To the Editor of the St. James’s Gazette
Sir,—In treating this subject I propose to myself four things:—
(1.) To prove that the existing method of assigning prizes in lawn tennis tournaments is, except in the case of the first prize, entirely absurd;
(2.) To prove that the existing method of scoring in matches leads, in many cases, to an unjust result;
(3.) To suggest a method for conducting tournaments which, while requiring less time than the present, shall give more equitable results;
(4.) To suggest a better method of scoring in matches.
(1.) To prove the absurdity of the present method of assigning prizes in lawn tennis tournaments will not need many words. Suppose there are 32 competitors and 4 prizes. On the 1st day, these contend in 16 pairs: on the 2nd day, the 16 winners contend in 8 pairs, the losers being excluded from further competition: on the 3rd day, the 8 winners contend in 4 pairs: on the 4th day, the 4 winners (who are now known to be the 4 prize-men) contend in 2 pairs: and on the 5th day, the 2 winners contend together, to decide which is to take the first prize and which the second—the 2 losers having no further contest, as the 3rd and 4th prize are of equal value.
Now, if we divide the original list of competitors into 4 sections, we may see that all, that this method really does, is to ascertain who is the best man in each section, then who is the best in each half of the list, and then who is the best of all. The best of all (and this is the only equitable result arrived at) wins the first prize: the best in the other half of the list wins the second: and the best men in the 2 sections not yet represented by a champion win the other two prizes. If the original list had chanced to be arranged in the order of merit, the 17th best player will necessarily carry off the 2nd prize, and the 9th and 25th best the 3rd and 4th! This of course is an extreme case: but anything within these limits is possible: e. g. any competitor, from the 3rd best to the 17th best, may, by the mere accidental arrangement of names, and by no means as a result of his own skill, carry off the 2nd prize. As a mathematical fact, the chance that the 2nd best player will get the prize he deserves is only 16-31ths: while the chance that the best 4 shall get their proper prizes is so small that the odds are 12 to 1 against its happening!
(2.) To prove that the existing method of scoring in matches leads, in many cases, to an unjust result, let us suppose a “set” to mean “the best of 5 games,” and a “match” “the best of 5 sets.”
Suppose A and B to play the following 23 games:—BAABB | AAA | BAABA* | B*ABAB | BAABA. Here A wins 13 games to 10, and also wins the match. But, by simply transposing A*, B*, we get BAABB | AAA | BAABB | AABA | BBAAB |, the last game of the original series not being played. Here A still wins 12 games to 10: yet he loses the match!
(3.) The method for conducting tournaments, which I have to propose, involves two departures from the present method. First, I propose to make a “match” last only half a day (the necessary reduction in the number of games I will discuss in section 4): secondly, I propose to give only 3 prizes. The rules for a tournament of 32 players would be as follows:—
(a.) The tournament begins in the middle of the 1st day, so that there is only one contest that day—the 32 players being arranged in 16 pairs.
(b.) A list is kept, and against each name is entered, at the end of each contest, the name of any one who has been proved superior to him—whether by actually beating him, or by beating some one who has done so (thus, if A beats B, and B beats C, A and B are both “superiors” of C). So soon as any name has 3 “superiors” entered against it, it is struck out of the list.
(c.) For the 2nd day (morning) the 16 unbeaten men are paired together, and similarly the 16 with 1 superior (the losers in these last-named pairs will now have 3 superiors each, and will therefore be struck off the list). In all other contests they are paired in the same way: first pairing the unbeaten, then those with 1 superior, and so on, and avoiding, as far as possible, pairing two players who have a common superior.
(d.) By the middle of the 3rd day the unbeaten are reduced to two, one of them is certainly “first-prize-man.” These two do not contend in the afternoon contest that day, but have a whole-day match on the 4th day—the other players meanwhile continuing the usual half-day matches.
(e.) By the end of the 4th day, the “first-prize-man” is known (by the very same process of elimination used in the existing method): and the remaining players are paired by the same rules as before, for the 2 contests on the 5th day. In some cases the 2nd and 3rd prizes will both be decided by the middle of the 5th day. If, in section (a), the tournament were begun in the morning, the two men named in section (d) being still allowed a whole-day match, nothing would be gained in time, as the tournament would still take 4½ days, while much would be lost in interest, as the first-prize would be settled in 3 days.
These rules will, I think, be sufficiently illustrated by going through a tournament of 16; and if the reader will draw up for himself these Tables, in blank, and fill them up, column by column, according to the following directions, he will easily understand the working of the system.
Let the players be arranged alphabetically, and call them A, B, C, etc., and let their relative skill be represented by the following numbers:—
| A | B | C | D | E | F | G | H | J | K | L | M | N | P | Q | R |
| 6 | 10 | 13 | 5 | 15 | 9 | 7 | 12 | 1 | 14 | 3 | 2 | 8 | 16 | 11 | 4 |
These numbers will enable the reader to decide which will be the victor in any contest: but of course they are not supposed to be known to the Tournament Committee, who have nothing to guide them but the results of actual contests. In the following tables, “I. (e)” means “first day, evening,” and so on: also a player, who is virtually proved superior to another, is entered thus “(A).” The victor in each contest is marked *.
| I. (e) | II. (m) | (e) | III. (m) | (e) | IV. (m) | (e) | ||||||||||||||
| A | } | * | A | } | D | } | * | D | } | D | } | D | } | D | } | |||||
| B | D | * | G | J | J | * | M | * | L | * | ||||||||||
| C | } | F | } | J | } | * | A | } | M | } | * | L | } | * | ||||||
| D | * | G | * | R | M | * | R | R | ||||||||||||
| E | } | J | } | * | A | } | * | G | } | L | ||||||||||
| F | * | M | F | R | * | |||||||||||||||
| G | } | * | N | } | H | } | L | } | * | |||||||||||
| H | R | * | N | * | N | |||||||||||||||
| J | } | * | B | } | * | M | } | * | ||||||||||||
| K | C | Q | ||||||||||||||||||
| L | } | E | } | B | } | |||||||||||||||
| M | * | H | * | L | * | |||||||||||||||
| N | } | * | K | } | ||||||||||||||||
| P | L | * | ||||||||||||||||||
| Q | } | P | } | |||||||||||||||||
| R | * | Q | * | |||||||||||||||||
| I. (e) | II. (m) | (e) | III. (m) | (e) | IV. (m) | (e) | |
| A | … | D | … | M (J) out | |||
| B | A | (D) | L out | ||||
| C | D | B (A) out | |||||
| D | … | … | … | … | J | M | L out |
| E | F | H (G) out | |||||
| F | … | G | A (D) out | ||||
| G | … | … | D | R (J) out | |||
| H | G | … | N (D) out | ||||
| J | … | … | … | … | First | ||
| K | J | L (M) out | |||||
| L | M | (J) | … | … | … | … | Third |
| M | … | J | … | … | … | Second | |
| N | … | R | (J) | L out | |||
| P | N | Q (R) out | |||||
| Q | R | … | M (J) out | ||||
| R | … | … | J | … | M | L out |
In contest I. (e), we see that A beats B, and so on: hence we enter A as a “superior” to B, D to C, and so on. For contest II. (m), we pair the winners A, D, and so on: and then the losers, B, C, and so on. After it, we enter the actual superiors, D, C, etc.: we then find that, A having a superior D, and B a superior A, B has a virtual superior D: and so on. Having done this, we see that C, E, K, P, have three superiors each, and must be struck out. For contest II. (e) we should avoid pairing F and H, because they have a common superior: and the same may be said of N, Q. After it, we strike out B, F, H, Q. On the 3rd day, as there are only 2 unbeaten left, they contend during the whole day, the others having half-day contests: L and N have to be paired, even though they have a common superior. After the morning contest, we strike out A, G, N. For the evening, D and J are still contending: so that M and R must be paired, though they have a common superior; and L is “odd man.” After the evening contest, J is seen to be “first-prize-man.” After contest IV. (m), we strike out R; and we see that M is “second-prize-man.” After contest IV. (e), we strike out D, and give L the 3rd prize.
If this tournament were fought by the present method, the 4 prize-men would be D, G, J, R: D would get the 2nd prize, and G and R the 3rd and 4th: i. e. the 5th best man would get the 2nd prize, and the 7th and 4th best the other two.
(4.) To make “matches” more equitable, I propose to abolish “sets,” and make a “match” consist of “games.” Thus instead of “best of five games = set; best of 5 sets = match” (i. e. he who first wins 3 games wins a set; he who first wins 3 sets wins a match), where a player may win with as few as 9 games, and must win with 13, I would substitute “he who wins 13 games, or who gets 9 games ahead, wins the match.” This, however is a short match. The London Athletic Club say “he who wins 6 games wins a set; he who first wins 3 sets wins a match.” Here a player may win with 18 games, and must win with 28: so that it might need as many as 55 games to decide a match. This again seems needlessly large. I am inclined for a compromise, and propose as follows: “For a whole-day, he who first wins 24 games, or who gets 16 ahead, wins the match: for a half-day, he who first wins 12 games, or who gets 8 games ahead, wins the match.” The proposed form of tournament, though lasting a shorter time than the existing one has a great many more contests going on at once, and consequently furnishes the spectacle-loving public with a great deal more to look at.—I am, Sir, your obedient servant,
Charles L. Dodgson
Student and late Mathematical Lecture of Christ Church, Oxford
July 30.