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 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.
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 on each turn.
GAME 1
Probability of winning: ½ - α
Probability of losing: ½ + α
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 – α. If not, the chance of winning is higher: ¾ - α.
GAME 2
Is the total amount of cash on hand a multiple of 3?
NO
Coin 2
Probability of winning: ¾ - α
Probability of losing: ¼ - α
YES
Coin 3
Probability of winning: 1/10 – α
Probability of losing: 9/10 + α
When α 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 ran computer simulations
of the games, demonstrating this counterintuitive result for 50,000 trials at α
= 0.005.
Alternating between the games produces a ratchet-like
effect. Imagine an uphill slope with its steepness related to a coin's bias.
Winning means moving uphill. In the single-coin game, the slope is smooth, and
in the two-coin 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 commented in the Dec. 23/30, 1999 Nature. The researchers suggested
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.
Original version
posted March 6, 2000
References:
Ball, P. 1999. Good news for losers. Nature Science Update (Dec. 23).
Blakeslee, S. 2000. Paradox
in game theory: Losing strategy that wins. New York Times (Jan. 25).
Bogomolny, A. 2001. Parrondo paradox. Cut the Knot! (June).
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. Random
fluctuations: Unsolved problems of noise. Nature 401(Sept. 2):24.
Peterson, I. 2000. Losing
to win. Science News 157(Jan.
15):47.
