Edge, Expectation and Kelly Criterion

In the course of your research you have probably come across the terms edge and expectation. You may also have heard of Kelly Criterion, a method for bet sizing that optimisies maximal investment growth. All of these terms are important for good money management.

Edge is just another (and easier to remember) term for mathematical expectation. The calculus of expectations is attributed to the Dutchman Christiaan Huygens, an early probability theorist. Expectation (also known as expected value) is defined as the weighted average of a variable. In gambling theory, expectation is the average expected rate of wealth accumulation. For any gamble, you are either going to win or lose your bet and so expectation is the sum of your average winnings plus your average losses, and is given by the following formula


where p is the probability of winning, profit is the profit from a £1 bet and loss is the loss of that £1 bet. Taking American roulette as an example we can determine the house edge in the long run. The wheel in American roulette has 36 numbers from 1 to 36, 18 of which are red and 18 are black. There are also two green numbers 0 and 00. Betting on the colour red or black will earn you even money on a winning bet. The house wins in the long run because of the two green numbers at the rate of


where (20 / 38) is the probability of someone not hitting their chosen colour (the 18 numbers of the other colour plus the two green numbers) and (1) is the unit bet. If the gambler should win then the probability of that ocurring is 1 - (20/36) and the house loses a unit bet (-1). We calculate the house edge to be


so that for every bet on black or red guarantees the house a profit of 5.3% in the long run. There is no bet on green and that is where the house gets its edge. If the house is winning 5.3% then the general public is losing 5.3% and no amount of Martingale betting is going to change that fact.

Kelly Criterion was developed in the 1950s by John Kelly, a colleague of the information theorist Claude Shannon. Using Shannon's information theory Kelly determined the optimal bet to maximise the growth of an investment. Kelly derived the following formula


where f is the fraction of your wealth to be invested, p is the probability of a winning bet and b is the net odds returned (also called yield) on a £1 winning bet. For the roulette example above the recommended fraction of your wealth to bet is


Two things to note here, firstly that the Kelly Criterion is telling you to invest a negative amount of your wealth. This is because the bet has negative expectancy therefore you should bet nothing in this instance. Secondly, if you calculate the numerator (top part) of the Kelly Criterion formula then (18/38) (1+1) - 1 = -0.053, which is the same as the expectation for the player. In other words p (b+1) - 1 is just another way of calculating expecation (or edge) and the Kelly Criterion can be re-written as 


And so, good money management first determines whether or not an investment strategy has an edge and then by using the predicted average yield you can calculate what percentage of your wealth to invest. For example, if the expectation for a given strategy is 0.056 and the average odds offered are 2.20 giving a yield to £1 of 1.20 then you should be investing 4.7% (0.056 / 1.20) of your net worth on each bet.

2 comments:

  1. Whilst the mathematical correctness of the kelly criteria is irrefutable. The big problem is calculating your edge for any one bet. Get this wrong and you get yourself in trouble. Fixed odds games like the casino games are fairly easy to calculate your edge. A horse race where the true price is only revealed at the off is another matter. For this reason, I've never really thought it is possible to reliably use kelly for horses and similar markets

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    1. For betting on horse races I would agree with you. Being a trader you can control the variance more so than you can with letting bets ride.

      You can also use fractional Kellys by dividing f by some denominator to give a value that you feel more comfortable with.

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