Scoring a Tournament.................................

Player A's first place doesn't look well-deserved, because:

• he took it mainly due to maximum victories over the outsiders

• his advantage over B is very small

• he lost to B 4:6

Scoring a Tournament

PEAKYEARING

Each of the duels constituting a tournament is scored in a uniform manner (independently from the others), making the win over a weak opponent equally valuable as a win over a strong one! Consequently, a player doing poorly against the top-ranked opponents can make up for this, and more, winning big against the weak opposition, scoring against them much enough to finish quite high overall (or even to win the whole event). That flaw has been well perceived and is often remedied by inviting the players of equal strength to a tournament. We can try something else, though...

Take a look at the „round robin” team tournament below (duels scored by distributing 10 Victors):

On completion of the tournament (alas, not earlier) it turns out that some of the duels were more important than the others. For example, the duel between C and D was hardly important since both C and D were very weak, the A's duel against B was more important than his duel against C, etc.

How to value the „importance” of a duel ?

The product of each player's playing strength seems to be the appropriate measure (the analogue of the Universal Law of Attraction), and the player's strength is, naturally, the place he took in the tournament. Thus, let's multiply the result of each duel by the strength of both players, ie the importance of the duel:

November 2018

The better way is to increment strengths (before multiplication) by 10% of the sum distributed Victors (here by 1). The aim is to avoid zero peakyeared points for both players and to get more harmonious their distribution.

This concept is not used yet here.

	A	B	C	D	Σ	In accordance with intuitive expectations, player B took the lead. Victors scored by A turned out to be less valuable than those scored by B.
A		4٠23٠24	10٠10٠24	10٠3٠24	5328
B	6٠24٠23		9٠10٠23	8٠3٠23	5934
C	0٠24٠10	1٠23٠10		9٠3٠10	500
D	0٠24٠3	2٠23٠3	1٠10٠3		168

To perceive this better let's adjust the numbers to more familiar sizes – let's multiply them by such coefficient that the sum of the results is, like before, 60 Victors. We'll get:

The results corrected in this way seem to be more fair than the original results !

Moreover – although winning big against the weak opponents has some value, in spite of that:

1) specializing in crushing weak opposition causes mannerism and wastes the potential of strong players,

2) the quality of play is variable ! and a player who was considered weak so far, in this particular tournament may play much better and there is no way to know this before the end of the tournament

3) finally, there is always a danger of deliberate losing – the curse of many competitions.

Those negative occurrences may be slightly reduced by justification.

Gradation of peakyearing

Was the players' strength allowed for to the right degree ? Maybe it was overemphasized ?

Let's introduce the parameter P, quantifying the degree of peakyearing – let's multiply the result of a duel by its importance raised to the power of P. The greater the P, the greater the weight of more important duels compared with that of the less important ones (for P = 0 there is no peakyearing, for P = 1 it's what we have been doing so far):

P = 0.2						P = 0.7						P = 1						P = 2
	A	B	C	D	Σ		A	B	C	D	Σ		A	B	C	D	Σ		A	B	C	D	Σ
A		5	11	9	25	A		9	12	5	27	A		11	12	4	27	A		17	8	1	26
B	8		10	7	25	B	13		11	4	28	B	17		10	3	30	B	26		7	1	34
C	0	1		7	8	C	0	1		3	4	C	0	1		1	2	C	0	1		0	1
D	0	2	1		3	D	0	1	0		1	D	0	1	0		1	D	0	0	0		0

Generality of peakyearing

Note that there is nothing specifically bridge-related in peakyearing !

Thus, it's applicable to any tournaments consisting of duels (basketball, chess) – and obviously to any bridge tournaments (individual, pairs, teams; IMPs or matchpoints). Moreover, it's not necessary that a tournament is played in the „round robin” format – the number of rounds played can be freely chosen.

Attention! 2019 Change of Algorithm

The „importance” of a duel = the sum of each player's playing strength raised to the power of P

P should be at least 1 – the best its value seems to be equal 2

The new algoritm is a more sensible. The calculations presented here were made according to the old algorithm but differences are small and the results are almost the same.

Questions

1)	Can peakyearing, as described above, be substantiated in some non-speculative manner ?
2)	Is there some optimum value for P or it has to be set following the intuition ?
3)	Should the prize pool be divided according to the peakyeared results (for P = ?)

TWO PROPOSALS FOR TOURNAMENTS

These are the outlines of ideas which can become the subject of mathematical speculations:

The Overlapping Tournament

As the tournament progresses the players who have been doing poorly so far lose motivation to play well, which, regardless of peakyearing, distorts the results. It can be mollified by setting up a special distribution of the prize pool.

The tournament consisting of, say, 8–rounds is treated as 8 sub-tournaments, each starting after the completion of the previous round – as illustrated below with the green stripes:

1	2	3	4	5	6	7	8		The results of each sub tournament are computed and the overall prize pool is divided among the sub tournaments according to a given rule (eg in the proportions shown in red). Each of the players participates in the prizes of each of the sub tournaments ! – therefore it's in player's best interest to play well till the end – despite the earlier misfortunes.
								8

1								7

	2							6

		3						5

			4					4

				5				3

					6			2

						7		1

The Foursome Tournament

This is a pyramid made of layers consisting of groups of 4 players : the highest layer – 1 group, the second from the top – 2 groups, the third highest – 4 groups... and so on, doubling the number of groups (the lowest layer may be imperfect).

Each round of the tournament is a round robin played inside the groups – the winner of a group advances to the higher layer, the runner-up stays in the same layer and the other two players drop one layer down.

Aside from the possibility to organize a very large tournament, the advantages of this system are numerous:

- Since the rounds are short, even two or three promotions from the very bottom are possible (the motivation is retained).

- New players can join the tournament any time – and be incorporated to the lowest, imperfect layer.

- A player's layer is his ranking – thus it makes sense to organize a perpetual tournament (lasting many years).

Technicalities:

- A pyramid can be built in 3 ways:
   1) gradually – starting with one layer forming a higher layer from the winners.
   2) arbitrarily – the stronger the player, the higher his original layer.
   3) by means of public auction for starting positions.

- The least important bottom layer can be set up and modified ad hoc.

- To determine the final outcome of the tournament, the final round should be played –
eg comprising the players from the top layer plus the four best players from the second layer.

The attractiveness of the Foursome Tournament is best seen if we compare it to the way the Polish Championships are played. Some 300 teams are divided into leagues – I , II and III. The groups within each of the leagues are quite numerous (16 teams), therefore it takes a whole year to determine who is promoted and who is relegated. As a result, a newly-formed team can join the championship not earlier than at the beginning of a new league season, and to become the champion of Poland it must play for 3 or 4 years !!


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