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There are two agents with utility functions $g_1$ and $g_2$, where the agent with function $g_2$ has higher (absolute) risk-aversion. The agents face a lottery $((q,x_1),((1-q),x_2))$, i.e. agents receive $x_1$ with probability $q$ and $x_2$ with probability $(1-q)$.

The expected value of the lottery for the agents is $$ q \cdot g_i(x_1)+(1-q) \cdot g_i(x_2), $$ where $i\in \{1,2\}$. How can I prove that, for any value of $q$, this expected value is less for the agent with higher risk aversion? It is known that $(i) E(g_i(x))\leq g_i(E(x))$, and $(ii) CE_2 \leq CE_1$, where $CE_i$ is the certainty equivalent of agents.

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