Balancing carry-over effects in round robin tournaments

Abstract

SUMMARY If team A played team B in its previous match of a round robin tournament and is now playing team C, team C is said to receive a carry-over effect due to team B. This paper investigates the construction of draws for round robin tournaments with the aim of distri- buting the carry-over effects due to any team as evenly as possible amongst the other teams. A balanced distribution is shown to occur when the number of teams is a power of two, and a method of construction of draws is given for these cases. It is conjectured that balanced draws do not exist for other numbers of teams, and the most effective method of construction yet found for this latter situation is presented. As considered here, a round robin tournament is made up of t teatns, which play every other team n times. The number of places in the draw is required to be even; if only (t - 1) teams enter the competition, the remaining place is called a bye, and a team drawn to play a bye has a rest day. For the purpose of this discussion, a bye may be regarded as an ordinary team. In each of the n rounds, 't matches are played on each of (t - 1) occasions, with a team meeting every other team once per round. The draw for every round subsequent to the first is the same as for the first round, except that venues may be altered, to allow for 'home' and 'away' matches, for example. Each team is considered to have an effect on its opponents which carries over to the next match. If team A meets team B in one match and team C in the next, then it is reasonable that team A's performance against team C will have been affected by team B. Particularly in body-contact sports, if team B is a strong, hard-playing side, then team A is likely to enter the match against team C bruised in both body and morale. Conversely, if team B is relatively weak, then team C can anticipate that team A will be confident and fit for their match. Team C is said to receive a 'carry-over effect' due to team B. The aim of this paper is to obtain draws, for various values of t, which spread as evenly as possible the carry-over effects of each team. No carry-over effect is present in the first match, so all teams will both pass on, and receive, n(t - 1) - 1 carry-over effects in all. A draw will be called balanced with respect to carry-over effects if every team receives carry-over effects n times from each of (t -2) teams, and (n - 1) times from the remaining team.