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 2ª or 3ª one gets the same bo­nus as for mak­ing 1ª, and for 5ª the same as for 4ª 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 4ªa 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

1

1 +

G

L

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 two-ta­ble 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

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 =

o

(

a

)

h

o

 

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 non-zero 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 dis­tribu­tional-strength 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 sin­gle-table 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.

 

 

 

 

 

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