A game on a graph 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 $m,\: 1\leq   m \leq 1000$ 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.
This problem is from the Turkey JBMO TST 2014. Could someone help? I have not gone anywhere with it. Thanks a lot.

This is not a full solution, but this may help:
The name of the structure we get at the end is tournament.
A transitive tournament is one where $a\to b$ and $b\to c$ implies $a\to c$, thus they give us a total ordering of the vertices.
Tournaments are only acyclic if they are transitive, in particular a tournament $T$ has no $3$-cycle if and only if it is transitive. Notice a transitive graph has no cycles. Now suppose a graph is not transitive, then there exists $a,b,c$ such that $a\to b, b\to c$ and $c\to a$.
So If Bob wants to win he has to build a transitive graph. Notice any tournament with outdegree sequence $(0,1,2\dots ,n-1)$ is transitive. Can Bob build this graph?
