A rich, honest, but mischievous father told his two sons that he had placed $10^n$ and $10^{n-1}$ in two envelopes respectively, where n ∈ $\{1,2,3,\ldots,10\}$. The father then randomly handed each son one of the two envelopes with a probability of $0.5$. After both sons opened their envelopes, his father privately asked each son whether he wanted to switch his envelops with the one his brother had. If both sons agreed, then the envelopes were switched. Otherwise, each son kept the original envelop he received.
Find all pure strategy Bayesian Nash Equilibria.
What I did was to formulate the problem.
Type Spaces: $T_1=T_2=\{10^n,10^{n−1}\}$
Action Spaces: $A1=A2=\{S,S'\}$ where $S$ is switch and $S'$ is don't switch.
Strategy Spaces: $S_i:T_i\rightarrow A_i$ where $i=1,2$
I got $(S', SS)$ and $(S', S'S')$ for my Bayesian Nash equilibria. It means if 1st son chooses not to switch then 2nd son's strategy is to switch when he has $10^n$ or $10^{n-1}$. The other will be 1st son choose not to switch and 2nd son's strategy is not to switch when he has $10^n$ or $10^{n-1}$.
What's puzzling is that I didn't use when $n=\{1,\ldots,10\}$. Is there anything wrong with my solution?
Both wanting not to switch in any circumstances is a Nash equilibrium: neither can do better by changing strategy.
One wanting not to switch and the other wanting to switch in any circumstances is not a Nash equilibrium: for example the first son could do better by wanting to switch when his envelope has $10^0$.
Consider these examples when looking for Nash equilibria pure strategies: