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I really need some help understanding how the subgame perfect equilibrium strategies can be found.

I understand how to get an outcome. To have something to talk about, consider this game here:

enter image description here

I find the following subgame perfect nash equilibrium outcome: First, both players play $\{1,1\}$. Then, player 1 plays $B$ and player 2 plays $R$, giving the payoffs $\{1,1\}$. This was done by using a tree, and employing backwards induction, iterating on each subgame.

However, this is the outcome. Using the three alone, what are the equilibrium strategies? How do I obtain them? What's a good, efficient way of doing this based on a three representation?

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    $\begingroup$ Jin5 you should get together with this other kid from your class. He is posting all the same questions (with different usernames) at the Economics.SE. $\endgroup$
    – Giskard
    Dec 20, 2016 at 19:43
  • $\begingroup$ It's not some other kid dude, it's me. You can't be that dumb .... $\endgroup$
    – Jaood
    Dec 22, 2016 at 20:21

1 Answer 1

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A strategy is just a specification of (possibly randomised) actions to take in every possible circumstance, even those that are ruled out by equilibrium. More formally, a strategy is a mapping from histories (descriptions of past events) into (probability distributions on) actions.

Hence, strategies in the first stage are as you describe. Both players choose $a_i = 1$.

Strategies in the second stage are just functions of what both players did in the first stage. Let's denote this function for player $i$ by $$s_i:\{-1,1\}^2\to A_i$$ where $A_1=\{T,B\}$ and $A_2=\{L,R\}$.

As you've observed, in equilibrium, $$ s_1(1,1) = B$$ $$ s_2(1,1) = R $$

Hence, all that's left is to define $s_i(-1,-1)$, $s_i(1,-1)$, and $s_i(-1,1)$ for $i=1,2$. Note that these have to be best responses in their respective subgames to form a subgame perfect equilibrium. (They can be defined arbitrarily for Nash equilibrium.)

For instance, one possible specification of subgame perfect equilibrium strategies would have that

$$ s_1(-1,-1) = B $$ $$ s_1(-1,-1) = L $$

Notice that the non-uniqueness of Nash equilibrium in the subgame after the first stage action profile $(-1,-1)$ gives us some leeway in defining subgame perfect equilibrium strategies here.

I'll leave the rest of the details to you.

As for your question on representing strategies, unfortunately I don't know of any specific way that's especially efficient.

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