Difference between action and strategy in game theory

Question

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.

Answer 1

As for the first question, in the book which I used for studying, the first players was always put at the left, and the second at the top, so following this notation, in the game 2 the payoff vectors should be (p2,p1).

As for the second question, the strategy set for a player is the set of all strategies that the player can choose, in this case, the set of strategies for the player 1 is {L,R}. A profile of strategies is a vector that has a strategy of each player, in this case: (L,L), (R,L), (L,R) and (R,R). I didn't use the word action, but I suppose it depends on the book that you're reading.