The question is wrong based on assumption. It assumes humans are predictable. First problem "50, 75, 63, then 57", this assumes there can only be one direction the algorithm can go. The number could also be 1. So let's assume worst case scenario if I'm rounding up - 50 ($5), 25 ($4), 13 ($3), 7 ($2), 4 ($1), 2 ($0)
So in this case I would lose $1. If I was rounding down like with a standard int algorithm I would not lose anything. But here's the thing. It assumes you are smart enough to know which number to pick, and as we've seen maths isn't really people's strong point, and all for the possibility of winning $1. But here's the real kicker, what if I don't strictly follow the median method but introduce randomness to reduce your trickery for a more equal random chance of winning or losing $5. You'll already be sweating when I don't pick 50 but 47 instead. It seems to me you are really the one that should not want to play this game.
Microsoft "geniuses" aren't the smartest of the bunch, so what would your response be to this answer where you pick the number but I pick the odds?