# Bounded rationality with Quantal Response Equilibrium model for an Extensive Form game

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?

• 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 Feb 9 '18 at 16:48
• Do you mean extensive form game, as in a game tree? – Herr K. Feb 9 '18 at 17:21
• 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 Feb 9 '18 at 17:33

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.

• 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 Feb 26 '18 at 13:57
• @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. Feb 26 '18 at 20:07
• @AmoghaVarsha: Please see my edited answer. – Herr K. Mar 3 '18 at 17:40
• @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. Mar 5 '18 at 14:51
• @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. Mar 14 '18 at 19:43