Lets say Agent 1 has a utility function that depends on the other person, i.e., $u_1(x_1-x_2)$, where $x_i$ is the choice of Agent $i$. Suppose the expected value of $x_2$ is denoted $E[x_2]$.

Can $u_1(x_1-E[x_2])$ be defined as Agent 1's expected utility from choosing some $x_1$?


1 Answer 1


No, $u_1(x_1-E[x_2])$ is agent $1$'s utility of the expected value, not the expected utility of the value.

Generally $u(E(x)) \neq E(u(x))$. A simple example:

Let $x$ take value $-1$ with probability $50\%$ and value $1$ with probability $50\%$. Let $u(x) = x^2$. Then $$E(x) = 50\% \cdot (-1) + 50\% \cdot 1 = 0$$ $$u(E(x)) = 0^2 = 0$$ $$E(u(x)) = 50\% \cdot (-1)^2 + 50\% \cdot 1 = 1$$

There are special cases where $u(E(x)) = E(u(x))$ does hold, e.g., this is always true when $u$ is an affine function. The equality never holds when $u$ is strictly convex, see Jensen's inequality.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.