The True Method of Assigning Prizes with a Proof of the Fallacy of the Present Method
Palmam qui meruit ferat
§ 1. Introductory
At a Lawn Tennis Tournament, where I chanced, some while ago, to be a spectator, the present method of assigning prizes was brought to my notice by the lamentations of one of the Players, who had been beaten (and had thus lost all chance of a prize) early in the contest, and who had had the mortification of seeing the 2nd prize carried off by a Player whom he knew to be quite inferior to himself. The results of the investigations, which I was led to make, I propose to lay before the reader under the following four headings:—
(a) A proof that the present method of assigning prizes is, except in the case of the first prize, entirely unmeaning.
(b) A proof that the present method of scoring in matches is constantly liable to lead to unjust results.
(c) A system of rules for conducting Tournaments, which, while requiring even less time than the present system, shall secure equitable results.
(d) An equitable system for scoring in matches.
§ 2. A proof that the present method of assigning prizes is, except in the case of the first prize, entirely unmeaning
Let us take, as an example of the present method, a Tournament of 32 competitors with 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 1st prize and which the 2nd—the two Losers having no further contest, as the 3rd and 4th prize are of equal value.
Now, if we divide the list of competitors, arranged in the order in which they are paired, into 4 sections, we may see that all that this method really does is to ascertain who is best in each section, then who is best in each half of the list, and then who is best of all. The best of all (and this is the only equitable result arrived at) wins the 1st prize: the best in the other half of the list wins the 2nd: and the best men in the two sections not yet represented by a champion win the other two prizes. If the Players had chanced to be paired in the order of merit, the 17th best Player would 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 pairs, 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 16-31sts; 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!
If any one thinks that, after all, we are merely introducing another element of chance into the game, and that no one can fairly object to that, let him try the experiment in a rifle competition. Let him interpose when the man, who has made the 2nd best score, is going to receive his prize, and propose that he shall draw a counter from a bag containing 16 white and 15 black, and only have his prize in case he draw a white one: and let him observe the expression of that rifleman’s face.
§ 3. A proof that the present method of scoring in matches is constantly liable to lead to unjust results
To prove this, let us suppose a “set” to mean “the best of 11 games,” and a “match” “the best of 5 sets”: i. e. “he, who first wins 6 games, wins a set; he, who first wins 3 sets, wins a match.”
Suppose A and B to play the following 50 games (“A2” means A wins 2 games, and so on):—
B2A5B4|A6|B3A5B2A*|B*A2B4A3B|B2A5B3A.
Here A wins 28 games to 22, and also wins the match.
But, by simply transposing A*, B*, we get
B2A5B4|A6|B3A5B3|A3B4A3|B3A5B3,
the last game of the original series not being played. Here A still wins 27 games to 22: yet he loses the match!
§ 4. A system of rules for conducting Tournaments, which, while requiring even less time than the present system, shall secure equitable results
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 5): 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 o£ the 3rd day the unbeaten are reduced to two, one of whom 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 was 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 take days, while much would be lost in interest, as the first prize would be settled in 3 days.
To illustrate these rules, I will give the complete history of a Tournament of 32 competitors, with 3 prizes. If the reader will draw out the following Tables, in blank, and fill them up for himself, referring, if necessary, to the accompanying directions, he will easily understand the workings of the system.
Let the Players be arranged alphabetically, and let the relative skill, with which they play in this Tournament, be:—
| A | B | C | D | E | F | G | H | J |
| 19 | 22 | 14 | 32 | 16 | 25 | 15 | 28 | 3 |
| K | L | M | N | P | Q | R | S | T |
| 10 | 8 | 1 | 29 | 4 | 12 | 2 | 17 | 23 |
| U | V | W | X | Y | Z | a | b | c |
| 26 | 11 | 20 | 31 | 13 | 18 | 6 | 24 | 9 |
| d | e | f | g | h | ||||
| 21 | 30 | 5 | 7 | 27 |
These numbers (“1” meaning “best”) will enable the reader to name 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 *: and ⊙ means “struck out.”
| I. (e) | II. (m) | (e) | III. (m) | (e) | IV. (m) | (e) | V. (m) | (e) | ||||||||||||||||||
| A | } | * | A | } | C | } | * | C | } | C | } | M | } | M | } | * | R | } | * | J | } | * | ||||
| B | C | * | G | M | * | V | * | f | f | f | f | |||||||||||||||
| C | } | * | E | } | M | } | * | V | } | J | } | * | J | } | * | J | } | J | ||||||||
| D | G | * | R | f | * | a | V | R | * | |||||||||||||||||
| E | } | * | J | } | V | } | * | A | } | L | } | R | } | * | ||||||||||||
| F | M | * | Y | J | * | g | * | g | ||||||||||||||||||
| G | } | * | P | } | a | } | G | } | R | } | * | |||||||||||||||
| H | R | * | f | * | L | * | c | |||||||||||||||||||
| J | } | * | S | } | A | } | * | R | } | * | ||||||||||||||||
| K | V | * | E | S | ||||||||||||||||||||||
| L | } | W | } | J | } | * | Y | } | ||||||||||||||||||
| M | * | Y | * | P | a | * | ||||||||||||||||||||
| N | } | a | } | * | L | } | * | g | } | * | ||||||||||||||||
| P | * | c | Q | T | ||||||||||||||||||||||
| Q | } | f | } | * | S | } | * | c | ||||||||||||||||||
| R | * | g | W | |||||||||||||||||||||||
| S | } | * | B | } | * | Z | } | |||||||||||||||||||
| T | D | c | * | |||||||||||||||||||||||
| U | } | F | } | * | g | } | * | |||||||||||||||||||
| V | * | H | B | |||||||||||||||||||||||
| W | } | * | K | } | F | } | ||||||||||||||||||||
| X | L | * | T | * | ||||||||||||||||||||||
| Y | } | * | N | } | d | } | * | |||||||||||||||||||
| Z | Q | * | h | |||||||||||||||||||||||
| a | } | * | T | } | * | |||||||||||||||||||||
| b | U | |||||||||||||||||||||||||
| c | } | * | X | } | ||||||||||||||||||||||
| d | Z | * | ||||||||||||||||||||||||
| e | } | b | } | |||||||||||||||||||||||
| f | * | d | * | |||||||||||||||||||||||
| g | } | * | e | } | ||||||||||||||||||||||
| h | h | * | ||||||||||||||||||||||||
| I. (e) | II. (m) | (e) | III. (m) | (e) | IV. (m) | (e) | V. (m) | (e) | |
| A | … | C | … | J (M) ⊙ | |||||
| B | A | (C) | g ⊙ | ||||||
| C | … | … | … | M | V (f) ⊙ | ||||
| D | C | B (A) ⊙ | |||||||
| E | … | G | A (C) ⊙ | ||||||
| F | E | (G) | T ⊙ | ||||||
| G | … | … | C | L (M) ⊙ | |||||
| H | G | F (E) ⊙ | |||||||
| J | … | M | … | … | … | … | R | … | Pr. III |
| K | J | L (M) ⊙ | |||||||
| L | M | … | … | … | g (f) ⊙ | ||||
| M | … | … | … | … | … | … | Pr. I. | ||
| N | P | Q (R) ⊙ | |||||||
| P | … | R | J (M) ⊙ | ||||||
| Q | R | … | L (M) ⊙ | ||||||
| R | … | … | M | … | … | … | … | Pr. II. | |
| S | … | V | … | R (f) ⊙ | |||||
| T | S | (V) | … | g ⊙ | |||||
| U | V | T (S) ⊙ | |||||||
| V | … | … | … | f | … | J (M) ⊙ | |||
| W | … | Y | S (V) ⊙ | ||||||
| X | W | Z (Y) ⊙ | |||||||
| Y | … | … | V | a (f) ⊙ | |||||
| Z | Y | … | c (V) ⊙ | ||||||
| a | … | … | f | … | J (M) ⊙ | ||||
| b | a | d (c) ⊙ | |||||||
| c | … | a | (f) | … | R ⊙ | ||||
| d | c | (a) | (f) ⊙ | ||||||
| e | f | h (g) ⊙ | |||||||
| f | … | … | … | … | … | … | M | R | J ⊙ |
| g | … | f | … | … | … | R (M) ⊙ | |||
| h | g | (f) | d ⊙ |
Directions for filling in the Tables:—
Tab. I. Day I (e). The names are written out alphabetically, and paired as they stand. The victors are marked with asterisks.
Tab. II. Day I (e). As B has been beaten by A, A is entered as his “superior”; C as D’s superior; and so on.
Tab. I. Day II (m). We first pair together all the unbeaten, A, C, E, G, &c. Then those who have one superior, B, D, F, H, &c.
Tab. II (m). We first enter the actual superiors, C, G, &c. Then, since A has a superior C, and B has a superior A, we see that B has a virtual superior C; and so on. We then see that D has 3 superiors, and must be struck out; and so with H, &c.
Tab. I. Day II (e) We first pair together all the unbeaten, C, G, &c. Then all with one superior, A, E, &c.; but when we come to J, L, we find we have a common superior; so we pair J with P, and L with Q. This series ends with an odd one, g, who must therefore be paired with the first of those who have two superiors each, F, T, &c.
Tab. I. Day III (m). Here, in pairing those with one superior, we again end with an odd one, g, who must therefore be paired with the first of those with two superiors, viz. T. We end with an “odd man,” c.
Tab. II. Day III (m). The unbeaten are now reduced to one pair, M, f, who therefore will do nothing this afternoon, but will have a whole-day contest to-morrow.
Tab. I. Day III (e). Those who have one superior are C, J, L, R, all with a common superior M; and then V, a, g, all with a common superior f. We therefore pair C with V, and so on, leaving an odd one R, who must be paired with the only one who has two superiors, viz. c.
Tab. II. Day III (e). Enter as usual.
Tab. I. Day IV (m). We pair the 2 unbeaten, M, f, for their whole-day contest. Then those with one superior.
Tab. II. Day IV (m). M and f are still contending. V and g are struck out.
Tab. I. Day IV (e). J and R must be paired together, though they have a common superior.
Tab. I. Day IV (e). M is First-prize-man.
Tab. I. Day V (m). R and f must be paired together, though they have a common superior. J is “odd man.”
Tab. II. Day V (m). R is now the only man with one superior, and is therefore Second-prize-man.
Tab. I. Day V (e). J and f contend for the Third prize.
If this Tournament were fought by the present method, the 4 Prize men would be C, M, V, f: f would get the 2nd prize, and C and V the 3rd and 4th: i. e. the 5th best man would get the 2nd prize, and the 14th and 11th best the other two.
§ 5. An equitable system for scoring in matches
In order to make “matches” more equitable, I propose to abolish “sets,” and make a “match” consist of “games.” Thus, instead of “best of 11 games = set; best of 5 sets = match” (i. e. he who first wins 6 games wins a set; he who first wins 3 sets wins a match), where a player may win with as few as 18 games, and must win with 28, I would substitute “he who first wins 28 games, or who gets 18 games ahead, wins the match.” I therefore propose as follows: “For a whole-day, he who first wins 28 games, or who gets 18 ahead, wins the match: for a half-day, he who first wins 14 games, or who gets 9 ahead, wins the match.”
§ 6. Concluding remarks
Let it not be supposed that, in thus proposing to make these Tournaments a game of pure skill (like chess) instead of a game of mixed skill and chance (like whist), I am altogether eliminating the element of luck, and making it possible to predict the prize-winners, so that no one else would care to enter. The “chances of the board” would still exist in full force: it would not at all follow, because a Player was reputed best, that he was certain of the I St prize: a thousand accidents might occur to prevent his playing best: the 4th best, 5th best, or even a worst Player, need not despair of winning even the 1st prize.
Nor, again, let it be supposed that the present system, which allows an inferior player a chance of the 2nd prize, even though he fails to play above his reputation, is more attractive than one which, in such a case, gives him no hope. Let us compare the two systems, as to the attractions they hold out to (say) the 5th best Player in a Tournament of 32, with 3 prizes. The present system says, “If you play up to your reputation, your chance of a prize is about th; and even if, by great luck and painstaking, you play 2nd or 3rd best, it never rises above a half.” My system says, “It is admitted that, if you only play up to your reputation, you will get nothing: but, if you play 2nd or 3rd best, you are certain of the proper prize.” Thus, the one system offers a chance of th, where the other offers nothing; and a chance of a half, where the other offers certainty. I am inclined to think the second the more attractive of the two.
If, however, it be thought that, under the proposed system, the very inferior Players would feel so hopeless of a prize that they would not enter a Tournament, this can easily be remedied by a process of handicapping, as is usual in races, &c. This would give every one a reasonable hope of a prize, and therefore a sufficient motive for entering.
The proposed form of Tournament, though lasting a shorter time than the present 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.