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On Designing a VP Scale

The essay that follows will probably not be of general interest, but its the sort of technical issue that some of us like to explore. 

 

International Match Points replaced total points as a scoring method for team events to reduce the impact of very large swings on the outcome of matches. Victory Points were introduced for much the same reason: to reduce the effect of lopsided wins in multi-team events.

 

In trying to compare the results among a large number of teams playing a significant number of matches – whether a round-robin or a Swiss – there is a fundamental first issue: should the winner be the team scoring the greatest number of net imps, with perhaps some limit on the margin in any single match? Or is winning matches important also, in which case the imps that determine whether a team wins or loses a match should be more important than imps that stretch a significant margin into an overwhelming one. The most severe form of the latter is win-loss scoring. Only the two imps that separate Win from Tie from Loss have any significance in determining the final result.

 

North America and the World Bridge Federation have come to different answers on this fundamental question. In North America many events are played on a near win-loss scale: one point for winning by three or more imps; three-quarters for winning by one or two, and a half for an exact tie. When the VP scale is extended from what is effectively a four-point scale to a twenty-point scale, successive victory points require more additional imps in the margin.

 

In contrast, the WBF scale is essentially linear with a cutoff: it takes the same number of imps (approximately) to get from 17VP to 18 as from 23 to 24. (A note: both the ACBL and WBF scales are effectively 20-point scales. The ACBL scale runs from 0 to 20 with a draw scored as 10-10; the WBF scale goes from 5 to 25 with a draw scored 15-15. The WBF allows the loser to score less than 5VP for overwhelming losses.)

 

There are advantages and disadvantages to both methodologies. The WBF scale essentially is a total-imps won/lost scale, making no differentiation between imps exchanged in close matches and those won in substantial wins and losses. Except in blowouts this means that the VP odds on bidding decisions and the imp odds are similar. For example, a 10imp gain in a 16-board match will gain 2VP half the time, 3VP half the time. A 6imp loss will lose 1VP half the time, 2VP the other half.

 

Using an ACBL type scale (the ACBL does not actually have scales for matches longer than 10 boards), if you were well ahead before the board another 10imps might gain only 1VP or 2 while a 6imp loss might cost the same amount. On the other hand if you were significantly behind, a 10imp gain would usually be worth 3VP, and a further 6imp loss might lose 1 or even no VP. In other words the state of the match affects the VP odds. If matches as such are important, this seems more realistic.

 

The second issue is whether to use the sort of “step function” that both the ACBL and the WBF currently use, or whether to make every imp count for something. In both the ACBL and WBF scales, particular imps are especially important because they take the team from one result to another, while others, most others, make no difference. The USBF decided to adopt a different approach: make every imp worth something.

 

Using the WBF methodology, in a 16-board match each imp could be worth ¼ of a VP until ten additional VP were earned at a win by 40 imps. Thereafter the loser would lose 1/6th of a VP for each additional imp in the losing margin until 5 more VP were lost.

 

The USBF uses a very different structure. As in the WBF scale, the first 20 imps in a 16-board match earn 5VP. But the next 20 earn about 3, and another 20 imps are required to reach 20-0. Another way to look at the design is that over any interval, the first third of the imps in that interval are rewarded with half the VP for the range. The USBF scale is posted in two-decimal accuracy.

 

Note: VPs for any margin can be calculated directly using the following procedure.

1.    Calculate 15 times the square root of the number of boards in the match. Call this B.

2.    Calculate the “Golden Mean” which is (square root (5) – 1)/2 approximately 0.618. Call this value Tau.

3.    Letting b = B/3, calculate Tau^1/b. Call this value R.

4.    For any imp margin M <= B (M less than or equal to B), the winner’s VP will be given by V = 10 + 10*(1-R^M)/(1-R^B). If M>B, the winner gets 20VP. The loser gets 20-Winner’s VP.

 

A win by b imps will get 15 VP, a win by 2b imps will get 18.09. 

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