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” tour­na­ment be­low (du­els scored by dis­trib­ut­ing 10 Vic­tors):

 

A

B

C

D

Σ

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

A

 

4

10

10

24

B

6

 

9

8

23

C

0

1

 

9

10

D

0

2

1

 

3

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:

 

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

 

23٠24

10٠10٠24

10٠3٠24

5328

B

24٠23

 

10٠23

 3٠23

5934

C

24٠10

23٠10

 

 3٠10

500

D

24٠3

23٠3

10٠3

 

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:

 

A

B

C

D

Σ

We can see that the sum of Vic­tors di­vided among both play­ers is no longer in­varia­bly 10, but it de­pends on the im­por­tance of the duel !

Eg: for A vs. B 28 Victors are di­vided, whereas for C vs. D only 1 Vic­tor.

Obviously, the propor­tions of Vic­tors at­trib­uted to the play­ers re­main con­stant.     Eg: 11:17 in A vs. B is the same pro­por­tion as 4:6 be­fore.

A

 

11

12

4

27 

B

17

 

10

3

30 

C

0

1

 

1

D

0

1

0

 

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

 

C

0

1

 

1

 

C

0

1

 

0

1

D

0

2

1

 

3

 

D

0

1

0

 

 

D

0

1

0

 

 

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.

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 !!

 

 

 

 

 

 

Pikier writ­ings

 

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