Prove that for every Nash equilibrium $\sigma^*$, the probability distribution $p_{\sigma^*}$ is a correlated equilibrium

Question

This is a classic theorem in game theory, that is left as an excersice in my textbook. Can anybody proove it? I can not thing of anything excpet from the definition of the correlated equilibrium in first place. Here is the theorem and the definition as well.

$\mathbf{Theorem:}$ For every Nash equilibrium $\sigma^*$, the probability distribution $p_{\sigma^*}$ is a correlated equilibrium.

$\mathbf{Definition:}$ A probability distribution $p$ over the set of action vectors $S$ is called a correlated equilibrium if the strategy vector $\tau^*$ is a Nash equilibrium of the game $\Gamma^*(p)$. In other words, for every player $i ∈ N$:

\begin{equation}\Sigma_{s_{-i}\in S_{-i}}p(s_i,s_{-i})u_i(s_i,s_{-i})\geq \Sigma_{s_{-i}\in S_{-i}}p(s_i,s_{-i})u_i(s^{'}_i,s_{-i}),\quad\text{$\forall s_i,s^{'}_i\in S_i$}\end{equation}

Every strategy vector $\sigma$ induces a probability distribution $p_{\sigma^*}$ over the set of action vector $S$. \begin{equation}p_{\sigma^*}(s_1,...,s_n)=\sigma_1(s_1)\times\sigma_2(s_2)\times...\times\sigma_n(s_n)\end{equation}

$\textit{Hint:}$ When we relate to a Nash equilibrium $\sigma^*$ as a correlated equilibrium we mean the probability distribution $p_{\sigma^*}$ given by the aforementioned equation:

Answer 1

A strategy profile $\sigma^*=(\sigma_i^*,\sigma_{-i}^*)$ is a Nash equilibrium if for all player $i$, \begin{equation} u_i(s_i,\sigma_{-i}^*)\ge u_i(s_i',\sigma_{-i}^*), \quad \forall s_i\in\mathrm{supp}(\sigma_i^*), \forall s_i'\in S_i. \end{equation} Rewriting this condition explicitly in terms of the probabilities over pure strategies: \begin{align} \sum_{s_{-i}\in S_{-i}}\sigma_{-i}^*(s_{-i})u_i(s_i,s_{-i}) &\ge \sum_{s_{-i}\in S_{-i}}\sigma_{-i}^*(s_{-i})u_i(s_i',s_{-i}), \quad \forall s_i\in\mathrm{supp}(\sigma_i^*), \forall s_i'\in S_i. \tag{1} \end{align} Let $p_{\sigma^*}(s_i,s_{-i})=\sigma_i^*(s_i)\sigma_{-i}^*(s_{-i})$ be the joint distribution implied by the equilibrium strategies. Then condition $(1)$ can be rewritten as \begin{equation} \sum_{s_{-i}\in S_{-i}}\underbrace{\sigma_i^*(s_i)\sigma_{-i}^*(s_{-i})}_{p_{\sigma^*}(s_i,s_{-i})}u_i(s_i,s_{-i}) \ge \sum_{s_{-i}\in S_{-i}}\underbrace{\sigma_i^*(s_i)\sigma_{-i}^*(s_{-i})}_{p_{\sigma^*}(s_i,s_{-i})}u_i(s_i',s_{-i}), \quad\forall s_i,s_i'\in S_i. \end{equation} This is the definition of correlated equilibrium.