Optimal Betting: The Kelly Criterion and Bankroll Management

The Kelly Criterion betting strategy requires strict conditions, but it is designed to optimize your betting strategy to make money. The theory is quite simple and easy enough for even beginners to get started with. You are basically looking for good value in bets, and on the basis of just how good that value is, you allocate a portion of your funds.

It cannot be applied in all scenarios, as you specifically need wagers in which you have a slight edge, something rarely found in casino games. This is not a be all end all solution to making money, but the principles of Kelly betting can help any gamers or sports bettors learn how to manage their bankroll better. We will show you how to use this strategy, and the areas in which you can tweak it to your liking. Plus, we will also analyze some tricks and ways through which you can integrate Kelly betting into games it is not even intended for.

Basics of Kelly Criterion Betting

The Kelly Criterion formula calculates how much of your stake you need to place, based on your mathematical advantage over the house. The formula is the following:

F = [(b x p) – q] / b

Where:

• F = how much of your bankroll you should play with
• b = the wager payout (payout – 1, the stake)
• p = the probability of winning
• q = the probability of losing (1 – p)

The model optimises your bankroll and shows you when to put more money on “safer” bets and risk less on “less safe” bets. You can calculate all the integers above except for p – the probability of winning. This is a figure based on how likely you are to win your bet. However, this is a number that is extremely difficult to predict. You will have to come up with some percentage of winning, or expected value.

Such as predicting that Barcelona has a 65% chance of beating Real Madrid in the next El Clasico. In sports betting, it is easier, as you can fish out golden opportunities where underdogs are underestimated, or the odds on favourites are inflated. In blackjack, it will require card counting. But to be perfectly accurate, running count is not enough. You will also need to check how many decks are left and have get lucky with the shoe.

Predicting Your Probability of Winning

Casino games are designed to give the house an edge. The chances of winning each bet, and the payout are not perfectly aligned. The house will take a small percentage from the payout, therefore meaning you will need to win more times than the maths suggests to be in a profit. This rules out using the Kelly betting system, as you need a positive expected value.

In all casino games, the RTP is always under 100%. This means, that over the course of time, you will lose your money, and the expected value is negative for games. Let’s look at an example using French Roulette, where we know the payout and the exact probability of winning. A straight bet on a number comes at odds of 35:1 (36x) and the chances of winning that bet are 1 in 37. Therefore:

• b = 35
• p = 1/37, or, 0.027
• q = 36/37, or, 0.973
• F = [(35 x 0.027) – 0.973] / 35 = -0.0008

A negative number basically means don’t stake any money. If you were using some trick to predict where the ball will land, possibly narrowing down the options to a quarter of the wheel, you would only be picking from 9 segments. That would mean:

• b = 35
• p = 9/37, or, 0.243
• q = 28/37, or, 0.756
• F = [(35 x 0.243) – 0.756] / 35 = 22.14%

And even then, when you have limited the options to just 9 pockets, the Kelly Criterion doesn’t tell you to go all in but to wager 22% of your bankroll. Though bear in mind, that predicting the spin of a ball in roulette is technically illegal. Casino dealers may ask you to leave the premises if you are found to be using predictive software.

Probability Predicting and Blackjack

Blackjack is a unique game, because technically you can count cards and open up the possibility to bet smarter when you recognise an advantage. Running count is the most popular card counting method, where you assign +1 to cards valued 2-6, 0 to cards valued 7-9, and -1 to cards 10 through Ace.

Basically, you keep tabs on the count by adding +1, 0, or -1 based on each card that is drawn. The running count can be put into a formula to count the True Count. The formula is the following:

Running Count / Remaining Decks = True Count

When the running count is negative, the shoe is working against you, and most experts will leave the table as there will be scarce opportunities to gain the advantage. What you want is a positive count – which means more 10s and Aces left in the shoe. The more you penetrate the shoe, the better the True Count, which means you need a sturdy bankroll to sustain long gaming sessions.

A positive true count gifts players the advantage, roughly at +0.5% for each +1 in the count. A count of +2 gives a player an advantage of around +1%. Winning in blackjack pays even money (b = 1), and normally, a basic blackjack strategy reduces the house edge to 0.5%, meaning the +1% gives you a positive EV of 0.5%.

• b = 1
• p = 0.505
• q = 0.495
• F = [(1 x 0.505) – 0.495] / 1 = 0.1%

With a true count of +2, the Kelly Criterion would have you betting 0.1% of your blackjack bankroll.

Predicting Probabilities in Sports Betting

The Kelly Criterion is perhaps the most popular, and easy to use, in sports betting. This is because the house has absolutely no control on what happens in a sports event, and oddsmakers are also taking risks by guessing the odds. They use software and metrics to figure out the chances of a bet winning or losing. And then, they add a little juice to ensure they get some profit, and the rest is on you to decide.

But sports statistics and data can only cover so much of what may or may not happen. Stats don’t use psychological factors, or take into account any external forces that can influence the outcome. This is where your keen sports knowledge can come into use. To effectively use the Kelly, you need to look for bets that have extremely good value. Such, that you see an opportunity to win money. It can happen in any of the following cases:

• The moneyline odds on two evenly matched teams are both very long
• An underdog is seriously underestimated, and their spread is set high
• A team is on a winning or losing streak
• The oddsmakers overestimate home advantage

There are really countless reasons why oddsmakers may get it wrong, and you have to do your homework and look for any discrepancies. You may find that an underdog is so heavily underestimated, that they get a huge positive line in the spread markets. Or, optimism bias leads to a team on a winning streak being given extremely short odds to win their next match. When that happens, you can be sure that the odds on the underdog will increase significantly, to bring a balance to the betting line.

Sports Betting Kelly Example

Now, let’s look at an example where we bet on a soccer game. Manchester City hosts Real Madrid, and the oddsmakers reckon home advantage will hurt Real’s chances. The moneyline odds are the following:

• Manchester City: 2.1
• Draw: 3.7
• Real Madrid: 3.3

The implied probability of Real Madrid winning are just over 30%, but you think the oddsmakers have underestimated the guests. Instead, you think Real Madrid has a 60% chance of beating Manchester City. In the Kelly formula, b would be 2.3 (3.3-1). The winning and losing probabilities would then be 0.6 and 0.4, respectively.

• b = 2.2
• p = 0.6
• q = 0.4
• F = [(2.2 x 0.6) – 0.4] / 2.2 = 41.8%

It is quite a bold assumption, as you think Real’s chances are double what the house reckons. The Kelly formula states you should play over 40% of your bankroll on the bet. If it pays off, you are coming out with a tremendous profit.

Full Kelly vs Half Kelly (and Other Denominations)

In betting circles, you will hear Kelly, Half Kelly, Quarter Kelly, and other denominations being thrown around. There is an extra integer you can add to the formula, to adjust the Kelly bankroll. Here is the entire formula:

F = {[(b x p) – q] / b} x K

Where:

K = a number from 0.1 to 1

We used full Kelly before (K =1), which is the most aggressive Kelly Criterion betting strategy. Should you use a Half Kelly, just add K = 0.5. In the above sports betting example with Real Madrid, the Half Kelly formula suggests you to bet 20% of your stake. While it is good for mitigating your losses, the Half Kelly, or any other denomination, does make winning a slow process. Just take the blackjack example, where your true count is +2. Half Kelly would have you betting 0.05% of your stake.

The Kelly Criterion formula as it is cuts your mathematical chance of losing. You can apply Half Kelly if you want to, or are not sure about the probabilities. Like, say if you are a bit hesitant about Madrid to beat City. But otherwise, reducing from 0.1% to 0.05% may also curb your chances of capitalising on a good shoe. And experienced blackjack players know the gambler’s remorse of not taking full advantage of a good shoe.

Concluding Kelly Criterion and Alternatives for Casino Games

Kelly Criterion is perhaps best applied to sports betting. There will be more discrepancies, but more importantly, you have more time to make your calculations. In blackjack, you can’t really stop a game for 2 minutes to quickly calculate how much to bet. Speed and quick decision-making are key to success.

Now we have ruled out Kelly Criterion betting for casino games. For games such as video poker, baccarat, slots, and roulette, it is just not possible with the negative EV.

However, there are loads of universal betting systems that you can use in these games. These range from the Martingale system (most aggressive) to sequences such as Paroli or D’Alembert (less aggressive). Plus, you can always brush up on your knowledge by checking our gaming guides, to find out which bets have the most house edge, whether it is possible to reduce the edge, and how to best apply your bankroll.