Aplicaciones de laParadoja de Parrondo

Miguel ArizmendiFac. IngenieríaUniversidad Nacional de Mar del PlataArgentina

Losing in order to win

Chess sacrifice bishop

N/2 Wrongs Make a Right

The Truel

Motores Moleculares

Parrondo’sParadoxical Games

Biology is wet and dynamic. Molecules, subcellular organelles and cells,

inmersed in an aqueous environment, are in continous riotous motion.

H.C. Berg – Random Walks in Biology

Games

The small size of molecular machines means that their physics is dominatedby thermal fluctuations – macroscopic intuition is of limited use

Length scales

Energy Scales

Thermal Energy

For proteins in water this energy is taken from collisions with water molecules

Life at Low Reynolds Number

Reynold’s number: Re = vL/

v speed of the object – L characteristic length -

ρ liquid density and η viscosity

Example: fish vs. bacterium

•fish of density approximately that of water ( = 1 gm/cc), length of 10 cm (L), moving at a velocity of 100 cm/sec (v) in

water ( = 0.01 g/cm sec), we calculate Re to be about 105.

•bacterium of the same density, length of 1 micron (L = 10-4 cm), moving at a velocity of 10-3 cm/sec through water,

we calculate Re to be 10-5.

What about Proteins?

• fish we calculate Re to be about 105.

• bacterium we calculate Re to be 10-5.

• protein: size ~ 6 nm, speed 8m/s in water

Re ~ 0.05

Overdamped

Quenched disorder effects on deterministic inertia ratchets

Games

•Processes like this with no memory are called Markov Processes

Random Walks and Diffusion

For short times and distances, diffusion is very fastK+ ion in water goes 1 micron in 0.25 ms, 0.1 mm in 0.25 sFor long time and distances, diffusion is very slow,K+ ion goes 1m in 8 years.

Why bother moving?Rickettsia (tifus)

~ 100 years for mitochondrion synthesized in spinal chord to get to foot synapse. .

Active Transport is necessary: molecular motors

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

Molecular Motor Model

Thermodynamics Second Law?Thermodynamics Second Law?

Molecular Motor Model

Can a Net Current J be obtained from Noise?

Thermal Ratchet Model

Feynman Lectures: Ratchet and Pawl

Maxwell’s Demon

i=1 i=2

Flashing Ratchet Current

- MA, JR Sanchez and F. Family: - MA, JR Sanchez and F. Family: PLAPLA 249,281 (’98)249,281 (’98) Physica A327Physica A327, 111 (2003), 111 (2003)

Flashing Ratchet Current / Entropy

Thermal Ratchet:Not a very good molecular motor

model

• Force against viscous loads ~ 2kT/l ~ 1pN << 4-5pN (measured value)

•Diffuses in the right direction half of the time 2 molecules ATP hydrolized in average/step.• 1 step/ATP hydrolized for kinesin

(J. Howard, Mechanics of Motor Proteins and the Cytoskeleton, Sinauer, 2001)

Highly diffusive, several ATP molecules hydrolyzed/step

What about Games?

win lose

¿Is X(t) a multiple of 3 ?

win lose win lose

Game A Game B

No Yes

: Player’s capital at -th run

(Fair games)

No Yes

Is X(t) a multiple of 3?

Average gain of a single player versus time with a value of The simulations were averaged over 50000 ensembles.

300

1

The player, with probability

)1(

Plays game A

Plays game B

Random case

Periodic case The player alternates between game A and B following a given Sequence of plays.

-Amengual y Toral: -Amengual y Toral: 'Transfer of information in Parrondo's games‘'Transfer of information in Parrondo's games‘, , Fluctuation and Noise LettersFluctuation and Noise Letters 5, L63 (2005)

The 2-girl paradox

Leunberger´s volatility pumping

How Often does the Parrondo EffectAppear?

G.C. Crisan, E. Nechita, M. Talmaciu, FNL 7, C19 (2007)

Game B (Capital dependent) bpwin :

cpwin :

Capital multiple of M ?

YES

NO

Notation: B: G(M,b,c), Original Parrondo: B: G(3,1/10-ε,3/4-ε)

How Often does the Parrondo EffectAppear?

Probability that two randomly-chosen losing games A=G(3,a,a), and B=G(3,b,c) generate the triplet (A,B,1/2A+1/2B) that completesParrondo’s Paradox : 0.0306%.Parrondo effect quite unusual!Highest probability: 0.0537% when the mixing parameter α=0.173 andM=4

Ensemble of interacting players. They chose either game A or game B randomly, i.e., with probability .

Cooperative games

Reversals of Chance

Ensemble of N interacting players. They choose either game A or game B randomly, i.e., with probability .

2

1winpGame A :

Game B:

winners number

w

Winning Probabilities

w > [2N/3] p1

[N/3]< w ≤[2N/3] p2

w ≤ [N/3] p3

Reversals of Chance

Juegos con Memoria

Matching Models

Consumers with specific wishes – Producers Employers – Job seekers Ph.D. Students - Supervisors

N men N women

Dating Game

Das and Kamenica IJCAI 2005, 947Two Sided Bandits and the Dating Market

N men N women

Statistical decision model of an agent trying to optimizehis decisions while improving his information at the same time.

Can Losers do Better?

Game A:

Random man j Best valued woman i

mod(n)1jj withmatch otherwise

man valued best notj ifotherwise

matchingP

)(

2/1)(

Woman chooses greedily best valued man

Fair game for every man

(sparkling personality)

New Rules for Dating Game

Game B:

Random man j Best valued woman i

)(N1)mod(j withmatch otherwise

man valued best notj if otherwise

Woman chooses greedily best valued man to match with

/ p and p ,p 32 1

Fair game for every man

Previous averagesparkling personality Trend follower?

New Rules for Dating Game

winners number

w

Winning Probabilities

w > [2N/3] p1

[N/3]< w ≤[2N/3] p2

w ≤ [N/3] p3

N=4

Results:Parrondo Effect in Total Matches

2/1

Results:Expected Payoff in Loser Matches

N=3N=3 – Mixing Probability of games A and B

2/1

No change for losers

N=3N=3 – Mixing Probability of games A and B

5/1

....ABABAB

....ABBBBA

Losers do worse!

Not losers also do worse!

Results: How Many Players?