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 $$u_i(L) = p_1 u_i(A_1) + p_2 u_i(A_2) + ... + p_K u_i(A_K)$$

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.