Losing to Win
It's a gift to born losers. Researchers have
demonstrated that two games of chance, each guaranteed
to give a player a predominance of losses in the long
term, can add up to a winning outcome if the player
alternates randomly between the two games.
This striking new result in game theory is now called
Parrondo's paradox, after its discoverer, Juan M.R.
Parrondo, a physicist at the Universidad Complutense de
Madrid in Spain (see http://seneca.fis.ucm.es/parr/GAMES/inbrief.html).
A combination of two losing gambling games
illustrates this counterintuitive phenomenon. The two
games involve tossing biased coins. In the simpler game,
the player gambles with a coin that's been loaded to
make the probability of winning less than 50 percent.
Winning means that the player receives $1 and losing
means that the player loses $1.
GAME 1
Probability of
winning 
Probability of
losing 
1/2  a 
1/2 + a 
The second, more complicated game, requires two
biased coins. One of the coins wins more often than it
loses, and the other loses more often than it wins. The
game is set up so that even though the winning coin is
tossed more often, this is outweighed by the much lower
probability of winning with the other coin.
Here's the rule for the two coins in the second game.
If the player's total amount of cash on hand is a
multiple of 3, the chance of winning is just 1/10 
a.
If not, the chance of winning is higher: 3/4  a.
GAME 2
Is the
total amount of cash on hand a multiple of
3? 
NO 
YES 
Coin 2 
Probability of winning 
Probability of losing 
Coin 3 
Probability of winning 
Probability of losing 
3/4  a 
1/4 + a 
1/10  a 
9/10 + a 
When a
is greater than zero, each game played repeatedly on its
own gradually depletes a player's capital.
However, if a player starts switching between the two
games, playing two turns of game 1, then two turns of
game 2, and so on, he or she starts winning. Randomly
switching between the games also results in a steady
increase in capital. Indeed, playing games 1 and 2 in
any sequence leads to a win.
Gregory P. Harmer and Derek Abbott of the University
of Adelaide in Australia have run computer simulations
of the games, demonstrating this counterintuitive result
for 50,000 trials at a
= 0.005.
Alternating between the games produces a ratchetlike
effect. Imagine an uphill slope with its steepness
related to a coin's bias. Winning means moving uphill.
In the singlecoin game, the slope is smooth, and in the
twocoin game, the slope has a sawtooth profile. Going
from one game to the other is like switching between
smooth and sawtooth profiles. In effect, any winnings
that happen to come along are trapped by the switch to
the other game before subsequent repetitions of the
original game can contribute to the otherwise inevitable
decline.
The same type of ratchet effect can occur in a bag or
can of mixed nuts, Abbott says. Brazil nuts tend to rise
to the top because smaller nuts block downward movement
of the larger nuts.
"There are actually many ways to construct such
gambling scenarios," Harmer and Abbott comment in the
Dec. 23/30, 1999 Nature. The researchers suggest
that similar strategies may operate in the economic,
social, or ecological realms to extract benefits from
what look like detrimental situations.
Unfortunately, Parrondo's paradox doesn't work for
the types of games played in casinos.
References:
Ball, P. 1999. Good news for losers.
Nature Science Update (Dec. 23). Available at http://helix.nature.com/nsu/991223/99122313.html.
Blakeslee, S. 2000. Paradox in game
theory: Losing strategy that wins. New York Times
(Jan. 25).
Harmer, G.P., and D. Abbott. 1999.
Losing strategies can win by Parrondo's paradox.
Nature 402(Dec. 23/30):864.
McClintock, P.V.E. 1999. Unsolved
problems of noise. Nature 401(Sept. 2):24.
Peterson, I. 2000. Losing to win.
Science News 157(Jan. 15):47.
Juan Parrondo's Web page is at http://seneca.fis.ucm.es/parr/.
Derek Abbott has a Web site at http://www.eleceng.adelaide.edu.au/Personal/dabbott/index.html.
Comments are welcome.
Please send messages to Ivars Peterson at ip@sciserv.org.
Ivars Peterson is the
mathematics/computer writer and online editor at
Science News (http://www.sciencenews.org/).
He is the author of The Mathematical
Tourist, Islands of Truth, Newton's
Clock, Fatal Defect, and The Jungles
of Randomness. He also writes for the
children’s magazine Muse (http://www.musemag.com/)
and is working on a book about math and art.
NOW AVAILABLE: Math Trek:
Adventures in the MathZone by Ivars
Peterson and Nancy Henderson. For children ages 10
and up. New York: Wiley, 1999. ISBN 0471315702.
$12.95 USA
(paper). 
