Well, just taking a sorta shot in the dark here, but I'd guess you'd need two tables like this:
Games:
Game Number
Player
Team(?)
Date Played
Voting:
User Voting
Vote
Player Voted For
Game Number
Timestamp Voted
That way you have track of the voting and the date/times. Maybe even a third table for players, just to keep track
Players:
Player ID
Player Name
Team
Then you could get rid of the team in the games table, and just link from the players table.
Just a guess. I'd have to dig into the logic of the program to really get a more accurate structure. Considering this is just the voting aspect of it, there's also going to need to be a place to enter the games, which makes me think you could instead of removing the team from the games table, remove the player. Then link from there to the players table by team to grab the list of players.
I think I over-thought this one again...
