## - Expected value and expected utility explained

## - Utility under uncertainty

## - How to calculate the optimal stake amount?

## The role of utility in profitable betting

Expected value, a concept that was first tackled by French scholars Pascal and Fermat in the 17th century as they attempted to decode a points game, illustrates the average winnings one might expect from a wager. It doesn't, however, offer much guidance on how much of one's capital should be put at risk per bet. This is where the notion of expected utility enters the picture.

## Expected value and expected utility explained

Expected value (EV) in betting is derived by taking your chances of winning (p) and multiplying it by the potential payout per bet, and then deducting the probability of a loss multiplied by the potential loss per bet. Since the probability of losing is the flip side of winning, expressed as 1 (or 100%) minus the probability of winning, this simplifies the calculation:

'o' signifies the European decimal odds offered by the bookie. For any gambler, expected value is crucial because it predicts whether they're likely to gain or lose money over time.

After determining the expected value, a gambler needs to decide how much of their funds to wager. The 18th-century mathematician Daniel Bernoulli argued that only the reckless decide on their stake based solely on the objective expected value, without considering the subjective impact of their bet, namely the appeal of the potential gain or loss. This subjective appeal is referred to as utility.

## Utility under uncertainty

Imagine being presented with two chests. The first holds €10,000 in cash. The second could contain either €20,000 or nothing, and it's a 50/50 chance. Which one would you pick?

Kelly's formula is designed to maximize a winning gambler's bankroll over time. This forms a typical utility dilemma. Mathematically, both chests have an expected value of €10,000. If you could play endlessly, your choice wouldn't matter. But you can only play once, making the law of large numbers irrelevant.

Choosing the first chest guarantees you €10,000. Opting for the second chest means your outcome is down to luck: win and pocket €20,000; lose, and you get nothing. Given these stakes, most folks opt for the sure €10,000.

From a utility standpoint, the surety of €10,000 far outweighs the risk of walking away empty-handed. Those preferring guaranteed outcomes over risky ones with the same mathematical expectation are exhibiting risk aversion.

## How to calculate the optimal stake amount?

Daniel Bernoulli posited that when faced with uncertainty, the typical rational response is risk aversion. He stated: “the utility derived from any small increase in wealth will be inversely proportionate to the quantity of goods previously possessed.” This means the more wealth you have, the less you value a little extra. This utility function, described as logarithmic, is also known as the diminishing marginal utility of wealth.

Although the Kelly Criterion can lead to notable fluctuations in returns, it allows winning bettors to maximize their bankroll in the long haul. This practical application of Bernoulli’s theory provides a framework for gamblers and investors alike, known as the Kelly Criterion.

John Kelly, while at AT&T's Bell Labs in 1956 dealing with long distance telephone noise issues, developed this, which gamblers and investors quickly adopted to optimize financial management and profit growth. While Kelly’s rationale differed from Bernoulli's, his criterion mathematically aligns with the logarithmic utility function. Essentially, it advises a bettor to bet a portion of their total wealth on a bet proportional to the expected value (EV) and inversely proportional to the success probability.

Remembering that EV = po – 1 (where p is the ‘true’ probability of success and o represents the decimal odds of the bet), we can determine the Kelly stake percentage (K) as:

## Utility maximization through the Kelly Criterion

Fundamentally, the Kelly Criterion aims to maximize expected logarithmic utility. One result of employing the Kelly Criterion is substantial fluctuation in returns, a trait that may not suit every bettor’s preference for utility. Moreover, its application necessitates accurate calculations of ‘true’ probabilities of outcomes.

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