Originally Posted by swe_suns
Does my solution to find a MSNE mean that there isn't one?
In this case yes, and in general as long as you didn't make any mistakes yes. Basically, in this particular game, in order to make our opponent indifferent between A and C, we must ourselves be using A and C in equal amounts. Having deduced that, we've effectively reduced the number of strategies to two-playing B or playing a 50-50 mix of A/C and we need some mixture of the two that works. However, against these two strategies, B is a dominant response. Hence there can be no mixed strategy equilibrium since no mixture will make your opponent indifferent among best responses (which is necessary for him to be willing to mix himself.)
In general, the easiest way to solve these is to immediately plug in the constraint (i.e. r=1-p-q). Now we have three expressions (involving two unknowns) that must all be equal. Select two pairs of them and we now have two equations with two unknowns, something we know how to solve. If no solution exists between 0 and 1, then something must be wrong with the original problem, i.e. that there is no mixed strategy equilibrium involving all three strategies.