
The flat slope is like game A, where the bias ε is like the steepness of the slope. Game B is like the sawtooth slope, where the difference between coin 2 and coin 3 is like the asymmetry in the tooth shape. In the brownian ratchet case, there are two types of slope, with falling particles, but when they are switched the particles go uphill. Similarly, two of Parrondo's games have declining capital that increases if the games are switched or alternated. The games can be thought of as being a discrete ratchet and are known collectively as a parrondian ratchet.
Game theory is linked to various disciplines such as economics and social dynamics, so the development of parrondian-like strategies may be useful, for example for modelling cases in which declining birth and death processes combine in a beneficial way.
Harmer, G. P., Abbott, D., Taylor, P. G. & Parrondo, J. M. R. in Proc. 2nd Int. Conf. Unsolved Problems of Noise and Fluctuations 11–15 July, Adelaide (eds Abbott, D. & Kiss, L. B.)(American Institute of Physics, in the press).
McClintock, P. V. E. Nature 401, 23–25 (1999).
Harmer, G. P., Abbott, D., Taylor, P. G., Pearce, C. E. M. & Parrondo, J. M. R. in Proc. Stochastic and Chaotic Dynamics in the Lakes 16–20 August, Ambleside, UK (ed. McClintock, P. V. E.) (American Institute of Physics, in the press).
Doering, C. R. Nuovo Cimento D 17, 685–697 (1995).
Rousselet, J., Salome, L., Ajdarai, A. & Prost, J. Nature 370, 446–448 (1994).
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Harmer, G., Abbott, D. Losing strategies can win by Parrondo's paradox. Nature 402, 864 (1999). https://doi.org/10.1038/47220
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