# Expected Utility And Calculating Bet Size

In a previous post, we talked about the importance of calculating EV – but how much should you bet on sports betting when you know it’s a profitable play?

To answer this question, we must first understand the utility concept.

## Expected Value

We recommend reading our previous post on calculating EV before going any further.

Come back to this one when you’re done:

## How To Calculate EV In Sports Betting

Now then

As you know, the expected value (EV) from a bet can be calculated by the following formula:

But this can be simplified.

Since the the Probability of Losing is 1 minus the Probability of Winning, we get the following:

where

EV is the most important number in sports betting.

It indicates if your bet is profitable.

Now, once you have calculated the EV of your bet, you must decide how much to bet.

This is where Daniel Bernoulli’s expected utility hypothesis comes into play:

In other words, you should not rash decisions on how much to bet based on the EV without thinking about its subjective consequences ie utility.

## Expected Utility

Let’s use a simplified form of the Monty Hall Problem.

Imagine you are standing in front of two doors.

Behind the first door is \$50,000.

Behind the second door is either \$100,000 or nothing.

You have an equal chance of picking either door.

Which one do you open? Mathematically speaking, both doors have the same EV.

If you were to open either door over and over again, it wouldn’t matter which door one you choose.

But in this scenario, you’re only allowed to make one choice.

If you open the first door, you’re guaranteed \$50,000.

If you open the second door, you’re relying on luck; get lucky and win \$100,000 or get unlucky and win nothing.

Given the large sums involved, most people would choose to take the guaranteed \$50,000.

Those that find more utility in certainties, such as choosing the first door, than from gambles, such as choosing the second door, demonstrate risk aversion.

Here’s what the typical risk aversion looks like for different types of investors:

Bernoulli investigated why people would prefer to take lower-risk bets, even if the alternative was theoretically more profitable.

If making as much money as possible wasn’t motivating their decisions, then what was?

Bernoulli solved this problem by switching to an expected utility way of thinking.

He proposed that the same amount of money changes in value depending on their existing wealth.

For example, \$70 is worth more to someone that makes \$10 an hour than it is to someone that makes \$10 a second.

This utility is logarithmic and is referred to as the diminishing marginal utility of wealth.

Compare the concept of utility to paying for insurance.

Most people prefer making regular insurance payments than risk not paying anything and facing a huge bill – even if the cost of the insurance payments ends up costing more in the long run.

Whether or not you decide to pay for insurance depends on its utility.

How To Calculate The Right Bet Size

This assumes that you’re not using a standard bankroll percentage sizing.

Now, moving away from the theory, a more practical application of this can be found in the form of the Kelly Criterion

While the motivation behind this application was completely different to that of Bernoulli, the Kelly Criterion is equal to the logarithmic utility function.

Bettors are required to wager a percentage of their net worth on a bet that is proportional to the EV and also inversely proportional to the probability of the bet winning.

Now recall the previous simplified EV calculation formula: EV = (P * O) -1.

We can now use that to create a bet size determined by the Kelly Criterion.

The formula is as follows:

In essence, Kelly bet sizes maximize the expected logarithmic utility.

However, Kelly is volatile, so it may not serve everyone’s needs.

Futher, Kelly requires you to have an accurate calculation of the probability of the bet winning.

But even so, the Kelly method ensures your bankroll is used to its full potential in the long run.