Slot Machines

# The Kelly Criterion

The Kelly Criterion was first introduced in a paper by John Kelly in 1956. His paper dealt with the fact that in probability theory, a formula could be devised to determine the optimal size for a series of bets to minimize risk and maximize expected return. This applies any time there is an overlay in the possible return when compared to the wager size.

A number of horse race betting systems have grown up around Kelly’s system and some stock investors use a similar theory when picking their stocks. Of course for the system to be successful, a gambler needs to be able to predict the outcome of an unknown event (such as a horse race) better than the average player. This is because the formula presented by John Kelly:

f = bp – q /b

is only viable when there is an overlay, meaning the odds are in the gambler’s favor. In the Kelly formula above, “f” equals the fraction of the bankroll to wager, “b” equals the net odds, “p” equals the probability of winning, and “q” equals the probability of losing. Using this formula, the gambler finds an acceptable bet size.

Kelly’s formula can be broken down to a simple to use Bet Size = Expected Net Winnings / Net Winning if you Win, or even easier, bet size = odds of winning over odds of losing. Here is an example:

There are ten cards remaining in a deck of cards, six are black, four are red, but the house offers to pay you even money (1 to 1) if you choose the next card’s color correctly. What do you do?

Choose and Wager Wisely

Well, aside from running to the nearest ATM, you obviously choose black, but how does the formula work? Well, six cards out of ten win and four cards out of 10 lose, so your odds of winning are .60 / .40 = .20 In this example you would then multiply your bankroll by .20 to figure your f or bet size. If your bankroll was \$2000, then \$2000 x .2 = \$400 and you would wager \$400. If you were to lose, and the casino offered the same bet with the same ten cards, you would recalculate to find your wager: .20 x \$1600 = \$320. If you had won and were offered another chance, the math would have been .20 x \$2400 = \$480.

A very good horse race handicapper might look at the tote board and see that a horse he had figured as a 25 to 1 long shot currently had odds of 30 to 1. Should he make a wager? Yes, according to the Kelly Criterion, because there is an overlay. The wager is not a favorite, but if the handicapper knows what they are doing, after a long series of bets, their bankroll should grow. Here are the numbers used in this formula:

Winning wager returns 30 to 1 plus the wager, so b = (31)

Odds of winning, 25 to 1, or 4 percent, so p = (.04)

Odds of losing are 96 in 100, so q = (.96)

Remember the formula is f = bp – q / b so f = (31)x(.04) – (.96) / 31 and this comes out to 1.24-.96 / 31 = .009 So, for a bankroll \$2000 x .009 = \$18 would be the correct bet.

Blackjack Players

Playing any game involves risk, but playing blackjack while using a card counting system has been proven to be a long-term winner for many players. Because a blackjack card counter increases their wagers when the odds are in their favor and lowers their wagers when the house has the edge, they are able to beat the game. Even with a simple plus minus count system.

To manage their bets and their bankroll, some players use the Kelly Criterion to calculate their wagers. This allows them to constantly bet more money as they are winning, maximizing any winning streak. One simple formula suggested by Edward O. Thorpe in his book Beat The Dealer suggested that a player can simply bet whatever percentage of their bankroll the deck currently offers.

If the player calculates a 1 percent edge with a \$2000 bankroll, the wager should be \$20. A 1.5 percent edge should carry a \$30 bet. During any session the player can recalculate their bankroll and recalculate their bets. With a bankroll that has grown to \$2200, a player could make a \$22 bet with a 1 percent edge.

Using this formula, a player with a \$2000 bankroll would start their wagers at \$5 and work up to about a \$50 bet. Knowing how much you can earn by counting cards would determine whether this line of play was worth your time.

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