1. Aplicaciones de la Paradoja de Parrondo Miguel Arizmendi Fac. Ingeniería Universidad Nacional de Mar del Plata Argentina

2. Losing in order to win Chess sacrifice bishop

3. N/2 Wrongs Make a Right

4. The Truel

5. Motores Moleculares

6. Parrondo’s Paradoxical Games

7. 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

8. Games

9. The small size of molecular machines means that their physics is dominated by thermal fluctuations – macroscopic intuition is of limited use Length scales

10. Energy Scales Thermal Energy For proteins in water this energy is taken from collisions with water molecules

11. 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

12. 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 10 5. 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.

13. What about Proteins? fish we calculate Re to be about 10 5. bacterium we calculate Re to be 10 -5. protein: size ~ 6 nm, speed 8m/s in water Re ~ 0.05 Overdamped

15. Quenched disorder effects on deterministic inertia ratchets

16. Games Processes like this with no memory are called Markov Processes

17. Random Walks and Diffusion For short times and distances, diffusion is very fast K + ion in water goes 1 micron in 0.25 ms, 0.1 mm in 0.25 s For long time and distances, diffusion is very slow, K+ ion goes 1m in 8 years.

18. Why bother moving? Rickettsia (tifus). ~ 100 years for mitochondrion synthesized in spinal chord to get to foot synapse. Active Transport is necessary: molecular motors

19. 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

20. Thermodynamics Second Law? Molecular Motor Model Can a Net Current J be obtained from Noise? Thermal Ratchet Model

21. Feynman Lectures: Ratchet and Pawl

22. Maxwell’s Demon

23. i=1 i=2

25. Flashing Ratchet Current

27. - MA, JR Sanchez and F. Family: PLA 249,281 (’98)Physica A327, 111 (2003) - MA, JR Sanchez and F. Family: PLA 249,281 (’98) Physica A327, 111 (2003) Flashing Ratchet Current / Entropy

28. 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)

29. Highly diffusive, several ATP molecules hydrolyzed/step

31. What about Games?

32. win lose ¿Is X(t) a multiple of 3 ? win lose winlose Game A Game B No Yes : Player’s capital at -th run (Fair games)

33. No Yes Is X (t) a multiple of 3?

34. Average gain of a single player versus time with a value of The simulations were averaged over 50000 ensembles. The player, with probability 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: 'Transfer of information in Parrondo's games‘, Fluctuation and Noise Letters -Amengual y Toral: 'Transfer of information in Parrondo's games‘, Fluctuation and Noise Letters 5, L63 (2005)

35. The 2-girl paradox

36. Leunberger´s volatility pumping

37. How Often does the Parrondo Effect Appear? G.C. Crisan, E. Nechita, M. Talmaciu, FNL 7, C19 (2007) Game B (Capital dependent) Capital multiple of M ? YES NO Notation: B: G(M,b,c), Original Parrondo: B: G(3,1/10-ε,3/4-ε)

38. How Often does the Parrondo Effect Appear? 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 completes Parrondo’s Paradox : 0.0306%. Parrondo effect quite unusual! Highest probability: 0.0537% when the mixing parameter α=0.173 and M=4

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

40. Reversals of Chance Ensemble of N interacting players. They choose either game A or game B randomly, i.e., with probability . Game A : Game B: winners number w Winning Probabilities w > [2N/3]p1p1 [N/3]< w ≤[2N/3]p2p2 w ≤ [N/3]p3p3

41. Reversals of Chance

42. Juegos con Memoria

43. Matching Models Consumers with specific wishes – Producers Employers – Job seekers Ph.D. Students - Supervisors N men N women Dating Game

44. Das and Kamenica IJCAI 2005, 947 Two Sided Bandits and the Dating Market N men N women Statistical decision model of an agent trying to optimize his decisions while improving his information at the same time.

45. Can Losers do Better?

46. Game A: Random man j Best valued woman i Woman chooses greedily best valued man Fair game for every man (sparkling personality) New Rules for Dating Game

47. Game B: Random man j Best valued woman i Woman chooses greedily best valued man to match with Fair game for every man Previous average sparkling personality Trend follower? New Rules for Dating Game winners number w Winning Probabilities w > [2N/3]p1p1 [N/3]< w ≤[2N/3]p2p2 w ≤ [N/3]p3p3

48. N=4 Results:Parrondo Effect in Total Matches

49. Results:Expected Payoff in Loser Matches

50. N=3 N=3 – Mixing Probability of games A and B No change for losers N=3 N=3 – Mixing Probability of games A and B Losers do worse! Not losers also do worse! Results: How Many Players?