Game Theory: meaning of notation $\text{X}_{i=1}^n A^i$ and $\times_{i\in I}A^i(s)$

I am reading a paper which discuss Game Theory and Nash equilibrium. What is the meaning of the symbol $$\text{X}_{i=1}^n A^i$$, as circled below:

I also found another paper which describe joint action $$a \in \times_{i \in I}A^{i}(s)$$:

Can anybody explain the meaning of the symbol $$\times_{i \in I}$$ highlighted?

• That means you have a cartesian product indexed by the set $I$. Jun 7 '20 at 17:50

The symbol $$\times_{i=1}^n A_i$$ is the Cartesian product of the sets $$A_1,\ldots,A_n$$. That is, the set of ordered tuples $$(a_1,\ldots,a_n)$$ such that $$a_1$$ belongs to $$A_1$$, $$a_2$$ belongs to $$A_2$$, and so on. Other texts use the notation $$\prod_{i=1}^n A_i$$ or the notation $$\times_{i\in I} A_i$$ when $$I = \{1,\ldots,n\}$$.
When $$A_i$$ is the set of actions for player $$i$$, elements of the cartesian product $$\times_{i=1}^n A_i$$ are called action profiles and specify one action for each player.