Drawing Arrows 1500 pts

There are $n \ge 1$ dots on a sheet of paper, initially with no arrows.

Alice and Bob take turns drawing an arrow from one dot to another distinct dot. Alice goes first. They must follow two rules:

• Each pair of dots may be connected by at most one arrow, regardless of its direction.
• No dot may have both an incoming arrow and an outgoing arrow.

The player who cannot draw a legal arrow loses.

Determine who has a winning strategy for each positive integer $n$.