calculations by a Spanish physicist now suggest that this paradox may have
a kernel of truth to it. The discovery by Juan Parrondo not only offers
new brain candy for mathematicians, but variations of it may also hold
implications for investing strategies.
Parrondo's paradox deals with two games, each of which results in steady
losses over time. When these games are played in succession in random
order, however, the result is a steady gain. Bad bets strung together to
produce big winnings — very strange indeed! To understand it, let’s switch
from a financial to a spatial metaphor.
Imagine you are standing on stair 0, in the middle
of a very long staircase with 1001 stairs numbered from -500 to 500 (-500,
-499, -498, ...-4, -3,- 2, -1, 0, 1, 2, 3, 4, ...,498, 499, 500).
You want to go up rather than down the
staircase and which direction you move depends on the outcome of coin
flips. The first game — let’s call it game S — is very Simple. You flip a
coin and move up a stair whenever it comes up heads and down a stair
whenever it comes up tails. The coin is slightly biased, however, and
comes up heads 49.5 percent of the time and tails 50.5 percent.
It’s clear that this is not only a boring
game but a losing one. If you played it long enough, you would move up and
down for a while, but almost certainly you would reach the bottom of the
staircase after a time.
gives you vertigo, you can substitute winning a dollar for going up a
stair and losing one for going down a stair.)
A More Complex
The second game — let’s continue to wax poetic and call
it game C — is more Complicated, so bear with me. It involves two coins,
one of which, the bad one, comes up heads only 9.5 percent of the time,
tails 90.5 percent. The other coin, the good one, comes up heads 74.5
percent of the time, tails 25.5 percent. As in game S, you move up a stair
if the coin you flip comes up heads and you move down one if it comes up
But which coin do you flip? If the
number of the stair you’re on at the time you play game C is a multiple of
3 (that is, ...,-9, -6, -3, 0, 3, 6, 9, 12,...), then you flip the bad
coin. If the number of the stair you’re on at the time you play game C is
not a multiple of 3, then you flip the good coin. (Note: changing these
odd percentages and constraints may affect the game’s
Let’s go through game C’s dance
steps. If you were on stair number 5, you would flip the good coin to
determine your direction, whereas if you were on stair number 6, you would
flip the bad coin. The same holds for the negatively numbered stairs. If
you were on stair number -2 and playing game C, you would flip the good
coin, whereas if you were on stair number -9, you would flip the bad coin.
Both Games Lead
to the Bottom
It’s not as clear as it is in game S, but game
C is also a losing game. If you played it long enough, you would move up
and down for a while, but you almost certainly would reach the bottom of
the staircase after a time.
Game C is a
losing game because the number of the stair you’re on is a multiple of
three more often than a third of the time and thus you must flip the bad
coin more often than a third of the time. Take my word for this or read
the sidebar to get a better feel for why this
So far, so what? Game S is simple and
results in steady movement down the staircase to the bottom, and game C is
complicated and also results in steady movement down the staircase to the
bottom. The fascinating discovery of Parrondo is that if you play these
two games in succession in random order (keeping your place on the
staircase as you switch between games), you will steadily ascend to the
top of the staircase.
Alternatively, if you play two games of
S followed by two games of C followed by two games of S and so on, all the
while keeping your place on the staircase as you switch between games, you
will also steadily rise to the top of the staircase. (You might want to
look up M.C. Escher’s paradoxical drawing, Ascending and
Descending, for a nice visual analog to Parrondo’s
Standard stock market investments
cannot be modeled by games of this type, but variations of these games
might conceivably give rise to counterintuitive investment strategies.
Although a much more complex phenomenon, the ever-increasing valuations of
some dot-coms with continuous losses may not be as absurd as they seem.
Perhaps they’ll one day be referred to as Parrondo profits.
Professor of mathematics at Temple
Allen Paulos is the author of several books, including A
Mathematician Reads the Newspaper and, most recently, I Think,
Therefore I Laugh. His “Who’s Counting?” column on ABCNEWS.com appears
on the first day of every month.