That's very interesting, and I believe you got the results you got, since a few simple computations on paper lead to the same result. The simple answer is that as long as you're giving up x% of your "score" and getting x% of the other guy's "score," your score moves up until the two sides are at parity. I think for a 4 state game, equilibirum is reached when all four states are at 25%, and so on, so the long-term probability of a particular mutation existing in a population (assuming no selection) is close to the 1 over the number of possible states for that gene. Of course, for low probability mutations, the drift toward equilibrium is slow. Why don't you pop in a low initial probability into your model (say, 1 in a thousand) and see how long it takes to get to equilibrium?
For a diploid population the drift toward equilibrium is either to 1 or 0 for a single allele.
Yes, that's what I realized after the fact. But I don't think there's any equilibrium int the instantaneous proportion; it will vary between 0 and 1 with an average of 1/2 although the distribution may not be uniform. Of course the total proportion will tend toward 1/2 - perhaps that's what you mean.
I'm sure you're right that the lower x is the more expected number of generations will need to pass before the mutation becomes prevalent. With a population around 1000 and rate of 1/1000 I ran it ten times. It took 347, 975, 775, 353, 262, 659, 609, 241, 79, and 204 generations before mutants exceeded 1/2 of the population. With the same population size and a rate of 1/10000 it took 1564, 3551, 4261, 3979, 1047, 7947, 3936, 8336, 747, and 1632 generations - about 10x. With 10000 and 1/10000 it looks like about another 10x. I'm guessing it is linear in both parameters.
I'd be happy to share the program with you, it's really simple.