Background

Details

• For each player, each chess game can result in a win (one point), loss (no points) or a draw (half point).
• In the first round, pairings may be made arbitrarily.
• In the second and later rounds, each pairing will ideally have the following properties:
  1. No two players will play each other more than once
  2. The players will have the same score
  3. Each player will alternate playing as black and white, or, if alternation does not work, each player should at least play as black and white an equal number of times
• If there is an odd number of players, in each round one player gets a bye. Each bye counts as a win, receiving one point. No player can receive more than one bye. In any given round, the lowest-ranked player who has not yet received a bye should receive the bye.

Assignment

• Your program will have two parts:
  1. A tournament simulator
  2. A pairing assigner
• The tournament simulator will be initialized with a number of tournament participants. Each participant should be represented by:
  • A name
  • Numbers of wins, losses and draws
  • For each round, whether the participant played white or black
  • Any other data you consider to be pertinent
• The tournament simulator will run for a pre-determined number of rounds. For each round, it will:
  • Request a set of pairings from the pairing assigner
  • Simulate a tournament by assigning wins, losses, and draws for each match
  • Update any statistics considered worthwhile to maintain
• The pairing assigner will assign pairings based on the properties given above. These properties are given in their order of importance. For each round, the pairing assigner will use the data maintained by the simulator in order to generate the pairings.
• The pairing assigner should also produce a totally ordered ranking of the players. Be sure to include a set of tie-breaking strategies to produce this, in the event that the win-loss-draw scores are insufficient.
• You may find elements of this assignment to be ambiguous in places. Handle the ambiguities as you see fit. Feel free to consult the instructor if you are confused about any particulars.

Analysis

• With respect to your implemented pairing assigner, prove that:
  1. If m ≥ log₂n, where m is the number of rounds and n is the number of players, there can exist no more than one undefeated player.
  2. No two players ever play each other more than once, as long as m < n
  3. No player receives more than one bye for the entire tournament.
• Again with respect to your algorithm, for each of the following state whether it is true or false, and prove your answer:
  1. Each player will alternate playing black and white on each round.
  2. If b is the number of games played as black and w is the number of games played as white, |b - w| &le 1.
  3. For some constant c_bw, |b - w| ≤ c_bw. If this is true, be sure to describe how c_bw is calculated. It may depend on the values for n and/or m.
  4. If x and y are the scores for two players who have been paired, |x - y| = 0.
  5. For some constant c_s, |x - y| ≤ c_s. If this is true, again be sure to describe how to calculate c_s.
• Run your simulator and pairing algorithm, with m = log₂n for all runs. Try 3 runs with each of 10, 50, and 100 players (for a total of 9 runs). Report each of the following:
  • Number of rounds played
  • Maximum number of games between any two players
  • Across all players, the minimum, maximum, average, and standard deviation of:
    • c_bw
    • c_s
    • Number of byes
  • Number of undefeated players
  • Any other statistics that can help describe the practical performance of your algorithm

Submission

• Your source code
• A word-processed report that includes:
  • A description of your algorithm
  • Your proofs
  • Graphs of your data
  • Textual analysis of your data
  • Conclusions
• You may use any text-preparation system you wish