In the normal-form game below, the strategy set is $\{L,R\}$ for each players and $p_1$ denotes player $1$'s payoff and $p_2$ denotes player $2$'s payoff. Are the two games exactly the same? Does the change make any difference?
Game $1$:
$\begin{matrix} &&& \text{ Player } 1 && \\ & &L & &R \\&L &(p_1, p_2) &&(p_1, p_2) \\ \text{Player } 2\\ &R &(p_1, p_2) & &(p_1, p_2) \end{matrix}$
Game $2$:
$\begin{matrix} &&& \text{ Player } 2 && \\ & &L & &R \\&L &(p_1, p_2) &&(p_1, p_2) \\ \text{Player } 1\\ &R &(p_1, p_2) & &(p_1, p_2) \end{matrix}$
My second question is: What is the difference between actions and strategies? Can you please explain using this example? I think the strategies are simply $\{L,R\}$, but what about the actions? Can we write actions properly for infinite games?
Note: There's a slight abuse of notation as I wrote $p_1$ and $p_2$ instead of $p_1^{(i,j)}$ and $p_2^{(i,j)}$ (or something else) for the payoffs.