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 people bet for win, n2 people 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 <= p / (1-p) + 1 when n2 == 0, we can likewise get M <= (1-p) / p + 1 To make it general, it makes sense to rewrite them like : N <= (p / (1-p) + 1) * (n2 / (n1+n2)), M <= ((1-p) / p + 1) * (n1 / (n1+n2)) **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.