Aumann's paper the notation 3

Question

I'm reading the Aumann's collected works and I have a notation question from his paper here on the page 77 namely what is $\alpha$ in his notation $$F_i(\sigma)=\sum_{s}h_i(s)\prod_{j\in N}\alpha_j(s_j).$$

Answer 1

I believe it is a typo and it should be $\sigma$ instead, as $\sigma_j(s_j)$ is the probability of player $j$ playing $s_j$, so that $\prod_{j\in N}\sigma_j(s_j)$ is the probability of player 1 playing $s_1$ and player 2 playing $s_2$ and so on, so that $F_i(\sigma)$ is the expected payoff for player $i$ of the mixed strategy profile $\sigma$. This is consistent with the rest of the paper.