Alice is the head of her high-school quiz bowl league. She is in charge of choosing who will play in the next quiz bowl tournament. She has executive authority about who plays when, and she can change her mind at any time, as long as it isn't for the game that is currently being played.
The rules of how she can put players on her team are as follows.
- She is in charge of a single team, which competes in every single round of the game.
- The team she is in charge of must have exactly 4 people playing at any time. Any person who is on the team that is not actively playing must sit out.
- As rounds progress, she can decide who is playing at any given time, unless its for the round she is actively playing on. If she does not like the line-up that is currently playing, she must wait until the round ends to change it.
- The entire tournament consists of an arbitrarily large number of rounds.
In general, this means that the order of events is
[Pick players] Play a round [Re-assign players] Play another round
and so on. Unfortunately for Alice, she does not know how many people will be on her team, and she does not know the skill level of any player. Thankfully, she can keep track of the scores of the players on her team during the rounds to determine who is answering the most, the second most, the third most, and so on.
Unfortunately, the teammates she has are pesky, and have strong feelings about how much each should be playing. Because of this, Alice has determined the following.
If Alice were to make the team consist of only the best 4 players, their team would win the tournament.
If Alice were to assign players completely at random, their team would lose.
The players on her team have reached a consensus - they are willing to play in unequal amounts, as long as the team selection is as fair as possible, while guaranteeing that they still win the championship.
How can Alice create the most fair teams, while ensuring victory?
Reminder: A team that is 100% fair, consists of completely random players, and a team that is 0% fair consists of only the best players.