Pick a number that is better than your friends Consider the following game.
There are $n$ players, each one has to pick a (real) number $x$ between $0$ and $100$.  There is one round to the game.  The winner is the person whose number is closest to $10\sqrt{\mu}$ where $\mu$ is the mean of the choices made by all the other players.     
How would one formulate the problem of choosing optimal play mathematically and is there an optimal strategy?
 A: This is a basic game theory problem.
You want to formulate each player's best response function, that is, describe each player's optimal action as a function of other player's actions.
In this case, that is remarkably simple.
Player $i$'s optimal pick as a function of other player's picks would solve the equation:
$$ x_i = 10\sqrt{\frac{1}{n}\sum_{j} x_j} $$
A Nash equilibrium is a tuple $(x_1, x_2, ..., x_n)$ where everyone is playing their best response. Taking a shortcut, let's look for symmetric equilibria: then $\mu = x$. Solving for the fixed point.
$$ x = 10 \sqrt{x}$$
Hence symmetric Nash Equilibria: $$x = 100  \quad \quad \quad \quad x = 0$$
If we all pick $100$, everyone splits the win and no one can do better by deviating. If we all pick $0$, everyone splits the win and no one can do better by deviating. Either outcome is what we might expect if everyone were perfectly rational. If you have alternative beliefs, you should play your best response function given those beliefs.
Small note: you need to assume $n > 2$ or it doesn't matter at all what anybody picks since both players would always be equidistant from the mean.
