In a normal form game $(N, A_i, u_i)$, consider a proper set $B_i \subseteq A_i$. This set satisfies the condition that for every action $a_i \in A_i$, there exists an action $a_i' \in B_i$ such that $u_i(a_i', a_{-i}) \geq u_i(a_i, a_{-i})$ for every action profile $a_{-i} \in A_{-i}$. What is the formal term for such a set $B_i$? Additionally, what term describes the set $B_i$ if it also satisfies the following criteria:
- For every $a_i \in A_i \setminus B_i$, there exists an $a_i' \in B_i$ for which $u_i(a_i', a_{-i}) \geq u_i(a_i, a_{-i})$ for all action profile $a_{-i} \in A_{-i}$ and the inequality is strict for some $a_{-i}$,
- For any $a_i \in B_i$, there is no $a_i' \in B_i$ that strictly dominates $a_i$, meaning $u_i(a_i', a_{-i}) \geq u_i(a_i, a_{-i})$ for some $a_{-i}$ and the inequality is strict for some $a_{-i}$.
I am currently using 'dominant strategy set' and 'strictly dominant strategy set' to describe these sets. Are these the formal names for sets with these properties? Is there any literature on these objects?