cricbaba:How to calculate the optimal betting stake?

Understanding the Kelly Formula and Risk Aversion in Betting

Cricbaba suggests that when people make decisions under uncertainty, they tend to avoid risk. He supports this idea with a hypothesis: “The benefit of gaining a small amount of additional wealth decreases as your existing wealth increases.” In other words, the more you own, the less each new dollar is worth to you. This concept is known as diminishing marginal utility of wealth, often represented with a logarithmic function.

What Is the Kelly Formula?
The Kelly Formula is a popular method for managing betting bankrolls. John Kelly developed it in 1956 while working at AT&T Bell Laboratories, originally to solve noise issues in long-distance telephone lines. However, gamblers and investors quickly adopted the formula because it offers a way to maximize long-term growth.

The basic idea behind the Kelly Formula is to determine the optimal percentage of your bankroll to bet based on the expected value (EV) and the odds of winning. It’s mathematically similar to the logarithmic utility function described by Daniel Bernoulli. In simple terms, the formula tells you to bet a percentage of your bankroll that’s directly proportional to the expected value of the bet and inversely proportional to the odds of success.

The Kelly Formula in Practice
To understand how the Kelly Formula works, let’s break it down:

– Expected Value (EV): This is calculated as \( p \times o – 1 \), where \( p \) is the true probability of success, and \( o \) represents the European odds.
– Kelly Bet Percentage (K): Given the expected value, you can calculate the percentage of your bankroll to bet using the following formula: \( K = (p \times o – 1) / (o – 1) \).

The Kelly Formula aims to maximize expected log utility, allowing bettors to grow their bankrolls as much as possible over the long term.

Challenges with the Kelly Formula
While the Kelly Formula can help maximize long-term returns, it does have some drawbacks:

– Large Swings in Returns: Because the formula encourages larger bets when the expected value is high, it can lead to significant fluctuations in your bankroll.
– Requires Accurate Probability Estimates: To use the Kelly Formula effectively, you need a precise estimate of the “true” probability of winning, which can be challenging to calculate.

Despite these challenges, the Kelly Formula is a useful tool for bettors who are looking to maximize their bankroll over time. It helps them manage risk and make informed betting decisions based on their analysis of odds and probabilities.