Alice, Bob and Cindy are playing a game of a circle. Firstly, Alice starts by drawing a point around the circle. Subsequently, being aware of Alice's decision Bob makes his move. Finally, Cindy puts a point around the circle being aware of Alice's and Bob's decisions. After all players fix their positions a point X drawn around the circle randomly. The winner of the game is the one whose position is the closest to the point X.
Question: How should Bob make his choice in order to maximize the probability of winning?
No matter how distance is measured (among those possibilities mentioned in a comment), he should put it opposite to that of Alice. The notion of 'probability' only makes sense though if Cindy is using a mathematically well-defined strategy.
Let the circle circumference go from $0$ to $1$. Say wlog. that Alice chooses $0$.
Also wlog. say that Bob chooses some $a\in[0,0.5]$. Then the optimal place for Cindy is $(a+1)/2$.
The range of Bob winning is $[a/2,((a+1)/2+a)/2]$ the length of which is $(a+1)/4$ which is maximized by $a=1/2$.