In a tennis tournament, there are 100 players. Each match is played between two players. The winner advances and the loser is eliminated. Assuming a minimum number of byes (games in which a player automatically advanced due to no opponent), how many matches need to be played to determine a winner?

For determined the number of match, we are going to divide the game in rounds, every round is when every player plays with another and one advance and the other loses. That means that every round, the number of players is reduced to the half approximately. So:

First Round: There are 100 players, so we can form 50 matches and 50 players advance to the second round

Second Round: now, there are 50 players, so we can form 25 matches and 25 players advance to the second round.

Third round: In this round we have 25 players, so we only can form 12 games with matches and 1 game with bye, so 13 players advance to the fourth round.

Fourth Round: Again we have an unpair number of player, so we can form 6 games with matches and 1 with bye, then 7 players advance to the next round.

Fifth Round: In this round we have 7 players, so we only can form 3 games with matches and 1 game with bye, so 4 players advance to the fourth round.
Sixth Round: we have 4 players, so we can form 2 matches and 2 players advance to the final round.
Final Round: We form 1 match to determined the winner

Finally the number of matches that need to be played to determine the winner is the sum of matches that we can form in every round. That is:

50+25+12+6+3+2+1=99