<p>Matrix looks correct. To final all pure strategy BNEs, you'll have to discuss cases based on the value of <span class="math-container">$p$</span>.</p>
<p>For example, if <span class="math-container">$p\in(0,1)$</span>, then <span class="math-container">$FT$</span> is player 2's unique best response to <span class="math-container">$F$</span>. Thus, to have a BNE, you'd want <span class="math-container">$F$</span> to be player 1's best response to <span class="math-container">$FT$</span> as well, meaning that you'd require <span class="math-container">$3p&gt;1-p$</span>, or <span class="math-container">$p&gt;\frac14$</span>. Hence, <span class="math-container">$(F,FT)$</span> is a BNE if <span class="math-container">$p\in(\frac14,1)$</span>.</p>
<p>You should be able to find other BNEs following a similar line of reasoning. (Hint: don't forget the edge cases where <span class="math-container">$p=0$</span> and <span class="math-container">$p=1$</span>.)</p>

CC BY-SA 4.0
2020-07-25T15:33:57.833Z
Ah so the idea would be to solve for $p$ in the columns $FT$ and $TF$ (the first one as you did in your comment above and the second one in a similar way). Then apply the same idea for the rows or am I wrong?
Justin Malik
CC BY-SA 4.0
2020-07-25T18:25:52.290Z
@JustinMalik: Yes that's right. Just note that $TT$ and $FF$ are also best responses for boundary values of $p$.
Herr K.
CC BY-SA 4.0
2020-07-25T18:58:29.627Z
Great, it is definitely much clearer now. Thank you so much.
Justin Malik
CC BY-SA 4.0
2020-08-02T06:40:52.927Z
@Tomcat: My understanding is that deriving a BNE from the ex ante perspective (i.e. before each player observes his type) is equivalent to deriving it from the ex interim perspective (i.e. after a player observes his own type but not the types of the others). [The extensive form representation of the game](https://i.sstatic.net/CGXVa.png) can actually be translated into the same normal form as the one given by OP.
Herr K.
CC BY-SA 4.0
2020-08-02T17:07:17.010Z
@Tomcat: The equivalence result is mentioned in Prop 8.E.1 of MWG (p.255), as well as in the textbook by Fudenberg and Tirole (1991, p.215). Regarding the game tree, I fail to see the issue you mentioned. In the tree I drew, player 1 does move at only one information set (the two decision nodes after Nature's move is connected by a dashed line representing player 1's information set). If you still think it's wrong, I would really appreciate that you draw a correct version of the tree so that I can learn from it.
Herr K.
CC BY-SA 4.0
2020-07-24T19:12:56.177Z
2020-07-24T19:12:56.177Z
Herr K.
<p>I'm doing a problem set on the subject of Bayesian Nash equilibrium. I'm asked to find the pure-strategy BNE of the following. I've calculated to matrix shown below. My first concern is if I've calculated the expected payoff matrix correctly, and second how do I find all of the pure-strategy BNE when the common prior is not concrete.
<a href="https://i.sstatic.net/xNVm4.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/xNVm4.png" alt="enter image description here" /></a></p>
<p><a href="https://i.sstatic.net/gEsJw.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/gEsJw.png" alt="enter image description here" /></a></p>

<game-theory><nash-equilibrium><bayesian-game><dynamic-games>
Pure-Strategy Bayesian Nash equilibrium with general common prior
Justin Malik
# Pure-Strategy Bayesian Nash equilibrium with general common prior

(Asked by: username_0 on 2020-07-24)

I'm doing a problem set on the subject of Bayesian Nash equilibrium. I'm asked to find the pure-strategy BNE of the following. I've calculated to matrix shown below. My first concern is if I've calculated the expected payoff matrix correctly, and second how do I find all of the pure-strategy BNE when the common prior is not concrete.
[![enter image description here](https://i.sstatic.net/xNVm4.png)](https://i.sstatic.net/xNVm4.png)

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

**Tags:** game-theory, nash-equilibrium, bayesian-game, dynamic-games


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## 1 answers

### Answer by username_1 on 2020-07-24. Score: 2 | Accepted answer

Matrix looks correct. To final all pure strategy BNEs, you'll have to discuss cases based on the value of $p$.

For example, if $p\in(0,1)$, then $FT$ is player 2's unique best response to $F$. Thus, to have a BNE, you'd want $F$ to be player 1's best response to $FT$ as well, meaning that you'd require $3p>1-p$, or $p>\frac14$. Hence, $(F,FT)$ is a BNE if $p\in(\frac14,1)$.

You should be able to find other BNEs following a similar line of reasoning. (Hint: don't forget the edge cases where $p=0$ and $p=1$.)

> **Comments:**
> **username_0**: Ah so the idea would be to solve for $p$ in the columns $FT$ and $TF$ (the first one as you did in your comment above and the second one in a similar way). Then apply the same idea for the rows or am I wrong?  
> **username_1**: @JustinMalik: Yes that's right. Just note that $TT$ and $FF$ are also best responses for boundary values of $p$.  
> **username_0**: Great, it is definitely much clearer now. Thank you so much.  
> **username_1**: @Tomcat: My understanding is that deriving a BNE from the ex ante perspective (i.e. before each player observes his type) is equivalent to deriving it from the ex interim perspective (i.e. after a player observes his own type but not the types of the others). [The extensive form representation of the game](https://i.sstatic.net/CGXVa.png) can actually be translated into the same normal form as the one given by OP.  
> **username_1**: @Tomcat: The equivalence result is mentioned in Prop 8.E.1 of MWG (p.255), as well as in the textbook by Fudenberg and Tirole (1991, p.215). Regarding the game tree, I fail to see the issue you mentioned. In the tree I drew, player 1 does move at only one information set (the two decision nodes after Nature's move is connected by a dashed line representing player 1's information set). If you still think it's wrong, I would really appreciate that you draw a correct version of the tree so that I can learn from it.  

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