MathNet 2012 Question 41

In a football tournament between $n \ge 4$ teams, each pair of teams play against each other exactly once. In the final scoreboard, the scores of the teams for the tournament are $n$ consecutive numbers. Find the maximum possible value for the score of the tournament winning team.

NOTE: a team gets 3 points when it wins a match, none when it loses, and 1 when the match ends in a tie.