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

Par Zero Deals

Par Score Stats

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			Makes North South West East Deal	NT 10 10 3 3 26	9 9 4 4 26	8 8 5 5 26	9 9 3 3 24	12 12 1 1 26
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			Makes North South West East Deal	NT 3 3 7 7 20	2 2 10 10 24	9 9 4 4 26	2 2 10 10 24	9 9 4 4 26
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 2 Par 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			Makes North South West East Deal	NT 9 9 4 4 26	9 9 4 4 26	5 5 6 6 22	10 10 2 2 24	4 4 9 9 26
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 1 Par 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			Makes North South West East Deal	NT 5 5 3 3 16	5 5 5 5 20	4 4 6 6 20	6 6 3 3 18	6 6 3 3 18
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 out Par 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			Makes North South West East Deal	NT 7 7 7 7 28	6 6 6 6 24	6 6 6 6 24	6 6 6 6 24	6 6 6 6 24
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 out Par 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			Makes North South West East Deal	NT 5 5 8 8 26	4 9 4 9 26	5 5 8 8 26	4 5 8 9 26	5 5 8 8 26
— — 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 out Par 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			Makes North South West East Deal	NT 0 13 13 0 26	12 13 1 0 26	1 2 12 11 26	1 2 12 11 26	0 0 13 13 26
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 out Par 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.

Summary	None Vul	Both Vul	N-S Vul	E-W Vul
Deals analyzed	10485760	10485760	10485760	10485760
Par plus N-S	5241511	5241511	5241511	5241511
Par plus E-W	5243995	5243995	5243995	5243995
Par zero	254	254	254	254
Unique scores	25	26	38	38
Minimum	0	0	0	0
Maximum	1520	2220	2220	2220
Mode	100	140	140	140
Median	400	600	400	400
Mean	372	530	441	441
Standard deviation	317	475	403	403

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

Score	Deals	Specific	At Least	At Most
0	254	0.0024	100	0.0024
70	544	0.0052	99.9976	0.0076
80	25534	0.2435	99.9924	0.2511
90	311654	2.9722	99.7489	3.2233
100	1703194	16.2429	96.7767	19.4662
110	471717	4.4986	80.5338	23.9649
120	426000	4.0627	76.0351	28.0275
130	396295	3.7794	71.9725	31.8069
140	1126959	10.7475	68.1931	42.5544
300	574095	5.4750	57.4456	48.0294
400	857017	8.1732	51.9706	56.2025
420	1172303	11.1800	43.7975	67.3825
430	635759	6.0631	32.6175	73.4456
450	940380	8.9682	26.5544	82.4137
460	440033	4.1965	17.5863	86.6102
500	42423	0.4046	13.3898	87.0148
800	92577	0.8829	12.9852	87.8977
920	231696	2.2096	12.1023	90.1073
980	379389	3.6181	9.8927	93.7254
990	354495	3.3807	6.2746	97.1062
1100	7832	0.0747	2.8938	97.1808
1400	11700	0.1116	2.8192	97.2924
1440	50858	0.4850	2.7076	97.7774
1510	87483	0.8343	2.2226	98.6117
1520	145569	1.3883	1.3883	100
U = 25	10485760	100	2600

Study 7A23   Main

Top   Par for the Course

Both Vul

Score	Deals	Specific	At Least	At Most
0	254	0.0024	100	0.0024
70	544	0.0052	99.9976	0.0076
80	25534	0.2435	99.9924	0.2511
90	311654	2.9722	99.7489	3.2233
110	1125546	10.7340	96.7767	13.9573
120	596794	5.6915	86.0427	19.6488
130	494735	4.7182	80.3512	24.3670
140	1626205	15.5087	75.6330	39.8757
200	280305	2.6732	60.1243	42.5489
500	574095	5.4750	57.4511	48.0239
600	857597	8.1787	51.9761	56.2025
620	1172303	11.1800	43.7975	67.3825
630	635759	6.0631	32.6175	73.4456
650	940380	8.9682	26.5544	82.4137
660	440033	4.1965	17.5863	86.6102
800	42423	0.4046	13.3898	87.0148
1100	92577	0.8829	12.9852	87.8977
1370	231696	2.2096	12.1023	90.1073
1400	96494	0.9202	9.8927	91.0275
1430	305448	2.9130	8.9725	93.9405
1440	339774	3.2403	6.0595	97.1808
1700	11700	0.1116	2.8192	97.2924
2000	14757	0.1407	2.7076	97.4332
2140	39650	0.3781	2.5668	97.8113
2210	83934	0.8005	2.1887	98.6117
2220	145569	1.3883	1.3883	100
U = 26	10485760	100	2700

Study 7A23   Main

Top   Par for the Course

N-S Vul

Score	Deals	Specific	At Least	At Most
0	254	0.0024	100	0.0024
70	544	0.0052	99.9976	0.0076
80	25534	0.2435	99.9924	0.2511
90	311654	2.9722	99.7489	3.2233
100	850820	8.1141	96.7767	11.3373
110	799092	7.6207	88.6627	18.9581
120	511183	4.8750	81.0419	23.8331
130	445768	4.2512	76.1669	28.0843
140	1376763	13.1298	71.9157	41.2141
200	140241	1.3374	58.7859	42.5515
300	287045	2.7375	57.4485	45.2890
400	506497	4.8303	54.7110	50.1194
420	741276	7.0694	49.8806	57.1887
430	329155	3.1391	42.8113	60.3278
450	503739	4.8040	39.6722	65.1318
460	221220	2.1097	34.8682	67.2415
500	464681	4.4315	32.7585	71.6731
600	295134	2.8146	28.3269	74.4877
620	388047	3.7007	25.5123	78.1884
630	286214	2.7295	21.8116	80.9179
650	402711	3.8406	19.0821	84.7585
660	215268	2.0530	15.2415	86.8114
800	67398	0.6428	13.1886	87.4542
920	136161	1.2985	12.5458	88.7527
980	211293	2.0150	11.2473	90.7678
990	179206	1.7090	9.2322	92.4768
1100	80221	0.7650	7.5232	93.2419
1370	86306	0.8231	6.7581	94.0650
1400	75703	0.7220	5.9350	94.7869
1430	104401	0.9956	5.2131	95.7826
1440	182287	1.7384	4.2174	97.5210
1510	45358	0.4326	2.4790	97.9535
1520	72677	0.6931	2.0465	98.6467
1700	7313	0.0697	1.3533	98.7164
2000	7979	0.0761	1.2836	98.7925
2140	13688	0.1305	1.2075	98.9230
2210	40037	0.3818	1.0770	99.3048
2220	72892	0.6952	0.6952	100
U = 38	10485760	100	3900

Study 7A23   Main

Top   Par for the Course

E-W Vul

Score	Deals	Specific	At Least	At Most
0	254	0.0024	100	0.0024
70	544	0.0052	99.9976	0.0076
80	25534	0.2435	99.9924	0.2511
90	311654	2.9722	99.7489	3.2233
100	852374	8.1289	96.7767	11.3522
110	798171	7.6120	88.6478	18.9641
120	511611	4.8791	81.0359	23.8432
130	445262	4.2463	76.1568	28.0896
140	1376401	13.1264	71.9104	41.2159
200	140064	1.3358	58.7841	42.5517
300	287050	2.7375	57.4483	45.2892
400	507378	4.8387	54.7108	50.1280
420	742218	7.0783	49.8720	57.2063
430	329950	3.1466	42.7937	60.3529
450	502760	4.7947	39.6471	65.1476
460	221243	2.1099	34.8524	67.2576
500	465264	4.4371	32.7424	71.6947
600	293876	2.8026	28.3053	74.4973
620	386854	3.6893	25.5027	78.1866
630	285765	2.7253	21.8134	80.9119
650	403453	3.8476	19.0881	84.7595
660	215379	2.0540	15.2405	86.8135
800	67602	0.6447	13.1865	87.4582
920	136400	1.3008	12.5418	88.7591
980	211021	2.0125	11.2409	90.7715
990	180089	1.7175	9.2285	92.4890
1100	79335	0.7566	7.5110	93.2456
1370	86315	0.8232	6.7544	94.0687
1400	75452	0.7196	5.9313	94.7883
1430	104952	1.0009	5.2117	95.7892
1440	181405	1.7300	4.2108	97.5192
1510	45237	0.4314	2.4808	97.9506
1520	72892	0.6952	2.0494	98.6458
1700	7444	0.0710	1.3542	98.7168
2000	8077	0.0770	1.2832	98.7938
2140	13686	0.1305	1.2062	98.9243
2210	40117	0.3826	1.0757	99.3069
2220	72677	0.6931	0.6931	100
U = 38	10485760	100	3900

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

© 2014 Richard Pavlicek