Let's break down the information step by step to determine the number of points each player scored:
1. Molly scores 6 points.
2. Nancy scores three times as many points as Molly, so Nancy scores 3 * 6 = 18 points.
3. Taylor scores four points fewer than Nancy, so Taylor scores 18 - 4 = 14 points.
4. Alana scores twice as many points as Taylor, so Alana scores 2 * 14 = 28 points.
Therefore, Alana scored 28 points in the basketball game.