1. Parrondo’s games as a discrete ratchet
Pau Amengual
Raúl Toral
Instituto Mediterráneo de Estudios Avanzados - IMEDEA
Universitat de les Illes Balears – UIB
SPAIN

2. Outline Introduction: flashing ratchet. Original Parrondo’s games.
Other classes of games. Cooperative games
Relation between Parrondo’s games and ratchets.
Coupled ratchets and coupled games.
Conclusions

3. 1.Introduction Brownian Motors The system must be out
System subjected
To thermal noise
Transport phenomena in small-scale systems
Two basic features are needed for the existence of directed transport :
The system must be out
of its equilibrium state
Breaking of thermal equilibrium: Accomplished either through
stochastic or periodic forcing : F(t)
Breaking of spatial
inversion symmetry
Ratchet potential : it consists of a periodic and
asymmetric potential

4. Flashing ratchet :
Potential switched on and off periodically or stochastically with a flip rate .
Ratchet Potential On
P(x) x x
Diffusion
P(x) x x
Ratchet Potential On
Net Motion
P(x)
VB(x) x x

5. 2. Original Parrondo’s games
Game A :
Game B (Capital dependent)
YES
Capital multiple of three ? NO
Both games are losing when played separatedly. Either periodic or random alternation between both games gives as a result a winning game .

6. Random case Periodic case Plays game A The player, with probability
Plays game B
Periodic case
The player alternates between game A and B following a given
Sequence of plays.
Average gain of a single player versus time with a value of
The simulations were averaged over ensembles.

8. 3.Other classes of games Parrondo’s games with self-transition
In this class of games a new probability is introduced : self-transition probability. The player has a probability distinct from zero of remaining with the same capital after a round played
p =9/20 - ε, r=1/10, p1=9/100 – ε, r1=1/10, p2=3/5 – ε, r2=1/5 and ε=1/500

9. Winning Probabilities
History dependent games
Game A :
Parrondo et al. PRL 85,
Game B:
Time step t-2
Time step t-1
Winning Probabilities
Losing Probabilities
Loss p1
1-p1
Win p2
1-p2 p3
1-p3 p4
1-p4
Simulations are carried out with ε = and averaging over ensembles. The probabilities are p1=9/10 - ε, p2 = p3 = 1/4 - ε,
p4 = 7/10 - ε

10. Cooperative games Capital redistribution between players
New versions for game A are presented :
Game A’ :
A player chosen randomly gives away one unit of capital to a randomly
selected player
Game A’’ :
A player chosen randomly gives away one unit of capital to ny of its nearest neighbours. Probability proportional to the capital difference.
Games B and B’ :
We will use either original game B, or the history
dependent game B’ with probabilities
p1 = ε p2 = p3 = ε p4 = ε ε = 0.01

11. Average capital per player
Time evolution of the variance of the single player capital distribution

12. Cooperative games Game A : Game B:
Ensemble of interacting players. They chose either game A or game B randomly, i.e., with probability  .
Game A :
Rules of chosing between probabilities for game B depend on the state of the neighbour’s player
Game B:
Player site
i-1
Player site i+1
Winning Probabilities
Losing Probabilities
Loser p1
1-p1
Winner p2
1-p2 p3
1-p3 p4
1-p4
The probabilities used for the simulation are
p = 0.5, p1 = 1, p2 = p3 = 0.16, p4 = 0.7

13. 4. Relation between Parrondo’s games and Brownian ratchet
Additive noise :
We have the following Master Equation for the evolution of the capital i of the player at the  th coin tossed
It can be rewritten as a continuity equation :
where
and

14. of fairness is fulfilled by the set of probabilities {pi,qi}, that is
For the original games we have (ri=0)
The current is then
We can define a potential in the following way :
This potential assures the periodicity of the potential when the condition
of fairness is fulfilled by the set of probabilities {pi,qi}, that is

15. Some examples of potentials :
Potential V(x) obtained from the probabilities
defining game B
Potential V’(x) obtained from the probabilities
of the combined game A and B

16. For the stationary case ( Ji = ctant and Pi() = Pi) we obtain
Plot of the current J vs the alternating probability between games 

17. is the numerical approx. to the integral using Simpson’s rule.
The solutions obtained for Pn and J are equivalent to the continuous solution of the Langevin equation with additive noise As
is the numerical approx. to the integral using Simpson’s rule.

18. Inverse problem
Solving the equation for the potential with the boundary condition gives
Last equation together with
can be used to obtain the
probabilities pi in terms of the potential Vi

19. Multiplicative noise :
Now , which corresponds to the case of non-null self-transition probability.
For this case
and
We can define an effective potential
with the same properties than
the previous one as
For the stationary case we obtain for the probability and the current :

20. These expressions can be compared with the continuous solutions of the Langevin equation with multiplicative noise
where

21. 5. Coupled ratchets and coupled games
Set of N players, we choose a random player and then :
Gives 1 coin to a random player
With probability
Plays game B
Master Equation for the joint PDF is :
Due to the constant transition probabilities, we can obtain the ME governing the evolution of one player performing the following sum

22. The result is
Solving the latter equation for the stationary case gives
With a current

23. Capital redistribution between neigbours
Comparison between the current obtained theoretically and numerically. The probabilities for game B are those of the original game. N = 50 players.
Capital redistribution between neigbours
Although the joint PDF function might be slightly different, the ME we obtain for a single player is the same as in the previous case, and so are the results.
From the discrete solution for the stationary Pn that we have obtained, we can derive its corresponding Fokker-Planck Equation and then the Langevin equation, giving

24. Conclusions
We have presented a consistent way of relating the master equation for the Parrondo games with the formalism of the Fokker–Planck equation describing Brownian ratchets.
This relation works in two ways: we can obtain the physical potential corresponding to a set of probabilities defining a Parrondo game, as well as the current and its stationary probability distribution.
Inversely, we can also obtain the probabilities corresponding to a given
physical potential
Our relations work both in the cases of additive noise or multiplicative noise
Our next goal is to ellaborate a relation between the collective games and their
associated collective ratchets.

25. Bibliography
R. Toral, P. Amengual and S. Mangioni, Parrondo’s games as a discrete ratchet, Physica A 327 (2003).
R. Toral, P. Amengual and S. Mangioni, A Fokker-Planck description for Parrondo’s games, Proc. SPIE Noise in complex systems and stochastic dynamics eds. L. Schimansky-Geier, D. Abbott, A. Neiman and C. Van den Broeck), Santa Fe, 5114 (2003).
P. Amengual, A. Allison, R. Toral and D. Abbott, Discrete-time ratchets, the Fokker-Planck equation and Parrondo’s paradox, Proc. Roy. Soc. London A 460 (2004).
R. Toral, Cooperative Parrondo’s games, Fluctuation and Noise Letters 1 (2001).
R. Toral, Capital redistribution brings wealth by Parrondoís paradox, Fluctuation and Noise Letters 2 (2002).
G. P. Harmer and D. Abbott, Losing strategies can win by Parrondo’s paradox, Nature 402 (1999) 864.
G. P. Harmer and D. Abbott, A review of Parrondo’s paradox, Fluctuation and Noise Letters 2 (2002).