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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???

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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.

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  • $\begingroup$ My problem is, if player 1 does not choose \textit{Ri}, then none of the subgame's NEs are SPNEs of the whole game. To find out what player 1 does, I have to use backwards induction (right?). But since the subgame allows for players to play pure strategies, player 1 does not know what outcome to expect of the subgame, so he cant choose. $\endgroup$
    – Andre
    Feb 23, 2020 at 17:07
  • $\begingroup$ Player $1$ does know what outcome to expect. Suppose we fix some NE of the subgame after $Ri$. Player $1$ computes whether $Ri$ or $Sa$ is optimal for him, and he does this under the assumption that the fixed NE will be played if he chooses $Ri$. In an SPNE with this NE of the subgame, he is acting optimally given that he expects this NE to be played if he chooses $Ri$, and, in fact, it will be played if he were to choose $Ri$. In other words, it is part of the definition of an SPNE that player $1$'s expectation over what will happen if he chooses $Ri$ are correct. $\endgroup$
    – induction_is_a_laddah
    Feb 23, 2020 at 19:07

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