Tournament Designs: Carry-Over

Julian D. A. Wiseman, March 2002

Abstract: The best n teams from two all-play-all leagues go to a final which is all-play-all except that games already played are carried over. These pages describe optimised tournament designs for such carry-over finals.

Contents: Introduction; Presentations and file formats; Technical notes (permutation score; left-right asymmetry measure).

Publication history: Earlier versions of some of the designs here have previously been made available in paper form by Dr Nicholas F. J. Inglis, for the use of ETwA and CUTwC tournament organisers. This is believed to be the only publication on the web. Usual disclaimer and copyright terms apply.

Introduction

Two leagues have played an all-play-all, and the top (say) eight teams from each league are to go through to a final. This final is to be a variant of an all-play-all. The scores from games between qualifying players are re-used; it remains only for each qualifier to play each of the other qualifiers whom they have not already opposed. So each qualifier plays each of the qualifiers from the other league. With eight teams going through this is done over eight rounds and at eight venues, with each team playing only once at each venue and only once in each round. On the right is the example in which eight teams from each league go to the final. Carry-over tournament designs are published for various numbers of players from 16+16=32 down to trivial sizes: 16+16=32, 15+15=30, 14+14=28, 13+13=26, 12+12=24, 11+11=22, 10+10=20, 9+9=18, 8+8=16, 7+7=14, 6+6=12, 5+5=10, 4+4=8, 3+3=6, 2+2=4 and 1+1=2. Also see the complete list of carry-over links.

The precise meaning of some of the terms can vary according to the game being played.

• A ‘player’ might actually be a pair or even a team of players. Players from the Roman league are labelled with capital roman letters: A, B, C, D, etc; those from the Italic league with lower-case italicised letters: a, b, c, d, etc.

• The specific meaning of the word ‘venue’ will depend on the context. In snooker or billiards, a venue refers to a particular table; in tiddlywinks (for which these tournament designs were originally devised), to a particular mat; in tennis to a court; in croquet to a lawn or to a set of balls; and in other games to a location or set of equipment as appropriate. Venues are always labelled with lower-case Roman numbers: i, ii, iii, iv, etc.

• Several games happen simultaneously during a ‘round’. Rounds are always labelled with an Arabic numeral: 1, 2, 3, 4, etc.

• A particular game might be described as ‘A:b’, meaning that player A is to play against player b (in a particular round and at a particular venue). However, ‘A:b’ is not necessarily the same as ‘b:A’. In tiddlywinks, the left player should take the dominant corners. In chess, left should play white, and in other games in which one player starts, left starts. In card games left deals first. In general, the left player should have the advantage, but each game will have its own meaning for left and right, in a manner to be determined by the tournament organiser. To encourage consistency, tournament organisers are invited to inform this author of any conventions which may be agreed.

Presentations and file formats

Each of these carry-over tournaments is published in a number of different presentations and file formats.

• A description of the game schedule is available in PDF, with versions optimised for A4, A3 and US Letter (USL).

• The same data is available as a plain text file.

• A score-sheet is available in PDF, again in A4, A3 and USL. Italic league players run horizontally, with their scores being entered in the lower-left part of the cell. Roman league players run vertically, with their scores being entered in the upper-right part of the cell. The diagonal of each cell shows the game description, as well as the round and venue.

• A more detailed score-sheet is available in PDF (in A4, A3 and USL), with extra columns and rows into which a player’s running total of scores may be entered. With a large number of players the extra columns can crowd the page. The running-total cells show the description of the game just played.

• Both the less- and more-detailed score sheets are also available in ‘blank variants’, which don’t have the game descriptions in the cells.

• Finally, each tournament is available in a machine-readable text format, with one row per game. Each row is in the form RR VV A:b, where RR is the round, VV is the venue (for ease of machine-readability exceptionally shown in ‘Arabic’ numerals rather than lower-case Roman numbers), and A:b is the game description. In this presentation (and in the other plain text file) the Italic league players are identifiable only by their case. It is from this file that the others are generated: readers sending improved tournament designs are asked to do so in this format.

Technical notes

There can be many ways to set out an carry-over tournament of any particular size. A particular one is chosen using some constraints and two optimisations.

• There are many ways to rearrange a carry-over, in that it is possible to rearrange the rounds and the players. What is wanted is for ‘important’ games to come late in the tournament, where important games are those between the best players. So the players are seeded, with A being the best-ranked player in the Roman league, and a the best in the Italic league, and we want games such as A:a to come late. The optimisation is then performed as follows. Assign numerical values to each player: A = a = 0, B = b = 1, C = c = 2, etc. Number the rounds starting at zero, so the round numbers are one lower than in the human-readable form. If the game X:y (or y:X) is in round r, then calculate the permutation score as the sum over all games of ( r / (1 + X*X + y*y) ) ^ 4, and maximise this. The maximisation is done by permuting rows and permuting players within each league. For all cases up to 9+9 players it is believed that the optimum has been found; beyond 9+9 players the design is likely to be near-optimal, but may not be the exact optimum.

• Which still leaves undecided the left-right assignments. We constrain the left-right assignments by imposing a symmetry condition: iff X:y then Y:x. (This condition must be this way round because of the case in which X=Y.) We also impose the condition that every player is to play left and right approximately equally (in a tournament with an odd number of players in each league, each plays an odd number of games, in which case there must be a one game mismatch).

• But that does not uniquely determine the left-right assignments. So consider the position of a player in a tournament: what would be to that player’s advantage? We know that the player is to play left and right approximately equally. How could these lefts and rights be distributed to maximum advantage? Clearly it is best to have the advantage in close games, where it might make a difference. Most players will lose against the best players, whether playing left or right; and will win against the bunnies, whether playing left or right. But against similar players an advantage is most valuable. So the tournament design should ensure that this advantage is as evenly spread as possible, and we start by calculating an asymmetry measure for each player. Assign the same numerical values to each player as above. For each game X:y add to X’s score 1/(1+|x-y|), and for each game y:X subtract this same term. This term is of largest magnitude for close games. Total these terms for each player, and square. If a player has a fair balance of left and right in close games, and hence in non-close games, this sum will be nearly zero. If a player has a mismatch, this squared-sum will be large. Now calculate the weighted total of these squared sums, where the weight is the number of players minus that player (‘n-X’), and call this the left-right asymmetry measure. (Because of the symmetry condition, it suffices to sum only over the players in one league.) So this left-right asymmetry measure has the largest weight on A’s squared sum, as fairness is most important amongst those with a large chance of winning. Minimise the left-right asymmetry measure (subject to the above constraints). There will be two minima, one being the reverse-everything opposite of the other. Choose the minimum containing A:a rather than a:A (which means that, if it is possible to compare A and a, then league identities should be chosen so that A is better than a, otherwise randomly). For all numbers of players up to 13, it is believed that the actual minimum of the left-right asymmetry measure has been found. For 14 or more players, the proffered design is at least close to the minimum.