New 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.
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. (If stair-climbing 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 Game 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
tails. 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 outcome.) 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 is. 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.
Connection
to Dot-Com Valuations? 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 paradox.) 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 University, John 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. |