Let's imagine designing an odds pattern for a game, in which players bet for win or lose. Suppose the probablity of winning is p, thus the probablity of losing is 1-p.

Now imagine n1 persons bet for win, n2 persons bet for lose, both ante is 1, and the odds for both are 1:M and 1:N, respectively.

The banker doesn't want even a penny out of his wallet, so it's reasonable we have , according to Mean Value Fomula: n1 * (M-1) * p + n2 * (N-1) * (1-p) <= n2 * p + n1 * (1-p)

specially, when n1 == 0: we have : n2 * (N-1) * (1-p) <= n2 * p , we get : N <= n2 * p / ( n2 * (1-p)) + 1

when n2 == 1, we can likewise get M <= n1 * (1-p) / (n1 * p) + 1

Here comes my question: the n1 and n2 are influenced by the M and N and P. However, the M and N relys on the n1 and n2. How to figure out what M and N should be chosen?

It seems we should have a transcendental value for n1 and n2. FYI, there is a restrition : n1 + n2 <= C - 2, C is a constant.