
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 topranked 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 welldeserved,
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 

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:

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 
2 

D 
0 
1 
0 

1 
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 bridgerelated
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 nonspeculative
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 subtournaments,
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. 








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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 runnerup 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 newlyformed 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 !!







