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Are Howell Movements Fair?

The short answer is no - but read on to find out exactly how unfair they are.

The basic idea of duplicate bridge is that you play the same hands as other people and your score on a hand is obtained by comparing your result with the results of other people playing the same hand. This is what happens in a complete Mitchell movement where the North/South pairs remain seated and the East/West pairs move from table to table until they have played at every table. All the North/South pairs have then played all the same hands as the other North/South pairs and all the East/West pairs have played the same hands as the other East/West pairs. We then have two winning pairs - one North/South and one East/West. You can't get a single winning pair since it makes no sense to compare the score of the winning North/South pair with that of the winning East/West pair since the scores were obtained on different hands against different sets of opponents.

If you want a single winning pair then you need everyone to play against each other. This is what happens in a complete Howell movement when you play against every other pair in the room. The trouble is, in the normal face to face situation, you can't play the same hands as the pairs that you are playing against since some pairs have to sit North/South and some East/West. To average things out, the movement will make you play some hands North/South and some East/West but it will be a different set of hands for every pair. Since most people find declarer play easier than defence this means that if you get to declare more hands than other people you will probably get a better result. You will generally declare more hands if you have more points so there should be a correlation between the average number of points you get and your final result. The question is - how strong an effect is this?

I was led to thinking about this by our poor result last night when we underperformed significantly. We only held an average of 18.63 points per hand between us instead of the 20 we might expect in a fair world. This meant that we only declared 9 of the 27 hands and hence defended twice as often as we declared. Could we write off all of our underperformance to this unjust points deficit? To check this I plotted all pair's final scores against the average number of points that they held as shown below.

Each red dot shows the average high card points held by each pair and the score that they achieved. It can be seen that most of the results below 20 points lie below the 50% horizontal line with only 1 above while the results above 20 points lie above the 50% line with only one below. This qualitatively shows the expected correlation of fewer points leading to a lower score.

The black line quantifies this effect. It's a 'least mean squares fit' to the data which is a way of drawing the 'best' line through the data. This has a slope of 1.56% per point which, for the people with only about 18 point,s means a 3% reduction in their score or, for those with 22 points, a 3% increase in their score. Unfortunately, while this is a significant effect, it still means we personally underperformed!

Finally, it's interesting to note that online bridge could provide a solution to the single winning pair problem. For a fair competition we want everyone to play the same hands against the same opponents. Each pair should therefore play every board in the same direction against a pair of robot opponents. This would then be a completely fair comparison. (Note, however, that bridge playing robots are often programmed to make random decisions in some situations. For a fair comparison, they should be programmed to bid and play identically in the same situation.)

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