Kelly’s criterion would have made them richer if they had used the martingale strategy.
Casinos are built on the foundation of mathematics, which they use to rob their customers of money. For years, mathematically inclined people have attempted to manipulate a system by exploiting its weaknesses using their knowledge of game theory and probability.
A funny example was when the American Physical Society hosted a conference in Las Vegas, Nevada, in 1986. The local newspaper ran the headline, “Physicists In Town, Lowest Casino Profit Ever.” It is said that the physicists were aware of the best strategy to beat any casino game, which was not to play.
A simple betting system that is based on probability can help you make money over the long term. However, there are some important caveats.
You can bet on either red or black when playing roulette. The payout is equal. If you bet $1 and you win, then you get $1. But if you lose, you lose your $1.) For simplicity’s sake, let’s assume you have a 50% chance of correctly predicting the colour. Real roulette tables include some green pockets where you can lose money, giving the casino a slight advantage. Let’s also assume that the table does not have a maximum bet.
Here is the strategy: Place $1 on any colour. If you lose, then double your bet and try again. Continue to double your bets ($1, $2. $4. $8. $16. etc.) until you are successful. If you lose your first two bets ($1 and $2) but win your fourth bet ($4), you will lose $3, but you’ll recoup that loss plus an extra $1 on your winning bet. If you win your first bet on the fourth, you will lose $7 ($1 + 2 + 4), but you’ll make a profit of $1 by winning $8. The pattern is repeated, and you always win a dollar. You can make $1 seem like a lot by repeating this strategy multiple times or starting with a larger initial stake. You will win $1,000 if you begin with $1,000 and double it to $2,000 then $3,000.
Some might argue that this strategy only makes money if you call the correct colour roulette. I, however, promised guaranteed profits. However, the chance that you will win at some point is 100 per cent. The probability of losing every bet decreases as you increase the number of rounds. Even in the realistic scenario where the house has a constant edge, this still holds. You will eventually win if there is a chance of winning. The ball cannot stay in the wrong colour forever.
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Should we all go on a road trip to Reno? Unfortunately, no. The martingale system was popularized in Europe in the 18th century. It still attracts bettors because of its simplicity and promise of riches, but it has flaws. Jacques Casanova de Seingalt was a notorious lothario who had many vices, including gambling. In his memoirs, he said, “I played the martingale but with such bad fortune that I soon found myself without a sequin.”
Can you see a flaw with the above reasoning? You have $7 and would like to make it $8. You can afford to lose your first three $1, $2, and $4 bets. The probability of losing three bets in a single row is one in eight. One-eighth (12.5%) of the time, you will lose $7, and the other seven-eighths, you will gain $1. The outcomes are equal: -1/8 x $7 + 7/8x $1 = 0.
This effect is the same for any starting capital. There’s a high chance to gain a little money and a low chance to lose it all. The martingale system is a good way to make a little money, but it can also be a bad one. The forces are balanced so that if many players use the strategy, their small wins and few large losses will average $0.
The argument is not over at $7. The idea, as I said before, is to play until you win. If you have lost three times in a row, then go to the ATM and place an $8 bet on a new spin. The guaranteed profit is dependent on your willingness to continue betting and the certainty of winning eventually with persistent play.
You only have a limited amount of money. You will soon find that you are betting a lot of money to cover your losses. You can’t generate wealth by taking a non-zero but small chance to risk your life for a dollar. You’ll eventually go bankrupt, and if it happens before you win the jackpot, you’re out of luck.
Finitude also breaks the martingale. Probability says that you will win eventually. But even if your purse was bottomless, you might die before you “eventually”. The pesky realities of real life interfere with our fun fantasies.
It might seem obvious, but you cannot force an edge in a casino. Astonishingly, we need to use arguments about mortality and solvency to eliminate it. Mathematicians live in a world of pencil and paper where they can freely roam across the entire universe.
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There is no winning strategy for games that have a 50 per cent chance of winning or less. What about games with better odds? What would you do with $25 if you could bet repeatedly on the outcome of a biased coin, which you knew was going to come up heads 60% of the time? You could either lose or win the same amount of money each time. Researchers conducted this exact experiment on 61 finance students, young professionals, and other participants. They let them play for 30 minutes and were shocked by the poor results. You can do it yourself.
Unbelievably, 28 per cent of players went bankrupt despite an advantage. And shockingly, two-thirds of them bet on the tails, which is not rational. The average participant won $91 in winnings (the maximum was $250). The researchers found that, over the 300 coin-tosses allowed, players who used the optimal strategies (described below) would have averaged more than $3,000,000 in winnings.
Players face a dilemma. If they bet too much, they run the risk of losing their entire bankroll in a few bad tosses. If you bet too little, you will miss out on the large advantage that the biased coin offers. Kelly’s criterion balances these opposing forces to maximize wealth. John Kelly, Jr., a scientist who worked at Bell Labs during the mid-20th century, discovered that to maximize profits, gamblers should wager a constant fraction of their money on each round.
In 1956, he published a paper that described his formula for the ideal fraction: 2 p– 1. p is your probability of winning (in the coin flip example, p=0.6). The experiment showed that betting 20 per cent of the available cash per flip was the best bet. The strategy increases the stakes if you win and reduces the bets as you lose money, so you are unlikely to go bankrupt.
Unlike the martingale strategy, the Kelly criterion is a proven method of quantitative finance. Professional card counters use it to determine the size of their bets during hot streaks.
Economists warn that while the Kelly criterion can be used to generate wealth, it is still a risky gamble. It assumes you have a good idea of your odds of winning, which is true for many casino games but not so much in ambiguous domains like the stock market. Kelly also asserts that you are most likely to increase your wealth in the coin toss test if you continue betting 20% of it. If you’ve got $1 million in your account, it is perfectly reasonable not to risk $200,000 on a flip of the coin. You will eventually have to take into account your level of risk aversion and make adjustments to your financial decisions.
If you are able to win bets, the Kelly criterion will prove more profitable than the martingale.