This may be a very simple and silly question, but since I am unfamiliar with Game Theory and am currently working towards a test for which my college lecturer gave very few materials to work on / learn with, I thought maybe someone on here could help.
My problem: I was looking at the exercise 3 with the game trees 1-3. I do understand that I need to solve for the SPNE by using backwards induction. However, I do not understand how the answer is supposed to be written down / how to interpret the sample solution provided.
Would anybody be so kind to help be out here or link some useful ressources with which I could broaden my knowledge a bit?
Exercise 3 game tree 1-2 Exercise 3 game tree 3 Exercise 3 sample solution provided
A strategy is a complete action plan for the entire game for a particular player. So for example, in your Exercise 3 Game Tree 1, Player 1 has a strategy set equal to $$S_1 = \{ Down, Up \},$$ while Player 2 has a strategy set equal to $$S_2 = \{ (left,down), (left,up), (right,down),(right,up)\}.$$ For Player 2, there are 4 strategies in total, and each strategy specifies what they will do at every node. There isn't one unified way of writing down a player's strategy, as long as you're able to articulate that a player takes an action at every node.
The set of SPNE is a set of strategy profiles, which are elements of $S_1 \times S_2$ (the Cartesian product). In other words, it's going to look like an ordered pair, where the first element is an element of $S_1$ and the second element is an element of $S_2$. In this case, there is only one SPNE given by $$(Up, (right,up)).$$ Here, $Up$ is the strategy that Player 1 is playing, and $(right,up)$ is the strategy that Player 2 is playing.