4.2a History

History

The history $h_t$ of a repeated game at period $t$ is the set of all possible actions taken by all players at times $\tau <t$. Formally, the history of a repeated game at time $t$ given the strategy profile of every stage game at $\tau <t$ can be expressed as
$h_t=s_2\times s_3\times \dots \times s_{t-1}$

Note! $1$ is simply $h_1=\varnothing$, since there is nothing occurring before stage $1$ of the game.

The history of period

Properties

At period $t\ge 2$, the set $h_t$ specifies $n^{t-1}$ actions.

Example

Infinitely Repeated Prisoner's Dilemma

Consider again the infinitely repeated Prisoner’s Dilemma of two players. At $t=1$, there is no history, so $h_1=\varnothing$. However, at $t=1$, players can play the strategies $s_i=\{D,C\}$. As such, the history at $t=2$ is $h_2=\{(D,D),(D,C),(C,D),(C,C)\}$. Then, at $t=3$, players can again play the strategies $s_i\{D,C\}$. This leads to
$h_3=\{\{(D,D),(D,D)\},\{(D,D),(D,C)\},\dots ,\{(C,C),(C,D)\},\{(C,C),(C,C)\}\},$
specifying a complete history of all possible paths of play up to $t-1$.