<p>There are three classes of equilibria of this game.</p>
<hr />
<p>The first class is sequential:
<span class="math-container">\begin{equation}
(s_1,s_2)=(y,r)
\end{equation}</span>
and the beliefs are
<span class="math-container">\begin{equation}
\mu_1(a)=\mu_1(b)=\mu_2(a\mid y)=\mu_2(b\mid y)=\frac12.
\end{equation}</span></p>
<p><a href="https://i.sstatic.net/xPkWF.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/xPkWF.png" alt="enter image description here" /></a></p>
<hr />
<p>The second class is not sequential, but weak perfect Bayesian:
<span class="math-container">\begin{equation}
(s_1,s_2)=(x,l)
\end{equation}</span>
and the beliefs are
<span class="math-container">\begin{equation}
\mu_1(a)=\mu_1(b)=\frac12,\quad\text{but }\mu_2(a\mid y)=1-\mu_2(b\mid y)=p&gt;\frac58.
\end{equation}</span></p>
<p><a href="https://i.sstatic.net/GgYrq.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/GgYrq.png" alt="enter image description here" /></a></p>
<hr />
<p>The third class is actually a boundary case of the second class, which permits P2 to use mixed strategies. This equilibrium requires P2's off equilibrium belief to be
<span class="math-container">\begin{equation}
\mu_2(a\mid y)=1-\mu_2(b\mid y)=\frac58,
\end{equation}</span>
so that P2 can play a mixed strategy that puts sufficiently high probability on <span class="math-container">$l$</span> (<span class="math-container">$\sigma_2(l)\ge\frac35$</span>). It is also necessary that P1 does not play <span class="math-container">$y$</span> with positive probability, because if they do, Bayes rule would kick in, requiring that P2's belief be <span class="math-container">$(\frac12,\frac12)$</span> instead of <span class="math-container">$(\frac58,\frac38)$</span> which rationalizes P2's mixed strategy.
<a href="https://i.sstatic.net/v9Att.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/v9Att.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2021-04-16T21:43:58.887Z
2021-04-17T03:06:46.717Z
2021-04-17T03:06:46.717Z
Herr K.
<p>I am working on the following game</p>
<p><a href="https://i.sstatic.net/98IAT.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/98IAT.jpg" alt="enter image description here" /></a></p>
<p>and I have <strong>to find all strong sequential equilibria here</strong>.</p>
<p>I determined that here any belief derived from a fully mixed strategy gives a distribution (1/2, 1/2) over the nodes in Player 2’s information set. Given this, player 2 will chose r and player 1 will chose y. I believe that this is the unique sequential equilibrium. But is it strong?</p>
<p>And <strong>are there any other Nash equilibria and if so why they are not sequential?</strong></p>
<p>Thanks!</p>

CC BY-SA 4.0
2021-04-16T20:09:15.040Z
What do you mean by "strong" SE? Also the belief you derive for P2 in a fully mixed equilibrium seems incorrect. The resulting strategies ($y$ for P1 and $r$ for P2) are not "fully mixed", and do not induce the $(\frac12,\frac12)$ belief for P2.
Herr K.
CC BY-SA 4.0
2021-04-16T20:12:18.133Z
@HerrK, well, strong is standart sequential equilibria, not weak.
And I don't see why they are not fully mixed
Rodger Haggings
CC BY-SA 4.0
2021-04-16T20:15:16.850Z
Aren't $y$ and $r$ pure strategies?
Herr K.
CC BY-SA 4.0
2021-04-16T20:16:18.493Z
@HerrK, yes they are
Rodger Haggings
<game-theory><nash-equilibrium><bayesian-game><signaling>
Strong sequential equilibria and the existence of others
Rodger Haggings
# Strong sequential equilibria and the existence of others

(Asked by: username_0 on 2021-04-16)

I am working on the following game

[![enter image description here](https://i.sstatic.net/98IAT.jpg)](https://i.sstatic.net/98IAT.jpg)

and I have **to find all strong sequential equilibria here**.

I determined that here any belief derived from a fully mixed strategy gives a distribution (1/2, 1/2) over the nodes in Player 2’s information set. Given this, player 2 will chose r and player 1 will chose y. I believe that this is the unique sequential equilibrium. But is it strong?

And **are there any other Nash equilibria and if so why they are not sequential?**

Thanks!

**Tags:** game-theory, nash-equilibrium, bayesian-game, signaling

> **Comments:**
> **username_1**: What do you mean by "strong" SE? Also the belief you derive for P2 in a fully mixed equilibrium seems incorrect. The resulting strategies ($y$ for P1 and $r$ for P2) are not "fully mixed", and do not induce the $(\frac12,\frac12)$ belief for P2.  
> **username_0**: @HerrK, well, strong is standart sequential equilibria, not weak. And I don't see why they are not fully mixed  
> **username_1**: Aren't $y$ and $r$ pure strategies?  
> **username_0**: @HerrK, yes they are  

---

## 1 answers

### Answer by username_1 on 2021-04-16. Score: 4 | Accepted answer

There are three classes of equilibria of this game.

---

The first class is sequential:
\begin{equation}
(s\_1,s\_2)=(y,r)
\end{equation}
and the beliefs are
\begin{equation}
\mu\_1(a)=\mu\_1(b)=\mu\_2(a\mid y)=\mu\_2(b\mid y)=\frac12.
\end{equation}

[![enter image description here](https://i.sstatic.net/xPkWF.png)](https://i.sstatic.net/xPkWF.png)

---

The second class is not sequential, but weak perfect Bayesian:
\begin{equation}
(s\_1,s\_2)=(x,l)
\end{equation}
and the beliefs are
\begin{equation}
\mu\_1(a)=\mu\_1(b)=\frac12,\quad\text{but }\mu\_2(a\mid y)=1-\mu\_2(b\mid y)=p>\frac58.
\end{equation}

[![enter image description here](https://i.sstatic.net/GgYrq.png)](https://i.sstatic.net/GgYrq.png)

---

The third class is actually a boundary case of the second class, which permits P2 to use mixed strategies. This equilibrium requires P2's off equilibrium belief to be
\begin{equation}
\mu\_2(a\mid y)=1-\mu\_2(b\mid y)=\frac58,
\end{equation}
so that P2 can play a mixed strategy that puts sufficiently high probability on $l$ ($\sigma\_2(l)\ge\frac35$). It is also necessary that P1 does not play $y$ with positive probability, because if they do, Bayes rule would kick in, requiring that P2's belief be $(\frac12,\frac12)$ instead of $(\frac58,\frac38)$ which rationalizes P2's mixed strategy.
[![enter image description here](https://i.sstatic.net/v9Att.png)](https://i.sstatic.net/v9Att.png)


---

