Hi,
I'm new to this forum so don't know quite what to expect yet! I've got an idea about being able to measure a team's form and I hope to be able to discuss it on this forum, to hear ideas, challenges, help etc from other forum members.
I'm a keen bridge player and the World Championships are on in Bali right now which I'm watching from afar via BBO/VuGraph. I am curious about the mathematics needed to estimate a team's form, it's "Form Rank". The idea is that amongst a closed set of teams, competing over a short period of time, in a very constant and stable environment, it can be used as a predictor of the next set of matches.
The tournament in question, the open teams is known as the Bermuda Bowl (BB for short), has the following format.
22 teams
Matches are always between 2 teams. Each team has 4 players of 2 pairs. One pair from each team plays a pair from the other, on two separate tables and they play the same hands at both table (called Duplicate Bridge)
Round Robin = 7 continuous days of 3 x round-robin rounds, so that each team plays 1 match with the 21 other opponent teams. Each match of 16 boards takes an indeterminate time about 2 hours long, so that each round of 11 matches starting at the same time, usually finishes in about 2.5 hours.
Knockout phase = after the 21 rounds, the top 8 teams by score are in the quarter finals. The top 4 teams, 1-3 choose, their opponent from the 5-8 teams, team ranked 4 gets the remaining team. From then on the scheduling proceeds as usual via knockout to a final. There are gold, silver, bronze medals. Losing semi-finalists playoff.
I'm sure others have looked at the mathematics of a competitor's Form (e.g. horse racing, football) extensively, because of betting, but I cannot find on the internet any information about it. Some pointers would be great!
Until then, here are some of my ideas about Form Rank. But first, what is generally understood to be a team's form? It's a measure of how well they are currently playing. It depends on their last performance especially P(1) , but also other performances P(i) with progressively less weight as P(i) goes from 1 to N. All other external influences equal (or disregarded), a performance is based on the form of the opponent and the score against them.
* the teams form a finite set, {T(i)}
* at the end of every round, k, each team, T(i) has a derived Form Rank, R(T(i),k) which is a real number, normalised, e.g. -1 to 1.
* the Form Rank of a team T(i) at round k, R(T(i),k) is a function of all the scores S(T(i),T(j),k) already played against opponents {T(j)} in rounds 1 to k-1 in the tournament. i.e.
R(T(i),k) = func( S(T(i),T(j),m), R(T(j),m) ) where k ∈ {1,n}, m ∈ {1,k-1}
As the influence on current rank of performances diminishes over time, I was thinking of a very simple decay function, f(x) = (1/2)^x. Apart from being easily explainable, each performance is half the weight of the next most recent one, it has the advantage as a weighting function because its sum over x ∈ {1,N) converges to 1 as N -> oo. e.g.
R(T(i),k) = Σ weight * score * rank
= Σ (1/2)^m * S(T(i),T(j),m) * R(T(j),m) where k ∈ {1,n}, m ∈ {1,k-1}
Here the scores S(T(i),T(j),m) ∈ (0,1), i.e. are normalized appropriately (in tournament Bridge opposing teams in a match the scoring is translated into IMPs, International Match Points, per board, then the total is converted via a set scale so that both teams will always share 20 VPs, Victory Points)
This is very simplistic and a first pass by me, although I am guessing this has all been done before...
Thoughts?
