How did von Neumann and Morgenstern use utility to represent a player’s preferences?

John von Neumann and Oskar Morgenstern addressed the limitation that individuals do not always maximize average monetary profit by developing a theory of utility to represent player preferences.1

To accomplish this, they established a set of axioms for rational decision-making—such as the requirement that preferences remain consistent (e.g., if one prefers tea to coffee and coffee to milk, they must prefer tea to milk). Based on these axioms, they proved that a utility function could be defined to assign numerical values to a player’s alternatives, reflecting their relative attractiveness.1

This approach, known as the von Neumann–Morgenstern utility function, incorporates the expected utility hypothesis. It demonstrates that when faced with outcomes involving chance, rational players seek to maximize the expected value of their utility rather than just monetary gain. By assigning subjective values to outcomes, this framework allows for the mathematical representation of different attitudes toward risk, including risk-averse, risk-neutral, and risk-loving behaviors.2