We have a multistage game where it is played by $I<\infty$ players. This game is played for $T$ rounds. If we suppose that $K$ players where $K<I$ are not going to continue playing in the game after some stage $\tau^*<T$, then what is the mapping that we should define that will model the number of players that are going to continue or stop the game?
For example, we have $I=4$, players, then either all players or some of them will play the game for every one of the $T$ rounds. Considering every player with his index, as $1$ for player $1$, $2$ for player $2$ and so on, at t=1, there are the following combinations:
The game will be played only by player $\{1\}$, or $\{2\}$, or $\{3\}$ or $\{4\}$ or $(\{1\},\{2\})$, or $(\{1\},\{3\})$, or $(\{1\},\{4\})$, or $(\{2\},\{3\})$ or $(\{2\},\{4\})$ or $(\{3\},\{4\})$ or $(\{1\},\{2\},\{3\})$, or $(\{1\},\{2\},\{4\})$, or $(\{2\},\{3\},\{4\})$, or $(\{1\},\{3\},\{4\})$, $(\{1\},\{2\},\{3\},\{4\})$ or by none of the players. What is this mapping and how does this change it at least one player must play the game ant any round until the terminal round $T$?
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.