For 3 players, let's set the chances of winning at 50%, 30%, and 20% for players A, B, and C respectively.
When A does win, what the first formula says is that B's chance of then getting 2nd place is (.30) / (1 - .50) = 60%. This makes sense because of the two remaining players, B and C, their "skill" still relates comparatively to what their chances of winning had been-- 30:20.
But to get a complete picture of your chances of second place, you must do the same thing with all other potential 1st place winners. So given that player C wins (20% of the time), you have (.30) / (1 - .20) = 37.5%.
Now your overall odds of getting second place is the sum of [ (odds of 2nd place given that player X wins) * (odds of player X winning) ]. In this example then, you end up with ((.60)(.50)) + ((.375)(.20)) = 37.5%.
Your overall equity then would be (.30)(1st place money) + (.375)(2nd place money) + (1 - (.30) - (.375))(3rd place money).