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Nine monkeys flipping coins


Imagine that instead of nine highly intelligent justices, the Supreme Court consisted of nine robed monkeys flipping coins. Cases are decided by the majority of coins showing either heads or tails. So how often will your favorite monkey justice be in the majority, purely by chance? Can we explain the actual win percentages for each justice we saw before?

There are 8!/(4!4!) = 70 possible ways to get exactly four heads out of eight coins (HHHHTTTT, HHHTHTTT, … , TTTTHHHH). These orderings are called permutations. If all coins are fair, then each of these has probability (½)^8, so the probability that the ninth vote is the deciding one is 70/256. [Note: In math ‘!’ is called a factorial and is shorthand for the product of all positive integers less than or equal to that number, so for example 4! = 4*3*2*1.]

First let’s assume that every coin is equally likely to come up heads or tails. If the other eight justices are evenly split 4-4 then you win every time, because you will cast the deciding vote. And if the other eight are not evenly split then you win as long as you side with the existing majority, the chance of which is just ½. The chance of a 4-4 split is 70/256 (see box), so the probability you are in the majority by chance alone is 70/256 + 1/2 * (1-70/256) = 0.6367188.

But we know that each justice has their own ideological disposition. So now let’s imagine that each monkey flips a biased coin, where the amount of the bias towards heads/tails represents how conservative/liberal the justice is. The Supreme Court Database has conservative/liberal classifications for each decision, and I used the individual votes on all cases since 2010 (when Elena Kagan was appointed) to calculate a coin bias for each justice.

The Court is not 9 monkeys flipping biased coins.

Expected vs Actual Win Percentages

Now how often a particular coin-flipping monkey should win will depend on the biases of the other eight coins. The calculation will be similar to before, but we need to know three things: our own coin’s bias, how often the other eight will be split evenly, and how often the majority will be conservative/liberal (it won’t be 50/50). The latter two can be calculated using a probability distribution called the Poisson-binomial distribution.

Sorting the current justices by their ideological bias, I’ve plotted both their actual win percentages (ignoring unanimous decisions) and their expected win percentage. The naive fair coin expectation is shown as a grey line. One observation is that some justices (Kennedy, Roberts) win much more than they are expected to by chance while others (Breyer, Ginsburg) win less often than they should by chance. We’ll return to this in a moment.

The Court rarely has 50/50 moderates.

The Court rarely has 50/50 moderates.

The other observation is there is a huge ideological gap between the conservative and liberal wings of the court. But in this respect is the court any more divided than it has been in the past?

Not really. I used the full data set to calculate ideology scores for every justice since 1946 and plotted those scores, color coding the justices by the presidential party who appointed them as I did for overall win percentage previously. It is clear that there has always been such an ideological gap, so the current one is nothing special. What I find interesting is that both Roosevelt and Eisenhower appointed justices of both ideologies.

So why do some justices do better/worse than the biased coins expect? The reason for this is because the justices are not completely independent like our imaginary coins are. Justice Scalia and Justice Thomas are both conservative justices, but they are not randomly conservative, rather tend to be conservative in the same way, and therefore on the same cases. In addition, justices may be persuaded by the arguments of their peers. We can estimate the correlation coefficients between each pair of justices to reveal how much more alike/different their voting patterns are than expected by chance, again taking into account their biases.

Recent correlation coefficients among the the current justices

Recent correlation coefficients among the the current justices

The correlation matrix of the current justices, sorted by ideology, reveals that the current court is highly structured. In fact the only non-significant (p > 0.05) correlations are between Justice Kennedy and each of Justices Thomas, Scalia, Roberts, and Breyer. This is I think what makes this court seem so divided, that nearly every member is strongly correlated (either positively or negatively) with nearly every other member. It also means that the court votes in blocks much more than coins would, making Justices Roberts and Kennedy more likely to be a deciding vote and thus increasing their win percentages.

As an aside, it is interesting to note that the correlation between Justices Kennedy and Scalia is actually very slightly negative, meaning they have disagreed recently a bit more than ideology would suggest.

Correlated votes over time

Correlated votes over time

Are the current justices more highly correlated than in the past? We can examine correlation coefficients for all members of the court since 1946, sorted by appointment date. By far the strongest negative correlation in this period was between William Douglas, the longest-serving member of the court’s history, and a young William Rehnquist. The strongest positive correlation was between William Brennan and Thurgood Marshall. So while Justice Ginsburg often agrees with Justice Kagan and rarely does so with Justice Thomas, those correlations are certainly not the strongest in the court’s recent history.

But the current court does seem qualitatively to have more/stronger correlations (higher density of darker colors) among justices than in the past. [We could do something more quantitative to test this, but this post is already too long as it is.] Now this doesn’t mean the court was never divided before, because it certainly was. Over time the correlations in voting patterns among court justices started numerous but moderate, became weaker for a time, and then became strong and plentiful leading to the court we see today.

5 Comments leave one →
  1. 2014.12.17 09:08

    Interesting stuff! Have you looked at individual judges’ voting tendencies with respect to time? e.g. Do young judges tend to vote more along party lines and become more moderate in later years (or vice versa, or no change) over their career?

    The last figure’s link might be broken:

    • 2014.12.17 09:33

      Thanks Michael. I tried to fix the figure link. I haven’t looked at how vote ideology changes over time, but others have.
      While I haven’t looked at their methods closely, I do love these figures because they track the same court seat with a color. So when one justice retires and is replaced by another of a vastly different ideology, it is really striking. The limit with these approaches is the number of cases in a year is not large, and the kinds of cases the court sees varies a lot, so interpreting fluctuations may or may not be meaningful.

  2. 2014.12.17 18:13

    I wonder how much strategic voting by some justices impacts their overall vote correlations/ideology. From time to time a justice who would otherwise be in the losing bloc will switch their vote to be in the winning bloc, primarily when they are assigned to write the opinion in that particular case. This is usually done to allow the defecting justice to moderate the actual holding in the written opinion.

    • 2014.12.18 05:20

      That’s a good point Jason. If anything, that kind of instance should ‘pull justices towards the center’ ideologically. So the ideology scores we calculate would be less radical than they might be otherwise.


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