How To Rate Bridge Players
and Other Paired Competitors
T. S. Michael
U. S. Naval Academy
Annapolis, Maryland
tsm@usna.edu
http://www.usna.edu/Users/math/tsm/
joint work with
Thomas Quint and Jeff Mortensen
University of Nevada
Joint National Mathematics Meetings, Baltimore, January 2014
T. S. Michael (U. S. Naval Academy)
How To Rate Bridge Players
January 2014 - Baltimore JMM
1 / 13
Summary
We rate individual bridge players using least squares
The resulting linear system has combinatorial interest
We use Moore-Penrose inverse for singular systems
We apply our results to real-world data
Our method can be used to rate other paired competitors
(doubles tennis)
T. S. Michael (U. S. Naval Academy)
How To Rate Bridge Players
January 2014 - Baltimore JMM
2 / 13
Current Method to Rate Bridge Players (ACBL)
Masterpoints
individuals accumulate points over their lives
ratings never decrease
Drawbacks
strong, infrequent players have low ratings
some weak, frequent players have high ratings
Issues
individuals play with partners
pairs compete against many pairs simultaneously
Data Available
list of partnerships
partnership scores over many sessions
T. S. Michael (U. S. Naval Academy)
How To Rate Bridge Players
January 2014 - Baltimore JMM
3 / 13
Partnership Multigraph G (for a year, say)
1
4
.62
.4
2
Players
Partnerships
Partnership scores
Usually,
3
.5
.5
.5
.5
.6
5
←→
←→
←→
vertices
edges
edge labels
0.3 ≤ partnership score ≤ 0.7
T. S. Michael (U. S. Naval Academy)
How To Rate Bridge Players
January 2014 - Baltimore JMM
4 / 13
Averaging Method
rating of
player i
xi
player
i
1
2
3
4
5
total score
si
1.12
1.62
2.50
1.00
1.00
T. S. Michael (U. S. Naval Academy)
degree
di
2
3
5
2
2
=
average
partnership score
=
si
di
rating
xi
0.56
0.54
0.50
0.50
0.50
1
4
.62
2
How To Rate Bridge Players
3
.5
.4
.5
.5
.6
January 2014 - Baltimore JMM
.5
5
5 / 13
Averaging Method (continued)
Solve the system: Dx = s
D = diag[d1 , d2 , . . . , dn ] = diagonal degree matrix
x = ratings vector
s = total score vector
Flaw: Too local
If we increase the score for the
partnership of players 1 and 2, then
1
4
.62
x1 and x2 increase
x3 should decrease, but doesn’t
T. S. Michael (U. S. Naval Academy)
2
How To Rate Bridge Players
3
.5
.4
.5
.5
.5
.6
January 2014 - Baltimore JMM
5
6 / 13
Least Squares Method (Quint)
Idea: Players with ratings xi and xj should get
partnership score =
xi + xj
2
Partnership scores p1 , p2 , . . . , pm are observations of the ratings
x1 , x2 , . . . , xn .
Choose ratings to minimize sum of squared errors
m X
xik + xjk 2
E=
pk −
.
2
k =1
sum is over all edges {ik , jk } of G
T. S. Michael (U. S. Naval Academy)
How To Rate Bridge Players
January 2014 - Baltimore JMM
7 / 13
Least Squares Method (continued)
Set derivatives of E equal to 0 to get a system of equations.
m
X
xik + xjk
∂E
0=
=−
pk −
∂xi
2
k =1
(i = 1, . . . , n).
i∈{ik ,jk }
sum is over all edges {ik , jk } incident with node i of G
..
.
Solve the system: (D + A)x = 2s
D = diag[d1 , d2 , . . . , dn ] = diagonal degree matrix
A = adjacency matrix of partnership multigraph G
x = ratings vector
s = total score vector
T. S. Michael (U. S. Naval Academy)
How To Rate Bridge Players
January 2014 - Baltimore JMM
8 / 13
Least Squares Method: Example






2
1
1
0
0
1
3
2
0
0
1
2
5
1
1
0
0
1
2
1
0
0
1
1
2






x1
x2
x3
x4
x5






 = 2




1.12
1.62
2.50
1.00
1.00

1





4
.62
2
3
.5
.4
.5
.5
.5
.6
5
x1 = 0.61 x2 = 0.69 x3 = 0.44 x4 = 0.52 x5 = 0.52
If we increase the score for the partnership of players 1 and 2, then
x1 and x2 increase
x3 decreases
x4 and x5 increase
Method is global. It accounts for strengths of partners.
T. S. Michael (U. S. Naval Academy)
How To Rate Bridge Players
January 2014 - Baltimore JMM
9 / 13
When Is D + A Singular?
The matrix D + A is the unsigned Laplacian matrix of the
(partnership) multigraph G. It is studied in graph theory.
Theorem
Let G be a connected multigraph with unsigned Laplacian matrix
D + A. The following are equivalent.
D + A is singular
nullity(D + A) = 1
G is bipartite
T. S. Michael (U. S. Naval Academy)
How To Rate Bridge Players
January 2014 - Baltimore JMM
10 / 13
Least Squares Method: A Singular Example






|
3
2
1
0
0
2
3
0
1
0
1 0
0 1
3 1
1 2
1 0
{z
singular
0
0
1
0
1






x1
x2
x3
x4
x5






 = 2




1.50
1.50
1.65
1.10
0.55






}
1
.4
2
.6
.5
3
.55
5
.5
.6
4
x = 0.38 x2 = 0.59 x3 = 0.68 x4 = 0.47 x5 = 0.42
|1
{z
}
use Moore-Penrose inverse
T. S. Michael (U. S. Naval Academy)
How To Rate Bridge Players
January 2014 - Baltimore JMM
11 / 13
Least Squares Method: Reno
We used our least squares method to rate the bridge players in the
Reno Bridge Club. The partnership multigraph G had
n = 182 vertices (players)
one giant (non-singular) component
I
x = 2(D + A)−1 s
sparse linear system
30 small bipartite components
I
x = 2(D + A)+ s
Moore-Penrose inverse
The computed ratings seemed reasonable to the players.
We can try the same least squares method to assign ratings to
individuals in any sport where players compete in pairs.
T. S. Michael (U. S. Naval Academy)
How To Rate Bridge Players
January 2014 - Baltimore JMM
12 / 13
References
Quint
A new rating system for duplicate bridge
Linear Algebra and Its Applications 2007
Michael, Mortensen, and Quint
Rating bridge players: The bipartite components
in preparation
T. S. Michael (U. S. Naval Academy)
How To Rate Bridge Players
January 2014 - Baltimore JMM
13 / 13