The paradox of parrondo games

A winning expectation is produced, even though both games A and B are losing when we play them individually. We develop an explanation of the phenomenon in terms of a Brownian ratchet model, and also develop a mathematical analysis using discrete-time Markov chains. Keywords: gambling paradox; Brownian ratchet; discrete-time M arkov chains 1. Introduction The study of probability dates back to the 17th century. It arises from games of chance, originating from the ancient game of throwing bones the forerunners of dice.

This dates back to correspondence between Pascal and Fermat in , when a problem was posed to Pascal by a French gambler. The erratic Brownian motion of dust particles or pollen grains in a liquid, due to collisions with the liquid molecules, is the classic example Hughes An analysis is then presented to explain the simulations using discrete-time Markov chains.

Harmer, D. Abbott and P. Taylor T2 T1 Figure 1. The ratchet and pawl machine. There are two boxes with a vane in one and a wheel that can only turn one way, a ratchet and pawl, in the other. Each box is in a thermal bath of gas molecules. The two boxes are connected mechanically by a thermally insulated axle.

The whole device is considered to be reduced to microscopic size so that gas molecules can randomly bombard the vane, to produce motion. This device was then later revisited by Feynman et al. In equilibrium, his equations of detailed balance made incor- rect use of energy probabilities rather than using crossing rate probabilities Abbott et al.

In both cases, net motion is strictly a non-equilibrium phenomenon. The focus of recent research is to harness Brownian motion and convert it to directed motion, or more generally, a Brownian motor, without the use of macroscopic forces or gradients. This shows how the mechanism of the ratchet potential works.

Recently, with the technology available to build micrometre-scale structures, many man-made Brownian ratchet devices have been constructed, and actually work Astumian ; Bier a.

There are several mechanisms by which directed Brownian motion can be achiev- ed Faucheux et al. The asymmetry of the potential Uon is determined by , where 0 1. In equilibrium, if the potential height is larger than the thermal noise, the particles are localized in a potential minima.

When the Uon is applied, the particles are trapped in the minima of the potential so the concentration of the particles is peaked. When Uon is switched on again there is a probability Pfwd , proportional to the darker shaded area of the curve, that some particles are to the right of L.

These particles move forwards to the minima located at L. Similarly, there is a probability Pb ck lightly shaded that some particles are to the left of 1 L, and move to the left minima located at L.

If the present capital is a multiple of M , then the chance of winning is p1 ; if it is not a multiple of M , the chance of winning is p2. It can be described mathematically by 1. We refer to capital and gain as if anyone playing these games is against a common opponent, the bank for example.

We will digress for a moment to discuss what constitutes a fair game. If the starting capital is a multiple of three, then we will lose a little, and vice versa. Before then we shall be a little looser with this terminology. We shall consider a game to be winning, losing or fair according to whether the probability of moving up n states is greater than, less than, or equal to the probability of moving down n states as n becomes large.

Using the above criterion, both game A and game B are fair when " is set to zero. This is true of game A because the probabilities of moving up and down n states are equal for all n. Consider the scenario if we start switching between the two losing games, play two games of A, two games Proc.

The values of a and b are shown by the vectors [a; b]. That is, the inset shows the outcome after the hundredth game is played. The result, which is quite counter-intuitive, is that we start winning. He is regularly invited to present plenary papers at international conferences. He has also acted on organising and program committees for many conferences. In , Peter became one of the five trustees of the Applied Probability Trust. By Catherine.

This is an example of one of these simulations:. Not at all. This paradox was discovered in the last nineties by the Spanish physicist Juan Parrondo and can help to explain, among other things, why investing in losing shares can result in obtaining big profits.

Profits are the result of humans gaming a system which abides by fungible rules; fungible by these same humans. Game B only tosses the coin 2 when profit is multiple of three. Otherwise, tosses coin 3 which is extremely biased to tail.