Question 83
Alice and Bob play a game on a complete graph $G$ with 2014 vertices. They take moves in turn with Alice beginning. At each move Alice directs one undirected edge of $G$. At each move Bob chooses a positive integer number $m$, $1 \le m \le 1000$ and after that directs $m$ undirected edges of $G$. The game ends when all edges are directed. If there is some directed cycle in $G$ Alice wins. Determine whether Alice has a winning strategy.