<p>You mostly got it, there are some minor mistakes.</p>

<ol>
<li>You cannot have the moving index $t$ of the Cartesian product in the superscript of said product
$$S_i={\sf X}^{t-1}_{t=1} A_i^{A^t}.$$</li>
<li>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.</li>
</ol>

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

CC BY-SA 3.0
2017-07-31T23:27:39.863Z
2017-08-01T07:38:14.767Z
2017-08-01T07:38:14.767Z
Giskard
<p>Lets say two players, $i=2$ compete in a standard prisoners dilemma game where the action space of each player $a_i \in A_i=\{C,D\}$</p>

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

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

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

<p>Hence the strategy set $S_i$ of player $i$ for the <em>entire</em> T-repetition game is the cartesian product</p>

<p>$$S_i={\sf X}^{t-1}_{t=1} A_i^{A^t}  $$</p>

<p>My question is this:</p>

<p>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:</p>

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

CC BY-SA 3.0
2017-07-31T14:14:02.100Z
What do you mean by strategy set $S_i$ at time period $t$? Is that the part of the strategy that decides everything until round $t$, or the part that decides round $t$ only?
Giskard
CC BY-SA 3.0
2017-07-31T19:11:52.473Z
Thank you for you comment. Sorry made the correction. The strategy set is for the entire game. I just want someone to confirm that the strategy set I wrote for the entire game is correct, ie a giant cartesian product.
jessica
<game-theory>
Game Theory: T-Fold Repetition Game
jessica
# Game Theory: T-Fold Repetition Game

(Asked by: username_0 on 2017-07-31)

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:

$$S\_i={\sf \cup}^{t-1}\_{t=1} H\_t\rightarrow A\_i $$

**Tags:** game-theory
> **Comments:**
> **username_1**: What do you mean by strategy set $S_i$ at time period $t$? Is that the part of the strategy that decides everything until round $t$, or the part that decides round $t$ only?  
> **username_0**: Thank you for you comment. Sorry made the correction. The strategy set is for the entire game. I just want someone to confirm that the strategy set I wrote for the entire game is correct, ie a giant cartesian product.  

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## 1 answers

### Answer by username_1 on 2017-07-31. Score: 3

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
$$
S\_i={\sf X}^{T-1}\_{t=0} A\_i^{A^t}.
$$


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