FCT18 Rules

1) (a) A Game is a single game of "Live960", i.e. Fischer Random Chess[6], on Chess.com[7], with time controls of "15|10", i.e. 15 minutes per player with 10 seconds added per move.

   (b) Unless otherwise agreed by both players:
       (i)   The player *lower* in the ladder starts the game by sending their opponent an invitation.
       (ii)  For their first Game against each other, the player *lower* in the ladder plays as White.
       (iii) For subsequent Games, the player who played as Black in their last game plays as White.
       (iv)  The winner, or White if the Game is drawn, submits the result.

   (c) When a Game is finished, one player must inform the Tournament Director(s) of the result, providing a game URL of the form <https://www.chess.com/live/game/2875606888>.

   (d) If the Game is drawn, both players gain 1/2 a Win; otherwise, the winner gains 1 Win.

   (e) If there is any dispute or uncertainty about the conditions or the result of a Game, both players must speak to the Tournament Director(s), who will give further instructions.

2) The Final consists of a best-of-5 match between two Finalists to determine the Champion:

   (a) The number of Final Wins of each player is recorded, initially set to 0 (separately from their preliminary Wins, which remain unchanged). The Finalists play Games until
       (i)  one player has at least 3 Final Wins, and
       (ii) one player has more Final Wins than the other.

   (b) (i)   The Finalist with the greater number of Final Wins is declared the Champion and given First Place in the tournament.
       (ii)  The other Finalist is given Second Place.
       (iii) Each other player receives a Placement equal to 1 plus the number of players in the Ladder with a higher Rank.

   (c) If one or both Finalists are removed from the tournament, the procedure in (5)(f) is followed to choose a new pair of Finalists and restart the Final.

3) Except by decision of the Tournament Director(s):

   (a) Any Game taking longer than 7 days to have a result submitted,
       (i)   from its announcement by the Tournament Director(s) if it is a preliminary game, or
       (ii)  from the announcement of the Finalists if it is the first game of the Final, or
       (iii) from the previous Final game, if it is a subsequent game in the Final,
       is given a Null result, with both players gaining 0 Wins.

   (b) Any player who cannot be contacted promptly after a Game of theirs is Annulled is removed from Ladder and disqualified from the tournament.

4) (a) The players are placed in a Ladder, initially in an arbitrary order from highest to lowest. In general, players thought to be more likely to win the tournament should initially be higher in the Ladder.

   (b) Each player has a separate number of Wins against each other player, a non-negative rational number which is initially 0. The number of Wins may have a fractional part of 1/2 if a player has been awarded 1/2 a Win for a draw, but is otherwise an integer.

   (c) Each player has a separate Score against each other player, an integer from 0 to 99. The Score of player P against player Q is given by the *width* of the *widest path* from P to Q, where:
       (i)   A *path* from P to Q is any sequence of 2 or more players whose first element is P and whose last element is Q.
       (ii)  The *width* of a path is the smallest *initial score* of R1 against R2 for any two players R1 and R2 in the path where R2 comes directly after R1 in the sequence.
       (iii) The *initial score* of R1 against R2 is given by 100A/(A+B+3) rounded *down* to the nearest integer, where A is the number of Wins of R1 against R2 and B is the number of Wins of R2 against R1. For this purpose, each player has 0 Wins against themselves.

       Remark: This system is similar to the Schulze[1] electoral method. Here, one player's *initial score* takes the role of the number of voters preferring one candidate over another. Calculating Scores from *initial scores* is the widest path problem[2], which can be readily solved using the Floyd-Warshall[3] algorithm.

       Remark: The *initial score* of P against Q is related to the percent chance that P wins a Game against Q. The formula in (iii) is similar to those used for the lower bounds of binomial confidence intervals[4].

   (b) Each player has a Rank, a positive integer related to their position in the Ladder.
       (i)   The Rank of the player at the top of the Ladder is 1.
       (ii)  If a player's Score against the player above them equals that player's Score against them, and this Score is *not* 0, the player's Rank equals that of the above player.
       (iii) The Rank of any other player is equal to 1 plus the Rank of the player above them.

5) (a) A number of preliminary Rounds are played. Rounds are numbered in order played, starting at 1.

   (b) If the number of players is odd, then before each Round, one player is selected to take a Bye, meaning they do not Participate in that round: among those players with the largest numbers of Completed[5] Games, the player highest, if the round number is *odd*, or lowest, if the round number is *even*, in the ladder takes a Bye.

      Remark: This means the top player has the privilege of joining the tournament fashionably late.

   (c) In each Round, Participants are Matched with each other by the following procedure:
       (i)   If the round number is *even*, go to (ii), or if the round number is *odd*, go to (iii).
       (ii)  Choose the bottommost unmatched Participant, and consider all Participants above. Go to (iv).
       (iii) Choose the topmost unmatched Participant, and consider all Participants below. Go to (iv).
       (iv)  Of the players under consideration, match the chosen player with the one who, in descending order of priority,
             1. has the smallest *match cost* with the chosen player, and
             2. has Completed[5] the least Games with the chosen player, and
             3. is closest in the Ladder to the chosen player,
             where the *match cost* of two players is given by G + D(D-1)/2, where G is the number of Completed[5] games between them and D is the difference between their Ranks.
       (v)   If there are any unmatched Participants left, go to (ii) if the last match was made via (iii), or go to (iii) if the last match was made via (ii); otherwise, the procedure is complete.

   (d) In each Round, each pair of Matched players plays a Game as defined in (1).

   (e) In each Round, after all Matches are complete, and before deciding any Bye for the next round, any player who has a higher Score, against any player above them in the Ladder than that player has against them is moved in the Ladder to directly above the highest such player.

   (f) After the 5th round:
       (i)   If there are *more than two* players of Rank 1, a special tie-breaking Round is held where each player of Rank 1 is Matched against each other player of Rank 1, and no other players participate.
       (ii)  Otherwise, if there is *exactly one* player of Rank 1 and *more than one* player of Rank 2, the procedure in (i) is done except with Rank 2 players.
       (iii) Otherwise, if the Tournament Director(s) deem it necessary, additional rounds are held with conditions specified by the Tournament Director(s).
       (iv)  Items (i), (ii) and (iii) are repeated until they do not apply.
       (v)   The top two players from the Ladder are selected as Finalists and participate in the Final, as defined in (3).

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[1] https://en.wikipedia.org/wiki/Schulze_method
[2] https://en.wikipedia.org/wiki/Widest_path_problem
[3] https://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithm
[4] https://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval
[5] A Game is Completed if it has a non-Null result recorded, i.e. if it contributes a Win to a player or 1/2 a Win to two players.
[6] https://en.wikipedia.org/wiki/Chess960
[7] https://www.chess.com