Thursday, January 2, 2014

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 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.

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.

Is the total amount of cash on hand a multiple of 3?

Coin 2
Probability of winning: ¾ - α
Probability of losing: ¼ - α

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


Ball, P. 1999. Good news for losers. Nature Science Update (Dec. 23).

Blakeslee, S. 2000. Paradox in game theory: Losing strategy that winsNew 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 paradoxNature 402(Dec. 23/30): 864.

McClintock, P.V.E. 1999. Random fluctuations: Unsolved problems of noiseNature 401(Sept. 2):24.

Peterson, I. 2000. Losing to winScience News 157(Jan. 15):47.

Sunday, October 20, 2013


Watch him pull a rabbit out of his hat? Not exactly.

Arthur T. Benjamin eschews the usual trappings of the magician's trade. Calling himself a mathemagician, he astonishes audiences with amazing feats of mental arithmetic. Behind the scenes, he reveals how you, too, can look like a genius without really trying.

Photo by Richard Haverty

A math professor at Harvey Mudd College, Benjamin has brought his particular brand of prestidigitation to a wide variety of appreciative audiences. He has appeared on television programs and performed in night clubs, classrooms, and banquet halls. He has even traded quips with comedian Stephen Colbert on The Colbert Report.

Benjamin's act is a striking demonstration of how miraculous fairly ordinary, simple mathematical machinery can appear when you don't really know what's going on under the hood.

One of my first encounters—many, many years ago—with arithmetic shortcuts was in a book titled Cheaper by the Dozen, the story of efficiency expert Frank B. Gilbreth and his family of 12 children. In a chapter about dinner conversation, Frank B. Gilbreth Jr. and Ernestine Gilbreth Carey write, "Also of exceptional general interest was a series of tricks whereby Dad could multiply large numbers in his head, without using pencil and paper."

For example, to multiply 46 times 46, you figure out how much greater 46 is than 25. The answer is 21. Then you calculate how much less 46 is than 50. The answer is 4. You square 4 and get 16. Putting 16 and 21 together gives the answer 2116.

That's the sort of venerable procedure that Benjamin uses to do fast multiplies and to square four-digit numbers faster than someone using a calculator. Turning a repertoire of such calculating tricks into a real show, however, requires developing a memory for numbers and learning how to calculate from left to right, performed at the speed of rapid-fire chatter.

Suppose you want to multiply 378 by 7. Starting from the left, you would get 2100 (300 x 7) plus something more. In the next step, 70 x 7 equals 490, which is added to 2100 to give 2590, plus something more. Finally, 7 x 8 equals 56, which is added to 2590 to give 2646.

One advantage of using this method is that you can start saying the answer while you're still calculating it, Benjamin remarks.

Here's a neat way to square two-digit numbers. Suppose the number to be squared is 37. That number is 3 less than 40, and 34 is 3 less than 37. Multiply 40 by 34 to get 1360, then add the square of the difference, 32 or 9, to get 1369.

The trick is to choose the difference so that the multiplication is easy. For example, to square 59, choose a difference of 1. Go up to 60 and down to 58. Multiply 60 times 58 to get 3480, then add 12 , to obtain 3481.

The proof is in the algebra: a2 = (a + d)(a - d) + d2 . The same idea can be extended to squaring 3-digit and 4-digit numbers.

Interestingly, when Benjamin performs calculations involving long sequences of digits, he relies on a phonetic code to remember the numbers. "There's no mental blackboard," he says. "It's much more an auditory process."

Benjamin turns sequences of digits into words that add up to some sort of crazy scenario. For example, he can mentally convert the sequence 9 6 4 8 3 7 5 4 8 3 1 2 7 5 9 6 into the words "pitcher fume color fume tinkle beach" as a handy mnemonic for (9 6 4) (8 3) (7 5 4) (8 3) (1 2 7 5) (9 6).

The underlying scheme assigns different consonants to different numbers, and the memorizer supplies the vowels: 1 (t, d), 2 (n), 3 (m), 4 (r), 5 (l), 6 (j, ch, sh), 7 (c, k, g), 8 (f, v, ph), 9 (p, b), 10 (z, s). This is the modern version of a number alphabet originally proposed by Pierre Hérigone and published in Paris in 1634. Three consonants—w, h, and y, spelling "why"—do not appear in the list.

For more on phonetic codes, see Benjamin’s article "A Better Way to Memorize Pi: The Phonetic Code," published in the February 2000 Math Horizons.

Benjamin can also handle magic squares, natural logarithms, cube roots, and much more. "As a kid, I liked to show off," he says. "Now I get paid to do this!" Besides, the techniques are so easy that even an elementary-school student can master them.

Benjamin describes and explains many of his mental calculation techniques in the book Secrets of Mental Math: The Mathemagician’s Guide to Lightning Calculation and Amazing Math Tricks (coauthored by Michael Shermer).

He also has a set of DVDs in The Great Courses series devoted to "Secrets of Mental Math." Benjamin’s "Mathemagic" performance, available as a TED video, has been viewed more than 5 million times.

Benjamin wrote about the mathematics underlying some of his mental feats in an article published in the January 2012 College Mathematics Journal—a special issue dedicated to Martin Gardner. Titled "Squaring, Cubing, and Cube Rooting," the article starts off:

"I still recall the thrill and simultaneous disappointment I felt when I first read Mathematical Carnival by Martin Gardner. I was thrilled because, as my high school teacher had told me, mathematics was presented there in a playful way that I had never seen before. I was disappointed because Gardner quoted a formula that I thought I had 'invented' a few years earlier. I have always had a passion for mental calculation, and the formula . . . appears in Gardner's chapter on 'lightning calculators.' It was used by the mathematician A. C. Aitken to square large numbers mentally."

Benjamin's article is reprinted in Martin Gardner in the Twenty-First Century, edited by Michael Henle and Brian Hopkins (MAA, 2012).

Original version posted October 5, 1998