# Analytically solvable examples of quantal response equilibrium

I have been reading up on quantal response equilibrium and am looking for some simple examples which can be solved analytically (in order to gain some intuition about how exactly QRE works). To be more concrete, I am willing to consider the logit version of the quantal response function $$\text{P(choose A)} = \frac{\text{exp}(\lambda\text{(expected payoff from A)}}{\sum_X\text{exp}(\lambda\text{(expected payoff from X)}}$$

and (somewhat) happy to further impose $$\lambda = 1$$ if this makes finding the equilibrium easier.

Does anyone have any ideas? Examples where QRE contrasts starkly with Nash equilibrium would be especially appreciated.

• How about the example I have in this answer:economics.stackexchange.com/a/20531/42? – Herr K. Mar 11 at 18:34
• Thanks for linking. Just to clarify, should I be able to find explicit expressions for $p$ and $q$ in your example? (Possibly after imposing $\lambda = 1$) – user17900 Mar 11 at 19:51
• NB I was not able to do this (I suspect that this is a general feature of these systems of exponential equations) – user17900 Mar 11 at 20:08
• Aren't equations 1 and 2 there expressions of p and q (after setting lambda=1)? Because of quantal reponse, p and q must be functions of each other. – Herr K. Mar 11 at 20:26
• Yes, I was just saying that I couldn't combine them to find explicit expressions for $p$ and $q$, i.e. find analytical solutions for $p$ and $q$. You are of course correct that these equations implicitly define $p$ and $q$. – user17900 Mar 12 at 12:35