Define the mapping that models the number of players that are going to continue or stop a game

I am not sure I get your question right. For simplicity, I will assume that you are talking about a repeated game. That is, in each round $t$ the players play the same stage game. Moreover, from your description it is unclear whether the number of players is deterministic in each round, random, or history dependent.

Let's start with deterministic. Denote by $\mathscr{T}$ the set $\{1,\ldots,T\}$ and by $\mathscr{I}$ the set $\{1,\ldots,I\}$. If the number of players in each round is deterministic, the mapping you are looking for is $\sigma:\mathscr{T}\rightarrow \mathcal{P}(\mathscr{I})$. That is, for each $t$, $\sigma(t)$ selects an element from the power set of $\mathscr{I}$.

If the number of players in each round is random, the mapping you are looking for is $\sigma:\mathscr{T}\rightarrow \Delta\mathcal{P}(\mathscr{I})$. That is, for each round $t$, $\sigma(t)$ selects a probability distribution over the power set of of $\mathscr{I}$.This is well defined as $\mathscr{I}$ is a finite set and so is its power set. If you want that at least one player participates in each round, you have to assume that $\sigma(t)$ always assigns a probability of $0$ to the empty set.

Now suppose the mapping you are looking for is history dependent. Let $A_i$ be the actions available in the stage game to player $i$ if he is playing in stage $t$. Denote by $(a_1^t,\ldots,a_n^t)$ the action profile of the players in stage $t$. If player $i$ was not playing in stage $t$, set $a_i=\text{'out'}$. Denote by $h^{t+1}=(a^1,\ldots a^t)$ the history of the game after stage $t$. Let $H^t$ denote the set of all possible histories at stage $t$ and by $\mathscr{H}$ the union $\bigcup_{t=1}^T H^t$. The mapping you are looking for is $\sigma: \mathscr{H}\rightarrow \mathcal{P}(\mathscr{I})$. That is, for each history $h^t$, $\sigma(h^t)$ selects an element form the power set of $\mathscr{I}$. Moreover, for a complete definition of the game, the action set available for player $i$ after history $h^t$, $A_i(h^t)$, is $A_i$ if $i$ is an element of $\sigma(h^t)$ and 'out' else.