## Digital filters and bridge rankings

Shaun Bullett forwarded to me this request from Mike Christie. If anyone would like to collaborate on this, please contact Mike directly: his contact details are below.

I have recently developed the method and software used by the English Bridge Union (EBU) to rank its 50,000 members by calculating a current grade whenever they play in duplicate bridge sessions. Grading or ranking systems are used in other activities such as golf, tennis, chess, and have been used in other bridge organisations. Although I have a PhD in pure maths and applied Kalman filters in some work for the MoD (35-40 years ago), I would welcome some input from an academic expert and would eventually like to publish an academic paper on the maths behind this grading system and its more interesting properties.

In duplicate bridge, each pair in a session gets a percentage score between 0 and 100% (average 50%) depending on how well they performed (and on their luck) in that session. In a field with 20 pairs a score of 62% is usually enough to top the session.

Duplicate bridge provides an additional challenge to a grading system as it is a partnership game and one tries to assess individual grades from pairs’ results relying on the fact that most players play with a variety of partners.

The problem is essentially one of state estimation, and I have used my knowledge of control theory, linear digital filters and Kalman filters to provide the mathematical foundations behind the methods that EBU had chosen, though other techniques such as graph and cluster theory would be helpful to analyse some aspects. One of the methods of grading in duplicate bridge is to use a proportion of the difference between the predicted and expected results in a session to update the current estimated grades, an exponential filter, whose basic properties for one dimensional systems are well known. However, the EBU chose a novel method which has more interesting properties from a mathematical viewpoint, a sort of feedback filter.

So I am looking for someone with a better understanding of how to analyse such filters than I have.

Details of the EBU’s system (NGS) can be found on their website https://www.ebu.co.uk/ngs/, but even the full guide is oriented to the practical details. I have various unpublished notes about some of my mathematical investigations.

Mike Christie
020-8444-3317
mob 07790 484269
mike99christie@btinternet.com