Give the von Neumann-Morgenstern utility function.

A utility function has expected utility propety if u(g)=∑p_{i}u(a_{i}) for any g∈G where (p₁°a₁,...,p_{n}°a_{n}) is the simple gamble induced by g.

The von Neumann-Morgenstern utility function is typically denoted as $�(�)$, where $�$ represents an outcome or a set of outcomes. This utility function satisfies the expected utility property, which states that for any gamble $�$ with associated probabilities ${�}_{�}$ and outcomes ${�}_{�}$, the utility of the gamble $�(�)$ is equal to the sum of the probabilities weighted by the utility of each outcome:

$�(�)={\sum }_{�}{�}_{�}\cdot �({�}_{�})$

Here:

• $�(�)$ represents the utility of the gamble $�$.
• ${�}_{�}$ is the probability associated with the outcome ${�}_{�}$.
• $�({�}_{�})$ is the utility of the outcome ${�}_{�}$.
• The sum is taken over all possible outcomes of the gamble.

This property essentially captures the notion that individuals make decisions based on the expected utility of the outcomes rather than the outcomes themselves.