Here's the problem that relates to a whole class of problems to which I am trying to figure out a general solution.
Given two players 1 and 2 who can select a number from the interval $[0, 1]$, define payoffs from a game as follows:
$\frac{s_i+s_j}{2}$ for $i
So the goal is to find such a combination of $s_i$ and $s_j$ that they are both best responses to each other, that is, there is no profitable deviation for either player. I have considered several cases, for example if $s_i
Symmetrically, the same argument applies when $s_i>s_j$. Then player $j$ can select a number arbitrarily smaller than $s_i$, thus increasing his/her payoff. That leaves us only the possibility of $s_i=s_j$ to investigate, and even then I managed to find only one set of responses ($\frac{1}{2}$ for each) that results into a Nash equilibrium.
Otherwise, say both players choose $0.4$. Then any player who chooses $0.5$ in response will get a profit of $1-\frac{.9}{2}$, exceeding the profits of $\frac{1}{2}$, thus there is a profitable deviation.
The question I have is how can I mathematically prove/show that ($\frac{1}{2}, \frac{1}{2}$) is the only Nash equilibrium (or are there more?). I managed to find those numbers by inspection, and would really like a more rigorous way of doing that since it would apply to all similar problems.