Arrowswitching

Background Information

Whenever you play a hand of duplicate bridge you are competing for Matchpoints. It is important to know against whom you are competing. In a Mitchell  movement this means.

• You are competing against the pair sitting at your table - in other words the better you do, the worse they do. The total matchpoints for which you are competing is the number of matchpoints in a 'TOP' - so if the hand is played 12 times, the top is 22 - and that is the level of competition between you and that pair on this board.
• You are competing aginst all other pairs who are playing the board in the same direction. You are competing for two matchpoints from each of them. If there are 12 rounds then the level of competition between you and each other pair is 24.
• You are NOT competing against the other pairs that are playing in the opposite direction at other tables - in fact you want them to do well so their opponents (against whom you are competing) do badly. So the level of competiion betwen you and those pairs is (-22) - two matchpoints a round for (Rounds - 1)
• If you add up the total competition then you see that there is no actual competition between you and the pairs who played the board in the opposite direction: (The +22 when you played against them is offsett by the -22 when they played against the other pairs)
• Since there is no actual competition between them then you can't rank the NS players and the EW players together - you have to separate them. The net effect is that you create a '2-winner' Mitchell.
• One corollory of this is that it doesn't matter how strong the opponents are that you are playing - you are not competing with them. In fact the stronger the people you play against, the better for you.

The organisers, however, may request the Director to organise a movement in which there is only ONE winner. To do that the Director must arrange the movement so that all pairs compete against each other. This means that each pair must sometimes play in the same direction as all the other pairs for some of the boards. This means that for some boards they must sit EW rather than NS (or vice versa). You can either do this by having the NS pair at the table move physically to the EW direction OR by rotating the board 90 degress (by convention clockwise), so that the NS pair pick up the EW hands - and the EW pair pick up the NS hands. Traditionally a board has an arrow on it pointing to North to make sure it is played the right way. Turning the board 90 degrees results in the arrow switching direction from the usual North to East. Hence the term - Arrowswitch.

How does this affect the level of competition?

In a perfect world, each pair would play in the same direction as all the other pairs, the same number of times. In fact there are very few movements that allow this to happen. The effect of this is that the total level of competition between each pair of pairs varies. A movement is said to be balanced if this different level of competition is kept to a minimum. (i.e. the variance of competition is as low as possible). You may think that half the rounds should be arrowswitched, so that players compete AGAINST the other pairs half the time and CO-OPERATE with the other pairs the other half. This is not the case because when they play AGAINST each other at the table, the level of competition is MUCH higher than when they are playing at other tables. (+22 compared with +-2 in the example above)

To show the level of competion, let us consider a straight-forward 8 table Mitchell movement with ONE round arrowswitched. There are several categories of competition depending on the direction and the play in the arrowswitch round..

e.g. for NS pairs.

• They compete against the other NS pairs on Six of the sets of boards (same direction) and co-operate on Two of the sets of boards (the two sets being played at the last round): Total competition is therefore (6 X 2) - (2 X 2) = 8
• They compete against 7 of the other East Wests as follows: They play against them once (competition = 7 X 2), on 5 sets of boards they co-operate (-5 X 2) and on two they compete (+2X2): Total competition is therefore (7X2) - (5X2) + (2X2) = 8
• Against ONE EW pair (the one they play on the last round), this is different: They play against them once (competition = 7 X 2) and for the other 7 rounds they co-operate: Total competition is therefore (7X2) - (7X2) = 0!

(From this you can see that in a Mitchell event with one arrowswitch you should aim to play against the strongest pair on the final round! )

Thus you can see that with ONLY ONE round arrowswitched, all pairs compete equally against every other pair but one.

Mathematical 'proof' of why just over 1 in 8 rounds should be arrowswitched

Suppose you have R rounds of which Q are arrowswitched, and T Tables.

The total masterpoints at stake is R(T-1)  : i.e the number of rounds multiplied by the top on each round.

There are 2T-1 pairs against whom any other pair is completing (2 at each table, less the pair).

So the AVERAGE number of masterpoints that each pair is competing for against each other pair = Total of points/ number of competitions  = R(T-1)/(2T-1)

Consider the NS line : Of the R Rounds Q are arrowswitched so the competition is (R - 2Q)  (rounds when competing) -2Q (the arrowswitched rounds).  = R -4Q

These should be equal to the average : Note we can write (2T-1) as 2(T-1) + 1 Therefore

R - 4Q = R(T-1)/2(.(T-1)+1) - If we ignore the (+1) then the T-1 cancel and the right hand side gets a bit bigger.

So R - 4Q < R/2 - and by rearranging we get R/2 < 4Q and thus Q (number of rounds to be arrowswitched) is just more than 1/8 approximately.

(Calculation assumes that the NS pairs do not play the same boards when arrowswitched. This is true when 1 round is arrowswitched (other than for a sharing table) and for most pairs of NS pairs this will be true when there is more than one arrowswitched round - only NS pairs next to each other for instance are affected if there are two arrowswitched rounds - which handles the case of 8< T < 17)

(A final note: it is impossible for a perfectly balanced competition based on a Mitchell movement, since the average number of masterpoints you should be competing for (R.(T-1)/(2T-1)) will only be an integer IFF 2T-1 is a factor of R and R cannot exceed T)