When I was studying Game Theory, I came across this equation: $$F_i(q_1, \ldots, q_n) = \sum_{s_1 \in S_1} \: \ldots \: \sum_{s_n \in S_N} \big\{ \prod_{j=1}^n q_j(s_j) \big\} \: \: f_i(s_1, \ldots, s_n)$$
How can one interpret this equation?
Additional Information: This is equation for a expected payoff function.
I think one can guess what the equation means. Presumably, the players are $1,2,...,n$, their pure strategies are denoted $s_1$,$s_2$, ...,$s_n$. The sets of pure strategies are $S_1,S_2,\ldots,S_n$. Mixed strategies are denoted by $q_i$, so that $q_i(s_i)$ is the probability with which player $i$ plays his or her pure strategy $s_i$. Because players choose their mixed strategies independently, the probability that we observe a strategy profile $(s_1,...,s_n)$ is just the product of the probabilities with which each player $i$ plays the pure strategy $s_i$ that is indicated for him or her in this profile, in other words, the probability of such a profile is: $$\prod_{j=1}^nq_j(s_j).$$ To get player $i$'s expected payoff we have to multiply each of these probabilities by the payoff that player $i$ gets when this strategy profile is played: $$\left\lbrace \prod_{j=1}^nq_j(s_j) \right\rbrace f_i(s_1,...,s_n)$$ where I assume that $f_i$ is player $i$'s payoff function. Finally, to get player $i$'s expected payoff we have to add up over all pure strategy profiles: $$\sum\limits_{s_1\in S_1} \ldots \sum\limits_{s_n\in S_n}\left\lbrace \prod_{j=1}^nq_j(s_j) \right\rbrace f_i(s_1,...,s_n)$$ which is the formula you asked about. Here, I assume that under the second summation sign in your formula there is a typo, and that it should read "$s_n\in S_n$" rather than "$s_n\in S_N$."