<p>I'm not sure what capacity allocation games you're applying QRE to. But here's a very stylized example where QRE is applied to an asymmetric game where the strategy spaces of the two players are (nominally) different: 
\begin{array}{c|cc}
&amp;L&amp;R\\\hline
T&amp;1,0&amp;0,9\\
D&amp;0,1&amp;1,0
\end{array}
This game can be easily represented in the extensive form as well.</p>

<p>In the standard formulation of QRE, each player $i$ plays a mixed strategy $\sigma_i$, where the probability of pure strategy $s_i$ being played is determined by the following formula: 
\begin{align}
\sigma_i(s_i)&amp;=\frac{\exp(\lambda u_i(s_i,\sigma_{-i}))}{\sum_{s_i'\in S_i}\exp(\lambda u_i(s_i',\sigma_{-i}))},
\end{align}
where $\lambda\in[0,\infty)$ measures the precision of the response; the larger the $\lambda$ the more precise the response. </p>

<p>Applying to the game above, let $p$ be the probability that player 1 chooses $T$, and $q$ the probability that player 2 chooses $L$. Note that $p$ and $q$ parameterize the two players' respective mixed strategy. Then, player 1's <strong>quantal response</strong> to <em>any given mixed strategy by player 2</em> (parameterized by $q$) is to play $T$ with probability $p$ and $D$ with $1-p$, where
\begin{equation}
p=\frac{\exp(\lambda\cdot (1q+0(1-q)))}{\exp(\lambda\cdot(1q+0(1-q)))+\exp(\lambda\cdot(0q+1(1-q)))}.\tag{1}
\end{equation}
Similarly, player 2's <strong>quantal response</strong> to <em>any given mixed strategy by player 1</em> (parameterized by $p$) is to play $L$ with probability $q$ and $R$ with $1-q$, where 
\begin{equation}
q=\frac{\exp(\lambda\cdot(0p+1(1-p)))}{\exp(\lambda\cdot(0p+1(1-p)))+\exp(\lambda\cdot(9p+0(1-p)))}.\tag{2}
\end{equation}</p>

<p>In a <strong>quantal response <em>equilibrium</em></strong>, $(\sigma_1,\sigma_2)$, the two player's strategies must be quantal responses to each other; that is,
\begin{align}
\sigma_1(T)&amp;=\frac{\exp(\lambda \cdot \sigma_2(L))}{\exp(\lambda\cdot\sigma_2(L))+\exp(\lambda(1-\sigma_2(L)))}&amp;\sigma_1(D)&amp;=1-\sigma_1(T)\\[12pt]
\sigma_2(L)&amp;=\frac{\exp(\lambda\cdot(1-\sigma_1(T)))}{\exp(\lambda\cdot(1-\sigma_1(T)))+\exp(9\cdot\lambda\cdot\sigma_1(T))}&amp;\sigma_2(R)&amp;=1-\sigma_2(L)
\end{align}</p>

<p>In the following graphs, the dashed lines plot the best responses of each player; the solid curves represent the quantal responses under different levels of precision. That is, the solid curves plots equations $(1)$ and $(2)$ above. The intersection of the two solid curves is the quantal response equilibrium point, described by the last set of equations above.  </p>

<p><a href="https://i.sstatic.net/grfzm.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/grfzm.png" alt="enter image description here"></a>
<a href="https://i.sstatic.net/MBf5s.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/MBf5s.png" alt="enter image description here"></a>
<a href="https://i.sstatic.net/4dL9x.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/4dL9x.png" alt="enter image description here"></a></p>

CC BY-SA 3.0
2018-02-26T13:57:24.310Z
K, the equations for σ1(T) and σ2(L) are dependednt on each other, hence how would you calculate their values without knowing the value of the other? Also these equations show no dependency on the payoff of the strategy as opposed to the definition in the first expression.
Amogha Varsha
CC BY-SA 3.0
2018-02-26T17:18:02.713Z
@AmoghaVarsha: They *are* supposed to depend on each other, otherwise we wouldn't have an *equilibrium*. I'm not sure I understand your second question. $\sigma_1(T)$ and $\sigma_2(L)$ both depend on payoffs; it's just that the payoffs are $1$ and hence omitted (note that there's a $9$ in the denominator of $\sigma_2(L)$).
Herr K.
CC BY-SA 3.0
2018-02-26T18:10:48.683Z
I understood the second part now, but how do you calculate the values if one depends on another, which one should be calculated first?
Amogha Varsha
CC BY-SA 3.0
2018-02-26T20:07:01.390Z
@AmoghaVarsha: The expressions I gave in the post are for the equilibrium point (the *E* in QRE). If you want to plot quantal response in general (without the *E*), then treat $\sigma_2(L)$ in the $\sigma_1(T)$ expression as a variable $q$, and $\sigma_1(T)$ in the last equation as $p$. For $\sigma_1(T)$, you can plot it against all possible values of $q$, i.e. $[0,1]$, to get player 1's quantal response function to player 2's (mixed) strategy. Symmetrically for $\sigma_2(L)$. QRE occurs where the two quantal response curves intersect; that is, the equilibrium is a *fixed point*.
Herr K.
CC BY-SA 3.0
2018-03-03T15:31:35.607Z
K This is a request, I am finding it really hard to comprehend the use of the expression for σ1(T) , could you just explain how to calculate with a random example with values? Thank You
Amogha Varsha
CC BY-SA 3.0
2018-03-03T17:40:21.273Z
@AmoghaVarsha: Please see my edited answer.
Herr K.
CC BY-SA 3.0
2018-03-05T14:05:40.273Z
How would I have to do it, if there are more than two strategies? Would'nt it get too complicated?
Amogha Varsha
CC BY-SA 3.0
2018-03-05T14:51:05.970Z
@AmoghaVarsha with two players you solve a system of two equations to get the equilibrium point. With n players you simply solve a system of n equations, provided that each player's strategy can be parameterized by one single parameter.
Herr K.
CC BY-SA 3.0
2018-03-06T13:11:23.407Z
K Can you suggest a tool to plot graphs from the values derived from solving the equations?
Amogha Varsha
CC BY-SA 3.0
2018-03-14T18:48:17.033Z
How would I plot a graph if there are are more than two strategies(with probabilities p,q,r etc) to find the equilibrium?
Amogha Varsha
CC BY-SA 3.0
2018-03-14T19:43:53.337Z
@AmoghaVarsha: If the total number of strategies is 3, you can do it in a 3-dimensional plot. If there are more than 3 strategies, then it's basically impossible to visualize them in the 3D world that we live in. Also, if you have a specific question about visualization tools, you should post it as a separate question instead of asking it here in the comment area.
Herr K.
CC BY-SA 3.0
2018-02-09T22:15:25.310Z
2018-03-03T17:46:41.857Z
2018-03-03T17:46:41.857Z
Herr K.
<p>I am working on my Master thesis based on bounded rationality in behavioral game theory models, I wanted to know if QRE(Quantal Response Equilibrium) can be applied to games where players have different strategy sets as opposed to what I have seen in resource allocation games, if yes, how?</p>

CC BY-SA 3.0
2018-02-09T16:48:52.750Z
Hi! Welcome to Econ.SE! What is an EFG? It's probably best not to use acronyms in the title of a question. Even if it makes the title long, it's probably an improvement.Good luck with your thesis!
jmbejara
CC BY-SA 3.0
2018-02-09T17:21:51.940Z
Do you mean *extensive* form game, as in a game tree?
Herr K.
CC BY-SA 3.0
2018-02-09T17:33:04.680Z
Hey! yeah exactly! but whatever solutions I went through were mostly for capcity allocation games where the action sets for all the players are the same. I would want to apply it for an attacker vs defender scenario where the strategies are different for each player
Amogha Varsha
<game-theory><behavioral-economics><bounded-rationality>
Bounded rationality with Quantal Response Equilibrium model for an Extensive Form game
Amogha Varsha
# Bounded rationality with Quantal Response Equilibrium model for an Extensive Form game

(Asked by: username_0 on 2018-02-09)

I am working on my Master thesis based on bounded rationality in behavioral game theory models, I wanted to know if QRE(Quantal Response Equilibrium) can be applied to games where players have different strategy sets as opposed to what I have seen in resource allocation games, if yes, how?

**Tags:** game-theory, behavioral-economics, bounded-rationality

> **Comments:**
> **username_1**: Hi! Welcome to Econ.SE! What is an EFG? It's probably best not to use acronyms in the title of a question. Even if it makes the title long, it's probably an improvement.Good luck with your thesis!  
> **username_2**: Do you mean *extensive* form game, as in a game tree?  
> **username_0**: Hey! yeah exactly! but whatever solutions I went through were mostly for capcity allocation games where the action sets for all the players are the same. I would want to apply it for an attacker vs defender scenario where the strategies are different for each player  

---

## 1 answers

### Answer by username_2 on 2018-02-09. Score: 2 | Accepted answer

I'm not sure what capacity allocation games you're applying QRE to. But here's a very stylized example where QRE is applied to an asymmetric game where the strategy spaces of the two players are (nominally) different:
\begin{array}{c|cc}
&L&R\\\hline
T&1,0&0,9\\
D&0,1&1,0
\end{array}
This game can be easily represented in the extensive form as well.

In the standard formulation of QRE, each player $i$ plays a mixed strategy $\sigma\_i$, where the probability of pure strategy $s\_i$ being played is determined by the following formula:
\begin{align}
\sigma\_i(s\_i)&=\frac{\exp(\lambda u\_i(s\_i,\sigma\_{-i}))}{\sum\_{s\_i'\in S\_i}\exp(\lambda u\_i(s\_i',\sigma\_{-i}))},
\end{align}
where $\lambda\in[0,\infty)$ measures the precision of the response; the larger the $\lambda$ the more precise the response.

Applying to the game above, let $p$ be the probability that player 1 chooses $T$, and $q$ the probability that player 2 chooses $L$. Note that $p$ and $q$ parameterize the two players' respective mixed strategy. Then, player 1's **quantal response** to *any given mixed strategy by player 2* (parameterized by $q$) is to play $T$ with probability $p$ and $D$ with $1-p$, where
\begin{equation}
p=\frac{\exp(\lambda\cdot (1q+0(1-q)))}{\exp(\lambda\cdot(1q+0(1-q)))+\exp(\lambda\cdot(0q+1(1-q)))}.\tag{1}
\end{equation}
Similarly, player 2's **quantal response** to *any given mixed strategy by player 1* (parameterized by $p$) is to play $L$ with probability $q$ and $R$ with $1-q$, where
\begin{equation}
q=\frac{\exp(\lambda\cdot(0p+1(1-p)))}{\exp(\lambda\cdot(0p+1(1-p)))+\exp(\lambda\cdot(9p+0(1-p)))}.\tag{2}
\end{equation}

In a **quantal response *equilibrium***, $(\sigma\_1,\sigma\_2)$, the two player's strategies must be quantal responses to each other; that is,
\begin{align}
\sigma\_1(T)&=\frac{\exp(\lambda \cdot \sigma\_2(L))}{\exp(\lambda\cdot\sigma\_2(L))+\exp(\lambda(1-\sigma\_2(L)))}&\sigma\_1(D)&=1-\sigma\_1(T)\\[12pt]
\sigma\_2(L)&=\frac{\exp(\lambda\cdot(1-\sigma\_1(T)))}{\exp(\lambda\cdot(1-\sigma\_1(T)))+\exp(9\cdot\lambda\cdot\sigma\_1(T))}&\sigma\_2(R)&=1-\sigma\_2(L)
\end{align}

In the following graphs, the dashed lines plot the best responses of each player; the solid curves represent the quantal responses under different levels of precision. That is, the solid curves plots equations $(1)$ and $(2)$ above. The intersection of the two solid curves is the quantal response equilibrium point, described by the last set of equations above.

[![enter image description here](https://i.sstatic.net/grfzm.png)](https://i.sstatic.net/grfzm.png)
[![enter image description here](https://i.sstatic.net/MBf5s.png)](https://i.sstatic.net/MBf5s.png)
[![enter image description here](https://i.sstatic.net/4dL9x.png)](https://i.sstatic.net/4dL9x.png)

> **Comments:**
> **username_0**: K, the equations for σ1(T) and σ2(L) are dependednt on each other, hence how would you calculate their values without knowing the value of the other? Also these equations show no dependency on the payoff of the strategy as opposed to the definition in the first expression.  
> **username_2**: @username_0: They *are* supposed to depend on each other, otherwise we wouldn't have an *equilibrium*. I'm not sure I understand your second question. $\sigma_1(T)$ and $\sigma_2(L)$ both depend on payoffs; it's just that the payoffs are $1$ and hence omitted (note that there's a $9$ in the denominator of $\sigma_2(L)$).  
> **username_0**: I understood the second part now, but how do you calculate the values if one depends on another, which one should be calculated first?  
> **username_2**: @username_0: The expressions I gave in the post are for the equilibrium point (the *E* in QRE). If you want to plot quantal response in general (without the *E*), then treat $\sigma_2(L)$ in the $\sigma_1(T)$ expression as a variable $q$, and $\sigma_1(T)$ in the last equation as $p$. For $\sigma_1(T)$, you can plot it against all possible values of $q$, i.e. $[0,1]$, to get player 1's quantal response function to player 2's (mixed) strategy. Symmetrically for $\sigma_2(L)$. QRE occurs where the two quantal response curves intersect; that is, the equilibrium is a *fixed point*.  
> **username_0**: K This is a request, I am finding it really hard to comprehend the use of the expression for σ1(T) , could you just explain how to calculate with a random example with values? Thank You  
> **username_2**: @username_0: Please see my edited answer.  
> **username_0**: How would I have to do it, if there are more than two strategies? Would'nt it get too complicated?  
> **username_2**: @username_0 with two players you solve a system of two equations to get the equilibrium point. With n players you simply solve a system of n equations, provided that each player's strategy can be parameterized by one single parameter.  
> **username_0**: K Can you suggest a tool to plot graphs from the values derived from solving the equations?  
> **username_0**: How would I plot a graph if there are are more than two strategies(with probabilities p,q,r etc) to find the equilibrium?  
> **username_2**: @username_0: If the total number of strategies is 3, you can do it in a 3-dimensional plot. If there are more than 3 strategies, then it's basically impossible to visualize them in the 3D world that we live in. Also, if you have a specific question about visualization tools, you should post it as a separate question instead of asking it here in the comment area.  

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