Consider a 2-player game: You and a robber. The robber tells You to give him all your money, otherwise he will kill You. However, the robber could be a 'Good' person (i.e. he would not kill You anyway) with probability $\mu$ or a 'Bad' person (i.e. he would kill You if you do not give him Your money, and not kill You if you don't) with probability $1-\mu$. And the robber knows about the type he is. How does the extensive form game look like (the payoffs are not of importance)?
What I tried so far:
- Nature determines the type of the robber. Then the robber should either kill You or not kill You. But then this sequence is incorrect: You should first decide to give money or not before the robber can choose to kill you or not.
- Nature determines the type of the robber, and the subsequent information set belongs to You. So right after Nature determines the type of the robber, You can make the choice to give the money or not. And the robber is next to move after the decision of You. If I do it this way, then I am not sure whether the extensive form is correctly defined (because in general Nature assigns the type of the robber and then the robber should move instead of You).
Indeed, your comment points out how it is different from the Beer-Quiche game. Fortunately, though, it's not that different. You can represent it in much the same way by just tinkering with the information sets (as you allude to in the question) as I've done below. In the typical setup, the second-mover must decide without knowing the type of the first player, hence his information sets. Here, we just draw the dotted line between the two decision nodes of the victim to represent they are both within the same information set. We no longer have to connect nodes for the robber because he has all the information at each of his nodes.
Analysis of this game (i.e. finding equilibria) will actually be quite straightforward. You can essentially solve out each of the four corners as the robber just best responds in each case (and has all the relevant information). Then, anticipating the robber's play in each of the four corners, the victim just has to compare the expected utility of giving or not giving the money, a simple calculation. So, while this looks similar to a beer-quiche game, it won't have any of the interesting pooling and separating notions. That's not to say it's not worthwhile to look at -- if this is the situation you want to model, then this is the situation you want to model!