1
$\begingroup$

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:

enter image description here

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

enter image description here

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

$\endgroup$
1
  • 3
    $\begingroup$ That means you have a cartesian product indexed by the set $I$. $\endgroup$
    – John Douma
    Jun 7 '20 at 17:50
0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.