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 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 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.