The contest analysis method is a useful vehicle for studying the
justices of the U.S Supreme Court--constantly scrutinized by the
public in attempts to predict their votes on issues. Now, for the
first time, a method is available for attaching a precise number to
the strength of each justice and another number to the amount of
support one justice gives another. To be sure, it has always been
possible to simply count the cases in which the various pairs of
justices are in agreement, but that sort of enumeration does not take
into account whether the justices are on the winning side of the
case. This is an important consideration because the winning (or
majority) side must be stronger, in some sense, so agreement on that
side should carry more weight than agreement on the losing
side.
What this contest-analysis method does is include the
nine justices and all of the cases into one huge contest, with the
justices making up a team which is pitted against a team composed of
the cases. Each case has a winning side and a losing side, and the
actions of the justices affect each side.(The '99-'00 session had 77
cases and the '00-'01 session had 85.) Only the participants on one
side are of interest (the justices), but both sides must be
considered to solve the problem. The justices receive credit from the
cases in which they are on the majority side and the cases receive
credit from the justices that are on the losing side--the more
justices on the losing side, the stronger the case. This credit is in
the form of points that make up a scoreboard, and this scoreboard can
be analyzed by the contest algorithm to produce strength and
interaction information for the justices and also for the cases. A
justice gains strength by supporting stronger cases and a case gains
strength when stronger justices are on its losing side. The workings
of the algorithm are discussed elsewhere on this site: It produces a
set of vectors in a vector space of many dimensions (a dimension and
a 'strength' vector for every contest participant) with the vector of
each participant having components in the directions of participants
on the other side. The vectors quantify the interactions among
participants.
Mr. Thomas C. Goldstein, a Washington attorney frequently involved
in Supreme Court cases, compiled the data. His data are more detailed
than necessary for this analysis, since he distinguishes three levels
of agreement (plus full disagreement) among justices on each case and
I only use agreement on the judgment. (Justices sometimes agree on
the judgment for entirely different reasons, while being basically in
disagreement.) The cases are numbered either 1-77 or 1-85, depending
upon the session, and the justices are numbered 1-9. The numbers
assigned to the justices are as follows:
1 - Rehnquist
2 - Stevens
3 - O'Connor
4 - Scalia
5 - Kennedy
6 - Souter
7 - Thomas
8 - Ginsburg
9 - Breyer
Below is an array for each session showing (with a
'1') which justices voted with the majority on each case. In the
second session, four votes were not cast; asterisks mark those
places.
SESSION OF '99-'00
Justice
1 2 3 4 5 6 7 8
9
1 2 3 4 5 6 7 8 9
Case
Case
1 0 1 1 1 1 1
1 1 1 41
0 1 0 1 0 1 1 0 1
2
1 1 1 1 1 1 1 1
1 42 1 0
1 0 1 1 1 0 0
3 0 1 0 0 1 1
1 1 1 43
1 0 1 1 1 1 1 0 0
4
1 1 1 1 1 1 1 1
1 44 1 1
1 0 1 1 0 1 1
5 1 0 1 1 1 0
1 0 0 45
1 1 1 0 0 1 0 1 1
6
1 1 1 1 1 1 1 1
1 46 1 1
1 1 1 1 1 1 1
7 1 1 1 1 1 1
1 1 1 47
1 0 1 1 1 0 1 0 0
8
1 1 1 0 1 1 0 1
1 48 1 0
1 1 1 0 1 0 0
9 1 0 1 1 1 0
1 0 0 49
0 1 0 1 1 1 1 1 0
10
1 1 1 1 1 1 1 1
1 50 1 1
1 1 1 1 1 1 1
11 1 0 1 1 1 0
1 0 0 51
1 0 1 1 1 0 0 0 1
12
1 1 1 1 1 1 1 1
1 52 1 1
1 1 1 1 1 1 1
13 1 0 1 1 1 0
1 0 0 53
1 1 1 1 1 1 1 1 1
14
0 1 0 1 1 0 1 1
1 54 0 1
1 1 1 1 1 1 1
15 1 1 1 1 1 1
1 1 1 55
1 0 1 0 0 1 0 1 1
16
1 1 1 1 1 1 1 1
1 56 0 1
1 0 0 1 1 1 0
17 1 1 1 1 1 1
1 1 1 57
1 1 1 0 1 1 1 1 1
18
1 0 1 1 1 0 1 0
1 58 1 1
1 1 1 1 1 1 1
19 1 0 1 1 1 1
1 0 1 59
1 0 1 1 1 0 1 0 0
20
1 1 1 1 1 1 1 1
1 60 1 1
1 1 1 1 1 1 1
21 1 1 1 1 1 1
1 1 1 61
1 1 1 1 1 1 1 1 1
22
1 0 1 0 0 1 0 1
1 62 1 0
1 1 1 0 1 0 0
23 1 1 1 1 1 1
1 1 1 63
0 1 1 0 1 1 0 1 1
24
1 1 1 1 1 0 1 0
1 64 1 0
1 1 1 0 1 0 0
25 1 1 1 1 1 1
1 1 1 65
1 1 1 1 1 1 1 1 1
26
1 0 1 1 1 0 1 0
0 66 0 1
0 1 1 1 0 1 1
27 1 1 1 1 1 1
1 1 1 67
1 1 1 1 1 1 1 1 1
28
1 1 1 1 1 1 1 1
1 68 1 1
1 1 1 1 1 1 1
29 1 1 1 1 1 1
1 1 1 69
1 1 1 1 1 1 1 1 1
30
1 0 1 1 1 0 1 1
0 70 1 1
1 0 1 1 0 1 1
31 1 1 1 1 1 1
1 1 1 71
0 1 1 1 1 1 1 1 0
32
1 0 1 1 1 0 1 0
1 72 1 0
1 1 0 1 1 0 1
33 1 1 1 0 1 1
1 1 0 73
1 0 1 1 1 1 1 1 1
34
1 0 1 1 1 1 1 0
1 74 1 0
1 1 1 0 1 0 0
35 0 1 1 0 1 1
0 1 1 75
0 1 1 1 1 1 1 1 1
36
1 1 1 1 1 1 1 1
1 76 0 0
1 1 1 1 1 1 1
37 1 1 1 1 1 1
1 1 1 77
0 1 1 0 0 1 0 1 1
38
1 1 1 1 1 1 1 1 1
39
0 1 0 1 0 1 1 0
1 62 52
71 63 69 59 66 55 59 Votes with majority
40 1 0 1 1 1 0
1 1 1
SESSION OF '00-'01
Justice
1 2 3 4 5 6 7 8
9
1 2 3 4 5 6 7 8 9
Case
Case
1 1 1 1 1 1 1
1 1 0 46
0 1 1 1 0 1 1 1 1
2
1 1 1 1 1 1 1 1
1 47 1 0
1 1 1 0 1 0 0
3 0 1 1 1 1 1
1 1 1 48
1 1 1 1 1 1 0 0 1
4
1 0 1 1 1 0 1 0
0 49 0 1
1 1 0 1 0 1 1
5 0 1 1 0 1 1
0 1 1 50
1 1 1 1 1 1 1 1 1
6
0 1 0 0 1 1 0 1
1 51 0 1
0 1 0 1 1 1 1
7 1 1 1 1 1 1
1 1 1 52
1 1 1 1 1 1 1 1 1
8
1 1 1 1 1 1 1 1
1 53 0 1
0 1 0 1 0 1 1
9 0 1 1 0 1 1
0 1 1 54
0 1 1 1 1 1 1 1 1
10
0 1 1 0 0 1 0 1
1 55 1 1
0 0 1 1 0 1 1
11 0 0 1 1 0 1
1 1 1 56
1 1 1 1 1 1 1 1 1
12
1 1 1 1 1 1 1 1
1 57 1 1
1 1 1 0 1 1 1
13 1 0 1 1 1 0
1 0 0 58
1 1 1 1 1 1 1 1 1
14
1 0 1 0 1 0 1 1
1 59 1 1
1 1 1 1 1 1 1
15 1 0 1 1 1 0
1 1 1 60
0 0 1 0 1 1 1 1 0
16
1 0 1 0 1 1 0 1
0 61 1 0
1 1 1 1 0 1 1
17 1 0 1 1 1 0
1 0 0 62
1 0 1 1 1 0 1 0 0
18
1 1 1 1 1 1 1 1
1 63 1 1
1 1 0 0 0 1 1
19 1 0 1 1 1 0
0 1 0 64
0 1 1 0 1 1 0 1 1
20
1 1 1 0 1 1 1 1
1 65 1 0
1 1 1 1 1 1 0
21 1 0 1 0 1 1
1 0 0 66
1 0 1 1 1 0 1 0 0
22
0 1 1 0 0 1 0 1
1 67 1 1
1 1 1 1 1 1 1
23 1 1 1 1 1 1
1 1 1 68
0 1 1 0 0 1 0 1 1
24
1 1 1 1 1 1 1 1
1 69 1 1
1 1 1 * 1 1 1
25 1 1 1 1 1 1
1 1 1 70
0 1 1 0 0 0 1 1 1
26
1 1 1 1 1 1 1 1
1 71 1 1
1 0 1 1 0 1 1
27 1 1 1 1 1 1
1 1 1 72
1 1 1 1 1 1 1 1 1
28
1 1 1 1 1 1 1 1
1 73 1 1
1 1 1 0 1 0 0
29 1 0 0 1 1 1
1 0 0 74
0 1 1 0 0 1 0 1 1
30
1 1 1 1 1 1 1 1
1 75 1 1
* 1 1 1 1 1 1
31 0 1 1 0 1 1
0 1 1 76
0 1 1 0 1 1 0 1 1
32
1 1 1 1 1 1 1 1
1 77 0 1
1 0 1 1 0 1 1
33 0 1 1 0 1 1
0 1 1 78
0 1 0 1 1 1 1 1 1
34
0 1 0 1 0 1 0 1
1 79 1 1
1 1 1 1 1 1 1
35 1 1 1 1 1 1
1 1 1 80
1 1 1 1 1 1 1 1 1
36
0 0 1 1 0 1 1 1
1 81 1 1
1 1 1 1 1 1 1
37 1 1 1 0 1 1
0 1 1 82
1 1 1 1 1 1 1 1 1
38
1 1 1 1 1 1 1 1
1 83 1 1
1 1 1 1 1 1 1
39 1 1 1 1 1 1
1 0 1 84
1 0 1 1 1 1 1 0 1
40
1 0 1 1 1 0 1 0
0 85 1 0
0 1 1 1 1 0 0
41 0 1 0 1 1 1
1 1 1
42 1 1 1 0 1 1
0 1 1
58 63 72 62 70 70 61 71 69 Votes with majority
43 0 0 0 1 0 1
1 1 1
44 0 1 0 1 0 1
1 1 1
45 1 * * 0 1 1
1 1 1
The problem is solved in a mathematical space in which there is a dimension (or direction), for each of the participants. Each participant has a vector with which he 'attacks' his opposition, which the justice can do in 77 directions (for the first data set) or 85 directions (for the second data set) and the case can do in 9 directions. The algorithm output is a specification of magnitude and direction for a set of 86 or 94 vectors. There is also a 'defense' vector for each participant, equal in magnitude to the attack vector, but generally different in direction. Only the attack vector goes into the determination of the influence of a justice, and this influence quantity is also called the applied strength of the participant. The applied strength is found by first computing the vector sum of all attack vectors in the problem--the contest vector--and then computing the projection of a particular participant's attack vector upon the contest vector. This gives a scalar quantity for comparing participants with each other. A small-size example of the procedure is given elsewhere on this site, with a complete explanation.
The computed applied strength numbers (scaled to a sum of 9) for
the justices are the following:
'99-'00 SESSION
'00-'01 SESSION
1 -
0.978
0.810 (Rehnquist)
2 -
0.756
0.870 (Stevens)
3 -
1.261
1.176 (O'Connor)
4 -
1.013
0.948 (Scalia)
5 -
1.177
1.090 (Kennedy)
6 -
0.961
1.097 (Souter)
7 -
1.103
0.898 (Thomas)
8 -
0.812
1.090 (Ginsburg)
9 - 0
939
1.022 (Breyer)
O'Connor has the most
strength in both sessions. Unanimous decisions have no strength and
therefore do not contribute to the strengths of justices. There are
many unanimous decisions, and they can be deleted from the data set
at the beginning without changing the results.
It is often of
interest to measure the agreement between pairs of justices in their
voting. For this, we compute the cosine of the angle between the
attack vectors of pairs. Perfect agreement gives parallel vectors
(cosine equal to 1) and complete opposition in voting gives
orthogonal vectors (cosine equal to 0). The cosines for the justices
in the two sessions under study here are given below.
'99-'00 SESSION
2 3
4 5 6
7 8 9
1 .212 .888 .685 .763 .428 .692
.370 .483
2 - .447 .388 .495 .803
.472 .782 .719
3 -
.684 .811 .627 .731 .601 .610
4
- .805 .409 .879 .309 .505
5
- .504 .803 .543 .535
6
- .519 .800 .790
7
- .393 .456
8
- .718
'00-'01 SESSION
2 3
4 5 6
7 8 9
1 .255 .700 .693 .837 .390 .695
.339 .296
2 - .590 .413 .465 .761
.301 .816 .874
3 -
.604 .773 .612 .639 .682 .630
4
- .633 .521 .799 .486 .490
5
- .612 .700 .542 .482
6
- .489 .893 .829
7
- .432 .416
8
- .906
Pairs with bigger numbers
support each other, and both sessions show
Stevens/Souter/Ginsburg/Breyer (numbers 2, 6, 8, and 9) forming a
group of mutual support on one side of issues and
Rehnquist/Scalia/Kennedy/Thomas (numbers 1, 4, 5, and 7) on the other
side. There is a good deal of consistency from one session to the
next. It is, of course, possible to combine the two sessions and
compute everything for the double data set.
The thinking among analysts of Supreme Court behavior holds that a
so-called 'swing justice' has a lot of extra power in determining
judgments. A justice not too closely aligned with a particular
philosophy might be the one to determine which way a judgment goes in
many cases. O'Connor is famous for being the swing justice in this
period of the supreme court; her strength numbers in the two sessions
analyzed here bear that out. She shows a strength greater than that
of the weakest justice by factors 1.67 and 1.45 in the two sessions.
This Supreme Court analysis was done to demonstrate how the
general-purpose contest algorithm can be applied to real-life data in
a problem of some current interest to the social-choice community. It
showed from observable data only--no judgment calls, no side
analyses--just how the Supreme Court justices relate to each other.
Alan E. Johnsrud
9 September 2001 (revised 23 October 2009)
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