Distinguish between utility function and expected utility function. Discuss the assumptions associated with the Von Neumann Morgenstern Utility Function.

 Utility function and expected utility function are both used to describe the preferences of individuals when making choices under uncertainty.

A utility function is a mathematical function that assigns a numerical value to each possible outcome of a decision. It represents the individual's preference for each possible outcome, with higher values indicating greater preference or satisfaction. Utility functions are used to analyze choices in situations where outcomes are known with certainty.

On the other hand, an expected utility function takes into account the probabilities of each possible outcome, as well as the individual's preference for those outcomes. It calculates the expected value of the utility function across all possible outcomes, taking into account the likelihood of each outcome. Expected utility functions are used to analyze choices in situations where outcomes are uncertain or risky.

The Von Neumann-Morgenstern (VNM) utility function is a widely used expected utility function that represents the preferences of individuals under conditions of risk. It is based on the following assumptions:

1. Completeness: The individual has a complete and transitive preference relation over all possible outcomes.

2. Continuity: The individual's preferences are continuous and can be represented by a continuous utility function.

3. Independence: The individual's preferences are independent of irrelevant alternatives. This means that the relative ranking of two outcomes is unaffected by the presence or absence of other outcomes.

4. Rationality: The individual's preferences satisfy the axioms of expected utility theory, meaning that they prefer outcomes with higher expected utility.

The VNM utility function is widely used in decision-making under risk, such as in finance, insurance, and economics. However, it has been criticized for its assumptions, particularly the assumption of independence, which has been shown to be violated in some situations. Despite these criticisms, the VNM utility function remains a widely used tool for analyzing decision-making under risk.

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