Frederick Mann forwards a long article from an unspecified URL
about Parrondo's paradox. Some of the material about Brownian
ratchets was also covered in July's Scientific American.
A few comments:
>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....
>
>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.
This result has been widely reported as "if you can switch between
two losing games you can make them winning games". The problem with
this interpretation lies in the last sentence above, where the rules
depend on how big your pool of money is. No real games behave this way.
This is the crux of the unintuitive result.
I came up with a trivial example of two losing games which become winners
if you can switch between them. Game 1: if your cash balance is even,
you win 1. If it's odd, you lose 2. Game 2 is the opposite: if your
balance is odd, you win 1; if it's even you lose 2.
Both games 1 and 2 are obvious long term losers. You win at most 1
the first time, then you lose 2 forever. But by alternating between
them, game 1 when your balance is even, game 2 when it's odd, you win
every time.
Does this prove anything? Not really. These aren't games as we usually
think of them, because real games don't have this peculiar dependency
on the size of your wallet. That's the only reason we are able to get
this odd result. But if we accept games that work this way, my games
show the result far more clearly.
>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.
This strategy of portfolio rebalancing is nothing new, indeed it is
almost universally recommended, although of course not at the time
scale described here. Conventional investment advice is to divvy up
your investment assets into groups, decide on a percentage investment
in each group, and to periodically rebalance your portfolio when the
percentages get out of line due to price changes.
This works especially well in cyclical market conditions, where the
"hot" investment varies from time to time. The rebalancing makes
you automatically tend to buy low and sell high.
The main alternative strategy is buy and hold. In the long run, if one
type of investment drastically outpaces others, then if you had even a
small part of your pool invested in it initially, its share will grow
over time and eventually you gain much of the benefit of its growth.
In practice people often pick a mix of these two strategies, doing
some rebalancing but allowing their portfolio to gradually become more
concentrated in particular investments which continue to do well over
a long period.
>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 Scientific American article discussed this; see also
http://www.eurekalert.org/pub_releases/2001-06/aaft-wpn062701.php which
presents a more conventional view of the kinesin motor.
This raises a question with regard to nanotech designs. Suppose it
turns out that nature is widely using Brownian motors. Does this
suggest that the current approach to nanotech by Drexler et al is
fundamentally misguided? His designs are based closely on macroscale
machinery: conveyor belts, presses, bearings. Few of them depend in any
way on Brownian motion. In fact his term "eutectic" means that all the
molecules are on controlled paths. The only place you have Brownian
effects is when they have to interface to the messy outside world.
But once material has been palletized and brought inside, it is controlled
just like the assembly line at Ford.
It could be that this whole approach is wrongheaded and unlikely
to succeed. We may have to rethink how nanotech should work.
Maybe probability needs to be an essential step. Maybe the assembly
line needs to take two steps backwards for every three steps forward.
It has been known for some time that this can lead to more efficient
designs, by exploiting thermodynamic reversibility (see threads from
last month on reversible Brownian computers).
All those Drexlerian gears and arms may be obsolete before they are
ever used. Truly effective nanotech might do better to copy nature and
adopt probabilistic, Brownian methodology.
Hal
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