I have a beauty contest question in which players must guess a number between $0$ and $5$. The closest score to ($p\times\text{average score}$) wins. Winners take 1 and losers take 0, whilst players who tie will both win. I have to represent this game in normal form, does anyone know how to do this for more than $2$ players, i.e for $n$ players?
Thanks all.
If $\ g_i\ $ is the guess made by player $\ i\ $, for $\ i=1,2,\dots, n\ $, the payoff, $\ P_j\left(g_1, g_2, \dots, g_n\right)\ $, to player $\ j\ $ is given by
$P_j\left(g_1, g_2, \dots, g_n\right)$ $$= H\left(\min_{1\le i\le n}\left(\left\vert\, g_i -\frac{p\sum_{k=1}^n g_k}{n}\right\vert\right)-\left\vert\, g_j -\frac{p\sum_{k=1}^n g_k}{n}\right\vert\right)\ ,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$ where $\ H\ $ is the Heaviside step function, defined by $$H\left(x\right) =\left\{ \begin{matrix} 0 & \mbox{ if } x < 0\ \ \\ 1 & \mbox{ if } x \ge 0\ \ . \end{matrix}\right. $$