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Bridge Scoring by Alvin Roth and Jeff Rubens

Here is a sample bridge score sheet.

WE	THEY

Contract	Trick Score for odd tricks
Notrump	40 points for the first trick and 30 points for all subsequent tricks
Hearts or spades	30 points per trick
Clubs or diamonds	20 points per trick

As you can see, tricks are worth the most in notrump and the least in the minor suits. The reason that only tricks bid for and made are scored below the line is to reward accuracy and penalize errors during the bidding. Remember that in order to score a game you must get 100 points below the line. Let's see how that fiendish horizontal line works in practice by looking at a few examples:

1. You bid two clubs and take eight tricks. You have made your contract, so you score 40 points below the line (20 for each odd trick). You have not made a game, which would require 100 points below the line, but you do have a start towards one; as you did exactly what you promised (took eight tricks with clubs as trumps), all of your points count towards game. Since you have scored part of a game, this result is called a part-score or partial.

WE	THEY

40

2. You bid one heart and take nine tricks, so you have made your contract with two extra tricks to boot. Only tricks contracted for during the bidding are scored below the line; you bid only one heart and therefore score just 30 points below the line. The 60 points for the two overtricks are placed above the line and do not count towards game(100 points below the line). This is not a major disaster, however, for even if you had bid three hearts and taken the same nine tricks you would have scored only 90 points below the line--a greater total, to be sure, but still not a game. And the 60 points above the line count towards your final score.

WE	THEY
60
30

3. You bid two notrump and take eleven tricks. You score 70 points below the line (40 for the first odd trick at notrump, and 30 for each additional trick) and 90 points above the line for the three overtricks. This is perhaps a most unfortunate result! Had you been less conservative during the bidding and gone on to three notrump, then made the same eleven tricks, you would have made a game, scoring 100 points below the line (and 60 points above the line). As it is, you have settled for a part-score--useful, but not nearly so good as a game.

WE	THEY
90
70

4. You bid four hearts and take ten tricks. You score 120 points below the line--game! The horizontal line drawn underneath your score signifies a completed game. A line was not drawn under the part-score results in the preceding examples because game had not been finished, and it is possible to score a game by combining the results of two or more part-scores into a total of 100 or more points. However, you must "complete" a part-score before the opponents score a game. When a game is scored, a new horizontal line is drawn and both sides start over in quest of game. Thus, if one side scores a game, the value of the other's part-score towards a game is cancelled (though the points remain on the score sheet). We'll see an example of "killing a part-score" in the sample rubber later on.

WE	THEY

120

    A side that has already scored one game is said to be vulnerable (otherwise it is not vulnerable, or nonvulnerable). Bidding and making four hearts, as in this example, makes your side vulnerable.

    5. You bid four spades and take nine tricks. Oops! You have failed to fulfill your contract, so you must be penalized. You receive no score at all (even though you have taken three tricks in excess of book), and the opponents are awarded a penalty (placed above the line in the "They" column). Penalties are scored as follows:

PENALTIES
    If you are not vulnerable (have not won a game), the opponents collect 50 points for each trick you are short of your contract (undertrick). [Thus, the scale for one undertrick, two undertricks, etc. is 50, 100, 150, 200, . . . ]

    If you are vulnerable, the opponents score 100 points for each trick you are short of your contract. [100, 200, 300, 400, . . . ]

    If you are not vulnerable and the contract has been doubled, the opponents receive 100 points for the first trick you are short of your contract, 200 points for each of the next two additional undertricks, and 300 points for each additional undertrick thereafter. [100, 300, 500, 800, 1100, . . . ]

    If you are vulnerable and the contract has been doubled, the opponents receive 200 points for the first trick you are short of your contract and 300 points for each additional undertrick. [200, 500, 800, 1100, . . . ]

    If the final contract has been doubled and redoubled, the penalties are twice as great as those when the contract is (only) doubled.

    Take a moment to inspect the scale of penalties, for those who bid too much can be severely punished. Fortunately, excess conservatism is also punished in an equally important (although less obvious) way, for those who do not bid enough miss games and lose chances to win rubbers. The need for bidding accuracy is one of the many features that add to the fascination of bridge.

    To complete example 5, let's suppose that your side is not vulnerable. The result would look like this:

WE	THEY
	50

6. On the first deal of a rubber, your side bids five clubs and makes twelve tricks, so you score 100 points below the line and 20 points above the line and have one game in the bag. (A new horizontal line is drawn.) On the next deal, you try to capture the rubber by bidding four spades (a bid of three spades would enable you to score only 90 points below the line no matter how many tricks you took and would make it impossible to score a game on that deal), but you have overestimated your prospects and take only seven tricks. The opponents score 300 points (above the line)--1OO points for each vulnerable undertrick. On the third deal, things get even worse. You bid three notrump, the opponents double, and you take only six tricks. Disaster! You are three tricks short of your contract, are vulnerable (have one game) and have been doubled. The opponents collect 800 points (200 for the first undertrick and 300 for each of the next two undertricks). At this point, the unhappy state of affairs looks like this:

WE	THEY
20	800 300
100

MAKING A DOUBLED CONTRACT
If you are fortunate enough to make a doubled contract, your trick score is doubled, and you get a special bonus of 50 points (above the line). Overtricks, however, are scored in a different way. If you are not vulnerable, doubled overtricks count 100 points each (regardless of the strain of the contract); if you are vulnerable, doubled overtricks are worth 200 points. Like any score involving extra tricks, such bonanzas are placed above the line. If you make a redoubled contract, the trick score is redoubled. (Thus, one spade redoubled is a game contract!) The score for an overtrick is twice what it would be if the contract were (only) doubled. The special 50-point bonus for making a doubled contract is also doubled, to 100.

7. You bid two hearts, the opponents double, and you take nine tricks. A great many good things are about to happen. First of all, your score below the line is doubled; instead of receiving the usual 60 points for making two hearts, you score 120. This is very significant, for you now have more than the required 100 points below the line and have therefore made a game. In addition, you get a bonus for making an extra trick. Let us suppose that you are not vulnerable, so you receive 100 points for the overtrick (instead of the usual 30). Finally, you get the special bonus of 50 points for making a doubled contract. The happy result looks like this:

WE	THEY
50 100
120
Mr. Smith	9
Mrs. Smith	Q
Mr. Jones	6
Mrs. Jones	9
Charlie	3

Mrs. Smith and Mrs. Jones, who have drawn the two highest cards, are partners (hearts outrank diamonds, so Mrs. Jones' card is higher than Mr. Smith's), and Mr. Smith and Mr. Jones, who have drawn the next two highest cards, are partners. Only four people can play at a time, so Charlie, who drew the lowest card, must sit out the first rubber. When the rubber has ended, Charlie will replace Mr. Jones (who drew the second-lowest card), and the two players to Charlie's right will switch seats. This last step may seem mysterious, but is an easy way of ensuring that everyone plays (once) with everyone else as a partner. For the third rubber, Mr. Jones will return and replace Mr. Smith (who drew the third-lowest card) and the two players to the right of Mr. Jones will change seats. This process continues until the fifth rubber is completed (with Mrs. Smith sitting out) or until one of the players withdraws (whereupon new cards are drawn and the process begins all over again).

DEAL 1

In our sample rubber, the players are engaged in a bridge battle of the sexes. On the first deal the women bid four diamonds, are doubled, and take only seven tricks (in bridge parlance, "go down three"). The men collect 500 points (100 points for the first undertrick plus 200 points for each of the next two undertricks for a doubled, nonvulnerable contract), and Mr. Smith, who is keeping score, enters this result with a satisfied air as follows:

WE	THEY
500

DEAL 2

The women seek revenge, and double Mr. Jones in a contract of two clubs. They have miscalculated, however, and he makes his contract with an overtrick (taking nine tricks in all). He has earned for his partnership 80 points below the line (double the usual 20 points per trick with clubs trumps). Above the line, the men score 100 points for a nonvulnerable doubled overtrick and 50 points for making a doubled contract.

WE	THEY
50 100 500
80

DEAL 3

Mrs. Smith plays a four-spade contract and makes eleven tricks. Her side scores 120 points below the line and 30 points above for the overtrick. Since the women have landed a game, the men's part-score is obliterated for game-making purposes (the points remain on the score sheet). They will need a full 100 points to register the next game. The score:

WE	THEY
50 100 500	30
80	120

DEAL 4

The men bid three notrump but take only seven tricks, two short of their contract. The women score 100 points above the line (50 points per undertrick for a nonvulnerable contract).

WE	THEY
50 100 500	100 30
80	120

DEAL 5

Mr. Jones plays a three-heart contract and takes exactly nine tricks. He has just made his contract and scores 90 points below the line.

WE	THEY
50 100 500	100 30
80	120
90

DEAL 6

Mr. Smith plays a one-notrump contract and makes eight tricks. He scores 40 points below the line and 30 points above the line. Since the men's total below the line (uninterrupted by an enemy game) is above the minimum of 100 points, they have scored game by combining two part-scores. Both sides are now vulnerable, and the next side to score a game will win the rubber bonus.

WE	THEY
30 50 100 500	100 30
80	120
90 40

DEAL 7

The men arrive at two diamonds, but Mr. Smith can take only five tricks. The women score 300 points above the line (100 per vulnerable undertrick).

WE	THEY
30 50 100 500	300 100 30
80	120
90 40

DEAL 8

Mr. Jones plays in six clubs and takes all thirteen tricks. The men score 120 points below the line and have therefore netted the crucial second game. They score 20 points above the line for the overtrick, 750 points for bidding and making a vulnerable small slam (not a grand slam bonus, because although they took all 13 tricks they did not bid seven), and 500 points for winning the rubber when the opponents have made a game. In addition, Mr. Jones proudly announces at the end of the deal that he held in his own hand the ace, king, queen, and ten of trumps (clubs) and is therefore entitled to the 100-point bonus for honors. The final score:

WE	THEY
100 500 750 20 30 50 100 500	300 100 30
80	120
90 40
120

2380	550

The rubber is over, and the total score for each side is tabulated. The men have registered a total of 2,380 points, while the women have scored 550, so the men win a net of 2,380 minus 550 or 1,830 points. These results are now entered on the back score, a place for keeping a record of each individuars performance. It is customary to round off scores to the nearest hundred for convenience before entering them on the back score; since 1,830 is closer to 1,800 than 1,900, the men have "won an 18 rubber."

	1	2	3	4
Mr. Smith	+18
Mrs. Smith	-18
Mr. Jones	+18
Mrs. Jones	-18
Charlie	0

    It is quite possible for the side that scores two games and collects the rubber bonus to lose; one way this could happen would be if it paid out a substantial amount in penalties.

Four-Deal Bridge

    The traditional form of bridge described above, usually called rubber bridge because the unit of play is one rubber, has largely been supplanted by a modified form. This newer version is often called four-deal bridge because each unit of play, called a chukker (from a word meaning "wheel," also used as the name of a period of play in polo), consists of exactly four deals, one dealt by each player. The four-deal variant offers several significant advantages, including predictability of the time required for one unit of play (rarely more than half an hour) and the regular changing of partners.

    Almost the only difference between the forms is that in four-deal bridge there is no rubber bonus and the vulnerability (for scoring purposes) depends not on which side has made a game but only on the deal number. On the first deal, neither side is vulnerable; on the fourth (i.e., the last) deal, both sides are vulnerable. There is no universal treatment for the second and third deals. In the original form, often called Chicago (can you guess where it was invented?), only the dealer's side is vulnerable on the second and third deals. In a more recent variant, usually considered an improvement, sometimes called the Cavendish form (after a famous bridge club where it was first used), only the dealer's opponents are vulnerable on the second and third deals. Since there is no rubber bonus, if a side makes a game it receives an immediate bonus: 300 for a game made when nonvulnerable; 500 for a game made when vulnerable.

The only other important differences in four-deal bridge are these:: If a deal is passed out, it is redealt by the same player with the same vulnerability conditions.; If, on the fourth deal, a side makes a part-score contract that does not complete a game, it receives a bonus of 100. (There is no bonus for an active part-score that was made prior to the fourth deal.)

Duplicate Bridge
One of the game's greatest attractions is duplicate bridge, in which two (or more) sets of people are given the opportunity to play the same deal. This has the obvious advantages of removing much of the luck of the deal and giving players with poor cards as much interest in the proceedings as those with strong ones (because you can win if you do better with your cards, weak or strong, than others do). It also provides less obvious, but equally enjoyable, opportunities for interesting comparisons and effective learning. It is not easy to find places on the earth where one can go and not be able to find a regularly scheduled duplicate bridge game. (We think some parts of Antarctica qualify, but we would not be much surprised to learn otherwise. Even prisoner-of-war camps have held duplicate tournaments.) Special methods are used to preserve the arrangement of the cards. Typically, each deal is preserved in a mechanical device, called a board, usually made of metal or plastic, that has four pockets for the hands. Instead of tossing played cards into the middle of the table, duplicate participants put them face down nearby, and keep the trick score by pointing a played card lengthwise towards themselves if they won the trick, or towards the oppoents otherwise. In duplicate bridge, each deal is score as a separate unit, with the dealer and vulnerability determined in advance (and usually indicated on the board in which the cards are carried). Each of the sixteen combinations of a dealer and a vulnerability condition is equally likely to be encountered. A side making a game contract gets a bonus of 300 or 500, depending on whether it is nonvulnerable or vulnerable (as in four-deal bridge). Since part-scores do not carry over from one deal to the next, a side making a part-score contract receives a bonus of 50 points. The only other scoring difference is that at duplicate, there are no honor bonuses.

Capsule Summary: Scoring
TRICK SCORES are scored below the line; all others above.
When a GAME is made, a new horizontal line is drawn and each side needs 100 below the line to score the next game.

TRICK SCORES (multiply by 2 if doubled, by 4 if redoubled)

Strain	Notrump	Major suit ( or )	Minor suit ( or )
Odd-trick score	40 (first)	30	20
	30 (others)

BONUSES

Rubber bonus	If opponents are not vulnerable, 700 If opponents are vulnerable, 500
Unfinished rubber	For the only game, 300 For the only part-score in an unfinished game, 100
Honors in one hand	Four honors in the trump suit, 100 Five honors in the trump suit, 150 Four aces at notrump, 150

Slams	Not vulnerable	Vulnerable
Small slam	500	750
Grand slam	1000	1500

For making a doubled (or redoubled) contract, 50 (or 100)

OVERTRICKS		Not vulnerable	Vulnerable
	Undoubled	Trick value	Trick value
	Doubled	100	200
	Redoubled	200	400

UNDERTRICKS		Not vulnerable		Vulnerable
		First	Others	First	Others
	Undoubled	50	50	100	100
	Doubled	100	*	200	300
	Redoubled	200	**	400	600

* 200 for the second, 200 for the third, 300 for each subsequent
** twice the value when doubled

Review Quiz

1. A rubber is described below, deal by deal. Enter the score for each deal on a bridge score sheet. When the rubber is completed, determine how many points are won by the victors.: a. "They" bid two clubs and take nine tricks.; b. "We" bid four hearts and take ten tricks.; c. "They" bid four diamonds, are doubled, and take five tricks.; d. "We" bid three notrump and take eight tricks.; e. "We" bid two clubs, are doubled, and take six tricks.; f. "They" bid four hearts and take nine tricks.; g. "They" bid six hearts and take twelve tricks; declarer announces that she held the ace-king-queen-jack of hearts in her own hand (but not the heart ten).; h. "We" bid two hearts and take twelve tricks; declarer announces that he held the ace-queen-jack of hearts and dummy held the king-ten of hearts.; i. "We" bid one notrump and take seven tricks.

2. How would the results of deals a., b., c. and d. be scored in a chukker of four-deal bridge? (Assume the Chicago form, and that a player from "We" deals first.)

3. Assuming the same vulnerability conditions as in question 2., how would the results of deals a., b., c. and d. be scored at duplicate bridge?

Solutions

1. To save space, only one diagram is shown. The letters in parentheses would not ordinarily appear, and are used to identify the deal in question.: a. Score 40 points below the fine (20 per trick) and 20 points above the line for the overtrick in the "They" column.; b. Score 120 points below the line in the "We" column, and draw a line beneath the score to indicate a game. The part-score in the "They" column is thereby cancelled, though the 40 points remain.; c. "They" have been set ("gone down") five tricks, doubled and not vulnerable; "We" score 1100 points (100 for the first undertrick, 200 for the second and third undertricks, and 300 for each subsequent undertrick) above the line.; d. For one undertrick, vulnerable, score 100 points above the line in the "They" column.; e. "We" must pay out 500 points (for doubled, vulnerable, undertricks, score 200 for the first one and 300 points for each subsequent one).; f. Score 50 points, above the line in the "We" column for one nonvulnerable undertrick.; g. "They" score 180 points below the line (game), 500 points above the line for bidding and making a nonvulnerable small slam, and 100 points above the line for honors.; h. "We" score 60 points below the line and 120 points above the line. No game or slam has been made, since the bid was only two hearts. Declarer's announcement gains nothing (except perhaps some blank stares), for honors must be held in one hand to receive the bonus.; i. "We" score 40 points below the line and now have game; since it is "our" second game, "We" have won the rubber and receive a bonus of 500 points (not 700, as the opponents have made a game).

"We" total 1,990, and "They" score 1,440, so "We" win 550 points. (For purposes of entering the results on the back score, results ending in 50 are rounded up, so "We" win a "6 rubber.")

WE	THEY
500 (i) 120(h) 50 (f) 1100(c)	100 (g) 500 (g) 500 (e) 100 (d) 20 (a)
120 (b)	40 (a)
	180 (g)
60 (h) 40 (i)

1990	1440

2. On deal a. [neither side vulnerable], on which "They" bid two clubs and take nine tricks, "They" get 40 below the line and 20 above.
   On deal b. ["They" vulnerable], on which "We" bid four hearts and take ten tricks, we score 420 (300 for the nonvulnerable game and 120 for tricks), wiping out the opponents' part-score. To save effort, most players would write all 420 points below the line. Once the score below the line is more than 100, the exact amount does not matter.
   On deal c. ["We" vulnerable], on which "They" bid four diamonds, are doubled, and take five tricks, "We" score 1100 for five doubled undertricks.
   On deal d. [both sides vulnerable], on which "We" bid three notrump and take eight tricks, "They" score 100 for one undoubled, vulnerable undertrick.

3.a. "They" score 110 (60 for tricks, 50 for part-score bonus).
   b. "We" score 420 (300 for game, 120 for tricks).
   c. "We" score 1100 (five undertricks).
   d. "They" score 100.