If you are keen on winning consistently in sports betting, it is very important that you use a betting strategy that has a positive EV or Expected Value, that is the correct estimation of the average amount you’ll win per bet. However, what’s the maximum capital you must risk on each bet for achieving these maximum profits? We’ll need to throw light on and comprehend the utility concept in order to understand this better.
The concept of Expected Value or EV was first explored by two mathematicians of French origin known as Fermat and Pascal in the 17th century. They did it while trying to solve a game of points problem. This concept tells us about the average amount we can expect to win from a bet. However, it doesn’t delve into the amount of capital that a sports bettor must risk on his bet. And this is where the concept of Expected Utility or EU comes into the picture.
Expected utility (EU) and Expected Value (EV) explained
You can easily calculate the Expected Value on your bets by multiplying your winning probability (p) with the money you stand to win per bet, and subtracting the losing probability multiplied by the money you stand to lose per bet. As the losing probability is equal to 1 (in other terms 100%) minus the winning probability, we can arrive at the simplification below:
EV = po – 1
The ‘o’ here is a representation of the European decimal odds that are provided by the bookmaker. This Expected Value figure is extremely important for any sports bettor, as it informs about whether he’ll be losing or making money from his betting endeavours in the long term.
Once a sports bettor has arrived at the Expected Value, he must figure the amount he is willing to risk on the bet. The widely known 18th-century mathematician named Daniel Bernoulli figured that it’s only unwise people who make decisions about the amount they must risk on bets based on objective Expected Value, not paying any heed to a bet’s subjective consequences (or the actual desirability of the thing that’s to be lost or gained). Utility or Expected Utility is the term used for this subjective desirability.
Utility in case of uncertainty
Let’s say you are provided with 2 chests. While the first one has £ 10,000 cash, the second one may have either nothing or £ 20,000 cash. You are unsure which one has what, but both the options are equally likely. Now, you are asked to select one chest from the two. Which one will you go for?
It’s a classic example of a utility puzzle. Mathematically speaking, both the chests have the same amount of Expected Value, which is £ 10,000. If you were to repeat this game multiple times endlessly, it’d make no difference to the chest you had picked. But, you can play this game only once. There is no application of the law of large numbers here.
If you choose the first chest, £ 10,000 is guaranteed. But if you go for the second one, it becomes nearly a matter of chance. If it’s your lucky day you may end up with £ 20,000, else you may get nothing. Unsurprisingly, considering the amount of money at stake, majority of people opt for the certainty provided by the first chest.
However, looking at it from the perspective of utility, the certainty of getting £ 10,000 is definitely far better compared to the risk of getting nothing. People who are able to find higher utility in certainties compared to gambles with the same mathematical expectation demonstrate what is known as risk aversion.
How to find out the optimal betting amount?
Bernoulli reasoned that the normal rational behaviour of majority of people when asked to take decisions under uncertain circumstances, is risk aversion. As per the quantification of his hypothesis, the utility arising from any minor increase in wealth is inversely proportional to the amount of goods possessed previously. Putting it another way, the more amount of wealth you are already in possession of, the lesser utility you’ll perceive from gaining more of it. This utility function is also logarithmic, and is commonly referred to as diminishing marginal utility of wealth.
Another major practical application of this Daniel Bernoulli theory can be seen in a money management plan referred to as Kelly Criterion by many sports bettors. This plan was a creation of John Kelly while he was working with AT&T Bell Labs back in the year 1956 on a problem related to long distance phone noise. It didn’t take much time for investors and gamblers to lap up this plan as an effective means of optimising their profit growth in money management.
Although the motivation behind Kelly’s work was completely different from Bernoulli, his criterion was mathematically equal to logarithmic utility function. In practical terms, it encourages a sports bettor to stake a percentage of its total wealth on a bet which is both directly proportional to EV or Expected Value and inversely proportional to the chances of success.
EV = po - 1
(where ‘p’ stands for actual probability of success, and ‘o’ stands for the bet’s decimal odds)
We can figure out the Kelly bet percentage or ‘K’ using the following formula:
K = (po – 1) / (o – 1)
Basically, Kelly criterion maximises the expected logarithmic utility. An important consequence of using Kelly criterion in betting endeavours is major volatility in the returns, something which may not be in best interest of everyone when it comes to utility. In addition, its correct usage requires precise estimations of actual probabilities of the outcomes.
Despite the fact that the use of Kelly Criterion may lead to major volatility and returns, it allows winning sports bettors to make maximum profits and increase their bankroll in the long term. Obviously, in order to do so, it’s important for a sports bettor to opt for a bookmaker which doesn’t frown upon money management strategies such as Kelly. More importantly, the bookmaker must not restrict the sports bettor’s betting activities simply because he wins more regularly. In that regard, Bet365 is one of the best bookmakers you can work with.