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???