Parrondo's Paradoxal Games

A paradox?

The physicist Juan M.R. Parrondo is the inventor of the paradox of the same name. One will find an English talk on his personal page. This is a relatively complex game presented as a succession of throws of not balanced coins. It is the combination of play A, simple throw of a coin #1, and play B where one lance either the coin #2 or the coin #3:

Fig1. Games A and B

The two games A and B

A is a throw of a coin where Head is winnning with the probability p₁=1/2-e, the gain is then of 1 euro. Tail is a loss (of 1 euro) with the probability 1-p₁=1/2+e .
B is a little more complicated, if the capital is a multiple of 3, then Head wins with the probability p3= 1/10-e, if not Head wins with the probability p₂=3/4-e, (gain or loss of 1 euro).

Wen e = 0, the play A, alone, is fair. The play B become fair when the n of plays tends to infinity. (Click on 'calculate A' or on 'calculate B').
A and B, alone, are lose when e > 0.

Click on 'simule A' or on 'simule B' to begin simulations.
The almost exact values (the rounding errors are often inevitable), are obtained then by clicking the button [ Calculates ]

Combinations of the two plays

Fig. 2 Average profits in B+, (AB)+, (AAB)+ (e=0)

When one uses combinations repeated like (AABB)+ or (AAABBAB)+, one observes that the game becomes gaining for certain these combinations, which can seem against-intuitive!

Obviously the play is paradoxical only seemingly, the results observed are calculated easily and the 'paradox' is explained by the no-commutativity of the product of certain matrix (of transition).
Who would think of finding paradoxical that a matrix product M×N is different from N×M?

One can conceive easily besides other plays, simpler, having the same type of behavior [EZ].

Simulations and calculations

You can modify certain parameters and calculate the average profit while carrying out with the choice:

a certain number of simulations of throws of coins
calculation by means of the matrices of transition.

Parrondo's games

Examples

Which is the most advantageous word?

Simulations :   AAAAA, AAAAB, AAABB, AABAB, AABBB, ABABB, ABBBB, BBBBB

Simulations :   AAAAAA, AAAAAB, AAAABB, AAABAB, AABAAB, AAABBB, AABBAB, ABBAAB, AABBBB, ABABBB, ABBABB, ABBBBB BBBBBB

Reiterated Fibonacci's words, (calculus) :   BA, BAB, BABBA, BABBABAB, BABBABABBABBA, BABBABABBABBABABBABAB BABBABABBABBABABBABABBABBABABBABBA

A et B chosen randomly

Simulations are carried out by choosing A randomly with the probability p indicated in x-coordinate. The probabilities are p₁=1/2-e, p₂=3/4-e, p₃=1/10-e. The two cases correspond to e=0.001 and e=0.005.
Each point indicated on the image required 10⁷ randomly choose numbers. (Fig. 3)

The optimal value of p seems close to the value p=0.4145 calculated by Doron Zeilberger and indicated in its paper, (with e=0.001 it seems).
C program

Fig. 3 Simulations

Words of Fibonacci (not repeated).

The figure (Fig 4.) indicates the average profits corresponding to words of Fibonacci of increasing sizes, from M₆=BABBABABBABBA to M₃₀ when e=0.
The length of the word M₃₀ is 1346269 and the average profit is 0.070423168.

These profits seem to remain lower than those obtained by concaténant several words of Fibonacci M₅=BABBA. One arrives in this case, always for e=0, with average profits from approximately 0.07567.

Fig 4.Fibonacci's words

References, resources, links

Page personnelle de Juan M.R. Parrondo

Brownian Ratchets, Parrondo's Games and more... Greg Harmer - Two games with unusual properties were originally devised by Prof. Juan MR Parrondo as a pedagogical example of a Brownian ratchet.

Parrondo's Paradox Dan Vellerman and Stan Wagon. Two losing games can combine into a winning game. wagon93.nb (317.8 KB) - Mathematica Notebook

Etats d'A M E - Le paradoxe de Parrondo Journal de l'Association des Mathématiciens de l'École Polytechnique Fédérale de Lausanne - Numéro 9 - Oct. 2002

Hasard mathématique et chaos biologique Hervé Ratel - Qui perd gagne - Sciences & Avenir Avril 2000 -- N° 638. Tentez votre chance à un jeu de hasard. Le plus souvent, vous perdez. Jouez à deux jeux de hasard, alternativement et de façon aléatoire : surprise, vous gagnez ! Ce paradoxe éclaire les mécanismes apparemment chaotiques, et pourtant bien huilés, des cellules ou des protéines.
Le paradoxe sert à expliquer, comment le caractère chaotique du mouvement brownien dans les cellules peut promouvoir l'évolution. Jusqu'à présent on pensait toujours que ce désordre empêchait toute amélioration des structures.

Parrondo Paradox cut-the-knot.com Alexander Bogomolny

Brownian Motor Roland Ketzmerick, Matthias Weiß, Franz-Josef Elmer, Roland Ketzmerick, Franz-Josef Elmer, Franz-Josef Elmer

On-line Simulator for Parrondo's Paradox Lee Spector (lisp + cgi program ?)

[EZ] Remarks On the PARRONDO PARADOX By Shalosh B. EKHAD and Doron Zeilberger

Parrondo's Paradox Eric Weisstein (MathWorld)

Winning With Losing Games   By John Allen Paulos Special to ABCNEWS.com. A New Paradox in the World of Probability
The Paradox of Parrondo's Games   Peter Taylor
On Parrondo's Paradox   - Optimal Adaptive Strategies for Games of the Parrondo Type - by Sven Rahmann (technical report, MATLAB functions, documentation)