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Decisions handed down by the nine judges of the U.S. Supreme Court give rise to legal and political interpretations. Studies suggest that the justices are heavily influenced by political viewpoints, be they left-wing, right-wing, conservative, or liberal. But as Lawrence Sirovich, a mathematician at Mount Sinai School of Medicine in New York City, showed, it is possible to make a completely dispassionate, objective mathematical analysis of the verdicts. In an article published in the Proceedings of the National Academy of Sciences, he examined close to 500 rulings made by the Supreme Court presided by Chief Justice William Rehnquist between the years 1994 and 2002.

In principle, one can imagine two different kinds of courts and all variations in between. At one extreme, there is a bench made up of omniscient judges, who—were they to exist—know the absolute truth and therefore render their decisions completely unanimously. This court would be mathematically equivalent to a panel of judges who are completely influenced by economic and political considerations. On the assumption that they are influenced in the same manner, they too would cast identical votes. In both of these cases it would suffice to have one single judge, since the eight colleagues would simply be clones and therefore superfluous.

In contrast to this efficient but boring scenario, there is the situation, at the other end, where judges comply with the platonic ideal of making decisions completely independently of one another. They do not let themselves be swayed by political pressures, lobbyists, or their own colleagues. In this type of court none of the judges would be replaceable.

Of course, variations in between these two extremes are also possible. To find out which scenario exists in reality, Sirovich analyzed the so-called entropy of the judges’ decisions. Entropy is a term from thermodynamics, which the mathematician Claude Shannon applied to information theory in the 1940s, that refers to the amount of “disorder” that exists in a system. When molecules are set within a crystal lattice—say, in ice—order is large and entropy is small. One would never be surprised to encounter a molecule at a certain point on the lattice. On the other hand, the random motion of molecules in a gas corresponds to disorder and hence to more entropy.

In information theory, entropy refers to the amount of information that exists in a signal. Applying the term to the decisions of judges implies the following: If all judges make exactly the same decision, order is at a maximum and entropy at a minimum. If, on the other hand, the judges make independent, random decisions, entropy is large. For this reason entropy can also be used as a measure of the information content of the decisions: If judges make identical decisions, the information content is small. It would suffice to obtain the ruling of a single judge, and the opinions of the other judges would be superfluous.

Sirovich calculated the information content of the 500 rulings of the Rehnquist court by computing the correlation between the decisions handed down by the different judges. It turned out that the rulings were far removed from a random distribution. Close to half of the decisions were unanimous, not necessarily because the judges were being influenced by any outside forces but rather because many cases turned out to be straightforward from a legal point of view. Added to this, Justices Scalia and Thomas voted identically in over 93 percent of the cases that reached them. Whenever one of these two justices reached a decision, it was more than likely that the other justice would make the same decision. It was also common knowledge that in many cases Justice Stevens’s decisions deviated from the majority, and this too gradually ceased to surprise anyone.

Sirovich’s work showed that, on average, 4.68 judges (yes, we are speaking fractional judges here) made their decisions independently of one another. In other words, they act as “ideal” adjudicators. This implies that the other 4.32 judges are actually superfluous, since their rulings were, in general, determined by the other judges. The figure of 4.68 is encouraging, says Sirovich, since it proves that the majority of the judges are in large measure independent of one another. They are not influenced by particular worldviews nor are they pressured by each other to make a unanimous decision. While one may believe that the ideal number of independent judges would be nine, this is, in fact, not true. Nine independent judges would imply that each decision is handed down randomly. This would only be possible, however, if the judges completely ignored the facts before them and the legal arguments. Obviously it would not be in the interest of justice or in keeping with the spirit of the law to have judges who could be replaced by random number generators.

Then Sirovich moved on to a different mathematical calculation. The question he now asked was how many judges would have been necessary to hand down 80 percent of the verdicts in the same manner that the Supreme Court of nine judges had done. Each ruling of a nine-justice court (yes, no, yes, yes, no, …) can be considered a point in nine-dimensional space. But since the rulings of certain judges correlate to some degree, this space can be reduced. To see how much the space can be reduced, Sirovich utilized “singular value decomposition” from linear algebra. This is a technique that has been successfully applied to such diverse subjects as pattern recognition and analyses of brain structure, chaotic phenomena, and turbulent fluid flow. Sirovich’s analysis showed that 80 percent of the rulings could be represented in a two-dimensional space. Hypothetically, then, only two virtual judges would have been required—albeit composed of different parts of the nine existing judges—in order to hand down four-fifths of all rulings.