Who's Counting: Winning at Losing Games

— There's an old story about a store owner who loses money on each individual sale but somehow makes it up in volume of sales.

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.

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