FW: Parrondo's Paradox

From: BigBooster (fm1@amug.org)
Date: Tue Jul 10 2001 - 16:35:11 MDT

>Published by the New York Academy of Sciences
>Playing Both Sides
>Parrondo's paradox shows that you can win at two losing games by switching
>between them. The result has surprising implications for the origins of
>By Erica Klarreich
>After a startling run-up, the stocks of Internet and technology companies
>have suffered a precipitous decline in the past year. A portfolio equally
>invested in such technology giants as America Online, Dell Computer, Lucent
>Technologies and Qualcomm, Inc., would have lost half its value between
>March 1 and December 1, 2000.
>But investors can remain fully invested in the market and make money even if
>they own only poorly performing stocks - provided their portfolio includes
>several different losers. On the surface, it might seem that if owning one
>sinking stock were a bad idea, owning several would be insane.
>Mathematicians, physicists and biologists have proved otherwise.
>January/February 2001
>Brownian Motion
>Flashing Ratchet
>Playing the Market
>Nature Exploits Brownian Ratchets
>Alfredo Castañeda, Our Trauma, 1999.
>The key to such a counterintuitive investment strategy is to take advantage
>of random fluctuations in the prices of the stocks from day to day and from
>week to week. Even though all the general trends may be downward, the price
>of virtually any declining stock can also rally briefly if the market
>momentarily believes the stock is cheap enough to buy. If investors can
>manage to sell that stock during the brief price uptick that results from
>such market demand, and then shift those funds to another stock that is in a
>slump, they can overcome the general losing trends in both stocks. It is not
>an unproblematic strategy to pull off - transaction fees could gobble up
>profits, or a crash could drop prices monotonically for every stock in the
>portfolio. Yet the alternative, holding on to a losing stock as its value
>progressively slips away, offers no winning possibility at all, and could
>lead to far greater losses.
>No one yet has found a foolproof way to play such odds profitably and
>reliably with a real losing stock portfolio. But it turns out, in a
>surprising number of circumstances, that it is quite possible to, in
>essence, "time the market." In other words, even when one is faced with a
>number of scenarios for which the odds of success are unfavorable - whether
>in games of chance, in choosing between competing political parties or in
>circumstances involving the second law of thermodynamics - it is possible to
>play two of the scenarios against each other to create a single winning
>The workings of that paradoxical result were elucidated recently by the
>Spanish physicist Juan M.R. Parrondo of Complutensian University in Madrid.
>Parrondo showed that what was once an obscure curiosity in physics could be
>relevant to daily life - and much more besides. Some investigators think
>Parrondo's paradox may hold the key to certain biological processes -and
>perhaps to the development of life itself.
>Brownian Motion
>Parrondo, as I have implied, was not THE FIRST to notice the paradox that
>bears his name. In 1992 the French physicists Armand Ajdari of the Paris
>School of Industrial Physics and Chemistry and Jacques Prost of the Curie
>Institute, also in Paris, discovered a microscopic engine that possesses an
>unusual property. If left on, the engine pushes particles to the left, but
>if turned on and off repeatedly, the engine moves particles to the right
>instead. The engine, which is now known as a flashing ratchet, operates by
>exploiting the random jiggling of particles known as Brownian motion.
>In 1827 the English botanist Robert Brown observed that microscopic pollen
>grains suspended in water are in constant, chaotic motion. Those motions are
>the by-product of ceaseless random collisions with individual water
>molecules. Each individual particle - the invisible water molecules as well
>as the pollen grains visible in the microscope - acts in strict accordance
>with Newton's laws of mechanics. Yet taken together, the motions of the
>particles are so diverse as to be effectively random. Predicting the
>behavior of particles subject to the bumps and grinds of so many collisions
>is as frustrating and, ultimately, impossible, as predicting the outcome of
>a random coin toss.
>Investigators frequently treat Brownian motion as noise, regarding it as an
>undesirable but unavoidable by-product of any machine that does useful work
>on the molecular scale. Engineers, however, have recently harnessed the very
>randomness of Brownian motion to generate orderly movement. Machines that
>can bring about such directed motions are known as Brownian ratchets, so
>named for the simple mechanical device that combines an asymmetric sawtooth
>wheel and a spring-loaded pawl, or rocker arm, to restrict rotary motion to
>one direction.
>In a mechanical ratchet, the pawl follows the contour of the toothed wheel.
>The teeth themselves are gently sloped on one side, and the pawl is mounted
>so that it slides smoothly over them when the wheel turns in one direction.
>On the other side of the teeth, however, the points fall away sharply, so
>that when a force tries to spin the wheel in the opposite direction, the
>pawl jams against a tooth, preventing the wheel from turning. Thus even when
>the ratchet is subject to forces that would move the wheel back and forth,
>the action of the pawl and the teeth allows the wheel to move in only one
>direction. Such asymmetry is also an essential part of the design of
>Brownian ratchets, including the one that inspired Parrondo's paradox.
>To see how a group of objects in random motion can be ratcheted
>preferentially in one direction over another, imagine a collection of
>electrically charged particles distributed randomly along a gentle slope.
>Assume the particles are all in Brownian, random motion, which makes them
>jiggle with equal probability to the right and to the left along that line,
>with a slight drift leftward, say, down the slope. One can then superpose
>along the slope an electric potential that looks much like the teeth of a
>saw: periodic energy peaks rise steeply on the left side of the tooth and
>descend gently down its right side. (Engineers have built such potentials,
>for example, by depositing asymmetric electrodes along a line and then
>running a current through them.)
>When current is turned off in such an apparatus, the charged particles
>simply jiggle in place along the line, with only a slight net downward
>drift. When the current is turned on, however, and the sawtooth potential is
>imposed, the particles are drawn toward the points of lowest energy,
>clumping together in the troughs of the potential.
>Now switch the current off a second time, so that the particles resume their
>random Brownian jiggling. Since the troughs of the potential are offset to
>the right, as the particle drifts, it has a better chance of crossing the
>peak of the potential to its immediate right than the one to its immediate
>left. Switch the current on a second time, and most of the particles that
>drifted to the left will be swept back into the same trough from which they
>started. But some of the particles that drifted to the right will have
>drifted so far that they will be drawn to the next trough over. If such a
>potential is switched on and off, or "flashed," at the proper frequency, the
>charged particles in the flashing ratchet will march up the slope,
>overcoming gravity.
>Flashing Ratchet
>Although investigators could model the flashing ratchet mathematically, the
>mathematics for doing so is formidable. The complexity of the problem madeit
>appealing only to a small coterie of physicists and mathematicians.
>"We've known for several years that there was this really counterintuitive
>phenomenon in physics," says Charles R. Doering, a mathematician at the
>University of Michigan in Ann Arbor, who coined the term "flashing ratchet"
>in the mid-1990s. "Parrondo's insight was to formulate the flashing ratchet
>in terms of games, which everyone can understand."
>Since the motions of microscopic particles are as random as a coin toss,
>Parrondo knew he could translate the mechanics of the flashing ratchet into
>the language of game theory - and in the process reduce its mathematical
>content to simple fractions. "I thought that it would be really nice to
>explain something so surprising just with three simple numbers, such as 1/2,
>3/4 and 1/10," says Parrondo. "I love simple things."
>Parrondo invented two coin-tossing games to embody the essential elements of
>the flashing ratchet, and summoned his collaborator, the engineer Derek
>Abbott of the University of Adelaide in Australia, who was delighted with
>Parrondo's innovation. Together, they worked through the details over
>coffee, scribbling on napkins in a Madrid café. One coin-tossing game would
>be set up to mimic the Brownian motion of a particle on a slight incline
>when the potential of a flashing ratchet was turned off, providing a slow,
>steady drag on winnings. The second game would mimic the sawtooth bias of
>the flashing ratchet with the potential turned on. When he got back to
>Australia, Abbott and his colleague Gregory P. Harmer began computer
>simulations to see what would happen if they played the games a hundred
>times. Sure enough, when they played either one of the games for a long
>time, they lost steadily.
>The outcome of a single coin toss in either of the two games is identical:
>win the toss and gain a dollar, lose the toss and give back a dollar. The
>player begins with some random number of dollars. In the first game the
>player tosses a coin weighted so that the odds of winning are slightly less
>than 50 percent. In the second game two weighted coins are in play: the odds
>of winning with the first coin are nearly 75 percent in the player's favor;
>the odds of winning with the second coin are less than 10 percent. The
>player gets to toss the favorable coin whenever his pool of money is not a
>multiple of three. He must toss the unfavorable coin whenever his pool of
>money is a multiple of three.
>It is obvious that the first game is a loser, but the long-term bias against
>the player in the second game takes a bit of explanation. Since the chances
>are two-to-one against beginning the game with a quantity of dollars that is
>a multiple of three, most players would begin with the coin weighted in
>their favor. Only in one start out of three would the player have to begin
>with the coin weighted heavily against him. So on the initial toss, the
>rough odds of winning are: (2/3 x 0.75) + (1/3 x 0.1) or about 53 percent.
>Think of it as a "new customer bonus."
>Play the second game long enough, however, and it becomes no longer true
>that the pool of money is divisible by three just a third of the time. A
>player whose pool of money is not a multiple of three tends to increase his
>earnings by a dollar with each coin toss, until the earnings reach a
>multiple of three. But any dollar amount divisible by three creates a large
>barrier to advancement - 90 percent of the time a loss results. Hence after
>the first few plays the pool spends most of its time bouncing between
>amounts divisible by three and amounts just one dollar less. Although the
>random nature of the game means that for a single run of tosses the end
>result could be a tidy profit, play the second game enough times (Harmer and
>Abbott ran 50,000 simulations of 100 tosses each on their computer) and the
>trend shows up clearly as a gradual but inexorable loss.
>Once the pool gets stuck at or just below some multiple of three, it proves
>difficult to extract oneself simply by playing the second game. But Parrondo
>saw that switching to the first game was a way to jump out of that rut.
>Although the chances of raising one's earnings above a given multiple of
>three with two coin tosses are a measly 7 percent in the second game, the
>chances of doing so are nearly 25 percent in the first game. And once on the
>other side of that "energy barrier," switching back to the second game gives
>good odds of increasing the pot to the next multiple of three.
>Alternating between the two games - two coin tosses in the first game
>followed by two coin tosses in the second game - turns two losing
>propositions into one winning strategy. Again, the results of executing that
>strategy on any single run of tosses can vary wildly, but over the course of
>50,000 runs of 100 tosses each, the simulations showed a trend for
>substantial gains. Remarkably, even when the switches between games were
>made at random, the overall gains were similar.
>With Parrondo's strategy, a player can extract money from an opponent at no
>cost - "money for free," in the words of Abbott and Harmer. And with
>microscopic Brownian ratchets, there seems to be a similar magic at work.
>After all, the random motion of molecules in solution - analogous to the
>random outcomes of a coin flip - is translated into a steady flow in one
>direction. But there is a catch: the second law of thermodynamics. The
>second law states that a mechanical system cannot give rise to more order
>than is fed into it. In his discussion of the second law in his famous
>lecture series, the physicist Richard P. Feynman whimsically suggested that
>a microscopic ratchet-and-pawl system attached to a highly sensitive wind
>vane might draw enough energy from the random motion of air molecules in a
>room to winch up a flea. Feynman decided that his setup would violate the
>second law of thermodynamics - it would be a perpetual motion machine. The
>machine might indeed hoist the flea, he concluded, except that if the
>machine were really built on the molecular scale, the spring, too, would be
>subject to Brownian fluctuations, and so it would lift at random times. The
>ratchet could then turn in both directions.
>At first glance, the flashing ratchet also appears to defy the second law.
>But a closer look shows that here there is an outside source of order acting
>on the system: the energy needed to switch the potential on and off.
>Brownian ratchets do not provide energy for free; like all motors, they
>merely convert one form of order, such as chemical or electromagnetic
>energy, into another form: directed motion.
>But Brownian ratchets are unique in that noise is not a drawback for them
>but rather an essential ingredient. Many physicists had assumed that random
>motions would always hinder, not assist a machine; after all, random
>molecular motions are the essence of the disorderly state toward which the
>second law of thermodynamics inexorably pulls. But the flashing ratchet
>shows that such an assumption is unwarranted by the second law: the device
>would fail entirely if there were no Brownian motion to randomize the
>positions of the particles when the potential is turned off. "Noise is
>generally thought of as something undesirable, something to get rid of or to
>filter away," the biochemist Martin Bier of the University of Chicago has
>written. "To think of noise as something to exploit is a major shift of
>Playing the Market
>Interestingly, people already exploit switching strategies similar to
>Parrondo's in everyday life - though not, of course, because they are
>applying theoretical insights about Brownian motion. But whenever someone
>changes jobs to get a salary increase, for instance, or sells peaking stocks
>and buys slumping ones, Parrondo's paradox comes into play. Moreover, many
>ordinary events are governed by random-seeming fluctuations, from the ups
>and downs of the financial markets to the succession of political parties in
>power. Economists and mathematicians have begun to ask whether the
>principles of Parrondo's paradox could apply to decision making in those
>The physicist Sergei Maslov of Brookhaven National Laboratory in New York
>has shown that principles similar to Parrondo's paradox can be applied to
>the stock market. Maslov found a way that the value of a stock portfolio can
>increase even if all of its stocks decline in the long run: The entire
>portfolio must be sold periodically - even daily - and the proceeds must be
>quickly plowed back into repurchasing the stocks in the same proportions as
>they were in the original portfolio. The net effect of such periodic
>rebalancing is to apply the gains from stocks that are doing temporarily
>better than average (and have therefore increased their relative proportion
>in the portfolio) to buy more stocks that are temporarily underachieving.
>The beauty of Maslov's proposal is that one need not know anything about
>particular stocks; simply sell them all and then buy them back in a
>rebalanced portfolio. Unless the balance of investments is shifted in that
>way, the long-term prospects for a portfolio full of duds are "grim indeed,"
>says Maslov, since all of the stocks tend to decline with time. But Maslov's
>tactic can generate better returns than the "buy and hold" policy that is
>the current favorite of many economists. (It should be noted, however, that
>the declines in technology stocks mentioned at the beginning of this article
>might well overcome even Maslov's strategy.)
>Abbott and Parrondo suggest that switching from one option to another could
>also prove useful in applying government policies to quantities that
>fluctuate unpredictably; for example, if two policies each cause
>unemployment to rise, alternating between them might actually create jobs.
>Furthermore, Abbott notes, citizens could be, in essence, following
>Parrondo's principles when they vote political parties alternately into and
>out of office.
>In short, Parrondo's paradox demonstrates that two wrongs can sometimes make
>a right - and that a little indecision can be a good thing. "One message of
>the Parrondian philosophy is to not be a moderate or an extremist, but to
>maintain a dynamic between positions that may appear to be conflicting,"
>Abbott notes. "Parrondo gives us permission to be 'swing voters' in life's
>As powerful as the insights of Parrondo's paradox may be at the macroscopic
>level, randomly switching from one "game" to another may have its greatest
>impact at the microscopic scale. In the past few years engineers working
>with such tools as lasers and microscopic electrodes have built ratchets to
>a degree of precision that would have been unimaginable just fifteen years
>ago. Such microscopic ratchets can work with objects of a truly molecular
>size - less than one-hundred-thousandth the size of Feynman's fleas.
>The chemists Steven G. Boxer and Alexander van Oudenaarden of Stanford
>University have found that minuscule ratchets can sift through the jumble of
>proteins, lipids and carbohydrates that make up cell membranes. The two
>investigators observed that the ratchet allows particles to pass through it
>at different speeds, depending on their size and electrical charge. That
>property makes it possible to sort membrane molecules simply by passing them
>through a ratchet. Sorting membrane molecules could, in turn, become an
>important step in understanding just how particular drugs attach them-selves
>to receptors in cell membranes.
>Nature Exploits Brownian Ratchets
>Even as engineers are developing the technology to build and use minuscule
>Brownian motors, biologists are beginning to suspect that nature has been
>exploiting the properties of intricately designed Brownian ratchets since
>the beginnings of life. The ability to convert energy into directed motion
>is vital to living creatures; people, for instance, depend crucially on the
>muscular contractions that pump blood, draw breaths and move arms and legs.
>On a much smaller scale, cellular motors perform such chores as transporting
>organelles. The fuel for those motors is well known - the chemical energy
>stored in the molecule adenosine triphosphate, or ATP. But the inner
>workings of the motors themselves remain enigmatic.
>Some investigators have proposed recently that so-called motor proteins,
>which haul microscopic cargo, might operate in part by a ratchet mechanism.
>In 1993 a team of engineers at Harvard University and the Rowland Institute
>for Science in Cambridge, Massachusetts, tracked the movements of the motor
>protein kinesin. That protein moves by propelling itself along cellular
>"train tracks" called microtubules, and can transport cargo more than a
>hundred times its size - much like an ant carrying an enormous cake crumb.
>The team discovered that instead of moving smoothly along the microtubule,
>kinesin makes distinct jumps, each about eight nanometers long, as if it
>were climbing the rungs of a ladder.
>Those discrete jumps led Martin Bier and Dean Astumian, a colleague of
>Bier's at the University of Chicago, to hypothesize that a ratchet-like
>engine was at work. The microtubule, they proposed, could be electrically
>charged to resemble the sawtooth potential of the flashing ratchet. When the
>kinesin molecule is at rest, they suggested, it prefers to stay in one of
>the troughs of the potential. But when ATP is burned, the kinesin molecule
>temporarily gains enough energy to escape the trough. Brownian motion makes
>it drift a little to the right or left, so that when the momentary burst of
>energy subsides, the kinesin molecule may slide down into the nearest
>neighboring trough - say, to the right. Occasionally, the molecule may
>instead shift to the nearest neighboring trough on the left, but as with the
>flashing ratchet, the combination of jiggling motion and the shape of the
>sawtooth should give rise to a net long-term movement to the right.
>Astumian and Bier calculated the power output of such a ratchet, and their
>estimates came within an order of magnitude of the measurements made by the
>Harvard and Rowland Institute team. Since that time, several other models
>have been suggested that fit even more closely with experimental
>observation. Each model exploits a Brownian ratchet engine, along with other
>Engineers hope that the mechanism of kinesin action,when finally understood,
>will provide the blueprints for minute, man-made motors that could be useful
>in microelectronic devices. "Ion pumps and motor proteins are the smallest
>engines we know and they can no longer be thought of as wind-up toys," Bier
>has written. "If nano-technological devices are to be as efficient and
>reliable as biological molecules a good understanding of noise will be
>indispensable to their design."
>Beyond explaining the inner workings of living creatures, some investigators
>suggest that Parrondo's paradox may hold the key to explaining one of the
>oldest puzzles of nature: how life began. One central problem is to explain
>how the inchoate, randomly jiggling mass of particles that physicists call
>primordial soup could give rise to the complex molecules of life. But
>Parrondo's paradox demonstrates that processes leading to net losses can be
>combined to generate a positive outcome, and that randomness can facilitate
>that outcome. Abbott suggests that a mechanism such as the flashing ratchet
>might have harnessed forces that would otherwise have led to disorder, and
>channeled them toward the assembly of complex amino acids. Put in a slightly
>different way, processes that would, if left to themselves, tear up
>molecules essential to the chemistry of life, might have worked together
>instead to foster the synthesis of such molecules.
>Thus perhaps the emergence of order is not as unlikely as it has often
>seemed. The physicist Paul C.W. Davies, who examined the question of how
>life began in his book The Fifth Miracle, has observed:
>Nobody knows what the process was, but anything that clarifies how
>information and organized complexity can emerge from the randomness of
>molecular chaos will cast welcome light on this most profound of scientific
>mysteries. That is the significance of Parrondo's paradox, and the
>associated work of people like Derek Abbott.
>Erica Klarreich is a freelance writer and mathematician living in Santa
>Cruz, California.
>January/February 2001
>Brownian Motion
>Flashing Ratchet
>Playing the Market
>Nature Exploits
>Brownian Ratchets
>© 2000/2001 New York Academy of Sciences.
>All rights reserved.

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