Extensive form: backward induction & subgame perfect nash equilibria?

Question

I've been given the SPNE through backward induction, I just want to understand how to interpret the equilibria properly.

The SPNE: $(agi,de)$ and $(bgi,df)$

For instance, how would I interpret $(agi,de)$?

Please let me know if the following would be an accurate interpretation of the above:

• If A chooses a first, then B will play d.
• In the event that we end up on the right hand side of the extensive form, B will select e and A will subsequently select g. However, if given the opportunity, A will select i.

This is supposed to be a complete contingent plan of action for both players (i.e. if it were in matrix form then we would be looking at all possible strategies for all players).

I find the first part quite easy, i.e. If A selects a, then B will select d which can be represented by $(a,d)$. However, I am getting confused when I have to account for all possible strategies that could be played $\rightarrow$ $(a \color{blue}{gi},d \color{blue}{e})$. I would appreciate it if you could offer some advice.

Answer 1

Until the last paragraph your interpretation is correct. $(a,d)$ is not a strategy profile, it is merely a 'history', a way that the game can play out.

A strategy of a player is a function that chooses an action in each of her decision nodes. As you can see in your graph Player A has three decision nodes while B has two. Hence any strategy of player A will consist of three elements, while any strategy of B will consist of two elements. So $(agi,de)$ is indeed a strategy profile. The corresponding history, what will happen if the players play these strategies, is $(a,d)$. The corresponding payoff is (1,2).

I am unsure what you mean by having to account for all possible strategies. Do you want to see what it means when an equilibrium is not subgame perfect? That would be $(ahj,de)$. No player can gain by unilaterally switching to another strategy, but if we were to reach player A's second or third decision node his action would not be optimal. But $(ahj,de)$ can be an equilibrium because the history given by this strategy profile never reaches these nodes.

Do you want to find all the equilibria that are not subgame perfect? In that case you should write down all possible strategies: There are 2^3 strategies for A, 2^2 strategies for B. Make a matrix using these as row and column labels. (Not sure if you really want to do this, as it is rather large.) Then you calculate the history and payoff for each strategy profile, creating a payoff matrix. In this you will find many equilibria. The ones that are not $(agi,de)$ and $(bgi,df)$ are not subgame perfect.