Study 7A23 Main

# Par for the Course

by Richard Pavlicek

What is the “par score” for a bridge deal? How is it calculated? You won’t find the answers in the classic, “Bubba Watson on the Play of the Hand,” but you might find them here. Most experienced players have a general idea of the logic, but few understand it completely. This study describes the algorithm I use to calculate par scores, the special case of par zero deals, and statistics of actual par scores.

## Par Score Calculation

The first step to calculate the par score is to determine the highest makable contract by rank (not score) for each side. The side that makes the higher contract must be the board winner, because a plus score can be assured by bidding and making that contract or by defeating the opponents if they bid any further. In most cases this determines par immediately.

 None vul Q 4 K 7 5 3 2 K J 10 6 5 3 MakesNorthSouthWestEastDeal NT10103326 994426 885526 993324 12121126 K 9 6 3 J 5 2 J 10 9 6 7 2 J 8 5 Q 8 7 6 3 Q 8 4 A 9 A 10 7 4 2 A K 10 9 A Q 8 4 Par contr: 6 N or SPar score: NS +920

North or South can make 6 , either by ruffing two diamonds, or by establishing the long spade if the defense leads two rounds of trumps. No higher contract is makable by anyone, so this appears to be par.

### Profitable sacrifice?

The next step is to determine if the losing side has a profitable sacrifice at the vulnerability, and if so, that sacrifice becomes the par score for the winning side. On the previous deal the most East-West could win is five tricks in hearts, so a sacrifice in 6 would be down seven doubled (minus 1700) which is hardly profitable to put it mildly. Now consider this deal:

 Both vul A 9 7 5 9 5 4 6 3 K Q 7 3 MakesNorthSouthWestEastDeal NT337720 22101024 994426 22101024 994426 Q 10 6 4 2 K 8 A 9 8 2 A 2 K J 8 7 2 K Q J 10 7 J 8 6 3 A Q J 10 6 3 5 4 10 9 5 4 Par contr: 5 × S or NPar score: EW +500

East-West own the highest makable contract at 4 , but N-S can win nine tricks in hearts. Therefore, N-S do better to bid 5 , which when doubled surrenders only 500 instead of 620. Curiously, N-S could also sacrifice in clubs, but the score would be the same, which is all that matters.

### Higher scoring contract?

If the losing side has no profitable sacrifice against the highest makable contract, it is necessary to check the winning side for alternate contracts that score higher. Three situations must be considered:

 Highest make 5 or 5 → check for 4 NT, 4 or 4 Highest make 4 or 4 → check for 3 NT, 3 or 3 Highest make 3 or 3 → check for 2 NT

If a higher scoring contract exists, it does not necessarily become the par contract; a further check for sacrifices is required. The losing side may have a profitable save against the higher scoring contract but not against the higher ranking contract, in which case this new sacrifice becomes par, unless the winning side does better to take its score for the higher ranking contract. This is easier than it sounds, applying little more than common sense. Consider this deal:

 E-W vul K 7 2 J 10 5 A K 8 2 A 4 2 MakesNorthSouthWestEastDeal NT994426 994426 556622 10102224 449926 10 8 A K 9 7 4 4 K Q J 9 3 J 9 5 3 8 2 7 6 5 10 8 7 5 A Q 6 4 Q 6 3 Q J 10 9 3 6 Par contr: 4 × W or EPar score: NS +200

North-South own the highest make at 4 , against which E-W have no profitable sacrifice; but N-S also make 3 NT. Should the par score then be 400 for N-S? No, because if N-S bid 3 NT, E-W would have a profitable sacrifice in 4 , which becomes the par contract (doubled, down one). If E-W were nonvulnerable, 4 × would net only 100, so N-S do better to ignore it and take their 4 ; then par would be 130.

 Study 7A23 Main Top Par for the Course

## Par Zero Deals

Almost all practical cases of determining the par score are resolved by the preceding calculation, but a few loose ends still exist. What if neither side can make any contract? On the following deal no one can win more than six tricks in any strain. This is rare, but the obvious consequence is a par score of zero. A perfect passout!

 None vul 7 6 5 4 8 3 9 5 A K Q J 10 MakesNorthSouthWestEastDeal NT553316 555520 446620 663318 663318 A K Q 8 2 J 10 8 3 9 5 3 2 10 A K Q 9 7 2 7 4 2 8 6 4 J 9 3 10 6 5 4 A K Q J 6 7 Par contr: PassoutPar score: zero

Another possible situation, even more unlikely, is both sides making the same contract. Consider the following symmetric deal in which everybody makes 1 NT. Who deserves the 90 points? One possibility is to assign plus par to the dealing side, because they have the first opportunity to bid 1 NT, but this seems unfair.

 None vul 10 8 A 5 4 Q J 3 2 K 9 7 6 MakesNorthSouthWestEastDeal NT777728 666624 666624 666624 666624 A 5 4 Q J 3 2 K 9 7 6 10 8 K 9 7 6 10 8 A 5 4 Q J 3 2 Q J 3 2 K 9 7 6 10 8 A 5 4 Par contr: PassoutPar score: zero

My decision is to assign “par zero” to any deal in which the highest make for both sides is the same contract. Vulnerability does not matter. The deal is like a jump ball in basketball; whoever grabs the opportunity beats par.

### Enter the bizarre

Consider the following deal I composed for my Optimum Contract puzzle, where the object was to find the best contracts for N-S and E-W. Remarkably, 3 is the best contract for both sides. South and East both win nine tricks in spades against any defense. Hard to believe, but proof is in the play.

 None vul 9 J 10 9 A Q 4 3 2 A K Q 2 MakesNorthSouthWestEastDeal NT558826 494926 558826 458926 558826 — — J 10 9 8 7 6 5 8 7 6 5 4 3 A K Q 8 A K Q 8 7 K J 10 9 J 10 7 6 5 4 3 2 6 5 4 3 2 — — Par contr: PassoutPar score: zero

For the ultimate fantasy, witness this deal from my Fewest HCP To Make Notrump record book. With only 11 HCP, South is cold for 7 NT with any lead. Strange enough, but the eerie fact is that so is West. Therefore, to keep in line with my arbitrary rule, par remains at zero. Bid 7 NT first to claim your prize!

 None vul — — 8 7 6 5 4 3 2 7 6 5 4 3 2 MakesNorthSouthWestEastDeal NT01313026 12131026 12121126 12121126 00131326 K K Q J 10 9 8 7 6 5 4 3 2 — — — — A K Q J 10 9 A K Q J 10 9 8 A Q J 10 9 8 7 6 5 4 3 2 A — — Par contr: PassoutPar score: zero

Somehow, I wouldn’t predict a passout. East will surely bid 7 , and South will bid 7 not realizing he has left West a chance to be a hero. But hey, maybe East will try to be the hero. In any case, both sides making 7 NT may be the short route to the looney bin.

 Study 7A23 Main Top Par for the Course

## Par Score Stats

Source for this par-score analysis was my database of 10,485,760 random deals, each of which is solved 20 times (4 hands x 5 strains) with double-dummy play. Separate analyses were made for each of the four vulnerability conditions, because this affects the number and value of par scores. Results are summarized below.

Par scores are considered as absolute values, regardless of which side is plus, or neither side if par zero. Note that the first four rows of the table are the same for each column, since vulnerability has no effect on which side is plus to par. The number of deals with N-S plus or E-W plus (Rows 2 and 3) are equal in theory, but only bear close proximity in this small sample. Yes, 10+ million is small, indeed infinitesimal in relation to 53+ octillion possible bridge deals.

SummaryNone VulBoth VulN-S VulE-W Vul
Deals analyzed10,485,76010,485,76010,485,76010,485,760
Par plus N-S5,241,5115,241,5115,241,5115,241,511
Par plus E-W5,243,9955,243,9955,243,9955,243,995
Par zero254254254254
Unique scores25263838
Minimum0000
Maximum1520222022202220
Mode100140140140
Median400600400400
Mean372530441441
Standard deviation317475403403

In duplicate bridge there are 205 possible absolute scores, but only 25 to 38 of them (depending on vulnerability) are possible as par scores. For instance, common bridge scores of 50, 150 and 170 can never be par scores.

The following four tables show the breakdown of absolute par scores by vulnerability condition. Percents of par score occurrence are shown in three ways: for the specific score, at least that score, and at most that score. To find the chance of multiple scores (e.g., 110 to 140) add the specific percents for each score.

None Vul
ScoreDealsSpecificAt LeastAt Most
02540.00241000.0024
705440.005299.99760.0076
8025,5340.243599.99240.2511
90311,6542.972299.74893.2233
1001,703,19416.242996.776719.4662
110471,7174.498680.533823.9649
120426,0004.062776.035128.0275
130396,2953.779471.972531.8069
1401,126,95910.747568.193142.5544
300574,0955.475057.445648.0294
400857,0178.173251.970656.2025
4201,172,30311.180043.797567.3825
430635,7596.063132.617573.4456
450940,3808.968226.554482.4137
460440,0334.196517.586386.6102
50042,4230.404613.389887.0148
80092,5770.882912.985287.8977
920231,6962.209612.102390.1073
980379,3893.61819.892793.7254
990354,4953.38076.274697.1062
110078320.07472.893897.1808
140011,7000.11162.819297.2924
144050,8580.48502.707697.7774
151087,4830.83432.222698.6117
1520145,5691.38831.3883100
U = 2510,485,7601002600

Both Vul
ScoreDealsSpecificAt LeastAt Most
02540.00241000.0024
705440.005299.99760.0076
8025,5340.243599.99240.2511
90311,6542.972299.74893.2233
1101,125,54610.734096.776713.9573
120596,7945.691586.042719.6488
130494,7354.718280.351224.3670
1401,626,20515.508775.633039.8757
200280,3052.673260.124342.5489
500574,0955.475057.451148.0239
600857,5978.178751.976156.2025
6201,172,30311.180043.797567.3825
630635,7596.063132.617573.4456
650940,3808.968226.554482.4137
660440,0334.196517.586386.6102
80042,4230.404613.389887.0148
110092,5770.882912.985287.8977
1370231,6962.209612.102390.1073
140096,4940.92029.892791.0275
1430305,4482.91308.972593.9405
1440339,7743.24036.059597.1808
170011,7000.11162.819297.2924
200014,7570.14072.707697.4332
214039,6500.37812.566897.8113
221083,9340.80052.188798.6117
2220145,5691.38831.3883100
U = 2610,485,7601002700

N-S Vul
ScoreDealsSpecificAt LeastAt Most
02540.00241000.0024
705440.005299.99760.0076
8025,5340.243599.99240.2511
90311,6542.972299.74893.2233
100850,8208.114196.776711.3373
110799,0927.620788.662718.9581
120511,1834.875081.041923.8331
130445,7684.251276.166928.0843
1401,376,76313.129871.915741.2141
200140,2411.337458.785942.5515
300287,0452.737557.448545.2890
400506,4974.830354.711050.1194
420741,2767.069449.880657.1887
430329,1553.139142.811360.3278
450503,7394.804039.672265.1318
460221,2202.109734.868267.2415
500464,6814.431532.758571.6731
600295,1342.814628.326974.4877
620388,0473.700725.512378.1884
630286,2142.729521.811680.9179
650402,7113.840619.082184.7585
660215,2682.053015.241586.8114
80067,3980.642813.188687.4542
920136,1611.298512.545888.7527
980211,2932.015011.247390.7678
990179,2061.70909.232292.4768
110080,2210.76507.523293.2419
137086,3060.82316.758194.0650
140075,7030.72205.935094.7869
1430104,4010.99565.213195.7826
1440182,2871.73844.217497.5210
151045,3580.43262.479097.9535
152072,6770.69312.046598.6467
170073130.06971.353398.7164
200079790.07611.283698.7925
214013,6880.13051.207598.9230
221040,0370.38181.077099.3048
222072,8920.69520.6952100
U = 3810,485,7601003900

E-W Vul
ScoreDealsSpecificAt LeastAt Most
02540.00241000.0024
705440.005299.99760.0076
8025,5340.243599.99240.2511
90311,6542.972299.74893.2233
100852,3748.128996.776711.3522
110798,1717.612088.647818.9641
120511,6114.879181.035923.8432
130445,2624.246376.156828.0896
1401,376,40113.126471.910441.2159
200140,0641.335858.784142.5517
300287,0502.737557.448345.2892
400507,3784.838754.710850.1280
420742,2187.078349.872057.2063
430329,9503.146642.793760.3529
450502,7604.794739.647165.1476
460221,2432.109934.852467.2576
500465,2644.437132.742471.6947
600293,8762.802628.305374.4973
620386,8543.689325.502778.1866
630285,7652.725321.813480.9119
650403,4533.847619.088184.7595
660215,3792.054015.240586.8135
80067,6020.644713.186587.4582
920136,4001.300812.541888.7591
980211,0212.012511.240990.7715
990180,0891.71759.228592.4890
110079,3350.75667.511093.2456
137086,3150.82326.754494.0687
140075,4520.71965.931394.7883
1430104,9521.00095.211795.7892
1440181,4051.73004.210897.5192
151045,2370.43142.480897.9506
152072,8920.69522.049498.6458
170074440.07101.354298.7168
200080770.07701.283298.7938
214013,6860.13051.206298.9243
221040,1170.38261.075799.3069
222072,6770.69310.6931100
U = 3810,485,7601003900

Mathematical check total of At Least + At Most columns = 100(U+1).

 Study 7A23 Main Top Par for the Course