How to find Nash equilibrium in a tree diagram?

So i have this question with the answers.

I know that the outcome with backward induction is (3,1) if p is smaller than 2/3 and (1+3,3-p) if x is greater than 2/3.

However I am having some problems writing the game is strategic form. Why is (1+3p,3-p) repeated so many times? Where is player 1 and player 2 in the strategic form? What do the rows and columns represent?

How do i write this tree game in strategic form?

Moreover, how do i find the NE as the answer sheet states from the strategic form of the game?

First you have to know what the term strategy means in game theory: Think of it as a game plan for one player where he writes down what he will do in every situation which could possibly occur. An example of a strategy for player $$I$$ would be: "I take $$a$$ and then if player $$II$$ takes $$c$$ I take $$g$$ or if player $$II$$ takes $$d$$ then I take $$l$$". Now every row corresponds to one strategy for player $$I$$ and every column corresponds to a strategy for player $$II$$.

The entries $$(1+3p, 3-p)$$ correspond to cases where $$I$$ chooses $$b$$. It is confusing that there is $$4$$ times the same row as there is only one strategy for player $$I$$ to get to this result: namely the strategy of choosing $$b$$ in the beginning. However the answer sheet probably thought of the $$4$$ strategies $$b, g, i$$, $$b, g, l$$, $$b, h, i$$, $$b, h, l$$. This does not make a lot of sense because when player $$I$$ chooses $$b$$ in the first place he does not have to be concerned with whether he should choose $$g$$ or $$h$$ (since this situation cannot occur once he chooses $$b$$).

How could you have found this strategic form yourself? First you write down all the strategies for $$I$$ and all strategies of player $$II$$. Then for each pair of strategies (one of each player) you determine what the outcome will be and write it in the corresponding cell.

How can you find the NE? You have to look for an entry in the matrix where no player would want to change strategy. Assuming $$p < 2/3$$ for example, entry in the second row and first column is a NE. This is because player $$II$$ would not want to change his strategy knowing player $$I$$ chooses his 2nd strategy. Also player $$I$$ would not want to change his strategy knowing player $$2$$ chooses his first strategy.

• I get that the strategies for player I are agi,agl,ahi,ahl and then b. Then for player II i get that the stragies are c and d? Is this right? – Sara Saletti Jul 1 '19 at 14:08
• And for example , player 1 has this strategy: agi: what would it be in strategic form? The first row ? But why ? ag=(3,1) and i=1,2? – Sara Saletti Jul 1 '19 at 14:11
• This wouldnt make sense since acg would then be a strategy which equalss (3,1) – Sara Saletti Jul 1 '19 at 14:12
• Another way to ask my question: what does the first row of the strategic form mean: ie, what does (3,1) and on the right (1,2) mean in terms of strategies? – Sara Saletti Jul 1 '19 at 14:16
• You got the strategies right yeah. the pairs of numbers in the matrix are the outcomes for both players if they follow the corresponding strategies i.e. if player $I$ follow his first strategy and player $II$ follows his first strategy then the outcome will be $3$ for player $I$ and $1$ for player $II$ – araomis Jul 1 '19 at 14:33