Martingale Theory

In the long run, the wheel will land half on black and half on red.

In probability theory, a Martingale is a sequence of random variables such that the the conditional expected value of the next observation, given all the past observations, is equal to the last observation. In common language, it is known as a negative progression theory that can best be understood by the repeated flips of a coin, where each independent flip of the coin has a 50-50 probability of turning up heads but a decreasing probability of turning up heads time after time after time. In other words, the probability of such a sequence, or run of heads, decreases over time while the probability of each independent outcome remains 50-50. The theory is not only mathematically true, statistically true and true by the laws of probability but has been applied in many other areas from roulette to biodiversity within an ecosystem.

The Law of Large Numbers

The Law of Large Numbers is the underlying rationale behind the Martingale theory. The chance of a large difference between the percentage of successes (if you're betting heads will turn up) and the likelihood of success gets smaller and smaller as the number of coin flips grows. But while flipping the coins 1 million times, nothing prevents the difference within a sequence of coin flips from being large (like 10 heads in a row). In fact, there are theorems of probability that dictate this to be true. Take six flips of the coin. Each flip has a 50-50 chance of coming up heads and the odds of six heads in a row are shown mathematically as 1/2 x 1/2 x 1/2/ x 1/2 x 1/2 x 1/2 = 1/64. However, the law of large numbers shows that when probability is 50-50, it will be born out statistically that when the numbers are large enough, the number of heads and tails will even out. In other words, you'll have runs or bunches of runs of tails, too, that will balance out over the million coin flips (see Resources).

Betting

Do the math with a roulette wheel. There are 32 slots--16 red and 16 black. The odds on any given spin of the wheel are 50 percent the ball will fall in a red slot and 50 percent it will fall in a black slot. But there are times when it will turn up black six times in a row followed by a red or three reds or another black.

Using the Martingale theory, betting roulette is a risky one because you're betting on the runs of the same color. The betting scheme works like this. If you bet 1 chip on red and win, you bet 1 chip again on red, and repeat this until you lose. When you lose you double your bet and continue doubling your bet until you win. When you win, you've won back all your lost bets plus one chip and you start the process all over.

This assumes several things to at least break even. One, that you have an unlimited bankroll and, two, that there is no house limit on bets. In other words you could hit the limit before recouping your losses on a bad run. To win, you have to know on a good run when to quit while you're ahead.

Negative Progression

It's ironic that the Martingale theory was developed in the 1700s to show the mathematical impossibility of winning at gambling in games whose odds are 50-50. When playing any even game of chance the number of independent events always diminish to zero. Gamblers turned the theory on it's head and employed it as a way to maximize winning while minimizing losses.

Biodiversity

The unified neutral theory of biodiversity and biogeography by ecologist Stephen Hubbell utilizes the Martingale theory to explain the diversity and relative abundance of species in ecological communities, although his theory assumes that the differences between members of an ecological community of similar species are neutral--that is, they have no interactive effect on the species being studied.

Hubbell extrapolated on Martingale's theory, using an amoeba as an over-simplified example. It has a 50-50 chance of splitting. If it splits, it wins and now a second amoeba has the same 50-50 odds and on and on, with each amoeba having the same probability of success but a diminishing likelihood as we saw above. The theory has been applied successfully to many groups of species, including tree species, bacterial populations, moths, certain bird populations and some plants. The Martingale theory is applied in economics and marketing practices, as well.

See the Sequential Runs Yourself

For fun, you can repeatedly reproduce a Martingale random sequential series in an Excel spreadsheet. If you're handy with Excel or similar spreadsheet software, here are instructions to get a feel for the uneven equality in sequential numbers in the short term and how they average out in the long term. Enter 0.0 in the A1 (top left) cell, and in the cell below it (A2) enter =a1+NORMINV(RAND(),0,1). Now copy that cell by dragging and create 300 to 1,000 copies. This will create a Martingale series with a mean of 0 and standard deviation of 1. That represents either heads or tails. With the cells still highlighted create a chart of these values using the chart creation tool. You can recalculate hitting F9 and check out the new sequences.

References

• Academic Dictionaries and Encyclopedias: Random Variables
• Berkeley: Interactive Law of Large Numbers Tool
• Kanzen: Explanation of Matingale Theory of Probability using Roulette