### Section 3: Kelly Criterion

Much of the following section will likely be familiar territory to readers who have spent time in private equity or as a part of a hedge fund, as the principles of optimal investment and growth are very similar in sports betting. Just like a portfolio of stocks, the long-term sports bettor has one goal: to capture the greatest long term-risk adjusted reward.

The first step towards accomplishing this goal is to utilize the Kelly Criterion (KC). The mathematics behind the KC were first alluded to by Daniel Bernoulli in the early 1700’s and were later separately explained and applied to finance and gambling by John Kelly Jr. in 1956. Like most brilliant formulas, the KC is simple: it states that when one is presented with an opportunity to invest in a series of positive expectation wagers, whether they are Dr. Bob’s sports bets, a card counter’s blackjack hands, a hustler’s unfair coin flips or dice rolls, a guru’s stock prices, or anything else, using the KC to determine the percentage of one’s bankroll to invest in each wager will result in the greatest long term growth.

The KC maximizes long term growth by finding the optimal percentage of a bankroll that should be invested which will cause the greatest expectation of the logarithm of the outcomes. Described less cryptically, this means that the single greatest expected growth can be achieved by investing the portion of your bankroll defined as (O*w-l)/O, where O are the odds received, w is the probability of winning and l is the probability of losing (1-w).

For example, consider the millionaire’s coin from the Primer, which will come up heads 57% of the time, and pays out even money. In this case, your odds are 1:1, your w is .57 and your l is .43, so the bettor should wager (1*.57-.43)/1 is .14 or 14% of your bankroll. Thus, the KC dictates that you should continuously invest 14% of your current bankroll on the next flip. Whether you have won or lost several in a row, if you keep investing 14% of your current bank roll on the next flip, you will grow your initial bankroll at an optimum (unbeatable) rate.

If you are rolling a fair die but are getting paid 10:1 when the side you pick comes up (instead of 5:1) then your KC would be calculated as ((10*.1666-.8333)/10) or 8.33%. Optimal Kelly growth dictates that you would wager 8.33% of your current bankroll per roll of the die.

In sports betting, we are typically getting laid 10:11 odds, so we have to bet 11 to win 10. If we know the likelihood of a particular team covering the spread, we can decide exactly what the optimum bet size is. If we have a 55% chance of winning, then we should invest ((.909*.55-.45)/.909) or 5.5% of our bankroll. If we are particularly certain about a game, and estimate a 59% win probability, then we should invest ((.909*.59-.41)/.909) or 13.9% of our bankroll.

Before we continue, note that these examples and the following simulation make two major assumptions: (1) that we can quantify our edge exactly, and (2) that the events take place sequentially. In real life, it is very difficult to perfectly quantify our edge, and we must often bet on several events at once (a week or a particular time block of games), and therefore the KC requires much modification from its pure form. These issues are discussed in Sections 5 and 6, but for now we continue on with a ‘perfect world’ (where all information is known and all events are sequential) simulation.