# Definition of subgame perfect Nash equilibrium

Take a two-stage game with complete information and simultaneous actions in each state:

(1) Player 1 and 2 simultaneously choose action $$a_1\in A_1$$ and $$a_2\in A_2$$ respectively.

(2) Player 1 and 2 observe the outcome of the 1st stage $$(a_1, a_2)$$, then simultaneously choose action $$a_3\in A_3$$ and $$a_4\in A_4$$ respectively.

Payoffs are $$u_i(a_1, a_2, a_3, a_4)$$ for $$i = 1,2$$.

As equilibrium concept I use subgame perfect Nash equilibrium. I find it by backward induction:

(A) find the functions $$a^*_3(\cdot)$$ and $$a^*_4(\cdot)$$ such that $$\forall (a_1,a_2)\in A_1\times A_2$$

$$\begin{cases} a_3^*(a_1, a_2)\in argmax_{a_3(\cdot)}u_1(a_1, a_2, a_3(a_1, a_2), a_4(a_1, a_2))\\ a_4^*(a_1, a_2)\in argmax_{a_4(\cdot)}u_2(a_1, a_2, a_3(a_1, a_2), a_4(a_1, a_2))\\ \end{cases}$$

(B) find $$(a_1^*, a_2^*)\in A_1\times A_2$$ such that $$\begin{cases} a_1^*\in argmax_{a_1}u_1(a_1, a_2, a^*_3(a_1, a_2), a^*_4(a_1, a_2))\\ a_2^*\in argmax_{a_2}u_2(a_1, a_2, a^*_3(a_1, a_2), a^*_4(a_1, a_2))\\ \end{cases}$$

Question: a subgame perfect Nash equilibrium is $$\{a_1^*, a_2^*, \underbrace{a^*_3(\cdot), a^*_4(\cdot)}_{\text{Functions}}\}$$ or $$\{a_1^*, a_2^*, \underbrace{a^*_3(a_1^*), a^*_4(a_2^*)}_{\text{Point in the image set of the functions a^*_3(\cdot), a^*_4(\cdot)}}\}$$ ?

• The equilibrium MUST include the functions and not just the point. If you only specified, e.g., $a^*_3(a_1^*)$ then how could you evaluate if a deviation from $a_1^*$ when it is not specified what happens afterwards (and thus you don't know the payoff)? Jun 5 '19 at 15:26

An equilibrium consists of a profile of strategies, which specifies an action for every player at each possible contingency. Since each action profile $$(a_1,a_2)$$ is a contingency, the SPE must include functions $$a^*_3(a_1,a_2), a^*_4(a_1,a_2)$$ that specifies what to do at those contingencies.
• Thanks. I forgot to ask one minor thing and I don't feel it is worth to be part of a separate question. Would you be so kind to reply? My question is: let $\{a_1^*, a_2^*, a_3^*(\cdot), a_4^*(\cdot)\}$ be a subgame perfect Nash equilibrium of the game above; take $(a_1^*, a_2^*, a_3^*(a_1^*), a_4^*(a_2^*))$; is there any specific terminology to indicate this last vector of points? I would say that "it is part" of a subgame perfect Nash equilibrium, but I'm wondering whether I can refer to it in a more technical way. Thanks again. Jun 6 '19 at 11:36