Redistributing Money in Monopoly Is there a class of functions that satisfies the following properties? * $\lim_{x \to -\infty}f(x)=-k, \lim_{x \to \infty}f(x)=k$ * $f(x)<f(y) \Longleftrightarrow x<y $ * $f(x_1)+f(x_2)+\ldots+f(x_n)=0 \Longleftrightarrow x_1+x_2+\ldots +x_n=0$ I wanted to find such a function after playing a game of monopoly, in which we wanted to redistribute income fairly when the game began getting out of hand. Specifically, each of the above properties correspond to * Bounding the final cash values in $(-k, k)$ * Preserving the ordering of cash values (i.e. If I had more than you before the redistribution, I would have more than you after) * Eliminating the need for money to be taken or returned to the bank. Help my monopoly game redistribute money fairly!

There is no function satisfying the first and third conditions simultaneously. If a function satisfies the first condition, then there exists $A>0$ such that $|f(x)+k|A$. This means, for all $x_1,x_2>A$, $$ f(x_1) + f(x_2) + f(-x_1-x_2) > \frac{2k}3 + \frac{2k}3 - \frac{4k}3 = 0, $$ contradicting the third condition.

Note that this proof doesn't require $f$ to be increasing or even continuous.