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?
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.