<p>An equilibrium consists of a profile of <em>strategies</em>, which specifies an action for every player at each possible contingency. Since each action profile <span class="math-container">$(a_1,a_2)$</span> is a contingency, the SPE must include functions <span class="math-container">$a^*_3(a_1,a_2), a^*_4(a_1,a_2)$</span> that specifies what to do at those contingencies. </p>

CC BY-SA 4.0
2019-06-06T11:36:49.920Z
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.
Star
CC BY-SA 4.0
2019-06-06T15:10:41.563Z
@user3285148: The vector of points you mentioned is called the *equilibrium outcome* of the SPE.
Herr K.
CC BY-SA 4.0
2019-06-05T14:38:01.893Z
2019-06-05T15:12:01.077Z
2019-06-05T15:12:01.077Z
Herr K.
<p>Take a two-stage game with complete information and simultaneous actions in each state: </p>

<p>(1) Player 1 and 2 simultaneously choose action <span class="math-container">$a_1\in A_1$</span>  and
<span class="math-container">$a_2\in A_2$</span> respectively.</p>

<p>(2) Player 1 and 2 observe the outcome of the 1st stage
<span class="math-container">$(a_1, a_2)$</span>, then simultaneously choose action  <span class="math-container">$a_3\in A_3$</span> and
<span class="math-container">$a_4\in A_4$</span> respectively.</p>

<p>Payoffs are <span class="math-container">$u_i(a_1, a_2, a_3, a_4)$</span> for <span class="math-container">$i = 1,2$</span>. </p>

<hr>

<p>As equilibrium concept I use subgame perfect Nash equilibrium. I find it by
backward induction: </p>

<p>(A) find the <strong>functions</strong> <span class="math-container">$a^*_3(\cdot)$</span>  and <span class="math-container">$a^*_4(\cdot)$</span> such that <span class="math-container">$\forall (a_1,a_2)\in A_1\times A_2$</span></p>

<p><span class="math-container">$$
\begin{cases}
a_3^*(a_1, a_2)\in argmax_{a_3(\cdot)}u_1(a_1, a_2, a_3(a_1, a_2), a_4(a_1, a_2))\\
a_4^*(a_1, a_2)\in argmax_{a_4(\cdot)}u_2(a_1, a_2, a_3(a_1, a_2), a_4(a_1, a_2))\\
\end{cases}
$$</span> </p>

<p>(B) find <span class="math-container">$(a_1^*, a_2^*)\in A_1\times A_2$</span> such that
<span class="math-container">$$
\begin{cases}
a_1^*\in argmax_{a_1}u_1(a_1, a_2, a^*_3(a_1, a_2), a^*_4(a_1, a_2))\\
a_2^*\in argmax_{a_2}u_2(a_1, a_2, a^*_3(a_1, a_2), a^*_4(a_1, a_2))\\
\end{cases}
$$</span> </p>

<hr>

<p>Question: a subgame perfect Nash equilibrium is
<span class="math-container">$$
\{a_1^*, a_2^*, \underbrace{a^*_3(\cdot), a^*_4(\cdot)}_{\text{Functions}}\}
$$</span>
or
<span class="math-container">$$
\{a_1^*, a_2^*, \underbrace{a^*_3(a_1^*), a^*_4(a_2^*)}_{\text{Point in the image set of the functions $a^*_3(\cdot), a^*_4(\cdot)$}}\}
$$</span>
?</p>

CC BY-SA 4.0
2019-06-05T15:26:53.050Z
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)?
Bayesian
<game-theory><dynamic-games>
Definition of subgame perfect Nash equilibrium
Star
# Definition of subgame perfect Nash equilibrium

(Asked by: username_0 on 2019-06-05)

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
$$
\{a\_1^\*, a\_2^\*, \underbrace{a^\*\_3(a\_1^\*), a^\*\_4(a\_2^\*)}\_{\text{Point in the image set of the functions $a^\*\_3(\cdot), a^\*\_4(\cdot)$}}\}
$$
?

**Tags:** game-theory, dynamic-games
> **Comments:**
> **username_1**: 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)?  

---

## 1 answers

### Answer by username_2 on 2019-06-05. Score: 2 | Accepted answer

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.

> **Comments:**
> **username_0**: 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.  
> **username_2**: @user3285148: The vector of points you mentioned is called the *equilibrium outcome* of the SPE.  

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