Game Theory: T-Fold Repetition Game

Question

Lets say two players, $i=2$ compete in a standard prisoners dilemma game where the action space of each player $a_i \in A_i=\{C,D\}$

Let say the game is repeated $T$ times and all information of past histories is common knowledge.

the strategy function for any player $i$ at time period $t$ is $$s_i^{t}:H_t \rightarrow A_i$$

where the history set at time $t$ is $H_t=A^t=(\{C,D\} \times \{C,D\})^t$

Hence the strategy set $S_i$ of player $i$ for the entire T-repetition game is the cartesian product

$$S_i={\sf X}^{t-1}_{t=1} A_i^{A^t} $$

My question is this:

Did I define the strategy set (above equation) of the player for the $T$ repetition game correctly? The reason why I ask if because lots of books define the strategy set differently, as the union of all histories mapped into the action space of the player, as such:

$$S_i={\sf \cup}^{t-1}_{t=1} H_t\rightarrow A_i $$

Answer 1

You mostly got it, there are some minor mistakes.

1. You cannot have the moving index $t$ of the Cartesian product in the superscript of said product $$S_i={\sf X}^{t-1}_{t=1} A_i^{A^t}.$$
2. At the start there have been no moves, so in round 1 you should map the empty history $H_0 = A^0$ to $A_i$, not $H_1 = A^1$ as you seem to have done.

Considering these I think the strategy set of $i$ is $$ S_i={\sf X}^{T-1}_{t=0} A_i^{A^t}. $$