# Comparing Nash equilibrium and Pareto optimal actions

Suppose that $(x_{i}, x_{j})$ identify actions for two players $(i,j)$. If we define Pareto optimal actions by $$h(x_i) +h(x_j)- \eta[p(x_i)+p(x_j)]=2\gamma$$ and Nash equilibrium actions by $$h(x_i) +h(x_j)- \eta[p(x_i)+p(x_j)]=\gamma$$ where $h'(.)<0$, $p'(.) \leq 0$ and $\gamma, \eta >0$ are constants.

The aim is to compare whether Pareto optimal actions $X^p\equiv (x_i ^p,x_j ^p)$ are greater than those that identify Nash equilibrium $X^n\equiv (x_i ^n,x_j ^n)$, i.e., whether $x_k ^p\leq x_k ^n$ for $k={i,j}$. If they can be comparable, please suggest how to prove that. If relevant, players can be assumed identical.

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