
Scoring a Deal 

Introduction
A bridge duel consists of a series of deals played
by two pairs: North – South and West – East.
A (bridge) score is the amount of points attributed
on a deal to one of the pairs (North – South or West–East) for making
its own contract or defeating that of the opponents.
The score is independent
of the results on previous deals – each deal is scored by its own virtue
!
(formerly it
was different – the rubber style was prevailing – which nowadays
(and rightly so!) disappears).
The higher the contract – the less
often it will be bid and made – and therefore the greater (inversly
proportional) the reward for making it should be. Unfortunately –
this principle is very poorly implemented in the scoring system used today:
– the bonuses for making a contract are
awarded in an inconsistent way (eg for making 2♠ or 3♠ one
gets the same bonus as for making 1♠, and for 5♠ the
same as for 4♠ – a relic of the formerly predominant
rubber style).
– the bonuses are highly dependent
on the „suit” of the contract (eg for 4§ one
gets significantly less than for 4♠ – a relic of „lucky suits” in
card games) – although neither the luck nor the rules
of bridge play make any of the suits a privileged one.
Those flaws of The Scoring negatively influence
the style of bidding. Moving to Fibonacci Scoring described below would be very advantageous
for bridge:
Level of contract 
1 
2 
3 
4 
5 
6 
7 
Each
overtrick 50 
Bonus for making 
100 
200 
300 
500 
800 
1300 
2100 
There are neither
doubles nor redoubles in the bidding. Penalties for subsequent undertricks
are also calculated according to the table above (one down = 100, two down
= 300 (100+200), three down = 600 etc).
The question:
1) How to designate
the optimum bridge scoring.
Duplicate bridge
The shorter the duel – the more likely that one
of the pairs will be favored by luck, getting „stronger” cards,
and will win the duel despite playing poorly. To avoid this, any given deal is
played at many tables and the results thus obtained are compared with each
other.
The simplest
fair method of assessing the results obtained on a deal is
Comparison
„To the Average”:

Scores 
Balances 
Scores
– NS scores are custom treated as plus scores +400 = average of scores (mean score) Balances –
each pair is assigned its deviation from the average: NS[1] wins 400 (it scored
that many points above the average); WE[2] loses 200 (it
scored that many points below the average); etc. 

Table 
NS 
WE 
NS 
WE 

1 
800 

+400 
–400 

2 
600 

+200 
–200 

3 

200 
–600 
+600 


+400 

Flattening
A big balance on a single deal (eg 1500, 2000) causes
the distortion of subsequent play in the tournament. The beneficiary is
already assured of a good place overall (so he will be playing cautiously
from then on) and the looser on the deal can't hope for a decent place (so he
will be playing wildly, taking any kind of risks). Moreover, a big balance
is frequently caused by pure luck, as the luck factor plays a big role in
bridge after all (including duplicate bridge). It can be remedied by progressively
reducing absolute balances according to Caracalla
Formula:

1000 × Balance 

The balance of 50 is reduced to 48;
200 – to 167, 400 – to 286, 600 – to 375. Thus, this is a strongly progressive
reduction to the section [0,1000]. 

1000 + Balance 

The name „Caracalla
Formula” derives from the fictional anecdote in which it was proposed.
More
„artificial” flattening is in common use though – balances
are reduced (according to a special table – formula is not given) to integer numbers from the section
[0,24], and the new units are called match points (or imps –
from international match points). This method differs from Caracalla
flattening almost only by the scale used (the numerator is 30, instead
of 1000); it's less meaningful,
though, as it introduces a new, artificial unit.
Questions:
2) 
Can
the rationale and degree of flattening be justified ? 
3) 
The
Caracalla Formula can be interpreted as a taxation tariff. Is there a theory
of such tariffs ? 
4) 
What
is the mathematical construction principle of imps ? (no relevant information could be found!) 
5) 
Is
it possible to incorporate flattening into bridge scoring itself ? 
Comparison „To the Others”:
This is the second fair method of assessing the scores obtained on a
deal:

Scores 
Balances 
A pair is assigned the average of
its flattened gains over the other pairs playing the same line (ie
NS or WE): NS[2] scored 200 less than NS[1]
– after flattening –167 NS[2] scored 800 more than NS[3]
– after flattening +444 Thus, the NS[2] balance
is +139 ( (–167+444) / 2 ) 

Table 
NS 
WE 
NS 
WE 

1 
800 

+417 
–417 

2 
600 

+139 
–139 

3 

200 
–556 
+556 
One can
easily check that without flattening the following holds true:
{balance To the Others} = {balance To the Average} x {number of tables} / ( {number of tables} –1 )
which means that if no flattening were
used both methods would differ by scale only, making the use of the more
complicated method To the Others unpractical. It would be likewise if in the method To the Others we would take not the „average of flattened gains” but
the „flattened average gain” into account, since then scaling
the balances before flattening would be sufficient, and the final (flattened)
balances in both methods would be identical.
Let us examine, then, what is the role of taking the
„average of flattened gains” into account ?
Profitability

It's known that such is the lowest probability
of success to show net profit when the possible gain is G and the loss is
L if we fail.
As we can see, it depends exclusively on the gain – loss ratio. 
No doubt such probabilities should not
depend on the „room's behavior”, ie at how many tables the same
decision will be taken. We play duplicate not to guess what will happen at
other tables but to make the game scored in a more equitable manner.
Let's consider the simplest case – the same dilemma
has to be solved at each table: whether to stop in an unbeatable 2nt or to
risk 3nt (vulnerable) ?

9 tricks 
8 tricks 
A8 = average if 8 tricks are available, A9 = if 9 tricks are available Gain from 3nt if there are 9 tricks =
(600–A9) –
(150–A9) = 450 Loss from 3nt if there are 8 tricks =
(120–A8) –
(–100–A8)) = 220 
3nt 
600 –
A9 
–100
– A8 

2nt 
150 –
A9 
120 –
A8 
As we can see – no matter at how many tables
3nt was bid – Gain and Loss remain constant – thus the
minimum profitability of 3nt also is constant (in
this example = 32.84%). This will change, however, if the balances in
the table are flattened (since the flattening function is not additive)
– and the minimum profitability will depend on the averages A9 A8,
which in turn depend on the number of tables at which 3nt was bid (trials show that relative variance here can be as
great as 10%).
And now the balances in
the „To the Others” method:

9 tricks 
8 tricks 
H = the number of tables at which 3nt was bid ( Higher contract) L = the number of tables at which 2nt was bid ( Lower contract) f( ) = flattening
function (Caracalla or imps – at will! ) 
3nt 
L •
s(450) 
L •
s(–220) 

2nt 
H •
s(–450) 
H •
s(220) 
To make it clear, the sums of flattened gains and not
the averages were given.
The gain from 3nt if
there are 9 tricks = L • f(450) – H • f(–450) =
f(450) • (L + H)
The loss from 3nt if there are 8 tricks = H
•f(220) – L • f(–220) = f(220) • (L + H)
Thus, the minimum profitability for 3nt does not depend
on the number of tables at which it was bid.
Neither
does it depend on the shape of the flattening function – it can be any
!
Conclusion: The „To the Others” method is better.
Moreover,
it's more explicit to the players (as it compares scores to scores of the
others and not to abstract average) and more convenient for a twotable
duplicate (a duel of two foursomes, a very popular form of bridge) since calculating
of the averages is not necessary.
The Question:
6) 
The presented proof of stability of probabilities included only
the simplest of situations. It would be desirable to examine more complex
situations (more possible choices). 
Harmonization NApoleon Points
(NAPs, naps)
If a deal is „flat”(ie it's
clear that it will produce small balances) , there is little risk in not paying
full attention to the play; if a deal is „swingy” (ie it's clear
that the balances will be big) – the reverse holds true: the utmost
concentration is in order. This is the way to lower this disproportion:
The dispersion of each deal has to be
calculated (the standard deviation is the proper measure), and then
– the balances on each single deal should be rescaled in such a way
that dispersions of the deals are (more or less) harmonized.
Let's assume that:
a = average dispersion of balances – from all possible bridge deals = about 200 points
o = old dispersion of balances – the one calculated
from the scores obtained on the deal
n = new dispersion of balances – the one we want to get for the deal.
and carry on with harmonization by „moving
closer” the dispersion of the deal to average dispersion (a) using the Napoleon Formula
(the name taken from a fictional anecdote presenting
this method):
n = 

where h = the degree of harmonization of deals (a number from the section
[0,1] ) The greater this number
– the stronger the harmonization of the deals. 
This is the comparison of the new balances of two
deals (from the sample of
3000 deals we got a = 200):

Scores 
h = 0 
h = 1 / 2 
h = 3 / 4 
h = 1 

Table 
flat 
swingy 
flat 
swingy 
flat 
swingy 
flat 
swingy 
flat 
swingy 
1 
+120 
+1430 
+24 
+498 
+66 
+373 
+111 
+321 
+185 
+278 
2 
+110 
+660 
+15 
+12 
+42 
+9 
+69 
+8 
+115 
+7 
3 
+50 
+650 
–43 
+3 
–119 
+2 
–198 
+2 
–331 
+2 
4 
+100 
–200 
+5 
–514 
+14 
–385 
+23 
–332 
+38 
–287 

dispersions
= 
±26 
±358 
±72 
±268 
±120 
±231 
±200 
±200 
The value of h can be chosen at will (1/2 seems
to be perfectly reasonable).
The balances are calculated using
the „To the Others” method, thus – regardless of h –
minimal profitabilities are the
same.
Questions
7) 
How
to justify the Napoleon Formula ? (it was created
through mathematical speculation) 
8) 
Can
the optimal degree of harmonization be indicated ? 
9) 
Should
the degree of harmonization be dependent on the number of tables ? 
Matchpointing
The „To the Others” method
– but the gains over remaining tables are flattened in a very special
way, namely each nonzero value (regardless of its size!) is converted
to 1. Thus, the balances on the deal are numbers from the section [0,1], making
it possible to express them in easily perceived percentages.
Making all the gains equal is obviously contradictory
to the notion of playing for points; nonetheless, this method is very
popular – likely for its randomness (sic!) and... because of players'
habits. No doubt matchpointing should be replaced by Maximum Harmonization
(h=1).
Reckoned average
Unfairness of bridge played at a single table can be
greatly reduced by reckoning the average score on the deal had it been
played at many tables. Obviously, the average is attributed to the pair
holding stronger cards. Here is the mnemonical approximation of such assessment
(a tiny distributionalstrength factor is added):
Each
pair calculates the strength of its hands: in
honors – using Point Count (A=4 K=3 Q=2 J=1); in distribution – using
the formula: (total number of cards in the longest
suit + total number of cards in the shortest suit) –13. If the
total exceeds 20, this pair is assigned the average, calculated in the
following manner: for each point above 20 – 50 nonvulnerable,
60 vulnerable for each point above 30 – twice as many. 
Simple and easy to memorize. Playing singletable
bridge this method should be universal !
Questions:
10) 
To
elaborate the most accurate method of calculating the average, using
a computer program. 
11) 
To
find the estimated average for bridge played with Fibonacci Scoring. 







