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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 $x_1$?

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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.

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