Linear utility function

Definition

A utility function $u_i$ is called linear if for every lottery $L = [p_1(A_1), p_2(A_2), ..., p_K(A_K)]$, it satisfies

Notes

Note that $u_i$ is a linear function in the probabilities $(p_k)_{k=1}^K$.
If a utility function is linear, the utility of the lottery is the expected value of the outcomes.
A linear utility function is also known as a von Neumann-Morgenstern utility function.

References

1. M. Maschler, E. Solan, and Shmuel Zamir, Game Theory, Cambridge University Press, 2013, p. 14.