I think, my question is best explained with an example, but general answers are, of course, also appreciated: Dynamic game with n-players: First: Player 1 has to choose between a save option (Sa) and a risky option (Ri). If he plays Sa, the game automatically ends, and everybody receives a payoff of 0. If he plays Ri, the remaining players will play a Chicken game, in which they move simultaneously and Player 1’s payoff will depend on the outcome of the chicken game, which looks like this:
\begin{align} \Gamma &= \{N, S, u\} \\ N &= \{1, 2\} \\ S &= \{ \{S, D\}, \{S, D\} \} \\ u &= \{a, b, c, d\} \forall \{ a, b, c, d \in \mathbb{Z} : a < b < c < d \} \end{align}
\begin{align} u_1(D, D) = u_2(D, D) = a\\ u_1(S, D) = u_2(D, S) = b\\ u_1(S, S) = u_2(S, S) = c\\ u_1(D, S) = u_2(S, D) = d \end{align}
If it comes to a “crash” (because everybody plays D), Player one will receive -10, if all other Players play the Swerve (S) option Player one receives a payoff of 10. Here is my attempt at notating the entire game:
\begin{align} \Gamma &= \{N, A, X, E, \iota , u\} \\ N &= \{1, 2, ..., n\} \\ A_1 &= \{Sa, Ri\} \\ A_{-1} &= \{S, D\} \\ u &= \{a, b, c, d\}\ \forall \ \{ a, b, c, d \in \mathbb{Z} : a < b < c < d \} \end{align}
\begin{align} u_1(Ri, \nexists S_{-i}) = -10\\ u_1(Sa, \exists D_{-i}) = u_1(,Ri \exists S_{-i}) \land \exists D_{-i} = 0\\ u_1((Ri, \nexists D_{-i}) = 10 \end{align}
\begin{align} u_{-1}(D_i, \nexists S_{-i}) = a\\ u_{-1}(S_i, \exists D_{-i}) = b\\ u_{-1}(S_i, \nexists D_{-i}) = c\\ u_{-1}(D_i, \exists S_{-i}) = d \end{align}
Edit for clarification:
I know that the chicken-sub-game has one NE in mixed strategies and n-1 NEs in pure strategies. If the players in the subgame (player {2..n}) play pure strategies, every outcome is possible (right?). And there is no way of calculating what outcome will be reached to what likelihood (right?). Ergo player 1 does not know what strategy to choose. -> I cant know what the SPNE is. Right???
There is nothing in the definition of a subgame perfect NE that requires it to be unique.
The game you describe seems to have multiple SPNE. If you simply want to find one SPNE, pick one of the Nash equilibria of the subgame after player $1$ has played $Ri$ that you like. Then, find the best response of player $1$ if she anticipates that choosing $Ri$ will yield her the payoff from the chosen NE of the subgame. The best response of player $1$ together with the strategies played in the NE of the subgame form an SPNE.
To find another SPNE, you could pick another NE of the subgame and repeat the procedure.