Sigma Xi Research Showcase 2013

1. Statistical mechanics of SCOTUS
Edward D. Lee1,2
Chase P. Broedersz1,2
William Bialek1,2
1Department of Physics, Princeton U.
2Lewis-Sigler Institute for Integrative Genomics, Princeton U.

2. From complex to simple
Group behavior of social organisms can
manifest complex patterns at the group
level even though they might reduce
down to simple rules for individuals
[1—3]. For example, starlings seem to
interact only with local neighbors, and
these local interactions produce global
patterns like on the right. We call this
emergent collective phenomena
because such behavior is not explicitly
encoded in the behavior of individuals,
but arises from interactions.
Is it the case that decisions by groups
of people, despite apparent higher Flocks of starlings form complex patterns
order behaviors, are likewise reducible (http://webodysseum.com/videos/spectacu
to simpler rules? lar-starling-flocks-video-murmuration/).

3. US Supreme Court (SCOTUS)
We investigate the voting behavior of the SCOTUS. We explore the structure
of decisions to gain insight into how these decisions are reached. We show
results for the second Rehnquist Court (1994—2005, 𝑁 = 895 cases) and
discuss other “natural courts”—periods of time when member remain
constant—where relevant.
SCOTUS facts:
• highest court in the US government • write a majority and a minority
• nine Justices appointed for life opinion, legally clarifying their
• vote on constitutionality of legislative decisions, which can be
and executive actions supplemented with separate opinions
• usually hears appeals from lower • Justices must ultimately render a
courts' decisions, which Justices binary decision
affirm or reverse • the second Rehnquist Court is
• sometimes Justices are recused for typically considered as 4 liberals and
conflict of interest, sickness, etc. 5 conservatives
• decisions by majority vote

4. Previous approaches to SCOTUS voting
Although there is a long history of scholarly study of SCOTUS, nearly all approaches rely on
important assumptions—even though they may be justified. Some assume Justices vote
independently given ideological preferences [5]. Others, if including interactions, do not
include them in a general model of voting or posit their structure instead of deriving it from
data [6]. Most importantly, many draw on complex, underlying cognitive frameworks like
rationality or expression of internal beliefs, which are validated by predictions made of the
data [5—8]. However, it is impossible to validate all aspects of these complex models. Is
there a way to construct accurate models while abstaining from introducing complexities?
Characteristics of
previous approaches
• Decision-making
framework [7—8]
• Independent • Ideological liberal vs. • Externally posited
voters given conservative axis [5,7— interaction [6]
ideological 8]
preferences [5] • Subset of votes deemed How do we approach
relevant [5] the system with
• Not generally predictive minimal
Attitudinal Game theoretical assumptions?

5. Back to the data
The data, published by [4],
immediately reveals strong
structure in Court behavior.
For example, we see that in
the second Rehnquist Court,
44% of votes were unanimous.
Overall, when considering the
natural courts shown on the
right, 36% of votes are
unanimous on average. Only
10% of votes fall along the
liberal vs. conservative divide.
Does an independent model
support this observation? Distribution of voting data for natural courts starting in
given year. Blue, 0 dissenting votes; red, 1; yellow, 2;
green, 3; black, 4. Terms are the number of years that
the same members remained on the Court. The number
of votes on record for each set of years is in gray.

6. The simplest model
Independent voters
Each Justice 𝑖 has probability 𝑝 𝑖 of voting to affirm, so the probability of 𝑘 votes in
the majority out of 𝑛 Justices is
𝑛 𝑘 𝑛−𝑘
𝑛
𝑃 𝑘 = 𝑝 1− 𝑝 + 𝑝 𝑛−𝑘 1 − 𝑝 𝑘
𝑘 𝑛− 𝑘
An independent model fails to explain the distribution of votes in the majority.
This is not so surprising because we’ve
assumed that all higher order behaviors are
encoded within the first moments. For
example, for a unanimous vote (𝑘 = 9) to
occur in the independent model, all Justices
would happen to vote the same way, but this
happens much more rarely than observed,
yielding 0.5% of the observed value. Indeed,
interactions are crucial to SCOTUS voting
behavior.

7. More evidence for interaction
Justices take at least two votes for a case: an initial secret vote and a final
decision. Justices may attempt to persuade each other in between, but it is
difficult to measure such interactions partly because the first vote is secret.
According to data available on the Waite Court (1874—1887), 9% of final
votes had at least one dissenting vote while 40% had at least one in the initial
vote [5]. In general, Justices more often switch to the majority than the
reverse, suggestive of consensus-promoting interaction. Maltzmann et al.
show from memos that Justices strategically manipulate their communication
to attempt to influence the vote and written opinions of the Court [6].
Nonetheless, most models either treat the Justices as independent or do not
explicitly include interactions in a predictive framework. Indeed, it is difficult
to devise the right structure for interactions!
How do we account for interactions in a principled fashion?

8. Including interactions…
Given that Justice 𝑖 can vote in two ways, we represent his or her vote as
𝜎 𝑖 ∈ −1,1 . Then, the independent model is the simplest model that fits
average voting records { 𝜎 𝑖 } and all higher order correlations,
{ 𝜎 𝑖 𝜎𝑗 , 𝜎 𝑖 𝜎𝑗 𝜎 𝑘 … 𝜎 𝑖 𝜎𝑗 … 𝜎 𝑛 }, are reducible to { 𝜎 𝑖 }. In fact, all higher
order statistics are as random as possible given the individual means, so there
is no reason any higher order correlations should match that of the data. We
can generalize this idea to a 𝑚th order model that fits all correlations up to
order 𝑚 yet generates all > 𝑚 order correlations randomly. Since these
distributions are as random as possible given what is fit, that also means that
we make no further assumptions than what is given in the fitted correlations
or about how these distributions are generated.
With SCOTUS, we might expect that we need to account for the bloc behavior
(5 vs. 4) and unanimous behavior by including terms of the 4th, 5th and 9th
orders explicitly. However, let us take only the next step of fitting both 𝜎 𝑖
and 𝜎 𝑖 𝜎𝑗 .

9. …as maximizing entropy
The formalization for generating these distributions is called the principle of maximum
entropy [9]. Entropy is a measure of the randomness of a distribution. The entropy of
a probability distribution 𝑃(𝜎) of the votes of a set of 𝑛 voters 𝜎 = {𝜎1 , … , 𝜎 𝑛 } is
𝑆 𝑃 𝜎 =− 𝑃 𝜎 log 𝑃(𝜎)
𝜎
which we maximize while constraining 𝜎 𝑖 and 𝜎 𝑖 𝜎𝑗
𝑛 𝑛
1
𝑆 𝑃 𝜎 , ℎ 𝑖 , 𝐽 𝑖𝑗 = 𝑆 − ℎ 𝑖 𝜎𝑖 − 𝐽 𝑖𝑗 𝜎 𝑖 𝜎𝑗
2
𝑖=1 𝑖,𝑗=1
with Lagrange multipliers ℎ 𝑖 , 𝐽 𝑖𝑗 . The resulting model is known as the Ising model
1
𝑃 𝜎 = 𝑒 −𝐻(𝜎)
𝑍
𝑛 1 𝑛
𝐻 𝜎 =− ℎ 𝑖 𝜎𝑖 − 𝐽 𝑖𝑗 𝜎 𝑖 𝜎𝑗
𝑖=1 2 𝑖,𝑗=1
with a normalizing constant, the partition function 𝑍, and Hamiltonian 𝐻(𝜎).

10. Ising model
1 −𝐻(𝜎)
𝑃 𝜎 = 𝑒
𝑍
𝑛 1 𝑛
𝐻 𝜎 =− ℎ 𝑖 𝜎𝑖 − 𝐽 𝑖𝑗 𝜎 𝑖 𝜎𝑗
𝑖=1 2 𝑖,𝑗=1
The ℎ 𝑖 loosely refer to the “mean bias” of each voter 𝜎 𝑖 , and the 𝐽 𝑖𝑗
loosely refer to the interaction between them, or “couplings.” Since
votes, or “states” 𝜎, with a smaller 𝐻 are more probable, ℎ 𝑖 > 0
implies that 𝜎 𝑖 is more likely to take value 1. Also, 𝐽 𝑖𝑗 > 0 implies that
𝜎 𝑖 and 𝜎𝑗 are more likely to take the same value.
We can solve for the parameters ℎ 𝑖 and 𝐽 𝑖𝑗 such that our model fits the
given moments ⟨𝜎 𝑖 ⟩ and ⟨𝜎 𝑖 𝜎𝑗 ⟩.

11. Mapping spins
We have yet to define how the values of 𝜎 𝑖 correspond to
actual vote. It is not as simple as calling one value affirm and
the other reverse: the outcome of affirming or reversing
depends on how the case is posed. It is entirely possible that
affirming one case is a liberal decision and conservative in
another. What is the right dimension along which to orient the
𝜎 𝑖 ? We abstain from making a choice, and introducing
external bias, by symmetrizing the up and down votes such
that −1 and 1 are equivalent.
This keeps 𝜎 𝑖 𝜎𝑗 the same and fixes 𝜎 𝑖 = 0.
Correspondingly, ℎ 𝑖 = 0. We find that absence of a bias is a
reasonable assumption because bias is not the dominant term
for judicial voting behavior.

12. Model fit
Remarkably, the Ising model fits the
data well. One measure of the fit is to
consider the difference in entropy of the
𝑚th order model with the data
𝐼 𝑚 = 𝑆 𝑚 − 𝑆data [2]. As we increase 𝑚,
we capture more correlation and the
entropy of our models monotonically
decreases to that of the data, where
𝑆 𝑛 = 𝑆data . The furthest distance 𝐼1 is
called the multi-information. Our model
captures 90% of the multi-information
(right).
Thus, it nearly captures all the structure
in the data. It also follows − log 𝑃 𝜎 ∝ The model
𝐻(𝜎) for the most frequent states. The captures 90%
least fit states only appear one or twice of the multi-
on average in a bootstrap sample of the information.
data.

13. Implications of Ising model fit
The fit by the Ising model shows that higher order behaviors
like ideological blocs and unanimity can emerge from lower
order behaviors at the level of pairwise interactions between
individuals. Including higher order terms will result in a
marginal improvement in the fit.
This result is surprising because it suggests that higher level
coordination is not the dominant explanation of voting
behavior. Previously, scholars have pointed to the high level of
consensus in the Court to as evidence for a “norm of
consensus,” which seems analogous to an effective ninth
order term for behavior [6]. Our results point to a different
sort of decision-making structure.

14. Found coupling network
𝐶 𝑖𝑗 = 𝜎 𝑖 𝜎𝑗 − 𝜎 𝑖 𝜎𝑗 and 𝐽 𝑖𝑗 graphs. Justices with a liberal voting record are
colored blue whereas those with a conservative are colored red. Positive edges
are red and negative blue. Widths are proportional to magnitude. All 𝐶 𝑖𝑗 are
positive whereas some 𝐽 𝑖𝑗 are negative. Justices are initialed: John Stevens (JS),
Ruth Ginsburg (RG), David Souter (DS), Steven Breyer (SB), Sandra O’Connor
(SO), Anthony Kennedy (AK), William Rehnquist (WR), Antonin Scalia (AS),
Clarence Thomas (CT).

15. Understanding couplings
As a simple check, we see that the average 𝐽 𝑖𝑗 within
ideological blocs (blue to blue or red to red) are positive
while the average between (blue to red) is negative
(previous slide). The corresponding averages of 𝐶 𝑖𝑗 also
show this relative change although all 𝐶 𝑖𝑗 are positive,
obscuring the antagonistic tendency.
To better understand the distribution of 𝐽 𝑖𝑗 , we consider
the effective field on 𝜎 𝑖 from its neighbors.
𝑛
1
ℎeff =
𝑖 𝐽 𝑖𝑗 𝜎𝑗
2
𝑗=1
Note that it depends on the state of neighbors 𝜎𝑗 . Since
this distribution over all 𝜎 is symmetric around 0, we
Distributions 𝑃 ℎeff 𝜎 . Red histogram is
only show the positive half (right).
𝑖
We fix 𝜎 𝑖 = 1 and compare the shifts in the distributions distribution of fields from only conservative
Justices. Ordered from most liberal to most
of ℎeff of its neighbors 𝜎 , which we measure by taking
𝑗 𝑗 conservative voting record from left to right,
the mean over standard deviation 𝜇/Σ 𝑗𝑖 of 𝑃 ℎeff . In
𝑖 top to bottom. The more conservatively a
the absence of such fixing, 𝜇/Σ 𝑗𝑖 = 0 (next slide). Justice votes, the more the mean field due to
conservatives marches to the right.

16. Shifts in ℎ eff
𝑖
𝜇
= 0.8 Average shifts in
Σ 𝑗𝑖 distributions of
𝜇 ℎeff over
= 4.7 𝑖
Σ 𝑗𝑖 Liberals Conservatives 𝜇 ideological blocs
= 4.3 when holding one
Σ 𝑗𝑖 member of a bloc,
𝑖, at 1 at a time.
Average shift in liberals (𝑗) 𝜇
when holding = 0.8
conservatives fixed (𝑖) Σ 𝑗𝑖
As expected, ideological neighbors are much more affected by fixing 𝜎 𝑖 = 1
by a factor of 5-6, showing that ideological blocs are a natural division of the
Court. Overall, the Court always shifts in the same direction as the
perturbation. Thus, we find that higher order behavior as ideological blocs
and general unanimity are reflected in the couplings.
O'Connor and Kennedy shift conservatives (liberals) to 𝜇/Σ 𝑗𝑖 = 2.6 (1.4) and
𝜇/Σ 𝑗𝑖 = 3.1 (1.1), reaffirming their moderate credentials. Stevens, however,
has overall weakest connections with 𝜇/Σ 𝑗𝑖 = 0.36 (2.72) as if more isolated.

17. Caveats with couplings
We must be careful not to interpret the 𝐽 𝑖𝑗 literally as corresponding to
behavioral interaction on the Court. The distinction that we cannot
make, which is indeed impossible with this data set, is to explain the
underlying mechanism for correlations. We may find two Justices that
vote together too much for chance, but it could be the case that either
they collaborate to a large extent or that their perspectives have been
shaped by a similar background. The latter involves a hidden third
actor, but it is indistinguishable from the other with only the voting
record. In many ways, possible confounding factors that contribute to
𝐽 𝑖𝑗 reflect fundamental limitations of the data.
Our guiding principle is that we refrain from assuming anything beyond
what is already given from the data; other models do not have the
same claim minimal assumptions. Furthermore, we know from
anecdotal evidence that Justices persuade each other, so there are
certainly interactions captured by the 𝐽 𝑖𝑗 .

18. Probing influence
With this model of voting behavior, we can probe the
behavior of the system under perturbations.
The quantity of interest here is the majority outcome of
the court
𝑁
𝑖 𝜎𝑖
𝛾=
| 𝑖𝑁 𝜎 𝑖 |
because this is the decision rendered.
How sensitive is the average decision 𝜸 to a small
changes in the average behavior of a Justice 𝝈 𝒊 ?

19. Probing influence
Formally, this is the susceptibility of ⟨𝛾⟩
1 𝜕⟨𝛾⟩ 1
𝜓𝑖 = = 𝛾𝜎 𝑖 − 𝛾 𝜎 𝑖
𝜒 𝑖 𝜕ℎ 𝑖 𝜒𝑖
which we have normalized over
𝜕⟨𝜎 𝑖 ⟩
𝜒𝑖 =
𝜕ℎ 𝑖
to compare the Justices equally with respect to changes in
their averages. The values we find are
SO AK WR DS SB AS RG CT JS mean
𝝍𝒊 0.834 0.809 0.719 0.650 0.644 0.623 0.616 0.608 0.421 0.658
95% confidence
0.001 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002
interval
𝝍 𝒊 − ⟨𝝍⟩ 0.176 0.151 0.061 -0.008 -0.014 -0.035 -0.042 -0.050 -0.237

20. Are ideological medians influential?
The typical wisdom in the political science literature is that these
ideological medians are the most influential for Court decisions.
Basically, the argument is that voters who sit in the middle of a
unidimensional, symmetric preference space will be predictive of the
majority [10]. The relevant space is liberal vs. conservative as we have
confirmed with ℎeff . The Justices to whom the outcome is most
𝑖
sensitive here are the ideological medians SO and AK, in agreement
with the claim.
However, real systems may be complicated by interactions that
constrain how such a voter may cast her vote or how a majority forms
initially and persists. We do not find that it is the ideological medians
to whom the outcome of the court is most sensitive in general.
Importantly, our results are derived under minimal assumptions. We
do not assume ideological behavior—which though generally visible,
still has to be imposed by the observer—and we account for
interactions.

21. Isolating interactions
We can further inquire about the nature of influence by leveraging our found
interactions. A Justice could affect the outcome in two ways:
1. Change own vote
2. Impact on colleagues’ votes through interactions
How do we distinguish between this two kinds of impact accounted for by
𝝍 𝒊 ? We simulate how 𝜎 𝑖 might increase pressure on colleagues through
couplings by
1. increasing the average coupling with 𝜎 𝑖 such that neighbors’ effective
fields break symmetry around 0 to ⟨ℎeff ⟩ = 𝐽 𝑖𝑗 𝜖. However, this will also
𝑗
incur a shift to 𝜎 𝑖 ≠ 0, so
2. we add a compensating field ℎ′𝑖 to ℎeff to fix 𝜎 𝑖 = 0.
𝑖
The latter step ensures that we do not allow an average shift in an individual’s
vote to affect the outcome. We denote the resulting change from pushing on
𝜎 𝑖 ’s neighbors 𝛿𝛾 𝑖 .

22. Isolating interactions
AK SO DS SB RG WR AS CT JS mean
𝜹𝜸 𝒊 0.348 0.340 0.296 0.276 0.245 0.231 0.195 0.138 0.130 0.244
95% confidence
0.001 0.001 0.001 0.001 0.001 0.002 0.003 0.003 0.003
interval
𝜹𝜸 𝒊 − ⟨𝜹𝜸⟩ 0.104 0.095 0.051 0.032 0.001 -0.014 -0.049 -0.106 -0.114
Comparing with 𝜓 𝑖 …
AK and SO switch order and are relatively closer. Interactions may
differentiate between Justices for whom interactions are important. It
is not the case that Justices highest by 𝜓 𝑖 are also highest by 𝛿𝛾 𝑖
across all natural courts although it is here.
WR falls from 1st to 6th place. WR is the Chief Justice who is responsible
for enforcing procedural rules and has prerogative for assigning
opinions. Interestingly, WR is consistently low by 𝛿𝛾 𝑖 but rises in rank
by 𝜓 𝑖 only being appointed Chief Justice.

23. Isolating interactions
AK SO DS SB RG WR AS CT JS mean
𝜹𝜸 𝒊 0.348 0.340 0.296 0.276 0.245 0.231 0.195 0.138 0.130 0.244
95% confidence
0.001 0.001 0.001 0.001 0.001 0.002 0.003 0.003 0.003
interval
𝜹𝜸 𝒊 − ⟨𝜹𝜸⟩ 0.104 0.095 0.051 0.032 0.001 -0.014 -0.049 -0.106 -0.114
Comparing with 𝜓 𝑖 …
CT and JS are relatively much closer. CT and JS are the most extreme voters on
the conservative and liberal ends of the spectrum. The outcome is similarly
least sensitive to their couplings even though CT votes with the Court 80% of
the time and JS 72%. Thus, ideological hardliners are identified by a certain
voting pattern distinguishing between agreement with or concurrence with
the majority.
CT and AS are similarly biased ideologically, but AS seems to be more strongly
embedded in the interaction network.
All 𝛿𝛾 𝑖 > 0, reflecting the general tendency to consensus.

24. Conclusion
We propose deriving behavior from data instead of testing a hypothesized
framework. With this approach, we show that SCOTUS voting behavior can be
explained as behavior that emerges from pairwise interaction even though
higher order behaviors are manifest. This model shows the higher order
structures like ideological blocs and unanimity quite clearly through the
parameters.
We show how one can exploit the model of voting behavior by considering
the susceptibility of ⟨𝛾⟩ to shifts in average voting behavior. We also isolate
the shifts in ⟨𝛾⟩ specific to interactions and distinguish between Justices
similar by 𝜓 𝑖 along that second dimension of 𝛿𝛾 𝑖 .
However suggestive our results, the correspondence of parameters to real
behavior remains unclear. We hope to soon start a collaboration with political
scientists investigate whether an interpretable correspondence can be
established.

25. Works cited
1. W. Bialek, A. Cavagna, et al., PNAS 109, 4786 (2012).
2. E. Schneidman, M. Berry, et al., Nature 440, 1007 (2006).
3. I. Couzin, J. Krause, et al., Nature 433, 7025 (2005).
4. H. J. Spaeth, L. Epstein, et al., Supreme Court Database (2011).
5. A.D. Martin & K. M. Quinn, Pol. Anal. 10, 134 (2002).
6. L. Epstein, J. A. Segal, et al., Am. J. of Pol. Sci. 83, 557 (2001).
7. S. Brenner & R. H. Dorff, J. of Th. Pol. 4, 2 (1992).
8. F. Maltzmann & J. F. Spriggs II, et al., Crafting law on the Supreme
Court (2000).
9. E. T. Jaynes, Phy. Rev. 106, 620 (1957).
10. D. Black, J. of Pol. Econ. 56, 23 (1948).

26. Further bibliography

27. Acknowledgements
Funding from
NSF grant CCF-0939370
Dept. of Physics, Princeton University
Thanks to
Sigma Xi for hosting this showcase.