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: 6 North (S)Par 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: 5 × South (N) down 2Par 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: 4 × West (E) down 1Par 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: Passed outPar score: 0

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: Passed outPar score: 0

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: Passed outPar score: 0

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: Passed outPar score: 0

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 analyzed10485760104857601048576010485760
Par plus N-S5241511524151152415115241511
Par plus E-W5243995524399552439955243995
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 mulitple 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
80255340.243599.99240.2511
903116542.972299.74893.2233
100170319416.242996.776719.4662
1104717174.498680.533823.9649
1204260004.062776.035128.0275
1303962953.779471.972531.8069
140112695910.747568.193142.5544
3005740955.475057.445648.0294
4008570178.173251.970656.2025
420117230311.180043.797567.3825
4306357596.063132.617573.4456
4509403808.968226.554482.4137
4604400334.196517.586386.6102
500424230.404613.389887.0148
800925770.882912.985287.8977
9202316962.209612.102390.1073
9803793893.61819.892793.7254
9903544953.38076.274697.1062
110078320.07472.893897.1808
1400117000.11162.819297.2924
1440508580.48502.707697.7774
1510874830.83432.222698.6117
15201455691.38831.3883100
U = 25104857601002600

 Study 7A23   Main Top   Par for the Course

Both Vul
ScoreDealsSpecificAt LeastAt Most
02540.00241000.0024
705440.005299.99760.0076
80255340.243599.99240.2511
903116542.972299.74893.2233
110112554610.734096.776713.9573
1205967945.691586.042719.6488
1304947354.718280.351224.3670
140162620515.508775.633039.8757
2002803052.673260.124342.5489
5005740955.475057.451148.0239
6008575978.178751.976156.2025
620117230311.180043.797567.3825
6306357596.063132.617573.4456
6509403808.968226.554482.4137
6604400334.196517.586386.6102
800424230.404613.389887.0148
1100925770.882912.985287.8977
13702316962.209612.102390.1073
1400964940.92029.892791.0275
14303054482.91308.972593.9405
14403397743.24036.059597.1808
1700117000.11162.819297.2924
2000147570.14072.707697.4332
2140396500.37812.566897.8113
2210839340.80052.188798.6117
22201455691.38831.3883100
U = 26104857601002700

 Study 7A23   Main Top   Par for the Course

N-S Vul
ScoreDealsSpecificAt LeastAt Most
02540.00241000.0024
705440.005299.99760.0076
80255340.243599.99240.2511
903116542.972299.74893.2233
1008508208.114196.776711.3373
1107990927.620788.662718.9581
1205111834.875081.041923.8331
1304457684.251276.166928.0843
140137676313.129871.915741.2141
2001402411.337458.785942.5515
3002870452.737557.448545.2890
4005064974.830354.711050.1194
4207412767.069449.880657.1887
4303291553.139142.811360.3278
4505037394.804039.672265.1318
4602212202.109734.868267.2415
5004646814.431532.758571.6731
6002951342.814628.326974.4877
6203880473.700725.512378.1884
6302862142.729521.811680.9179
6504027113.840619.082184.7585
6602152682.053015.241586.8114
800673980.642813.188687.4542
9201361611.298512.545888.7527
9802112932.015011.247390.7678
9901792061.70909.232292.4768
1100802210.76507.523293.2419
1370863060.82316.758194.0650
1400757030.72205.935094.7869
14301044010.99565.213195.7826
14401822871.73844.217497.5210
1510453580.43262.479097.9535
1520726770.69312.046598.6467
170073130.06971.353398.7164
200079790.07611.283698.7925
2140136880.13051.207598.9230
2210400370.38181.077099.3048
2220728920.69520.6952100
U = 38104857601003900

 Study 7A23   Main Top   Par for the Course

E-W Vul
ScoreDealsSpecificAt LeastAt Most
02540.00241000.0024
705440.005299.99760.0076
80255340.243599.99240.2511
903116542.972299.74893.2233
1008523748.128996.776711.3522
1107981717.612088.647818.9641
1205116114.879181.035923.8432
1304452624.246376.156828.0896
140137640113.126471.910441.2159
2001400641.335858.784142.5517
3002870502.737557.448345.2892
4005073784.838754.710850.1280
4207422187.078349.872057.2063
4303299503.146642.793760.3529
4505027604.794739.647165.1476
4602212432.109934.852467.2576
5004652644.437132.742471.6947
6002938762.802628.305374.4973
6203868543.689325.502778.1866
6302857652.725321.813480.9119
6504034533.847619.088184.7595
6602153792.054015.240586.8135
800676020.644713.186587.4582
9201364001.300812.541888.7591
9802110212.012511.240990.7715
9901800891.71759.228592.4890
1100793350.75667.511093.2456
1370863150.82326.754494.0687
1400754520.71965.931394.7883
14301049521.00095.211795.7892
14401814051.73004.210897.5192
1510452370.43142.480897.9506
1520728920.69522.049498.6458
170074440.07101.354298.7168
200080770.07701.283298.7938
2140136860.13051.206298.9243
2210401170.38261.075799.3069
2220726770.69310.6931100
U = 38104857601003900

Check total of At Least + At Most should be 100(U+1). For example, if there are four unique scores (U = 4) with specific percents [abcd] then At Least column is a+b+c+d+b+c+d+c+d+d, and At Most column is a+a+b+a+b+c+a+b+c+d, so the sum of both columns is 5(a+b+c+d) or 500 percent.

 Study 7A23   Main Top   Par for the Course