Consider a game $G=(N, (A^i)_{i\in N}, (g^i)_{i\in N})$, $N=\{1,2,\dots,n\}$, $A=\Pi_{i\in N}A_i$ is the set of actions and $g^i:A\to \mathbb{R}$ is the payoff function. The latter can be extended from $\Delta(A)$, which is the set of (correlated) strategies, to the real line. If $S$ is a calition, it is a non-empty member of $2^N$ and $A^S=\Pi_{i\in S}A^i$ is the actions set of the memebers of the coalition where a memeber of $\Delta(A^S)$ is called an $S-$strategy profile. Also, $-S$ denoted the complementary coalition. Suppose that $U$ is the set of uncorrelated strategy profiles and $U^S$ the set of the uncorrelated $S$-strategy profiles .Given $q\in U$, we write $q = (q^S , q^{−S})$ where: $q^S\in U^S$ , $q^{−S} \in U^{−S}$. I am trying to understand the intuition of the following definition
$\mathbf{Definition:}$ An uncorrelated strategy profile $q\in U$ is a $k$-strong Nash equilibrium if and only if for all coalitions $S\subset N$ satisfying $|S|\leq k$ and for every uncorrelated $S$-strategy profile $p^S \in U^S$, there exists a player $i \in S$ such that
$$g^i(q)=g^i(q^S,q^{-S})\geq g^i(p^S,q^{-S})$$
$\textbf{Question:}$ I can not understand the intuition of the $k-$strong N.E. It is an equilibrium where up to $k$ players do not have an incentive to deviate from the strategy profile $q$. What confuses me is the part where it says ``there exists a player $i \in S$" and I wonder it is enough for the coalition to choose the strategy profile $q$ if and only if one member of her, say $i$ gets weakly better with respect to any other $(p^S,q^{-S})$ strategy profile?
