It looks like it should be 2/3*3/4 = 1/2 but there's something funny here.
If A, B and C are evenly matched, A beats B in a two way race 1/2 of the time and A beats C in a two way race 1/2 of the time.
But the probability of A beating B and C in a three way race is not 1/2*1/2 = 1/4. It is clearly 1/3 because they are all equally likely to win.
So how about the problem in the first post?
You can imagine a 22 faced dice. (I can't bring myself to say "die".) 12 faces labelled A, 6 labelled B, and 4 labelled C. The dice is rolled until one of the players gets their letter. In two way matches, A beats B 2/3 of the time and A beats C 3/4 of the time. In a three way match, A wins 12/22 of the time.
But I have to agree with Dason. Other scenarios give different results.