#King of the Tournament

In graph theory, a [tournament](http://en.wikipedia.org/wiki/Tournament_(graph_theory)) is a set of `N` vertices, where each each vertex is connected to each other vertex with a directed edge.  Furthermore, it has been proven that every tournament has at least 1 king.

We are going to be playing a round-robin tournament, where every player plays each player  exactly once each round.  Then, after the tournament, some of the players will be kicked out, and then another tournament will start.  This cycle will repeat until 1 player is victorious.

#The Tournament#
At the beginning of each tournament, each player will receive `N*5` points.  They will be randomly ordered from `1` to `N`.  (This ordering will be circular...imagine each player sitting in a circle, so the `N`th player is next to the first)

Then, each player will face every other player exactly once.

#The Battle#

For each battle, each player will decide how many points to use to fight the other.  These points will be subtracted from their total they received at the beginning.  If they both used the same amount of points, then the following will be the tie breakers.

1. The player with the least points
2. If the space between P1 to P2 is smaller than the space between P2 to P1, then P1 will win
3. The player who won last between the two
4. Multiply the round number by 2 (there are N/2 rounds).  The player whose turn number most immediately follows the round number will win

#End of the tournament#

The kings of the tournament are determined as followed:

A King must have *defeated* every other player.  `P1` has *defeated* `P2` if `P1` directly beat `P2` in battle, or if `P1` beat a `P3` in battle who beat `P2` in battle.

If a tournament ends, and every player is a king, then the players with the least amount of wins in the most recent tournament will be removed.  If all players have the same amount of wins, then the players with the least *total* amount of wins will be removed.  If this is the same as well, then every remaining player will be declared victorious.