# Expected Utility

Expected utility theory is a mathematical framework used in economics and finance to model decision-making under uncertainty. It was developed by the Swiss mathematician Daniel Bernoulli in the 18th century, but has since been formalized and expanded upon by other economists.

The basic premise of expected utility theory is that individuals make decisions by comparing the expected utility of different outcomes. Utility is a measure of an individual’s satisfaction or happiness and is often thought of as a subjective value. Expected utility refers to the expected value of the utility that an individual would receive from a particular outcome.

The expected utility of an outcome is calculated by multiplying the probability of the outcome occurring by the utility that would be received from that outcome. The decision-maker then chooses the option that maximizes their expected utility.

One of the key assumptions of expected utility theory is that individuals are rational, meaning they consistently make decisions that maximize their expected utility. Additionally, it is assumed that individuals have well-defined, stable preferences and that they can assign a numerical value to their preferences, allowing them to compare the expected utility of different options.

Expected utility theory can be used to model a wide range of decision-making situations, including investments, insurance, and risk management. In financial markets, for example, investors can use expected utility theory to make decisions about portfolio diversification, risk management, and investment strategy.

One of the strengths of expected utility theory is that it provides a formal framework for analyzing risk-taking behavior. In particular, it can be used to model how individuals trade off risk and reward when making decisions. For example, an investor may be willing to accept a higher level of risk if the expected reward is high enough.

Expected utility theory can also be used to model the effect of psychological factors, such as risk aversion and overconfidence, on decision-making. For example, if an individual is risk averse, they will be less likely to take on risky investments, even if the expected reward is high.

Despite its strengths, expected utility theory is not without its criticisms. One of the main criticisms of the theory is that it relies on several assumptions that may not hold in practice. For example, it assumes that individuals are rational, have well-defined, stable preferences, and can accurately assign numerical values to their preferences.

Additionally, expected utility theory has been criticized for its simplicity and its inability to fully capture the complexity of real-world decision-making. For example, it does not take into account the impact of emotions and biases on decision-making, and it assumes that individuals make decisions in isolation, rather than taking into account the impact of their decisions on others.

Despite these criticisms, expected utility theory remains a widely used framework for modeling decision-making under uncertainty. It provides a simple, yet powerful tool for analyzing risk-taking behavior and understanding how individuals make decisions in uncertain environments. Additionally, it has been successfully applied to a wide range of real-world situations and continues to be an important area of research in economics and finance.

The core concepts of expected utility theory involve preferences for one enterprise or venture over another when there are random prospects, with the enterprises or ventures being called “lotteries”.

This is commonly applied to gaming, but recent events involved financial ventures that were virtually gaming with risky investments based on probabilities of certain outcomes, such as mortgage failures. The outcomes of these high risk financial ventures had a huge impact on the world’s economies.

The probabilities are considered to be “objective”, or part of natural forces and not under any influence by the person. There is a matter of choosing among lotteries and trying to find the best choice. The “utility” is in the outcome or consequence of the choice. Given the probability that an outcome will be positive, the preference is for the lottery that has the best probability. However, preference can form over many lotteries or can be formed by participating in lotteries.

Expected utility theory originates with Daniel Bernoulli in 1738 and possibly earlier with Gabriel Cramer in 1728. Daniel Bernoulli attempted to solve his cousin’s “St Petersberg” paradox of infinite utility, which begins with people valuing the outcome from random ventures such as the toss of a coin that pays when “heads” comes up. Heads will come up with 50-50 odds of winning for each toss, even if the coins are tossed for an infinite period of time. There is an infinite 50/50 probability of winning, so people should find reasons to invest infinite amounts of money to play the random game of coin toss.

But Bernoulli found that wealth does not have a linear relationship to the utility that is related to wealth. In other words, there is over time, less and less of an increase in utility that relates to wealth.This is called “diminishing marginal utility”.

It is not the “win” or return that matters, it is the expected utility that matters when people assign value to a risky venture. Even though the expected return might be infinite, the expected utility causes a person to risk only a finite amount of money.

But that was not enough of a solution to the St Petersberg and similar paradoxes for John Von Neumann and Oskar Morgenstern . In 1944 the von Neumann-Morgenstern Expected Utility Theory brought in a new concept of of expected utility theory involving preferences for one enterprise or venture over another when there are random prospects, with the enterprises or ventures being called “lotteries”.

The probabilities are considered to be “objective”, or part of natural forces and not under any influence by the person. There is a matter of choosing among lotteries and trying to find the best choice. The “utility” is in the outcome or consequence of the choice. Given the probability that an outcome will be positive, the preference is for the lottery that has the best probability. However, preference can form over many lotteries or can be formed by participating in lotteries.

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