Suppose we have $4$ players. Each player picks a real number between $0$ and $100$ (both included). The player with the number closest to the average of the $4$ numbers picked wins. If there is a tie, the prize is split evenly. My goal is to find the Nash Equilibrium.
Players: $4$
Strategies: $S_{i}\in [0, 100]$ for $i=1, 2, 3, 4$
Payoffs: {$1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, 0$}
I let $c=\frac{n_{1}+n_{2}+n_{3}+n_{4}}{4}$ where $n_{i}$ are the numbers picked by the players respectively.
As player $1$, my goal is to minimize the following function:
$|c-n_{1}|=|\frac{-3n_{1}+n_{2}+n_{3}+n_{4}}{4}|$ and it is minimized at $n_{1}=\frac{n_{2}+n_{3}+n_{4}}{3}$
The same approach can be taken for any player $i$. How do I go about finding the Nash Equilibrium?