The Kelly Criterion is a math formula that tells you what fraction of your bankroll to wager on a given bet to attain the highest growth expectation for your bankroll.  It was developed in 1956 by physicist and Bell Labs research scientist John L. Kelly.

The formula is quite simple from a math standpoint:

f = (bp-q)/b

where

• f is the portion of the current bankroll to wager
• b is the ratio of profit to amount risked on the bet when you win
• p is the probability of winning the bet
• q is the probability of losing the bet (1-p)

If the calculated bet size is 0 or negative, you should not take the bet.  You will be losing money in the long run.

A simple illustration of this would be a situation where you are rolling a single die.  If the number rolled is a 6, you win 7 times the amount wagered.  If the number rolled is 1,2,3, 4 or 5, you lose your bet.  What is the optimal amount to bet on this game?

• b = 7
• p = .1667
• q = .8333

f = [(7)(.1667) - .8333]/7

f = 0.0477

Your Optimum bet size in this situation would be 4.77% of your bankroll.  As your bankroll grows the dollar amount you bet would also grow. The 4.77% would remain constant.

This formula can be applied to sports betting with great effect, but only if the bettor can accurately estimate the probability that his bet will win. The greatest risk of inaccurately assessing the win probability is that you may be overbetting your bankroll, which can lead to decreasing results over time. Consistently overbetting an advantage play will turn your advantages into an expectation of decreasing your bankroll. 

Let's look at an example of that.

We are offered a series of sports bets that we know with 100% surety have a 50% chance of winning.  However, we are getting these bets at +200.  We have  a 1 - 1 chance of success with a 2 - 1 payoff.  What a phenomenal opportunity!  We definitely want to make the most of this situation.  Kelly tells us to bet f = [(2)(.5) - .5]/2 or 25% of our bankroll on every bet.  But this is such a great opportunity, we decide to ditch Kelly and bet 60% of our bankroll on each bet. 

Let's start with a $100 bankroll and look at our expected results after 4 bets.  Our expectation is that we would win 2 bets and lose 2 bets.  We bet $60 on the first bet and we win, adding $120 to our bankroll.  Now we bet 60% of $220 or $132 on our second bet, and we win that one, too.  Our bankroll is now $484.  We bet 60% or $290.4 on the third bet and we lose.  We are down to $193.60.  We bet 60%, or $116.16 on the fourth bet and lose.  Our bankroll after 4 bets is now $77.44.  We have taken a great opportunity and turned it into a situation where we can expect to lose money because we are overbetting our bankroll. (Note that the sequence of wins and losses does not have any impact on the final expected bankroll.)

My example was very simplistic for illustrative purposes.  You will never get such a huge advantage on a sportsbook.  You can, however, get small advantages.  When you do it is important to not overbet them.  This is the reason that nearly all sports bettors lose money in the long run.  They bet too much on a small advantage.  They might know sports enough to gain an advantage, but they don't know how much to bet.  Consistently betting more than twice the Kelly value creates an expectation of negative bankroll growth.  (Betting exactly 2X Kelly gives a 0 expectation.)

For poker players I use the analogy of a drawing hand.  The turn card gives you a gut shot draw. Your opponent has a big pair.  If you river the straight your hand will win.  There are 3 bets in the pot and your lone opponent bets into it, giving you 4-1 pot odds to make the call.  If you call the bet you are overbetting the advantage that you have with a gut shot draw.  You will pick up some pots, but you have created the expectation of negative bankroll growth with this play over time.

Knowing the exact value of your advantage in a sports bet tends to be very difficult, so as a precaution against overbetting many bettors will bet half the Kelly recommended amount.  This Half Kelly method produces about 75% of the rate of return of full Kelly, while also diminishing volatility and preventing inadvertent overbetting when the estimated win probability (p) is too high. 

Kelly Criterion Example