There are $n\ge3$ players playing a game. In this game, one person will come out in first place, one in second, and so on. It's impossible to tie. The person in first place gets $n$ points, the person in second place gets $n-1$, and so on, so that the person in last place gets $1$ point. After playing this game some fixed number of times, the scores are tallied up and the winner is whoever has the highest score.
Can we fix a number of games (other than one game) in advance, so that a tie for first place is impossible? With two players this is easy - just play an odd number of games. But otherwise?