Harsanyi's utilitarian theorem

Assumptions

1. The society (or group) is rational and maximizes expected social welfare.
2. Each individual in the group is rational and maximizes his individual's expected utility.
3. If every individual in the group is indifferent between two probability distributions, then the group as a whole is indifferent as well.

Statement

John Harsanyi, through this theorem, proves that a social welfare function that satisfies these three assumptions, would be the weighted sum of expected individual utility functions.^[2]

Mathematically, the social welfare function ${\displaystyle W}$ social welfare function ${\displaystyle W}$ can be represented as a linear combination of individual utilities:^[3][4]$${\displaystyle W(x)=\sum _{i=1}^{n}a_{i}u_{i}(x)+c}$$

where:

• ${\displaystyle u_{i}(x)}$ is the von Neumann–Morgenstern (vNM) utility function of individual ${\displaystyle i}$.
• ${\displaystyle a_{i}}$ is the weight assigned to individual ${\displaystyle i}$, reflecting the individual's contribution to social welfare.
• ${\displaystyle c}$ is a normalization constant, that does not affect the ordering of social preferences.