I play many online games and there are always those messages:
"Player has joined the game" and "Player left the game".
Actually I can fully understand the "has joined" because it affects the game - he comes and he is there.
However, why isn't it "has left" as well?
The player leaves the current match, which is still running, and it has influence on the match - because he is missing.
Both are possible. I prefer what you do-- has left.