Contest Details
A game starts off with a certain number of players. This number must be a power of 2 ( 2, 4, 8, 16, 32, 64, ...), the reason being that after each round the number of players that advance is halved. Any power of two that gets halved again and again will always reduce to 1, that player being declared the winner. There are two stages to complete one single round: normalization and elimination. These terms will be made apparent very soon. When the game begins, everyone is placed in Group 0. Everyone battles a random opponent within their current group. All those players that win, stay in Group 0, all the losers get bumped down to Group 1. Then all players in Group 1 play a random opponent from Group 1, and all players in Group 0 play a random opponent from Group 0. The losers from Group 1 go down to Group 2, the winners stay in Group 1. The losers from Group 0 go down to Group 1, the winners stay in Group 0. This process is continued until Group 0 has only a single player left.
Before we continue, let us show an example that we will carry through in its entirety. We will begin a new game with 16 players. They all begin in Group 0.
Group 0: Andrew, Billy, Carl, David, Ethan, Fred, Gordon, Harry, Ichabod, John, Kevin, Larry, Moe, Ned, Oscar, Peter
Everyone will be paired with a random opponent, and they will battle. For each battle, there will be one winner and one loser. Let us say that the first eight players in our list win and the last eight lose. Our groups are then as follows:
Group 0: Andrew, Billy, Carl, David, Ethan, Fred, Gordon, Harry
Group 1: Ichabod, John, Kevin, Larry, Moe, Ned, Oscar, Peter
Now comes the second battle of the round. Everyone battles a random opponent within their own group. Let us say that John, Larry, Ned, and Peter won their respective battles. They will remain in Group 1, while the losers (Ichabod, Kevin, Moe, and Oscar) get bumped down to the newly created Group 2. Let us also say that the winners from Group 0 were Andrew, Carl, Ethan, and Gordon. They remain in Group 0, while the losers (Billy, David, Fred, and Harry) get bumped down to Group 1. The groups now look like this:
Group 0: Andrew, Carl, Ethan, Gordon
Group 1: John, Larry, Ned, Peter, Billy, David, Fred, Harry
Group 2: Ichabod, Kevin, Moe, Oscar
There are now 4 players in the first group (all of whom have a 2-0 record), 8 in the second (1-1), and 4 in the third (0-2). The third battle ensues, the winners of which are Andrew and Carl in Group 0; John, Ned, Billy, and Fred in Group 1; Ichabod and Oscar in Group 2. Practically speaking, those in the lowest groups play first in order to preserve the groupings from the previous rounds. The groups look as follows:
Group 0: Andrew, Carl
Group 1: John, Ned, Billy, Fred, Ethan, Gordon
Group 2: Ichabod, Oscar, Larry, Peter, David, Harry
Group 3: Kevin, Moe
Everyone in Group 0 has a record of 3 wins and 0 losses (3-0). Everyone in Group 1 has a record of (2-1), Group 2 has a record of (1-2), and Group 3 has a record of (0-3). One more battle takes place with the winner from Group 0 being Andrew; John, Ned, and Billy from Group 1; Ichabod, Oscar, and Larry from Group 2; and Kevin from Group 3. This marks the end of the first stage: normalization.
Group 0: Andrew
Group 1: John, Ned, Billy, Carl
Group 2: Ichabod, Oscar, Larry, Fred, Ethan, Gordon
Group 3: Kevin, Peter, David, Harry
Group 4: Moe
This is the end of normalization. If you know anything about a normal curve, plotting group population vs. group number will result in the semblence of such a curve. The other interesting thing to note is the numbers of the group populations: 1, 4, 6, 4, 1 are binomial coefficients.
So, we now have Andrew who has won 4 times, and Moe who has won 0 times. Everyone else has won either 3, 2, or only once. The number of wins becomes important now: to advance to the next round, a player needs to accumulate 5 wins. This means that Andrew needs win only one more time, but Moe needs to win 5 more times. Fred will need to win 3 more times, Harry needs to win 4 more times. Ned needs to win twice. This is the start of the elimination stage of the round. This is the only time when players will battle an opponent that might be in a different group than they are. The symmetry of the populations of the groups is the factor here. Andrew will play Moe until one of them accumulates 5 wins. John will someone in Group 3 (Kevin, Peter, David, or Harry) until one of them accumulates 5 wins. Each person in Group 1 will be paired with a person in Group 3. Group 2, being the middle group, will get opponents from amongst themselves. Whatever the battle is, Andrew (who has never lost) will need to win only once against Moe (who has never won). It is safe to say that Andrew will win and advance. Let us say that in the battle of John vs Kevin, John beats Kevin after only two battles (remember John needed only 2 more wins to advance). Ned beats Peter, Billy beats David, but Harry surprised everyone and beat Carl (by winning 4 times before Carl could win twice). The winners from Group 2 were: Ichabod, Larry, Ethan, and Gordon. All the winners advance, while all the losers are eliminated. The next round will began as follows:
Group 0: Andrew, John, Ned, Billy, Harry, Ichabod, Larry, Ethan, Gordon
The next round will began with this one group. Each person to advance has 5 victories, but a different number of losses. In Battle Elimination it's only the wins that matters.
