A famous problem: a lady is in the center of the circle lake, the monster is on the boundary of the lake. The speed of the monster is $v_m$, of swimming lady — $v_l$. The goal of the lady is to come to the shore without meeting the monster, the goal of the monster — to meet the lady.
If the monster cannot swim, the lady will always win if $\frac{v_m}{v_l} < 4.6033$. If this condition is not satisfied — the monster and the lady will follow the strategies which don't allow them neither win nor lose.
I was thinking about “real-life” extensions of this game (e.g. probabilistic) such that for any fraction of speeds either a lady or a monster can win (with non-zero probability for probabilistic extension). I always get a result that monster will never win. This fact fact is quite sad for the poor monster.
So, I will be happy if you can suggest me a non-trivial extension in which a monster cannot swim but can win (situation when they start at the same point e.g. is trivial).