Fairness in Matchpoint Duplicate Games

by Marvin French

Matchpoint pairs is a game of comparisons. You and partner bid and play a pair of hands, and your result is compared with others who held the same hands. It immediately becomes obvious that the fairest game is one in which pairs competing with each other (1) have played the same hands (hands, not boards), (2) have compared each of their results with all pairs competing with them, and (3) have met opponents of approximately equal ability.

Is there such a game? Yes. The lowly club non-championship game, with straight Mitchell movement, if completed (e.g., 13 rounds, 26 boards) is the perfect game. Pairs in each direction have: (1) played the same hands, (2) faced the same opponents, and (3) compared results with everyone else in their direction. Such fairness is also present in the qualifying sessions of pair championships, in which pairs are not ranked overall and qualifiers come from each field independently. While this game is not perfect (usually pairs miss one or two sets of boards), it is nearly so. An unavoidable flaw is that the two fields may meet pairs of significantly different abilities, thereby introducing an undesirable amount of luck.

Missing a few sets of boards is not a large contributor to unfairness, unless the number missed is excessive. The opinion of tournament director John Probst of London is that competing pairs should share in the play of at least 80 percent of the boards if the game is to be considered a fair contest. A Mitchell movement for 16 tables, 32 boards in play, meets this criterion when 26 boards are played, but for 17 tables, 34 boards in play, an additional two-board round must be played in order to meet this criterion, although no sponsor of such a game seems willing to do that. A 17 or 18-table game with 13 rounds, 26 boards played, cannot therefore be considered a fair contest.

The solution for 18 tables is to put players in two 9-table sections, each with 27 boards in play, three to a round, with across-the-field matchpointing (maximum score on a board 17). This is many more comparisons than an 18-table Mitchell with a “top” of only 12, and pairs play all boards. If boards cannot be preduplicated1, however, one set of boards must be missed and the total played will be only 24. Is that so bad? The two sections must be carefully balanced in regard to ability, to maximize criterion (3) in the first paragraph.

Note that a Mitchell movement is a two-winner game. The two fields, East-West and North-South, have (1) played entirely different hands, (2) faced entirely different opponents, and (3) have compared only with those in their field. That's two different games, not one, making overall ranking inappropriate. Pair championships that are held from club level up to national bridge pair championships do not recognize this principle, and regularly rank pairs overall, turning a two-winner game into a one-winner game with no justification. This is wrong.

If sponsors insist on having a one-winner game, with overall ranking, they should at least see to it that pairs compare with as many other pairs as possible. There are two ways to increase the number of mutual comparisons: (1) scramble the Mitchell fields, East-West and North-South, after each session (only half the pairs switching direction), and (2) switch the direction arrow to point in the opposite direction for some rounds (“Arrow Switch”). The two ways can be combined to further increase the fairness of the game.

For instance, in the two-session finals of a pair championship, have only half the pairs switch direction for the second session instead of having all pairs (except the disabled) do so. The result is that pairs compare with 1/4 of the others twice, 1/2 once, and 1/4 never. That is much better than 1/2 twice and 1/2 never. If the arrow is switched for the last two rounds, comparisons become even more distributed, and moreover it produces a one-winner movement. Arrow switching also has the non-intuitive effect of tending to balance out the inevitable inequality of field strengths.

A rule of thumb is to switch the arrow on approximately 1/8 of the total rounds, which means switching on just one round (preferably the last), for Mitchell movements comprising 11 or fewer tables, and on two rounds (preferably the last two) for sections of 12 or more tables. This is a common practice in England, or at least in London.

Another approach to arrow-switching is favored by the Scandinavians. Instead of aiming to balance out the effect of uneven strength in the two Mitchell fields, the arrow is switched on a greater number of rounds at various points in the game in order to provide more equality of comparisons among pairs. While this is a reasonable goal, such switching complicates the game. Moreover, there seems to be no agreement as to when the arrow should be switched for various numbers of tables. Besides, as an authority once said, too many arrow switches spoil the balance of strength in the comparison groups for each board.

Another step toward fairness is use of the Web movement, now employed in finals of the American Contract Bridge League's major pair championships, for sections consisting of 14 or more tables. It makes possible the limitation of boards to the convenient 26 number even with 14 or more tables, but requires two sets of boards for each Web section, which in turn requires preduplication of boards. For instance, an 18-table section can play 26 boards with 17 top, instead of the common setup in which pairs play 26 boards out of 36, with 12 top. It is obviously more fair if pairs play the same boards and compare results on each with all others in their field. Reducing the number of qualifiers in order to have 13-table sections is another solution, but that often means qualifying an insufficient number of pairs.

The Web movement should be used more widely, but is rather unwieldy if the number of, tables in sections is not known well ahead of time so that the boards can be placed as required. Also, an even number of tables is highly desirable because the movement gets complicated with an odd number. Note that this is a two-winner movement, which means that overall ranking is wrong even though the American Contract Bridge League does that in the finals of national pair championships.

The seven-table Howell is often praised as an ideal game because all pairs meet and pairs play every board. However, pairs have played different hands (hands, not boards) and have not compared with the other pairs an equal number of times (5, 6, or 7 times). A Howell with an even number of tables can be "perfect" if completed, since pairs will compare with other pairs an equal number of times. However, ranked pairs will not have played all the same hands, so it really isn't perfect. Also, completing such Howell movements requires the play of an awkward number of boards (e.g., 22 or 33 for six tables, 15 or 30 for eight tables).

A common Howell game that is one of the worst is that produced by use of table guide cards that direct each pair's location on each round. These are usually “Three Quarter” movements that result in a horrendously bad balance of comparisons. That is, pairs compare with the other pairs a vastly unequal number of times. To bring about more equality of comparisons, a one-winner movement requires that there be some bye stands (holding boards not played for a round) between some of the tables. Also, individual pair guide cards are necessary to tell pairs where they play, which boards they play, and which pair they face, for every round. It is understandable that club directors don't like this arrangement (too much work and chance for mixup), but as long as players are unaware of, or don't care about, the unfairness, unfair games will continue to be the rule.

Let's say you have 8 tables and want to play a one-winner movement. The table guide cards (usually a 3/4 movement) are so very bad that it is much preferable to use the guide cards that the scoring program ACBLscore can print out. The 3/4 movement has pairs comparing with other pairs on anywhere from 3 to 9 sets of boards, while the “perfect” 8-table Howell movement (all pairs compare an equal number of times and all pairs meet) requires the playing of 30 boards, there is no great harm in just not playing the last two rounds, resulting in a 26-board game. Missing four boards and not playing two of the other pairs is of minor importance, and the comparisons, while not perfect, are fairly well-balanced, each pair comparing with the others on 5, 6, or 7 sets of boards. The 6-table movement on table guide cards is equally bad, and the game can also be greatly improved with guide cards in a truncated Howell movement.

The comparison balance for one-winner 4, 5, or 7-table Howell games is satisfactory, and in fact is perfect for the 4-table game.

When the number of tables is 9 or more a Mitchell movement is the way to go, because the 3/4 movements have comparisons that are too unbalanced and there is no remedy. While Mitchell pairs compare with only 1/2 of the other pairs, at least they compare with that half an equal number of times. But, as I said above, this is a two-winner game. If overall ranking is a must, an arrow switch should be used on the last round or two.

1Before the widespread use of duplicating machines at tournaments, players would manually duplicate one round of boards and therefore could not play them.